Decoherence and Beyond

Decoherence and Beyond - Decoherence and Beyond by Robin...

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Unformatted text preview: Decoherence and Beyond by Robin Jack Blume-Kohout B.A. (Kenyon College) 1998 M.A. (University of California, Berkeley) 2000 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, BERKELEY Committee in charge: Professor Raymond Chiao, Chair Doctor Wojciech H. Zurek Professor Birgitta Whaley Professor Daniel Stamper-Kurn Spring 2005 The dissertation of Robin Jack Blume-Kohout is approved: Chair Date Date Date Date University of California, Berkeley Spring 2005 Decoherence and Beyond Copyright 2005 by Robin Jack Blume-Kohout 1 Abstract Decoherence and Beyond by Robin Jack Blume-Kohout Doctor of Philosophy in Physics University of California, Berkeley Professor Raymond Chiao, Chair I investigate the ways in which information about an open quantum system is recorded in its environment. The underlying themes are classicality and its symptom, objectivity. How decoherence leads to redundantly recorded information is of particular importance. Redundancy is introduced as a concrete manifestation of objectivity, and a set of tools for identifying redundancy is presented. Several variants of spin bath and oscillator bath models for decoherence are analyzed in terms of redundancy. Redundant information storage is shown to be a common feature of decoherence models. Predictability and amplification, two related symptoms of classicality, are also explored. I introduce the operator sieve, an improved algorithm for implementing the predictability sieve, and apply it to the problem of connecting redundancy and predictability. This leads to a general theory of information-preserving structures, which unifies pointer bases with decoherence-free subspaces/subsystems in an algebraic framework. As a first step in exploring amplification (and its connections with redundancy), a simple analytic model of an unstable environment is solved. I show that a single amplifying degree of freedom can produce more (and faster) decoherence than an bath of infinitely many oscillators. 2 Professor Raymond Chiao Dissertation Committee Chair i To Asher Peres, who I’d like to think would have liked this. 1934 - 2005 ii Contents List of Figures List of Tables vi ix I Decoherence: Introduction, Implications, Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 4 9 10 10 10 11 11 12 1 Introduction 1.1 Background: decoherence . . . . . . . . . . . . . . 1.2 Introduction to objectivity and redundancy . . . . 1.3 Overview of original material . . . . . . . . . . . . 1.3.1 Introducing Redundancy . . . . . . . . . . . 1.3.2 Redundancy in spin-bath systems . . . . . . 1.3.3 Redundancy in quantum Brownian motion 1.3.4 Pointer bases and the operator-sieve . . . . 1.3.5 Amplification . . . . . . . . . . . . . . . . . II Redundancy: a Detailed Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 15 15 17 18 21 22 22 22 23 24 26 27 30 39 42 2 Introducing Redundancy 2.1 Redundancy and its relevance . . . . . . . . . . 2.1.1 The redundancy programme . . . . . . . 2.1.2 Computing quantitative redundancy . . 2.1.3 Identifying qualitative redundancy . . . 2.2 Characterizing randomly-selected states . . . . 2.2.1 The uniform ensemble . . . . . . . . . . 2.2.2 Partial information plots . . . . . . . . . 2.2.3 Conclusions . . . . . . . . . . . . . . . . 2.3 Decoherence and branching states . . . . . . . . 2.3.1 The branching-state ensemble . . . . . . 2.3.2 Numerical analysis of branching states . 2.3.3 Theoretical analysis of branching states 2.4 Conclusions and extensions . . . . . . . . . . . 2.4.1 The next steps . . . . . . . . . . . . . . iii 3 Dynamical redundancy in spin systems 3.1 Overview . . . . . . . . . . . . . . . . . . . . 3.1.1 Our model universe . . . . . . . . . . 3.2 Interaction-only models . . . . . . . . . . . . 3.2.1 Results . . . . . . . . . . . . . . . . . 3.2.2 Theory . . . . . . . . . . . . . . . . . 3.2.3 Discussion . . . . . . . . . . . . . . . . 3.3 Quantum-measurement models . . . . . . . . 3.3.1 Results . . . . . . . . . . . . . . . . . 3.3.2 Theory and discussion . . . . . . . . . 3.4 The dynamical-system model . . . . . . . . . 3.4.1 Results . . . . . . . . . . . . . . . . . 3.4.2 Theory . . . . . . . . . . . . . . . . . 3.4.3 Discussion . . . . . . . . . . . . . . . . 3.5 Multiple-measurement interactions . . . . . . 3.5.1 Results . . . . . . . . . . . . . . . . . 3.5.2 Discussion . . . . . . . . . . . . . . . . 3.6 Dissipative models . . . . . . . . . . . . . . . 3.7 Conclusions, discussion, and future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 43 44 46 47 50 54 55 56 61 63 63 65 67 67 68 69 71 72 76 76 77 78 80 82 82 83 88 91 96 96 97 99 100 103 104 104 105 105 106 112 4 Redundancy in quantum Brownian motion 4.1 Introduction and background . . . . . . . . . . . . . . 4.1.1 General properties of the QBM model . . . . . 4.1.2 Practical issues in simulation of QBM . . . . . 4.1.3 Relevant known properties of QBM models . . 4.2 Simulation results: redundancy of stored information . 4.2.1 Methods and parameters . . . . . . . . . . . . 4.2.2 Partial information plots . . . . . . . . . . . . . 4.2.3 Total and non-redundant information . . . . . 4.2.4 Redundancy . . . . . . . . . . . . . . . . . . . . 4.3 Theoretical analysis of a quantum-measurement model 4.3.1 Initial conditions . . . . . . . . . . . . . . . . . 4.3.2 Derivation of Gaussian decoherence factors . . 4.3.3 Structure and entropy of decohered states . . . 4.3.4 The ohmic quantum-measurement model . . . 4.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . 4.4 Timescales . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 The redundancy timescale . . . . . . . . . . . . 4.4.2 The decoherence timescale . . . . . . . . . . . . 4.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . 4.5 Local information . . . . . . . . . . . . . . . . . . . . . 4.6 Concluding remarks, future directions . . . . . . . . . .. .. .. .. .. .. .. .. .. for .. .. .. .. .. .. .. .. .. .. .. .... .... .... .... .... .... .... .... .... QBM .... .... .... .... .... .... .... .... .... .... .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III Objectivity and Decoherence: the Bigger Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 117 117 119 119 120 121 5 Pointer bases, operators, and algebras: the operator-sieve algorithm 5.1 The Predictability Sieve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Operator-Sieve Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Ensembles, basis identification, and the density superoperator . . . 5.2.2 The Operator-sieve Algorithm . . . . . . . . . . . . . . . . . . . . 5.3 Simple Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv 5.4 5.5 5.6 5.3.1 Perfect Pointer Bases . . . . . . . . . . . 5.3.2 Overcomplete Pointer Bases . . . . . . . . 5.3.3 Incomplete Decoherence . . . . . . . . . . Things Algebraic: Pointer Algebras and Noiseless Application to a spin bath model . . . . . . . . . 5.5.1 Two mechanisms of einselection . . . . . . 5.5.2 How to analyze spin bath models . . . . . 5.5.3 Results . . . . . . . . . . . . . . . . . . . 5.5.4 Discussion . . . . . . . . . . . . . . . . . . Conclusion: Implications and Applications . . . . ... ... ... Stuff ... ... ... ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 123 123 125 127 128 130 131 133 134 135 135 136 137 137 141 142 142 145 148 154 157 157 158 160 162 164 165 166 6 Amplification in an upside-down oscillator 6.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . 6.1.1 Sensitivity and chaos . . . . . . . . . . . . . . . . . . . 6.2 Analysis of the Unstable Oscillator . . . . . . . . . . . . . . . 6.2.1 Obtaining the Coefficients of the Master Equation . . 6.2.2 Master Equation for the Inverted Harmonic Oscillator 6.3 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Unitary evolution . . . . . . . . . . . . . . . . . . . . . 6.3.2 Diffusive terms . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Entropy of the Reduced Density Matrix . . . . . . . . 6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Applying these results to experiments 7.1 Experimental detection of information . . . . . . . . . . . . . . 7.2 Detecting and characterizing information-preserving structures 7.3 Detecting and characterizing redundancy . . . . . . . . . . . . 7.3.1 How to measure redundancy generically . . . . . . . . . 7.3.2 Systems suited to exploring redundancy . . . . . . . . . 7.4 Experimental implications of amplification . . . . . . . . . . . . 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusions and further work 167 8.1 What has been accomplished . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.2 The next steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Bibliography A Supporting Material on Redundancy A.1 Details on quantifying redundancy . . . . . . . . . . . . . . . A.1.1 The importance of tensor product structures . . . . . A.2 Properties of QMI: the Symmetry Theorem . . . . . . . . . . A.3 Miscellaneous approximations for both ensembles . . . . . . . A.3.1 Useful properties of the uniform ensemble . . . . . . . A.3.2 Useful properties of branching states . . . . . . . . . . A.3.3 Perfect states . . . . . . . . . . . . . . . . . . . . . . . A.4 QMI for the uniform ensemble: Page’s mean entropy formula A.5 Entropy of a near-diagonal density matrix . . . . . . . . . . . A.6 Probability distributions for additive decoherence factors . . . 174 184 184 186 187 187 188 188 188 189 189 193 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v B Supporting Material on Spin Bath Dynamics B.1 Overview of simulation techniques . . . . . . . . . . . . . . . . . . . . . . . . B.2 Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Pliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.1 Calculating pliability for the interaction-only model . . . . . . . . . . B.3.2 The effect of an environment’s dynamics on its pliability . . . . . . . . B.3.3 Redundancy in simple spin models . . . . . . . . . . . . . . . . . . . . B.4 Detailed R(t) plots for quantum-measurement models . . . . . . . . . . . . . B.5 Dependence of redundancy on complete decoherence . . . . . . . . . . . . . . B.5.1 Branching-state models (interaction-only and quantum-measurement) B.5.2 Dynamical-system and multiple-measurement models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 196 198 199 200 203 212 216 220 220 221 225 225 227 231 C Supporting Material on Quantum Brownian Motion C.1 Quantum Brownian motion: Details of Rδ ’s dependence on various parameters . . . C.2 Quantum Brownian motion: Differential information . . . . . . . . . . . . . . . . . . C.3 Quantum Brownian motion: Cyclic PIPs . . . . . . . . . . . . . . . . . . . . . . . . . D Supporting Material on Pointer algebras 235 D.1 The Gell-Mann matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 vi List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 4.1 4.2 4.3 Objectivity in the C-NOT model of decoherence . . . . . . . . . . . . . . . . . Venn diagram for classical information . . . . . . . . . . . . . . . . . . . . . . . Different locality (tensor product) structures for the “universe” . . . . . . . . . Three kinds of partial information plots (PIPs) . . . . . . . . . . . . . . . . . . PIPs for uniform ensembles (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . PIPs for uniform ensembles (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent environments in the uniform ensemble . . . . . . . . . . . . . . . . . Scaled partial information plots (SPIPs) for the uniform ensemble . . . . . . . PIPs for the branching ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent environments for the branching-state ensemble . . . . . . . . . . . . Scaled partial information plots (PIPs), and the two regimes of information (linear and exponential) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PIPs for “thermal”, or geometric, states . . . . . . . . . . . . . . . . . . . . . . Redundancy in branching states . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytical approximations to branching-state PIPs . . . . . . . . . . . . . . . . Specific redundancy in branching-state ensembles . . . . . . . . . . . . . . . . . Quantum Darwinism in action . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... ... ... ... ... ... ... ... gain ... ... ... ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 19 20 21 23 24 25 26 28 29 30 31 32 36 40 41 48 49 50 51 57 58 59 60 64 65 69 70 71 72 73 85 87 89 Time-series PIPs for the interaction-only model . . . . . . . . . . . . . . . . . . . Simulation data: Redundancy vs. time in the interaction-only model . . . . . . . Time-averaged redundancy vs. Number of environments (interaction-only model) Relative Timescales for decoherence and redundancy . . . . . . . . . . . . . . . . Time-series PIPs for dynamically decoupled models . . . . . . . . . . . . . . . . . Time-series PIPs for super-pliable models . . . . . . . . . . . . . . . . . . . . . . Simulation data: Redundancy vs. Time (quantum-measurement models) . . . . . Specific redundancy in quantum-measurement models . . . . . . . . . . . . . . . Time-series PIPs for the dynamical-system model . . . . . . . . . . . . . . . . . . Simulation data: Redundancy vs. Time (evolving-system model) . . . . . . . . . Time-series PIPs for a dipole multiple-measurement model . . . . . . . . . . . . . Simulation data: Redundancy vs. Time (Z-Y multiple-measurement model) . . . Simulation data: Redundancy vs. Time (dipole multiple-measurement model) . . Time-series PIPs for dissipative models . . . . . . . . . . . . . . . . . . . . . . . Summary of all spin bath models . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical QBM PIPs (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical QBM PIPs (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total information available from QBM environments . . . . . . . . . . . . . . . . . . vii 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 5.1 5.2 5.3 5.4 5.5 6.1 6.2 6.3 6.4 6.5 6.6 6.7 Non-redundant information in QBM environments . . . . . . . . . . . . . . . . . Redundancy (of a fixed fraction) in QBM environments . . . . . . . . . . . . . . Redundancy (of the redundant information) in QBM environments . . . . . . . . Dependence of redundancy on initial squeezing . . . . . . . . . . . . . . . . . . . Illustration of how coherent states evolve under x-conditional Hamiltonians . . . Theoretical PIPs and non-redundant information for quantum Brownian motion Initial decoherence rates in QBM . . . . . . . . . . . . . . . . . . . . . . . . . . . Local information vs. frequency and time . . . . . . . . . . . . . . . . . . . . . . Theory predictions for local information. . . . . . . . . . . . . . . . . . . . . . . . Local information for a free particle system . . . . . . . . . . . . . . . . . . . . . Local information for a low-frequency underdamped oscillator . . . . . . . . . . . Local information for an ultra-high-frequency oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 92 94 95 98 102 106 107 108 109 110 111 122 124 125 131 133 143 146 147 150 151 152 155 Simple ensembles of post-decoherence states . . . . . . . . . . . . . . . . . . . . . . . Post-decoherence state ensembles for dissipative processes . . . . . . . . . . . . . . . Post-decoherence state ensembles for strange/impossible processes . . . . . . . . . . Depolarization of the pointer basis in three environment-entangling models of decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum residual coherence in three environment-entangling models of decoherence Plots of “unitary” master equation coefficients for the IHE . . . . . Plots of diffusive master equation coefficients for the IHE . . . . . Plots of master equation coefficients for the SHO . . . . . . . . . . Plots of system entropy vs. initial configuration . . . . . . . . . . . Plots of system entropy vs. master eq. parameters . . . . . . . . . Dependence of system entropy on effective Lyapunov exponent . . Comparison of entropy growth to energy growth in the IHE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Useful information for computing ensemble properties of additive decoherence factors 194 B.1 B.2 B.3 B.4 B.5 B.6 B.7 B.8 B.9 B.10 B.11 B.12 B.13 B.14 B.15 B.16 B.17 Initial rise of redundancy in the interaction-only model . . . . . . . . . . . . . . . . . Initial rate of decoherence in the interaction-only model . . . . . . . . . . . . . . . . Distribution of frequency differences for a linear system . . . . . . . . . . . . . . . . Time dependence of multiplicative decoherence factor for spin-j environments . . . . Time dependent pliability in the interaction-only model . . . . . . . . . . . . . . . . Time-averaged additive decoherence factor vs. environment size . . . . . . . . . . . . The multiplicative decoherence factor in a general measurement model . . . . . . . . Additive decoherence factor for a generalized quantum-measurement model with spin1/2 environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additive decoherence factor for a generalized quantum-measurement model with spinj environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamical range (without pinning) of the d-factor for assorted environment sizes . . Multiplicative decoherence factor for D=3 . . . . . . . . . . . . . . . . . . . . . . . . Predicted values of dynamical (interaction-only) redundancy, for deficits of 10% and 1% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Predicted dynamical redundancy (quantum measurement model): spin-1/2 environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Predicted dynamical redundancy (quantum measurement model): spin-{1, 3/2, 2} environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation data: Redundancy vs Time (quantum-measurement model, N=12) . . . . Simulation data: Redundancy vs Time (quantum-measurement model, N=48) . . . . Simulation data: Redundancy vs Time (quantum-measurement model, N=128) . . . 198 198 203 204 205 206 207 209 210 211 212 213 214 215 217 218 219 viii B.18 B.19 B.20 B.21 The approach to Simulation data: Simulation data: Simulation data: complete decoherence with increasing N (interaction-only Entropy vs. Time (evolving-system model) . . . . . . . . Entropy vs. Time (Z-Y multiple-measurement model) . Entropy vs. Time (dipole multiple-measurement model) model) ..... ..... ..... . . . . 220 222 223 224 C.1 Ohmic QBM simulations: Effects of varying cutoff frequency, number of bands, coupling constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Ohmic QBM: Differential information (free particle system) . . . . . . . . . . . C.3 Ohmic QBM: Differential information (low frequency system) . . . . . . . . . . C.4 Ohmic QBM: Differential information (medium frequency system) . . . . . . . C.5 Ohmic QBM: Differential information (ultra-high frequency system) . . . . . . C.6 Ohmic QBM: Cyclic PIPs for a free particle system . . . . . . . . . . . . . . . . C.7 Ohmic QBM: Cyclic PIPs for a low-frequency underdamped oscillator . . . . . C.8 Ohmic QBM: Cyclic PIPs for a medium-frequency underdamped oscillator . . . C.9 Ohmic QBM: Cyclic PIPs for a high-frequency underdamped oscillator . . . . . and ... ... ... ... ... ... ... ... ... 226 227 228 229 230 231 232 233 234 ix List of Tables 2.1 3.1 7.1 Additive decoherence factors for small environments . . . . . . . . . . . . . . . . . . Interaction-only specific redundancy: theory vs. simulation . . . . . . . . . . . . . . 39 53 Eigenoperators of an experimental 2-qubit NMR process . . . . . . . . . . . . . . . . 161 x Acknowledgments This dissertation represents a tremendous amount of work, and required a tremendous amount of support from others. I owe the following people a great deal. Many members of the quantum science community have molded my thinking, and their efforts are reflected here. At Los Alamos, I am grateful to Juan Pablo Paz, Lorenza Viola (now at Dartmouth), Fernando Cucchietti, Augusto Roncaglia, Daniel James, Salman Habib, Peter Milonni, Tanmoy Bhattacharya, Howard Barnum, David Sigeti, Diego Dalvit, Raymond Laflamme (now at Perimeter) and Manny Knill (now at NIST). At the University of New Mexico, Carlton Caves, Ivan Deutsch, Sonja Daffer (now at Imperial), Joseph Renes (now at Erlangen), and Andrew Scott (now at Calgary). Around the world, Bill Wootters, Bill Unruh, Joseph Emerson, David Poulin, Harold Ollivier, Thomas Seligmann, Chris Fuchs, Hilary Carteret, Michael Nielsen, Jennifer Dodd, Michael Bremner, Chris Dawson, Kurt Jacobs, Barry Sanders, Steve Bartlett, Rob Spekkens, Asher Peres, Todd Brun, Nicholas Boulant, Daniel Gottesmann, and Dave Bacon. This dissertation is a Los Alamos Unclassified Report #LAUR-05-0669. I am very grateful to Los Alamos National Laboratory, as well as the ARDA Quantum Computing program, for supporting my research financially. The Kavli Institute for Theoretical Physics and the University of California (Berkeley) have also provided me with places to work, and I have been fortunate to receive travel funding from the U.S. National Science Foundation, NATO’s Advanced Study Institute program, and Dartmouth College. I am thankful for the patience shown by John Preskill, Ann Harvey, and the Institute for Quantum Information at Cal Tech. I could never have navigated all the administrative labyrinths without the help of Eleanor Alarid and Nancy Kurnath at LANL, and Donna Sakima and Anne Takizawa at UC-Berkeley. Like every scientist, I have stood on the shoulders of giants. This project – or anything like it – would not have been possible without the pioneers of the field. The list of philosophers (in the old sense of “wisdom lover”) who have inspired me stretches back at least to Niels Bohr and Albert Einstein, and includes Richard Feynman, Rolf Landauer, John S. Bell, Claude Shannon, Charlie xi Bennett, Asher Peres, and particularly John Archibald Wheeler. I have been incredibly fortunate in my advisors. Benjamin Schumacher (Kenyon College) got me started in this business, and showed me that quantum mechanics was really cool. Wojciech Zurek has provided everything that a graduate student could ask from an advisor – direction and vision, freedom to inquire, support, wisdom, and when it occasionally became necessary (as it probably does for every graduate student) a reminder to get back to work. Finally, while working on this dissertation, I have accumulated incalculable debt to a few individuals who kept me sane. In Los Alamos, Mouser Williams provided me a with a place to live during the worst times, and opportunities to commiserate about thesis writing. Chris Jarzynski and Gabe Rockefeller accompanied me on innumerable coffee runs (and listened to the associated rants). From a safe distance, Jasper Halekas, Colin McCormick, and Matt Jadud put up with frantic 3 A.M. emails, crushing depressions, and manic phone calls. My parents – to whom I am more grateful for everything that got me here than I can begin to say – accepted my vanishing off the face of the earth with remarkable equanimity. Last and most, I acknowledge the constant support and tolerance of my wife Meg. She has paid the greatest price of any other person for my finishing this dissertation, and has kept me focused, alive, and sane throughout. I couldn’t have done it without a meep. 1 Part I Decoherence: Introduction, Implications, Extensions 3 Chapter 1 Introduction As the 20th century began, most physicists believed they understood the Universe1 moderately well, and that the role of further research was to tidy up the details. Over the next 20 years, quantum mechanics utterly destroyed that complacency. It divided physical theories into incompatible “quantum” and “classical” regimes, and gave uncertainty a permanent place in the scientific canon. Since 1920, a great deal of intellectual effort has been expended in trying to understand the incompatibilities between these two regimes, and to reconcile them into a consistent theory of the Universe that surrounds us. A major accomplishment toward that goal was the establishment, in the last quarter-century, of decoherence as the process by which some classical features emerge from the quantum substrate. During the same time period, the emergence of quantum information theory has propelled the study of quantum mechanics into the limelight. Decoherence has emerged as the nemesis of things quantum, in addition to its original role as generator of the classical. As quantum technology enters the mainstream, questions about quantum physics that once seemed metaphysical are becoming urgent and practical. A qualitative understanding of the quantum-to-classical transition is no longer satisfactory; now we need to know exactly how and where it occurs. Decoherence happens because the environment somehow measures the system, but this insight is insufficient. We need to know what performed the measurement, what was measured, and how well it was measured. Beyond observing that “information leaked out of the system into the environment,” we would like to know: What information? Where did it go? How was it recorded? The urgent question for quantum technologists today is how to prevent classicality from destroying the coherence of a quantum information processing system; tomorrow, we may well need to enforce classicality, to prevent coherence from interfering with the next generation of nanotechnology. To achieve such control, we require a much more thorough understanding of decoherence and, more generally, the transfer of information between controlled systems and the greater Universe. In this thesis, we lay the groundwork for achieving this understanding and perform some of the first focused investigations.2 Unlike many projects, whose goal is to fill in a circumscribed niche in the scientific edifice, we are to some degree launching into the unknown here. The process has been described as “reconnaissance by force.” While several of the particular investigations described herein are reasonably complete, in a number of cases we have no choice but to conclude, “this is going to require further research.” Decoherence has been actively investigated for over 25 years now, and in the end has led not just to conclusions, but to more questions – some of which are pursued here. The work presented here is divided into three parts. The first is an introduction and an 1 The word “universe” will be used extensively to refer either to the real physical Universe in which we live or to a model system together with its environment. While the model universes that we study are intended to shed light upon the physical Universe, they remain distinct concepts. To avoid confusion, we consistently capitalize the real Universe, and not the model universes. 2 Prior work on objectivity and redundancy, in this context, exists in [160, 167, 106, 105] 4 overview of the field. We begin by considering the important background material of decoherence, from the early controversies over interpretation of quantum theory to recent information-theoretic work. An overview of the original research is then presented, to ease the transition from one technical section to another. The second part dives directly into the core concept of redundancy. We motivate its significance in the emergence of objectivity, define the tools and terminology for investigating redundancy, and study it quantitatively in models ranging from the trivial to the realistic. The two main environments found in decoherence models are spin baths and oscillator baths. We examine the emergence of redundant information storage in each of them. In the third part, we refine our focus, shifting it from redundancy to other processes important for objectivity. A new method for identifying pointer states is described, and we discuss how this operator sieve can be a powerful tool for understanding the dynamics of information. We apply the operator sieve method to the spin bath model examined previously. This example not only serves as proof of principle for the operator sieve, but also allows us to comment on the connection (in one particular context) between predictability and redundancy. Finally, we consider amplification, a route to objectivity that is in many ways parallel to redundancy, in the context of an unstable linear system. We conclude by outlining the next steps in this program of research, along with some more general questions to be explored. 1.1 Background: decoherence The problem: Quantum vs. Classical Quantum mechanics is exceptionally good at predicting the outcome of experiments. Unfortunately, classical mechanics is also very good at predicting the outcome of experiments. While quantum mechanics is undoubtedly the correct theory, the difficulty of “deriving” classical behavior from the quantum substrate has produced a great deal of angst among physicists. As Zurek [167] says, “[T]he principle of superposition,. . . thoroughly tested in the microscopic domain, bears consequences that defy classical intuition: It appears to imply that the familiar classical states should be an exceedingly rare exception. . . As Einstein noted, localization with respect to macrocoordinates is not just independent of, but incompatible with, quantum theory.” The problem, in other words, is not merely that the two theories predict different dynamics for physical systems. They don’t even admit the same underlying reality. This awkward state of affairs has been recognized since the inception of quantum theory. Schr¨dinger [127, 128, 129], Heisenberg [63], Bohr [21, 22, 23], and Einstein [43] wrestled with the o problem of interpreting quantum theory. The conventional resolution to the early debates over quantum theory’s interpretation and application is the Copenhagen Interpretation, named for Bohr [21, 23]. The Copenhagen Interpretation sidesteps the conflict between quantum and classical theory by erecting a wall between the two. Quantum theory is used to describe very small systems, classical theory is used to describe large systems (including measuring devices and observers), and the border region is explicitly “terra incognita.” Action is often used to discriminate between the two regimes: systems with a total action s ∼ are treated as quantum; those with s are considered classical. However, Bohr recognized that the boundary could not be fixed. While certain objects (e.g., measurement apparatus) are always treated as classical, almost any object could in principle be considered quantum. The Copenhagen interpretation has, for almost a century, been both highly functional and fundamentally unsatisfying. It provides a framework that predicts experiments well, but which appears arbitrary and incomplete. The most important alternative interpretation is the many worlds interpretation, due to Everett [44] (see also [147, 37, 155, 156, 38, 35, 36, 139]). The many-worlds interpretation views the Universe as fundamentally quantum. The measurement-induced collapse of 5 wavefunctions (in the Copenhagen Interpretation) is replaced with a branching process. The state vector of the Universe simply splits into multiple alternatives (coherently) whenever a measurement could have occurred. What the many-worlds interpretation cannot explain is classicality. It avoids the capriciousness of Copenhagen at the cost of failing to account for a classical world at all. In particular, many-worlds cannot explain the superselection of preferred states – what Einstein referred to as “Narrowness with respect to macrocoordinates” (see footnote in [167]). Macroscopic (i.e., classical) objects are almost always found3 in a very small and particular subset of the possible quantum states. The pointer of a measurement apparatus, in particular, is never found in a superposition of states corresponding to distinct outcomes. Instead, the apparatus is invariably found to be in a particular pointer state. Classicality is thus inextricably linked with the superselection of preferred states. The solution: Decoherence and Einselection Decoherence is the destruction, by the environment, of coherences between states of a quantum system4 . For an isolated quantum system, the superposition principle mandates that every superposition of classical states is equally valid. The interaction of a open system with its environment breaks the superposition principle. Interaction with the environment selects out a preferred basis of pointer states5 , by transforming any coherent superposition of pointer states into an incoherent mixture. If |ψ and |φ are pointer states, the process is: |ψ |φ + β |φ −→ |ψ −→ |φ −→ |α|2 |ψ (1.1) (1.2) (1.3) α |ψ ψ| + |β |2 |φ φ| By breaking the principle of superposition, decoherence can explain the appearance of preferred states. The emergence of the resulting superselection rules is known as Einselection (EnvironmentInduced Superselection [158]). Decoherence and einselection are dual to each other – in order for some states (the pointer basis ) to be einselected, all the other states (superpositions of the einselected states) must suffer decoherence. The observation that classical states and behavior emerge as a result of a system’s interaction with the environment is neither trivial nor obvious. The environment can (and does) produce other effects – in particular, dissipation. Dissipation is the destruction (by the environment) of distinctions between classical states of the system. When dissipation occurs, classical states which were initially perfectly distinguishable 3 A common convention in the decoherence literature is to speak of a system or apparatus being “found”, “observed”, “discovered” or “consulted”, and to note that the result of this operation always implies that the system/apparatus is in one of the pointer states. This language is convenient and intuitively appealing, but not precise. What we are attempting to convey with such language is predictability. In principle, a classical object could be measured in a non-pointer basis. The result of such a measurement will always be unpredictable. As such, it is useless to make the measurement. It is not even possible to prove that a measurement has been made – a measurement whose outcome does not correlate with anything is indistinguishable from a coin flip. Thus, whenever a system is verifiably “found” or “observed” to be in some state, that state must be a pointer state. 4 What a quantum “system” is, precisely, has become unclear in recent years. The traditional definition of “isolated quantum system” is: The set of states, operations, and measurements that can be performed on a particular Hilbert space. That is, given a system S , there exists a Hilbert space such that every measurement made on the system has probabilistic outcomes whose frequencies are predicted by well-defined functionals on a state which is either a ray in that Hilbert space or a statistical mixture thereof. In this view, a subsystem is represented by a Hilbert space whose tensor product with some ancillary Hilbert space yields the full system’s Hilbert space. We adopt this view throughout the dissertation. In Chapter 2, we outline the tensor product structure of our “universe.” Recent work – including, indirectly, Chapter 5 in this dissertation, has begun to motivate a more abstract view of subsystems. In this view (see e.g. [12, 11]), a subsystem corresponds to an algebra of observables, and its state is a functional on this algebra. This paradigm is not particularly relevant for the purposes of reading this dissertation. 5 The name originated in [157], where pointer states were introduced to describe the stable state of a measurement apparatus 6 evolve into the same state (or at least states that cannot be reliably distinguished). Dissipation typically affects all states, but does not produce einselection. It occurs in classical as well as quantum systems, whereas decoherence cannot even be formulated outside of a quantum context. In most quantum open systems, the environment produces both decoherence and dissipation; einselection initially identifies a metastable pointer basis, which slowly dissipates over a longer timescale. Decoherence is akin to measurement. The system’s state influences the environment, which becomes correlated with the system.6 By monitoring the system in this way, the environment effectively measures which pointer state the system is in. Unlike a traditional quantum measurement [142], no collapse or nonunitarity occurs – for the combined “supersystem”. The restricted dynamics of the system alone is nonunitary, which is what produces decoherence. Even by considering the reduced system alone, however, no collapse can be made to emerge. Instead, the system’s reduced density matrix is a mixture of all the possible measurement outcomes. Quantum information and decoherence Quantum information theory [102] has developed almost simultaneously with the theory of decoherence. Decoherence and einselection were originally studied to resolve the quantum-toclassical transition. Decoherence explains why most quantum states are never observed in the classical world, while einselection naturally identifies the states which are macroscopically stable. Quantum information theory, on the other hand, relies on the stability of exactly the arbitrary quantum states that decoherence destroys. From the perspective of quantum technology (e.g., quantum computation, quantum communication, or quantum cryptography), decoherence is the great enemy. By restoring classicality, it nullifies the power of quantum information. Quantum information and technology have contributed greatly to the interest in decoherence. In the first place, decoherence theory provides a quantitative estimate of how “large” a quantum system can be before it begins to suffer classicality. Understanding the mechanisms of decoherence helps to identify ways of extending the quantum regime (e.g., decoherence-free subspaces/subsystems [90, 74, 64, 47]). It has also contributed to the development of quantum error correction [76, 54], since decoherence is a primary source of errors in quantum computing and communication. Decoherence is integral to quantum measurements [158, 149], which in turn are an essential element of any quantum information processing technology. As Wheeler noted, “No [quantum] phenomenon is a phenomenon until it is a recorded (observed) phenomenon”[148, 149, 167]. At the end of the day, classicality (in the guise of measurement) is needed to obtain a useful result from any quantum process. The tools of quantum information theory have changed the way decoherence is viewed and studied. The observation that decoherence is like a measurement (above) has motivated an information-based view of decoherence. In measuring the system, the environment gains information about it. At the same time, information about the environment flows the opposite direction – into the system [163, 164]. The results presented in this thesis are largely motivated by the informationtheoretic perspective on decoherence. Models of decoherence and their structure Open quantum systems have been studied for a long time, but most of the early work (e.g., seminal work by Feynman and Vernon [45] on linear systems) focused exclusively on dissipation. The earliest papers to touch the connection between classicality and the environment are probably by Gottfried [56] and Zeh [155, 156]. Development of decoherence in its modern form is largely due 6 There is a reciprocal relationship between decoherence and dissipation: as Zurek points out, “Relaxation and noise are caused by the environment perturbing the system, while decoherence and einselection are caused by the system perturbing the environment.”[167] The two effects are connected; intuitively, this is obvious because unitarity forbids the actual loss of information. If information is lost from the system, it must be replaced by information about the environment (noise). The connection is quantified by a variety of fluctuation-dissipation theorems. 7 to Zurek [157, 158]. The recommended reference is the recent review paper by Zurek [167]; see also the book [72] by Joos et. al, and the review paper by Schlosshauer [126]. Decoherence models are almost always analyzed in terms of their effect on the central system’s reduced density matrix (ρS ). The evolution of an isolated system’s density matrix is unitary, ˆ ˆ ρS (t) = Ut ρS (0)Ut† . (1.4) Open systems evolve according to completely positive, trace-preserving maps (CP -maps) [81, 102], ˆ ˆ ρS (t) = S [ρS (0)] . (1.5) ˆ ˆ S is a superoperator – a matrix which acts on operators (e.g, ρS ). It must preserve the trace of ρS (otherwise probability would not be conserved), and also its positivity (i.e., the eigenvalues of ρS cannot be negative.7 Decoherence models are usually solved by finding a master equation. A master equation is an equation of motion for ρS , ∂ρS = f (ρS , t), ∂t (1.6) exactly as the Schr¨dinger equation is an equation of motion for |ψS . o Another important tool for the analysis of decoherence is the predictability sieve [162]. It provides a framework for identifying the pointer states of a system. Initial states of the system (|ψi ) are sorted according to the amount of entropy8 they develop in the course of decoherence. Those that remain the most pure (i.e., develop the least entropy) are the most predictable – e.g., if |ψ0 evolves into a pure state |ψt , then its evolution is completely predictable. These highly predictable states are the pointer states. A wide variety of models for quantum open systems have been considered and solved. Three types are particularly relevant to decoherence: • Quantum Brownian motion (QBM) models [45, 26, 58, 140, 66, 114] involve a bath (E ) of harmonic oscillators. The oscillators are coupled to a central system (S ) through position (ˆ). The solutions to QBM models usually take the form of a master equation, which yields x timescales for decoherence and dissipation. The environment monitors x, so superpositions of ˆ different x-eigenstates are particularly subject to decoherence. For instance, it has been shown ˆ [161, 160] that a state with position width ∆x decoheres on a timescale − τD 1 = γ ∆x λT 2 , (1.7) where γ is the dissipation rate (fixed by the coupling to E ), and λT is the thermal de Broglie wavelength of the system. A rough idea of how fast decoherence occurs can be gained from calculating τD for a 1 gram particle, at room temperature, with a coherence length of ∆x = 1 centimeter. Equation 1.7 predicts [167] a decoherence rate of 1040 γ . This is a substantial overestimate, as pointed out in [7]; the decoherence rate is limited by the highest frequency present in the environment. However, the fact remains that decoherence happens extremely quickly for macroscopic systems. Studies of QBM models have also identified coherent states, under a wide range of conditions, as the pointer states [169, 138, 49]. 7 Actually, complete positivity is stronger than positivity. If we add an ancilla (A) to the system (S ), then a completely positive map on S must preserve the positivity of any joint state ρSA . Examples are given in Chapter 5. 8 The entropy of a single quantum state is a measure of how unpredictable the most predictable measurement (of a non-degenerate observable) on that state is. It was defined by Von Neumann [142] as H (ρ) = Trρ log ρ, in analogy P to the classical thermodynamic entropy of an ensemble [112], H (p) = i pi log pi . 8 • Spin bath models ([118] and the extensive references therein) model the environment as a bath of low-dimensional quantum systems – e.g., spins. The key feature of spin bath models is that the environment modes are (1) localized, and (2) limited in number. In contrast, oscillator baths usually consist of infinitely many (weakly-coupled) modes, which are delocalized in space. In particular, spin baths describe the interaction of a nuclear spin, on a large molecule, with other spins on the same molecule (e.g., NMR quantum computers [136]). The model is also relevant to several solid-state implementations of quantum computation [102]. A simple model of a spin bath [158] has been used as an elementary but productive example of decoherence [106, 105, 20]. • Lindblad-type models, also known as dynamical semigroups, are a more abstract approach to decoherence models [79, 91, 53, 3, 49, 2]. The Lindblad approach assumes that the dynamics are Markovian, under which assumption the system’s dynamics can be condensed to a set of Lindblad operators. The Lindblad operators are then used to construct a completely positive, trace-preserving map master equation. The disadvantage of the Lindblad form is that the assumption of a strictly Markovian structure for the dynamics is incompatible with a number of physical models – for instance, quantum Brownian motion. “Environment as a witness” and “Quantum Darwinism” Since its inception, decoherence has been studied from the perspective of the central system (S ). The environment (E ) induces non-unitary dynamics (e.g., decoherence and dissipation), but is traced out and thrown away prior to analysis. The predictability sieve (see discussion above, and in Chapter 5) is an excellent example of this paradigm. It identifies a preferred basis (the pointer states) by identifying states (and information) that persist in S despite the environment’s influence Recent work has started to analyze the environment, instead of throwing it away. The focus is on the information (about S ) lost to decoherence, which according to quantum information theory must persist somewhere [122]. E takes on the role of a witness to the system’s evolution. This relatively new9 “environment as a witness” paradigm [165, 167, 106] is used to advance the information-theoretic view of decoherence [165], to address fundamental questions of quantum theory [60], and even to explore advanced error-correction techniques [57]. A central result of this paradigm is the observation that the environment does not treat every observable the same. Some information appears to be more “fit” than the rest. The fit information not only survives the interaction with E , but propagates throughout the environment. When an observer interrogates the environment-as-a-witness, the fittest information is easy to obtain. Information about incompatible observables is virtually impossible to obtain. The obvious analogies with natural selection have led to the name “quantum Darwinism” [167, 106] for the dynamics that produce this situation. Two simple examples of quantum Darwinism in practice are: • A light switch on a wall has two positions. Within the context of quantum theory, the switch could be described equally well by specifying its position, or by specifying the relative phase between the two possible positions. In practice, the relative phase property might as well not exist – states of definite phase cannot be prepared, and the phase cannot be reliably measured. Decoherence annihilates this property of the light switch. At the same time, information about position is propagated throughout the local environment, so that any observer standing nearby can collect a tiny fraction of the photons scattered from it, and determine whether the switch is on or off. • A heavy ion is placed in a harmonic potential (e.g., a Paul trap), and prepared in an superposition of two energy eigenstates. Subsequent measurements will show that the phase between 9 Although the core ideas of objectivity, redundancy, and the environment as a communication channel are present in early work on decoherence [158, 159], intensive research dates only to about 2000. 9 the eigenstates becomes indeterminate. Once this occurs, reconstructing the original phase is impossible. By monitoring low-energy modes of the surrounding electromagnetic field, however, it is possible to predict the result of a subsequent measurement of energy. A particle coupled weakly to a low-energy environment (see [113]) has its energy monitored by the environment, so that only measurements of energy are predictable. The predictable information about energy is available from many different modes of the decoherence-inducing field. These two cases span the size spectrum, from an ostensibly quantum system (suggested for use as a quantum computer) to an unquestionably classical one. What they have in common is that on some timescale, one particular observable distinguishes itself from the rest. By virtue of (a) retaining information that was encoded in it, and (b) propagating that information throughout its local environment, this “pointer observable becomes objective. 1.2 Introduction to objectivity and redundancy Objectivity, which has emerged from the environment-as-a-witness paradigm as a sign of classicality, is a central concept in this work. Our notion of objectivity is derived from Ollivier et. al. [106, 105], and summarized in [167]. A property (e.g., the state of a system) is objective if (and only if) many independent observers can determine it independently (i.e., without prior knowledge and without prior consultation), then agree about their observations when they meet for the first time. The state of a quantum system clearly fails this test – two independent observers, who measure the quantum system independently, will generally elect to measure incompatible observables. When they meet to discuss their results, they not only disagree, but have no common ground to argue on. Studies of quantum reference frames (e.g., [1, 9, 14, 16, 15]) begin to address this problem. However, the observers must collude in order to recognize and respect the reference frame. This violates the assumption of independent observers. By contrast, observers of a classical system not only agree on the properties of a system, but find it utterly impossible to measure incompatible observables in the first place. Objectivity is mandatory in the classical world. Thus, we regard the objectivity of a system’s important properties as necessary for that system to be effectively classical. In order for a property to be objective, it is sufficient that information about that property be recorded redundantly. Information is redundant if and only if it is contained independently in multiple disjoint subsystems. Several observers can then perform independent (commuting) measurements which reveal that information. The observers cannot be allowed to measure the system of interest directly, for direct measurements will either (a) yield information about different properties, or (b) not (in general) commute. If the observers agree ahead of time on what measurements they will make, then they can make commuting measurements – but this violates the assumption of complete independence. Thus, information about a property must be recorded outside the system. Decoherence suggests that we should look for it in the system’s environment. Experience indicates that properties of classical objects in our Universe do become objective. We propose to explain the emergence of objectivity by showing that decohering systems typically develop redundant correlations with their environments. These correlations can be exploited by many non-interacting observers to deduce the objective properties of the system. The first papers on objectivity and redundancy [106, 105, 167] demonstrated that only a single complete observable10 can be redundantly recorded, and that this observable must coincide with the system’s pointer observable (if one exists). It was also conjectured that redundant storage of information about the system, in the environment, is a generic feature of decoherence processes. Testing this 10 An observable is complete if and only if it has no degenerate eigenvalues, so that there is no ambiguity in defining its eigenbasis. Measurement of a complete observable completely fixes the system’s post-measurement state. 10 conjecture, by constructing quantitative measures of redundancy and applying them to decoherence models, is a major theme in this work. 1.3 Overview of original material This work is firmly rooted in the “environment as a witness” paradigm. In the five research chapters that follow, we address one central question: How does the environment record information about an open quantum system? Exploration of that question is guided by a hypothesis: Through decoherence, open quantum systems [frequently] develop objectivity, from which classicality emerges. Chapters 2-4 focus on redundancy as a signature of objectivity.11 We begin (in Chapter 2)by motivating the study of redundancy, defining the structure necessary to consider it, and introducing the tools used to analyze it. We also apply these tools to some simple ensembles of states. In Chapter 3, we explore the dynamical development of redundancy in spin bath models. Chapter 4 applies the same analysis to quantum Brownian motion (oscillator bath) models of decoherence. In Chapters 5 and 6, we consider other approaches to the central question. Chapter 6 pursues objectivity through the idea of amplification, wherein many observers’ access to the same information is ensured not by repeated copying, but by increasing the visibility of a single copy. In Chapter 5, we develop an enhanced version of the predictability sieve which is useful both for connecting redundancy with predictability and for understanding the structure of information storage. The capsule summaries below are the briefest possible guides to the respective research chapters. A more detailed summary of results may be found in Chapter 8. 1.3.1 Introducing Redundancy We accomplish three tasks in Chapter 2. First, we establish a set of tools for analyzing information storage. We establish basic structural conditions for the universe under study, introduce quantum mutual information as a measure of how much information an environment fragment has about the system, and construct two mathematical tools to help determine how redundantly information is stored. Second, we show conclusively that states selected at random, from the uniform ensemble of all possible states for the universe, do not display redundant information storage. Instead, randomly selected states hide or encode information about the system throughout the entire environment. Third, we introduce singly-branching states as an alternative ensemble of states that do display redundancy. Using the tools constructed previously, we show that these “branching states” definitely store information redundantly, and compute their average redundancy. The main purpose of Chapter 2 is to introduce redundancy, and its general properties. We show that the presence of redundancy causes the information about S to separate into three parts. Some information is highly redundant, objective, and apparently classical. An equal portion is very difficult to obtain, and represents purely “quantum” properties of the system. In between, there lies a thin layer of non-redundant information, which is obtained gradually in proportion to the fraction of E captured. 1.3.2 Redundancy in spin-bath systems 1 Chapter 3 is devoted to analyzing the dynamical emergence of redundancy in a spin- 2 1 system (i.e., a qubit) coupled to a bath of spin- 2 particles. We consider four different spin bath 11 Redundancy is quantitative – e.g., we can identify that there are approximately 50 copies of a given bit of information in a particular environment. The connection to objectivity is necessarily somewhat vague, since objectivity is binary – a property is either objective or it is not. We simply note that information with no redundancy is not objective, whereas massively redundant information is definitely objective. Locating the borderline is left to future study. 11 models: interaction-only, where the only terms in the Hamiltonian are Jz ⊗ Jz interactions between the central system and its environments; quantum-measurement, in which the spins that make up the environment have internal dynamics; dynamical-system, where the central system has an independent Hamiltonian; and multiple-measurement, in which multiple non-commuting interaction terms are introduced (e.g., Hint = Jz ⊗ Jz + Jy ⊗ Jy ). The interaction-only and quantum-measurement models are analyzed in considerable detail, and shown to generate massive redundancy as predicted in Chapter 2. We note that dynamical models have varying levels of redundancy, and introduce pliability to describe different environment’s tendency to record information about the system. Using pliability, we predict specific redundancy accurately in the interaction-only model. We also discuss the effect that dynamics of individual environment have on pliability and redundancy. The other two models (dynamical-system and multiple-measurement) have markedly different behavior. The redundancy that develops initially is eroded away by intra-environment entanglement mediated by Hsys or the additional measurement terms. We examine this decay and its dependence on interaction strength in detail. 1.3.3 Redundancy in quantum Brownian motion We conclude our examination of redundancy in Chapter 4, with a study of quantum Brownian motion (QBM). After reviewing the basic properties of QBM models and presenting an overview of the simulation techniques used, we demonstrate that information about a central harmonic oscillator is stored redundantly in a bath of many oscillators. The tools introduced in Chapter 2 are used to show that information storage in QBM models is different from spin bath models, as non-redundant information becomes significant for the first time. After confirming numerically that superpositions of classically distinguishable positions lead to massive redundancy, we construct a theoretical model that explains some of the observed features. We also examine how much information is stored in different frequency bands of the environment. The data demonstrate that the resonance between S and the appropriate bands of E is important, since resonant bands store the majority of information. Resonance produces dissipation, which destroys non-redundant information and makes the remaining information unambiguously redundant. 1.3.4 Pointer bases and the operator-sieve In Chapter 5, we present a new algorithm for identifying the pointer states of a decoherence process, the operator sieve. The algorithm is based on the observation that the evolution of an open system is represented by a superoperator. Furthermore, this superoperator must be trace-preserving and completely positive – it must preserve the positivity not only of local states, but also of states which are entangled with external systems. An equivalent condition is the existence of a Kraus representation [80]: ρ −→ S [ρ] = µν Mµν ρM† µν where µν M† Mµν = 1l µν (1.8) This places constraints on the geometrical structure of the post-decoherence states. This structure is best represented in terms of operators, and the linear Hilbert-Schmidt space on which they are supported. By focusing on the most predictable observables, instead of states, we connect pointer bases with decoherence-free subsystems and subspaces. All of these structures can be combined into a theory of information-preserving structures. A theorem due to Kribs is used to show that each perfect information-preserving structure corresponds to an associative operator algebra. By cataloging the 12 possible algebras for a given Hilbert-Schmidt space, we can list all possible information-preserving structures. We show that the possible structures in 2- and 3-dimensional Hilbert spaces have simple information-theoretic interpretations. After briefly discussing the extension of this theory to imperfectly preserved structures, we apply the operator sieve to the spin bath models examined in Chapter 3. We use the results to discuss the connections between redundancy and predictability. In particular, we can show that the decay of redundancy observed for dynamical-system and multiple-measurement models is not reflected in decoherence. The operator sieve yields the amount of (1) depolarization in the pointer basis, and (2) residual (off-diagonal) coherence. Neither depolarization nor residual coherence can explain the decay of redundancy. We conclude that redundancy and decoherence are influenced by different processes. 1.3.5 Amplification Objectivity can emerge from quantum systems not only through redundant records, but through amplification [167]. In Chapter 6, we propose an inverted harmonic environment (IHE) as a model of amplification. A complete analytic solution is presented for a harmonic oscillator system coupled to an IHE environment. The master equation for the model displays surprising divergences, not seen previously in master equations. We show that these divergences do not reflect any corresponding divergence in the physics of the model, but rather a breakdown of the master-equation paradigm itself. When all information about a particular observable of S is transferred into the environment, the master equation cannot exist. Finally, we demonstrate that the decohering power of the IHE is different from (and, in a sense, greater than) other models in the literature, because the IHE can quickly resolve arbitrarily small differences in position. The system’s entropy increases linearly with time, indicating that lower- and lower-order bits of xS are being resolved – i.e., amplified – by E . ˆ 13 Part II Redundancy: a Detailed Exploration 15 Chapter 2 Introducing Redundancy When many independent observers agree about a property – e.g., the state of a system – then it is objective. The observers’ independence is crucial. When many secondary observers are informed by a single primary observer, then it is only the primary observer’s opinion, not necessarily the property which he observed, that is objective (see Fig. 2.1). The state of an isolated quantum system cannot be objective, since each independent observer will measure it in a different basis1 , which invalidates the previous observers’ results. Classical theory, in contrast, permits observers to measure a system without disturbing it; thus, properties of classical systems are objective. Each observer can record the state in question without altering it, and afterward all the observers will agree on what they discovered. Of course, observers may obtain different information – e.g., one observer may make a more effective measurement than another – but not contradictory information. Objectivity thus provides an excellent criterion for exploring the emergence of classicality through decoherence. A quantum system becomes more classical as its measurable properties become more objective. The use of “measurable” is significant. Nothing can make every property of a quantum system objective, because some observables are incompatible with others. Two observers can never obtain [simultaneously] reliable information about incompatible observables (such as position and momentum) of the same system. Decoherence partially solves this problem by destroying all the observables incompatible with a system’s pointer observable. We are thus motivated to explore (a) how the pointer observable becomes objective, and (b) how decoherence and the emergence of objectivity are related. 2.1 Redundancy and its relevance At the core of our entire project is a simple observation: information about a system (S ) is obtained by measuring its environment (E ). Although the simplest theories of quantum measurement (Von Neumann [142], etc) presume a direct measurement on the system, real experiments rely on indirect measurements. As you read this chapter, you measure the albedo of the page – but actually, your eyes are capturing quanta (photons) of the electromagnetic field surrounding it. Your measurement of the page, which provides information about its contents, actually measures its environment. Information about the page is inferred from assumed correlations between text and photons. A similar argument holds for every physics experiment; the scientist gets information about S by capturing and measuring some portion of E . This “environment as a witness” paradigm [167, 106, 105, 165] is extraordinarily useful for understanding objectivity. In order to make independent measurements of S , multiple observers 1 Assuming that the observers are independent implies that they cannot collude and choose a particular measurement basis beforehand. 16 Independent "observers" (a) (b) Non−Independent "observers" S S A ε1 ε2 ε3 ε4 ε5 (c) Perfect C−NOTs ε1 ε2 ε3 ε4 (d) S S A ε1 ? ε2 ε3 ε4 Random Gates ? ? ? ? ? ε5 ε1 ε2 ε3 ε4 ? ? ? ? Figure 2.1: A very simple model of decoherence involves controlled-not (C-NOT) gates on qubits. The four circuits above, (a)-(d), illustrate why independent observers are important for ensuring objectivity. The nth observer has access to a fragment En of the full environment, from which he attempts to deduce the state of S . In circuits (a) and (c), each observer’s fragment interacts directly with S , whereas in (b) and (d) each fragment interacts indirectly with S , through an apparatus A. Circuits (a) and (b) assume perfect C-NOT gates, while in (c) and (d) the gates are unreliable. When the gates are perfectly reliable, indirect and direct interaction lead to the same end result. However, real interactions are always somewhat unreliable! This means that A is not a perfectly reliable witness. When all the fragments En are informed by an unreliable A (e.g., (d)), the state of S does not become objective. On the other hand, independent but unreliable observers (e.g., (c)) will eventually ensure that reliable information is redundantly recorded – i.e., the state of S is objective. For a review of quantum circuits, see [102]. must partition the environment into fragments (Ei ).2 Alice captures and measures E1 , Bob gets E2 , Charlie gets E3 , and so on. Alice’s ability to obtain information about S is limited by several factors: the degree of correlation between S and E1 , her knowledge of precisely how they are correlated, and her ability to make a good measurement on E1 . When these factors permit not only Alice, but also Bob, Charlie, etc. to each obtain some information (I ) about S , then we conclude that I is objective. We can make more or less optimistic assumptions about some of these factors – e.g., an observer’s ability to measure her Ei – but the degree of correlation between S and Ei is clearly 2 In this paper, we assume that measurements must, in order to be independent, be made on distinct Hilbert spaces. This motivates our division of E into fragments according to a tensor-product structure. However, we suspect that some powerful insights would result from considering other ways of defining “independent subsystems” of E . The work of Barnum et al [12, 11] on generalized entanglement is a possible starting point. 17 a limiting factor. An observer whose particular Ei is not correlated with S has no way to obtain information about S . That observer is irrelevant, and might as well not exist. We conclude that the absolute prerequisite for demonstrating a property’s objectivity is that information about it be recorded in many fragments – that is, redundantly – in the environment. The redundancy of some information (I ) is quantified by the number of environment fragments (Ei ) which can provide I . Redundancy is a natural measure of objectivity for the property described by I [167]. Classical properties are characterized by the effectively infinite redundancy of information about them. Suppose a coin is flipped. its orientation is recorded not just by the photon field surrounding the coin, but by many independent chunks of the field. Thousands of cameras, each capturing a tiny fraction of the trillions of photons that scatter off the coin, could each provide a record of how the coin lies. Eliminate the cameras, and we can still be sure that the coin’s orientation is recorded redundantly – many photons have independently recorded it. Thus, redundancy is not dependent on actual observers. Instead, it is a statement about what observers could do, if they existed. Since quantum systems evolve unitarily, and thus (in principle) reversibly, one could argue that the information about S recorded in E is not stable. By reversing the arrow of time, the environment could be made to “forget” the information it has about S . This idea has applications in quantum information science, in schemes to mitigate decoherence [57]. As redundancy increases, however, putting the information genie back into its bottle becomes practically impossible. When the environment contains a single copy of some information, we can imagine using quantum control to recapture it – but when ten, or a thousand, or 1023 copies have been made, the situation evolves from bad to hopeless. Redundancy provides a measure of how irrevocably a quantum system has decohered. 2.1.1 The redundancy programme In order to fully understand the role that redundancy and objectivity play in the emergence of classicality (and the destruction of quantum coherence), we’d like to answer the following questions: 1. Given a state ρSE for the system and all its relevant environments (the “universe”)3 , how do we quantify the redundancy of the information (about S ) in E ? 2. For complicated systems, with many independent properties, how do we distinguish what property a particular bit of information is about? and how redundant it is? 3. When information about an observable is redundantly recorded, is information about incompatible observables inaccessible? 4. Given some assumptions about the Universe, what states are typical (that is, likely to exist)? Do they display redundancy, and if so, how much? 5. What sorts of (a) initial states, and (b) dynamics lead to the dynamical development of redundancy? 6. Finally, do realistic models of decoherence generically lead to the massive redundancy we expect in the classical regime? This is an ambitious program, much like the investigation of decoherence itself when it began a quarter-century ago. The building blocks of this work – development, primarily, of the reasoning presented in this section and the previous one – have been laid, in recent years, by [164, 165, 167, 106, 168]. The first attempt to address items (1) and (3) appeared in [106], which also analyzed a particular simple model of decoherence numerically. 3 We distinguish the model universe under study from the Universe we live in by consistently capitalizing the latter. 18 In the current chapter, we answer (1) and (4) in detail, and consider (2) and (5) briefly. We focus on thoroughly understanding the redundancy properties of individual states, and averages over a few relevant ensembles. We view the development of this theory – the kinematics of redundancy – as essential for understanding the more interesting dynamics questions inherent in (5) and (6). Objectivity and redundancy require a new paradigm, above and beyond the one normally used for examining decoherence. Models of decoherence typically focus exclusively on the system of interest. The environment is important only inasmuch as it influences the system, and once the interaction has occurred, the environment is traced out and thrown away. Here, we are interested in the environment, and how the system affects it. This has farreaching implications both for our conclusions – which are qualitatively different from previous studies of decoherence – and for our methods. Previous work on decoherence focused on obtaining the system’s reduced density matrix (ρS ), and master equations that describe its evolution. Studying Quantum Darwinism (the process by which the “fittest” observables are reproduced throughout the environment, at the expense of the incompatible properties [167]), we use density matrices only as tools, and master equations not at all. We regard the work presented here as a natural extension of the decoherence program, but employing the environment as a witness – not just as a “sink” for the information lost to decoherence – is also in a sense “beyond decoherence.” This “environment as a witness” paradigm represents the next evolutionary stage of an ongoing investigation into how effective classicality can arise in open quantum systems. 2.1.2 Computing quantitative redundancy To compute the redundancy R of information I , we divide the environment into fragments (E = E1 ⊗ E2 ⊗ . . .), and demand that each fragment supply I independently. The redundancy of I is the number of such fragments into which the environment can be divided. A canonical example of a state that displays redundancy this way is a generalized GHZ state: |ψ SE = α |0 S |00000...0 E + β |1 S |11111...1 E (2.1) Each qubit of the environment clearly provides all the available information about the system, since by measuring any sub-environment we can determine the system’s state.4 In order to analyze arbitrary states, however, we need a metric for information and a protocol for dividing the environment into fragments. We use quantum mutual information (QMI) as our information metric. QMI is defined in terms of the Von Neumann entropy H = −Tr(ρ log ρ): IA:B = HA + HB − HAB (2.2) This is simple to calculate, and a reliable measure of correlation between systems. QMI is a generalization of the classical mutual information.5 Figure 2.2 provides a diagrammatic perspective of classical mutual information. Unlike classical mutual information, the QMI between two systems A and B is not bounded by the entropy of either system. In the presence of entanglement, the QMI can be as large as HA + HB , reflecting the existence of quantum correlations beyond the classical ones. The need for a protocol to divide the environment is less obvious. Allowing arbitrary tensor-product decompositions of the environment is equivalent to permitting arbitrary unitary transformations on the entire environment. This would make every state where S is entangled6 with 4 We must make the right measurement – in this case, one which distinguishes |0 from |1 – in order to get the information. In this work, the amount of information that one subenvironment has is always maximized over all possible measurements. 5 Classical MI is defined in the same way, but with Shannon entropy instead of Von Neumann entropy. See also Fig. 2.2, and the book by Cover and Thomas [32]. 6 In this work, we assume that the universe is in a pure state. Any correlation between S and E is due to entanglement. Similar conclusions seem to apply when the environment is initially mixed, but we have not investigated these cases exhaustively. 19 H(S) H(E) H(S|E) I(S:E) H(E|S) H(S,E) Figure 2.2: Venn diagram illustrating various forms of classical entropy and information. H (S ) is the entropy of the central system, H (E ) is the entropy of an environment (or fragment thereof), and H (S , E ) is the entropy of both at once (i.e., S ⊗ E considered as a single system). Their mutual information is I (S : E ). The mutual information is defined as I (S : E ) = H (S ) + H (E ) − H (S , E ). It measures the amount of entropy that would be produced by destroying correlations between the parts. This picture is only partially correct for quantum mutual information, however. E equivalent to a GHZ-like state (Eq. 2.1), and the concept of redundancy would be trivialized. A pre-existing concept of locality, usually expressed as a fixed tensor product structure or a set of allowable structures, is fundamental to redundancy analysis. Both decoherence and entanglement require a fixed division between the system and its environment (see Fig. 2.3a). Redundancy requires that the environment have an Nenv -part tensor product structure of the form E = E1 ⊗ E2 ⊗ E3 ⊗ . . . ENenv , (2.3) where the Ei are indivisible subenvironments (see Fig. 2.3b). Once the basic subenvironments are established, they may be rearranged into fragments (see Fig. 2.3c). We use the notation E{i1 ,i2 ,...im } to denote the fragment consisting of the m subenvironments {Ei1 , Ei2 , . . . Eim }. Frequently, only the total number of subenvironments is significant, and we write the fragment as E{m} . In general, not all of the available information about S (or any property of S ) is stored redundantly.7 Following the reasoning in [106], we demand that each fragment provide some large fraction, 1 − δ (where δ 1), of the available information about S . The available information is the entropy of ρS (HS ), and the precise magnitude of the information deficit δ should not be important. We denote the redundancy of “all but δ of the available information” by Rδ ; e.g., for a deficit of δ = 0.1, we are computing R0.1 or R10% . To compute Rδ , we start by defining Nδ as the number of disjoint fragments Ei such that IS :Ei ≥ (1 − δ )IS :E . We might just define Rδ = Nδ , except for two caveats. 1. Allowing a deficit (δ ) could lead to spurious redundancy. Suppose that N = 5 fragments exist, each of which provides full information. If δ = 0.5, then we might split each fragment 7 The GHZ state in Eq. 2.1 is the exception that proves the rule. Such states are measure-zero in Hilbert space. Perfect C-NOT interactions are required to make them. 20 E S (a) Decoherence Paradigm: Universe is divided into System & Environment E7 E6 E8 E1 S E5 E4 E2 E3 (b) Redundancy Paradigm: Environment is divided into Subenvironments } E {1,7,8 E8 E1 E7 E6 E {2,3} E2 E3 E4 S E5 Figure 2.3: Three ways to divide up the universe. In order to discuss decoherence, we must divide the universe into a system (S ) and an environment (E ) as in (a). To consider redundancy, we must further subdivide the environment into subenvironments, as in (b). The basic idea is that (1) no subenvironment can be further subdivided, and (2) it’s easier to measure one subenvironment than to make a joint measurement on several. In order to obtain fragments of the environment that contain sufficient and independent information, we combine subenvironments as in (c). The fragments’ information is sufficient to infer the system’s state by construction, and they are independent of each other because measurements on distinct subsystems of E are guaranteed to commute. } ,6 E {5 E {4} (c) Subenvironments are combined into Fragments that each have nearly-complete information. 21 2 1.5 I(m) 1 (b) (a) (c) 0.5 0 2 4 m 6 8 10 12 Figure 2.4: The three basic profiles for partial information plots (I vs. m). Curve (a) shows the behavior of independent environments; curve (b) shows redundantly stored information; and curve (c) illustrates the case where information is encoded in multiple environments. in half and obtain Nδ = 10 fragments that provide “sufficient” information. This would be a misrepresentation of the redundancy. To compensate for possible spurious redundancy introduced by δ , we replace Nδ with (1 − δ )Nδ . 2. Because of quantum correlations, IS :Ei can be as high as 2HS . We allow for this by assuming that one fragment’s information represents strictly quantum correlations, and throwing it away. This means replacing (1 − δ )Nδ with (1 − δ )Nδ − 1. Our resulting formula for Rδ is thus a guaranteed lower bound for the redundancy: Rδ ≥ (1 − δ )Nδ − 1, For a more detailed discussion of the reasoning involved, see Appendix A.1. (2.4) 2.1.3 Identifying qualitative redundancy Frequently, the actual amount of redundancy is not as important as the qualitative observation “there’s a lot of redundancy here.” Moreover, it’s useful to have a broader spectrum of information: e.g., how dependent is R on δ ? or why does a state have virtually no redundancy? For these purposes, we plot the amount of information about S supplied by a fragment of size m (IS :E{m} ), against m. Since there are very many fragments of a given size, we average IS :E{m} over a representative sample of fragments to obtain I (m). The plot of I (m) is called a partial information plot (PIP), since it shows the partial information yielded by a partial environment. We proved in [20] (see also Appendix A.2) that when the universe is in a pure state, the PIP must be anti-symmetric around its center (see Figure 2.4). Together with the observation that it 22 must be strictly non-decreasing (capturing more of the environment cannot decrease the amount of information obtained), this permits the three basic profiles shown in Figure 2.4. Redundancy (see Fig. 2.4b) is characterized by a rapid rise of I at relatively small m, followed by a long “classical plateau”, where all the available information has been obtained, and additional environments provide nothing new. Only by capturing all the environments can an observer can manipulate quantum correlations. The power to do so is indicated by the sharp rise in I at m ∼ Nenv . 2.2 Characterizing randomly-selected states Since redundant information appears to be a ubiquitous feature of the classical world, we might na¨ ıvely expect that states chosen at random from the Hilbert space of the universe would display massive redundancy. To test this hypothesis, we compute partial information plots for random states, averaging over the uniform ensemble. This was first done in [20]; we reproduce the main results here, and extend to arbitrary sizes of the system and environment. 2.2.1 The uniform ensemble For any [finite] D-dimensional Hilbert space, there exists a uniform (unitarily invariant) distribution over states. This is technically a Lebesgue measure on a compact manifold CP (D − 1); however, it is frequently referred to as Haar measure, since the measure is induced by the invariant Haar measure on the symmetry group U (D). Methods for averaging over the uniform ensemble fall into three categories: (1) explicitly parametrizing the measure for a fixed D; (2) numerical generation of random states using a normalized Gaussian distribution; (3) clever tricks involving symmetries of the space. We used methods (2) and (3) to produce PIPs. Numerical averaging is simple and effective for relatively small environments. Since the Hilbert space of the universe grows exponentially with the number of environments, this method becomes impractical somewhere around Nenv = 20 qubits. Fortunately, the average can also be performed analytically, by means of a formula for the average entropy of a subspace, conjectured by Page [110] and proved by Sen [132] and others [46, 125]. The details of this calculation are given in [20] and in Appendix A.4; here we present only the major results. All the results presented were obtained using the analytical formulae in Appendix A.4, and verified independently by numerical averaging. 2.2.2 Partial information plots Partial information plots (Fig. 2.5-2.8) demonstrate that typical states from the uniform ensemble do not display redundancy. Figure 2.5 illustrates typical behavior. As an observer captures successively more subenvironments (increasing m), virtually no information about S is gained (i.e., IS :E{m} remains nearly zero). Only when approximately Nenv of the subenvironments have been 2 captured, does the observer begin to gain substantial information. At that point, I rapidly rises through Hs , and on to nearly 2Hs . Information about S is encoded in the environment, in much the same way that a classical bit can be encoded in the parity of an ancillary bit string. In the classical case, however, every bit of the environment must be captured to deduce the encoded bit. This “anti-redundancy”, or encoding, phenomenon is related to quantum error correction [76, 54, 130]. Since any majority subset of the environments has essentially full mutual information with the system, the loss of any minority of them doesn’t affect the information stored in the rest. These states can be used as a quantum code which protects against bit loss – at least until 50% of the participating states are lost. The somewhat remarkable aspect is that generic states8 have this 8 e.g., states selected randomly according to Haar measure from the whole SE Hilbert space. 23 PIPs for Random States: qubit system 2 I (bits of information obtained) 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 12 m (# of subenvironments captured) Nenv = 2 Nenv = 4 Nenv = 6 Nenv = 8 Nenv = 10 Nenv = 12 Nenv = 16 14 16 Figure 2.5: Partial information plots (PIPs) illustrate how information about S is stored in E . PIPs plot the amount of information (I ) that can be obtained from a fragment of the environment (E{m} ), against the fragment’s size (m). A qubit system (DS = 2) is coupled to environments consisting of Nenv qubits (DE = 2). The PIPs are obtained by averaging I (m) over all states in the uniform ensemble. They display the same general profile regardless of the environment’s size. No significant information is obtained until almost half the subenvironments have been captured. Once m > Nenv , 2 however, virtually all possible information (both quantum and classical) is available. Because more than half the environment is required to obtain useful information, there is no redundant information storage in typical uniformly-distributed states. property – random states form a nearly-optimal error-correction code for bit-loss errors. Similar behavior was observed for classical codewords by Shannon[135]. Figure 2.6 extends this result to larger systems. The shape of the PIP continues to indicate that information is encoded; only the total amount of encoded information changes. In Figure 2.7, we keep the total Hilbert space dimension of the environment constant at D = 216 , but vary its division into subsystems. Except for the scaling of the m-axis, the resulting curves are identical. We conclude that the appropriate measure for “amount of environment” is not the number m of captured subsystems, but the fraction of the total Hilbert space that is captured. Accordingly, we rescale the axis in this fashion, and obtain the scaled PIPs shown in Fig. 2.8. 2.2.3 Conclusions We can summarize our results for uniform ensembles as follows. Most importantly, typical states selected randomly from the uniform ensemble display no redundant information 24 PIPs for Random States: larger systems 8 I (bits of information obtained) 7 6 5 4 3 2 1 0 0 2 4 6 8 10 12 m (# of subenvironments captured) 14 16 DS = 2 DS = 4 DS = 6 DS = 8 DS = 10 DS = 12 DS = 16 Figure 2.6: Partial information plots (PIPs) for different system sizes (DS ) reinforce the conclusions in Fig. 2.5. The environment is fixed (16 qubits), but DS is varied from 2 to 16. As DS increases, the total amount of available information grows as HS = log(DS ). However, information continues to be “encoded” in entangled modes of E , as seen previously in Fig. 2.5. The slope of the PIP at ∂I m = Nenv increases initially, then asymptotes at ∂m = 1 (see Appendix A.3.1 for a discussion). No 2 more than 1 bit of additional information can be gained, on average, from 1 bit of the environment. storage; instead, they display encoding or anti-redundancy. This is not to say that all states are nonredundant, merely that the redundant states are uncommon and not “typical”. As 1 we show in Appendix A.3.1, the average value of I (m) decreases exponentially away from m = 2 N . Thus, as N grows, the states where information is not encoded this way must be exponentially rare. This provokes a dilemma. Na¨ ıvely, it is tempting to conjecture that the complex dynamics of the Universe should lead to relatively random states. The properties of random states, however, are inconsistent with observations of the Universe around us. Massive redundancy is ubiquitous in the Universe; as you read this chapter, you obtain excellent information about it while measuring a tiny fraction of the available photons. We conclude that the uniform ensemble cannot describe typical states of the Universe. The interactions of systems with their environments must pick out some preferred subensemble of states that is characterized by greater redundancy. In the next section, we suggest and analyze such an ensemble. 2.3 Decoherence and branching states Since decoherence – the loss of information to the environment – is a prerequisite for redundancy, we look to decoherence theory for inspiration. The simplest models of decoherence [73] 25 PIPs for random states: equivalent environments I (bits of information obtained) 8 7 6 5 4 3 2 1 0 0 Dε = 2, Nenv = 24 Dε = 4, Nenv = 12 Dε = 8, Nenv = 8 Dε = 16, Nenv = 6 5 10 15 20 m (# of subenvironments captured) Figure 2.7: This plot shows PIPs for a single qubit system coupled to several equivalent environments with Dtotal = 216 . The individual environments, however, are {2,4,8, or 16}-dimensional, and their number is scaled appropriately. Most notably, the plots are essentially identical – only the m-axis changes. If we plot in terms of captured fraction of the environment, these plots are identical. are essentially identical to those of quantum measurement theory. A set of pointer states ({|n }) for the system are singled out, and the environment “measures” which |n the system is in, by evolving from some initial state (|E0 ) into a conditional (upon n) state (|En ). If ρS is written out in the pointer basis, the diagonal elements (ρnn ) of ρS remain unchanged. Coherences between different pointer states (e.g., ρnm ) are reduced by a decoherence factor γnm = En |Em . (2.5) Such models, when used with multiple environments, presume that (a) the environments Ei are initially unentangled, and (b) each environment “measures” in the same basis of the system (otherwise the pointer basis is destroyed). We simplify the model further, by assuming that the state of the universe is pure. Condition (a) above thus implies that the initial state of the universe is a product state: |Ψ 0 = |S ⊗ |E1 ⊗ |E2 ⊗ . . . |EN . (2.6) In this simple model, the environments do not interact with each other, and the system does not evolve. The state of the model universe at an arbitrary time t can be written rather simply. Letting the initial state of the system be |S = n sn |n , where each |n is a pointer state for the system9 , 9 i.e., in this case, an eigenstate of the interaction Hamiltonian. 26 Scaled PIPs: qubit system 1 fI (dimensionless) fI (dimensionless) 0.8 0.6 0.4 0.2 0 0 0.2 Nenv = 2 Nenv = 4 Nenv = 6 Nenv = 8 Nenv = 12 Nenv = 16 0.4 0.6 0.8 fcap (dimensionless) 1 1 0.8 0.6 0.4 0.2 0 0 Scaled PIPs: larger systems DS = 2 DS = 4 DS = 6 DS = 8 DS = 12 DS = 16 0.2 0.4 0.6 0.8 fcap (dimensionless) 1 (a) (b) Figure 2.8: Scaled versions (SPIPs) of the plots in Figs. 2.5,2.6. This form of the PIP is particularly useful for comparing environments with different numbers of subenvironments, and for computing Rδ , the redundancy for a given fraction 1 − δ of the total information. To estimate redundancy, simply draw a horizontal line at fI = 1−δ , and note the value of fcap where it intersects the PIP. This 2 provides a good estimate of 1/Rδ . It is not a perfect estimate for several reasons; most importantly, the PIP and SPIP plot the average I obtained from a given-sized fragment of the environment. This is not the same as the average fragment size (m) required to obtain I , since we average the same data over different variables. In these plots, of course, no redundancy is evident – we are looking ahead to the next section. we get: |Ψ t = n sn |n S (1) (2) ( ⊗ |En ⊗ |En ⊗ . . . |EnN ) , (2.7) ( where |Enj) is the state into which the j th environment evolves if the system is in state |n . The conditional states of the environments will not generally be orthogonal, except in highly simplified (e.g. C-NOT) models. 2.3.1 The branching-state ensemble We refer to these states (Eq. 2.7) as singly branching states, or simply as branching states. The wavefunction of a branching state has DS branches, in the Everett interpretation of quantum mechanics, each of them perfectly correlated with a particular state of the system. The subenvironments are not entangled with each other, only correlated (classically) via the system. A typical random state from the uniform ensemble, on the other hand, has Duniverse branches, with a new branching at every subsystem. The particular branching state that describes the universe at a given time will depend on the environment’s initial state, and on its dynamics. In this chapter, we sidestep the difficulties of specifying these parameters by considering the ensemble of all branching states. The conditional ( |Enj) are selected randomly from each subenvironment’s uniform ensemble. Each pointer state of the system is correlated with a randomly chosen product state of all the environments. The amount of “available” information is set by the system’s initial state vector (i.e., the sn coefficients). The environment obtains information about S through the [pre]-measurement interaction between them, which determines what observable the environment measures. If the system’s state is an eigenstate of the measurement observable, then the measurement has no effect. Such is a state is minimally measurable : no information (beyond that already possessed) can be gained by the measurement that Hint defines. More generally, a state’s measurability is the amount 27 of new information that measurement can provide. Decoherence can be viewed as a measurement whose outcome is not yet known; by measuring the environment, we determine which of the possible measurement outcomes occurred, and thus gain information. The post-decoherence eigenvalues of ρS are |sn |2 , and these determine the system’s entropy. We cannot consider all possible states, so we focus on generalized Hadamard states, where 1 ∀ n. (2.8) sn = √ DS These generalized Hadamard states are maximally measurable, and therefore show the effects of measurement most clearly. To verify that our results are generic, we also treat (briefly) another class of initial states. Generally, however, we focus on Hadamard-like states for quantitative results By examining the branching-state ensemble, we are not conjecturing that the Universe is found exclusively in branching states. Branching states form an interesting and physically wellmotivated ensemble to explore. We shall see that, unlike the uniform ensemble, the branching-state ensemble displays redundancy in way consistent with observations of the physical Universe. Our Universe might well tend to evolve into similar states, but we are not prepared to make such a conjecture. Characterizing the states in which the physical Universe (or a fragment thereof) is found is a substantially more ambitious project. 2.3.2 Numerical analysis of branching states We begin our exploration of branching states by examining typical PIPs, for various systems and environments. By averaging PIPs over the ensemble, the adjustable parameters are reduced to three: DS , DE , and Nenv . These PIPs confirm that information is stored redundantly. We proceed to examine a quantitative measure of redundancy (Rδ ), and its dependence on the parameters of the universe. Finally, we derive some analytical approximations, compare the results to numerical data, and discuss their implications. Partial information plots The partial information plots for branching states (Figs. 2.9, 2.10) display precisely the features of redundancy discussed in section 2.1.3. A rapid rise from I = 0 is followed by an asymptotic approach to I = HS , producing a classical plateau. We see that the degree of redundancy Nenv is determined almost exclusively by the initial rise at m 2 . As Nenv grows beyond a certain point, the interesting regimes at m ∼ 0 and m ∼ Nenv remain constant, and the classical plateau simply extends to connect them. Again, this is entirely consistent with classical experience. The initial bits of information that we gain about a system are extremely useful, but eventually we reach a point of diminishing returns where all the available information has already been deduced. In Section 2.2.2, we found that different environments were equivalent when their total Hilbert space sizes were the same (e.g., 16 qubits are equivalent to 8 ququits). For branching states, we find a similar behavior, even though the shape of the PIPs is very different. In Fig. 2.11, we examine scaled PIPs, which plot the fraction of total information obtained (fI ) against the fraction of the environment that has been captured. The SPIPs allow us to compare environments with different sizes. Figure 2.11b illustrates results for a 16-dimensional system, coupled to nine different environments, which yield virtually identical behavior. While the number and size of the subenvironments range widely, each environment has a total information capacity (c = log (dimH)) of 120 bits, and their scaled behavior is essentially identical. Non-Hadamard states: a necessary digression We also consider some states that are not maximally measurable – i.e., non-Hadamard states, where the sn vary. We consider states which are as different as possible from a generalized 28 Branching states (DS = Dε = 2) 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 m (# of env.) 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 Branching states (DS = 2, Dε = 5) (bits) {m} Nenv = 4 Nenv = 6 Nenv = 8 Nenv = 10 Nenv = 12 10 12 {m} (bits) Nenv = 4 Nenv = 6 Nenv = 8 Nenv = 10 Nenv = 12 2 4 6 8 m (# of env.) 10 12 IS:ε (a) Branching states (DS = 5, Dε = 2) 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 2 4 6 8 m (# of env.) 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 2 4 IS:ε (b) Branching states (DS = Dε = 5) (bits) {m} Nenv = 4 Nenv = 6 Nenv = 8 Nenv = 10 Nenv = 12 10 12 {m} (bits) Nenv = 4 Nenv = 6 Nenv = 8 Nenv = 10 Nenv = 12 6 8 m (# of env.) 10 12 IS:ε (c) IS:ε (d) Figure 2.9: Here we examine the raw (unscaled) partial information plots for branching ensembles. The average information I obtained by capturing m out of N environments is plotted for: (a) a qubit decohered by qubits; (b) a qubit decohered by qupents (D = 5); (c) a qupent decohered by qubits; and (d) a qupent decohered by qupents. In each case, as Nenv is increased from 4 to 12, redundancy appears, in the guise of a plateau centered at m = Nenv . As Nenv increases, I (m) at 2 m Nenv approaches its asymptotic form. We also note, in (c), the cause of the fact (mentioned in Fig. 2.13) that large systems display no redundancy at small Nenv . The available environment is barely sufficient to decohere the system, and so the environments hold essentially independent information. Hadamard state, defined by 1 sn ∝ √ . 2n (2.9) All the eigenvalues of ρS (after decoherence) are distinct. In fact, the spectrum of ρS is exactly that of a thermal spin – i.e., a particle with a Hamiltonian H = Jz , in equilibrium with a bath at finite temperature. For this reason, we refer to these states as thermal states. A simple calculation shows that the entropy of a decohered thermal state does not continue to increase as DS becomes larger. Instead, it asymptotically approaches H = 2 bits. This is exactly the entropy of a Hadamard state for DS = 4. Our general approach is to assume that the system’s maximum entropy determines its informational properties, so in the limit DS → ∞, thermal states should behave much the same as a DS = 4 Hadamard state. PIPs for thermal states with DS = 16, and Hadamard states with DS = 4, are shown in Fig. 2.12. The plots are very similar, confirming our conjecture that HS is the major factor in how 29 PIPs for a homogenous (DS = Dε) universe 2 1.8 1.6 1.4 / Hs IS:ε {m} 1.2 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 m (# of env.) DS = Dε = 2 DS = Dε = 3 DS = Dε = 4 DS = Dε = 5 12 14 16 Figure 2.10: When the system and its environments are the same size, 16 independent environments are sufficient to produce noticeable redundancy (R10% ∼ 5) – but not so much that the endpoint behavior of I (m) is obscured. The PIPs shown here, for D ∈ {2, 3, 4, 5}, illustrate that if DS and DE increase in lockstep, the redundancy profile remains largely unchanged. The exception, for D = 2, seems surprising given that its actual redundancy (see Fig. 2.13) is very near to that for D = 3. It occurs because a small minority of qubit environments provide anomalously little information about the system, and distort the averaged PIP. Note also that the statement that “the redundancy profile remains largely unchanged” as DS and DE are scaled is only a qualitative observation – the actual scaling is quite a bit more complex. information about S is recorded. A careful examination of Fig. 2.12 reveals that information about the thermal states is gained more slowly as m increases. However, the effect is quite small. For the rest of this discussion, we will specialize to Hadamard states for the sake of convenience. Redundancy: numerical values Branching states are natural generalizations of GHZ states, so we expect redundant information storage. Figure 2.13 confirms this for a wide range of parameters. The amount of redundancy is proportional to the size of the environment, which agrees with the classical intuition that very large environments should store many copies of information about the system. Larger subenvironments (measured by DE ) increase redundancy by storing more information in each subenvironment. Conversely, larger systems have more properties to measure, which in turn require more space for information storage. The total amount of redundancy is reduced for large DS . The other important feature of the plots in Fig. 2.13 is the relatively weak dependence 30 SPIPs vs. Nenv (DS = Dε = 3) 2 1.5 / Hs / Hs {m} {m} Equivalent Environments (branching states) 1.2 1 0.8 0.6 0.4 0.2 0 1 0 1 0.5 0 0 Nenv = 4 Nenv = 8 Nenv = 16 Nenv = 32 Nenv = 64 Nenv = 128 0.2 0.4 0.6 0.8 Fraction fcap of Env. captured ε = 16 x 30 ε = 12 x 33 ε = 10 x 36 ε = 8 x 40 ε = 6 x 46 ε = 5 x 51 ε = 4 x 60 ε = 3 x 75 ε = 2 x 120 0.02 0.04 0.06 0.08 0.1 fcap (fraction of ε) 0.12 0.14 IS:ε (a) IS:ε (b) Figure 2.11: Scaled partial information plots are well-suited to comparing redundancy in states with different environments. Here, a qutrit system is coupled to environments with: (a) different sizes; and (b) different sizes and different dimensions. Plot (a) shows the increase of redundancy with environment size. In plot (b) we see that the size Nenv and dimension DE of the environment can be scaled in such a way that the redundancy remains unchanged. The key quantity is the total N Hilbert space dimension ((DE ) env ) – or its logarithm, the information capacity (c). Plot (b) also illustrates the difference between the regime of linear information gain, for about the first 4% of the environment, and the exponential convergence to the “classical plateau” thereafter. of Rδ on the deficit δ . As δ is varied over a full order of magnitude (between 2% and 25%), Rδ changes by less than a factor of 2. Changing δ (the information deficit) does not affect the distinction between classical (massively redundant) and quantum (nonredundant) information. The details of computing the QMI for branching states are given in Section 2.3.3. 2.3.3 Theoretical analysis of branching states The numerical analysis above offers compelling evidence that information is stored redundantly in branching states, that the amount of redundancy scales with Nenv , and that Rδ is relatively insensitive to δ . We now construct a theoretical model to confirm these conclusions, and also to gain a better understanding of how redundancy scales with various parameters. We present two analytical approximations: one to model PIPs (i.e., I (m)) and one to model quantitative redundancy (by predicting m(I )). A simple algorithm for computing the mutual information between S and a fragment is central to both approaches. We begin by using the structure of a branching state to obtain such an algorithm. The structure of branching states To compute the mutual information IS :E{m} = HS + HE{m} − HSE{m} (2.10) between the system and a partial environment E{m} , we require the entropies of ρS , ρE{m} , and ρSE{m} . The structure of the states given by Equation 2.7 simplifies the partial-tracing, so that each of the relevant density matrices (regardless of its actual dimension) is rank-DS . That is, the reduced states for S , E{m} , and SE{m} are all “virtual qudits” with D = DS . These density operators “live” in totally different Hilbert spaces. However, by eliminating zero eigenvalues, each can be reduced to its DS -dimensional support. When this is done, each of 31 Non-Hadamard states: Dε = 2 4 3.5 3 (bits) 2.5 2 1.5 1 0.5 0 0 2 4 ‘‘Thermal’’ state, DS = 16 Hadamard state, DS = 4 6 8 10 m (# of env.) 12 14 16 (bits) 4 3.5 3 2.5 2 1.5 1 0.5 0 0 2 Non-Hadamard states: Dε = 3 {m} IS:ε IS:ε {m} ‘‘Thermal’’ state, DS = 16 Hadamard state, DS = 4 4 6 8 10 m (# of env.) 12 14 16 (a) Non-Hadamard states: Dε = 4 4 3.5 3 (bits) 2.5 2 1.5 1 0.5 0 0 2 4 ‘‘Thermal’’ state, DS = 16 Hadamard state, DS = 4 6 8 10 m (# of env.) 12 14 16 (bits) 4 3.5 3 2.5 2 1.5 1 0.5 0 0 2 4 (b) Non-Hadamard states: Dε = 5 {m} IS:ε IS:ε {m} ‘‘Thermal’’ state, DS = 16 Hadamard state, DS = 4 6 8 10 m (# of env.) 12 14 16 (c) (d) Figure 2.12: PIPs for non-Hadamard-like states. We construct “thermal” states of a large (DS = 16) spin, by assigning sn ∝ √1 n . This state decoheres into a density matrix whose spectrum matches 2 that of a thermal spin. The entropy of this density matrix is 2 bits (as opposed to 4 bits for a DS = 16 Hadamard state). Hadamard states with DS = 4 also develop 2 bits of entropy. We compare the PIPs for “thermal” DS = 16 states to PIPs for Hadamard DS = 4 states. The subenvironments’ size is varied from DE = 2 in plot (a) to DE = 5 in plot (d). These PIPs confirm that our observations apply to non-Hadamard states. the three ρ is spectrally equivalent to a partially decohered variant of the undecohered system state, which is |S0 S0 | = nm sn s∗ |n m m|. (2.11) In other words, we can obtain ρS , ρE{m} , or ρSE{m} by taking |S0 elements according to a specific rule (defined below). S0 | and suppressing the off-diagonal To do this, we first define the multiplicative decoherence factor, γ : γij = (k ) Ei (k) |Ei (k) . (2.12) 32 R10% (DS = Dε) 30 Rδ (redundancy) 25 20 15 10 5 0 0 10 20 30 40 50 Nenv (size of environment) 60 DS = Dε = 2 DS = Dε = 3 DS = Dε = 4 DS = Dε = 5 30 Rδ (redundancy) 25 20 15 10 5 0 0 R10%: Dependence on Dε (DS = 5) Dε = 2 Dε = 3 Dε = 4 Dε = 5 10 20 30 40 50 Nenv (size of environment) 60 (a) R10%: Dependence on DS (Dε = 4) 35 30 Rδ (redundancy) 25 20 15 10 5 0 0 10 20 30 40 50 Nenv (size of environment) 60 DS = 2 DS = 3 DS = 4 DS = 5 DS = 8 DS = 16 30 Rδ (redundancy) 25 20 15 10 5 0 0 10 (b) Rδ: Dependence on δ (DS = Dε = 5) R25% R10% R5% R2% R1% R0.5% R0.1% 20 30 40 50 Nenv (size of environment) 60 (c) (d) Figure 2.13: The amount of redundancy present in an assortment of branching-state ensembles. Each plot shows the ensemble-average of the redundancy (R), as a function of the number of environments. In every case, R increases linearly with the number of environments. The environments appear to be fungible; any n of them (where n is a function of the parameters) provide sufficient information. Plot (a) shows the redundancy typical of a D-dimensional system decohered by D-dimensional environments. The level of redundancy increases slightly with system size, but not dramatically. Plot (b) demonstrates the strong dependence of R on the sizes of the individual environments. As expected, larger environments store more information in proportion to their information capacity log(D), and produce correspondingly greater redundancy. Plot (c) varies the size of the system; we see a similar 1/ log(D) dependence here except for D = 2. Larger systems also display no redundancy for small Nenv , when the environment is too small. Plot (d) shows the effect of varying the information deficit (δ in Rδ ). Note that varying δ by a full order of magnitude (from 2% to 25%) changes R by less than 50%. For an analysis of the slopes of these lines (specific redundancy), see Fig. 2.15. Now, the three relevant density matrices are given by: i| ρS |j = = = (si s∗ ) j k∈E γij , γij , k∈E{m} (k ) (k ) (2.13) (2.14) (2.15) i| ρE{m} |j (si s∗ ) j (si s∗ ) j k∈E{m} i| ρSE{m} |j γij , (k ) 33 Each of the three appears to have been decohered by a different subset of the subenvironments: • ρS has been decohered by every environment, • ρE{m} has been decohered by all the environments in E{m} ,10 • ρSE{m} has been decohered by all the environments not in E{m} . An alternative (and highly useful) representation of decoherence factors is the additive decoherence factor, (k ) (k ) dij ≡ − log γij . (2.16) We can rewrite products of γ -factors as sums of d-factors: ei k (k) | ej (k) = exp − k dij (k ) (2.17) Since the d-factors are additive, the total decoherence factor of a fragment is a sum over subenvironments: dij {m} dij dij (E ) = k∈E{m} dij dij k∈E (k ) (k ) (2.18) (2.19) (S ) = = (SE{m} ) dij . k∈E{m} (k ) (2.20) The structure of branching states simplifies the mutual information calculation. Instead of calculating the entropy of very large density matrices, we need only compute H = −Trρ ln ρ for the three DS × DS operators ρS , ρE{m} , and ρSE{m} . This calculation can be done analytically for qubit systems (see [20] for extensive details). For larger DS , H must be computed numerically, a fairly easy task as long as DS is fairly small. Theoretical PIPs: averaging I (m) In this section, we derive a fairly simple analytic model to explain features of the PIPs presented earlier. As the number of subenvironments that contribute to decohering a particular density matrix ρ increases, the off-diagonal ρij rapidly decline to zero, while the diagonal terms remain fixed. We begin with a fully-decohered (diagonal) ρ0 , with eigenvalues λi = |si |2 and entropy H0 . The off-diagonal elements (ρij = γij si s∗ ) are a perturbation, where γij is the small parameter. j Our approach is to write ρ = ρ0 + ∆, where ∆ is a small off-diagonal perturbation to ρ0 . The entropy of ρ is expanded in a series around H (ρ0 ), as H (ρ) = = ≈ ≈ −Tr [ρ ln(ρ)] −Tr [(ρ0 + ∆) ln(ρ0 + ∆)] −Tr [ρ0 ln(ρ0 )] + O(∆) + O(∆2 ) + . . . H (ρ0 ) + O(∆) + O(∆2 ) + . . . (2.21) (2.22) (2.23) (2.24) 10 If this seems counter-intuitive, it may help to recall that for any bipartite decomposition of a pure state |Ψ AB , the reduced states ρA and ρB are spectrally equivalent. Thus ρE{m} is equal to ρS E , where E{m} contains all the { m} environments not in E{m} . 34 An intuitively appealing starting point is the MacLaurin expansion of H (x) = −x ln(x), which yields H (ρ0 + ∆) ≈ H (ρ0 ) − Tr [∆(1 − ln(ρ0 ))] − 1 ∆2 1 ∆3 ... + 2 ρ0 6 ρ2 0 (2.25) However, this approach immediately runs into problems. k+1 The problem is that the matrix quotient ∆ρk is not well-defined when the two matrices do not commute. Additionally, the 1st order term vanishes because ∆ is purely off-diagonal and 1l − ln(ρ) is purely diagonal (in the pointer basis), so their trace vanishes. The leading term is an ill-defined matrix quotient. The solution to this problem involves a more involved expansion of H (ρ) around ρ = 1l. The full derivation is quite arduous, and may be found in Appendix A.5. We end up with a series for H (ρ0 + ∆) which is equivalent to Eq. 2.25 for scalars, but involves (a) expanding ρ−k in a power 0 series, and (b) taking a totally symmetric product between ∆k+1 and the resulting power series. By inserting ∆ and ρ0 , we obtain (as the final result of Appendix A.5) the following expression for H (ρ0 + ∆): H (ρ) ≈ H (ρ0 ) − h(ρ0 ) = |γ |2 h(ρ0 ) 2 ∞∞ Tr [ρ0 (1l − ρ0 )p ] Tr [ρ0 (1l − ρ0 )n ] −1 . n+p+1 n=0 p=0 (2.26) (2.27) The factor of |γ |2 which appears here is an average over all i, j of the decoherence factors |γij |2 . This expression is exact to leading order – which is to say for sufficiently small |γ |2 . The complicated function h(ρ0 ) in Eq. 2.27 does not simplify in general. However, it is well approximated by the effective Hilbert space dimension of ρ0 . To see this, we consider the special 1 1l case where ρ0 has D identical eigenvalues, λi = D . When reduced to its support, ρ0 = D . The summation can be done explicitly: h(ρ0 ) = Tr D−1 1l(1l − D−1 1l)p Tr D−1 1l(1l − D−1 1l)n −1 n+p+1 n=0 p=0 ∞ ∞ ∞ ∞ (2.28) = n=0 p=0 ∞ ∞ (1 − D−1 )p (1 − D−1 )n −1 n+p+1 1 − D−1 n+p+1 1 − D−1 n+p (2.29) = n=0 p=0 ∞ −1 (2.30) = n+p=0 n+p −1 (2.31) (2.32) = D−1 Note that D appeared only based on the eigenvalue spectrum of ρ0 . In the example above, the entropy H0 of ρ0 is H0 = log(D). Since the total range of I (m) is proportional to H0 , a logical generalization of h(ρ0 ) = D − 1 is h(ρ0 ) ≈ eH0 − 1. (2.33) Numerical experimentation, particularly for DS = 2 (where an exact analytic result can be obtained) has confirmed that Eq. 2.33 is a good approximation everywhere, in addition to being exact for (1) maximally mixed states, and (2) nearly-pure states. Substituting Eq. 2.33 into Eq. 2.27 yields H (ρ) ≈ H (ρ0 ) − |γ |2 H0 e −1 2 (2.34) 35 The γij depend on the details of ψSE . However, when they are small enough to count as a perturbation on ρ, the Hilbert space of the environment is very large. The |γij |2 can then be treated as independent random variables, so |γ 2 | is equal to an average over the entire branching state ensemble: |γ 2 | = = = ψ |ψ ψ |ψ ψ | |ψ ψ Tr |ψ | Tr (1l1l) 2 DE (2.35) − = DE 1 This is the mean value of |γ 2 | from a single subenvironment. For a collection of m subenvironments, − m such γ factors are multiplied together, so the mean value of |γ 2 | becomes DE m . Putting this all together, the average entropy of a DS -dimensional system decohered by m DE -dimensional environments is eH0 − 1 −m H = H0 − DE , (2.36) 2 and the average mutual information between the system and m subenvironments is I (m) ≈ H0 − eH0 − 1 −(N −m) − DE m − DE env 2 Nenv = H0 + eH0 − 1 sinh m − ln(DE ) . 2 (2.37) Equation 2.37 is only a good approximation only near the classical plateau, where I H0 . Around m = 0 and m = Nenv , I rises linearly, not exponentially. Each subenvironment can provide only log2 DE bits of information, so until the information starts to become redundant, we’re in a different regime (see Fig. 2.11b). Once the information capacity of the captured environments (m log DE ) becomes greater than the amount of information in the system (H0 ), Eq. 2.37 becomes valid. It describes the slow approach to “perfect” information about the system, as m increases. Figure 2.14 compares exact (numerical) results for I (m) to the approximation in Eq. 2.37. Theoretical redundancy: averaging m(I ) The data in Section 2.3.2, and Figure 2.13, indicate strongly that the degree of redundancy in a branching state is proportional to the number of environments – not surprising, given that the environments measure the system independently from one another. Regardless of the total size of the environment, a certain number mδ of subenvironments is sufficient to acquire I = (1 − δ )HS . We define the specific redundancy r of a state in terms of the average chunk size mδ as rδ = = Nenv →∞ lim Rδ Nenv (2.38) (2.39) 1−δ mδ In this section, we use the specific redundancy to examine precisely how the system and subenvironment sizes affect information storage for branching states, and how important the deficit δ is. After deriving approximations to rδ , we compare the predicted results to numerical data. The theory for the average information yielded by m environments, computed in the previous section, could be used for an estimate of mδ , by solving Eq. 2.37 for m at I = (1 − δ )HS . 36 Exact PIP vs. Theory (DS = Dε = 2, Nenv = 8) 2 exact Theory Bits of info obtained 2 Exact PIP vs. Theory (DS = Dε = 2, Nenv = 32) exact Theory Bits of info obtained 1.5 1.5 1 1 0.5 0.5 0 0 1 2 3 4 5 6 m (# of environments captured) 7 8 0 0 1 2 3 4 5 6 m (# of environments captured) 7 8 (a) Exact PIP vs. Theory (DS = Dε = 4, Nenv = 8) 4 3.5 Bits of info obtained 3 2.5 2 1.5 1 0.5 0 0 1 2 3 4 5 6 m (# of environments captured) 7 8 exact Theory Bits of info obtained 4 3.5 3 2.5 2 1.5 1 0.5 0 0 1 exact Theory (b) Exact PIP vs. Theory (DS = Dε = 4, Nenv = 32) 2 3 4 5 6 m (# of environments captured) 7 8 (c) Exact PIP vs. Theory (DS = Dε = 16, Nenv = 8) 8 7 Bits of info obtained 6 5 4 3 2 1 0 0 1 2 3 4 5 6 m (# of environments captured) 7 8 exact Theory Bits of info obtained 8 7 6 5 4 3 2 1 0 0 1 (d) Exact PIP vs. Theory (DS = Dε = 16, Nenv = 32) exact Theory 2 3 4 5 6 m (# of environments captured) 7 8 (e) (f ) Figure 2.14: The approximation to I (m) that was calculated in Section 2.3.3 is illustrated for several universes. The approximation is virtually perfect near the classical plateau, leading us to conclude that we’ve effectively captured the underlying physics. In the small m regime, however, the rate of information gain is more nearly linear, and the approximation fails. Although it works well at m = 0 for DS = 4 (plots (c)-(d)), it fails spectacularly near m = 0 for large DS (plots (e)-(f )). In these figures, the solid lines represent the average information (I ) over many randomly chosen branching states; the error bars indicate the fluctuations (∆I ) over that ensemble. 37 The correct approach, however, is to compute the average m required to achieve a given I . In this section, we use a different approximation to compute a good estimate of the typical redundancy in branching states. Assuming that Nenv is very large, HSE = HS = H0 , so IS :E{m} = H (E{m} ). We take equation 2.34 as a starting point: 1 IS :E{m} ≈ HS − |γ |2 eHS − 1 . 2 Reducing the information deficit (I − HS ) to less than δHS requires |γij |2 (2.40) i=j DS (DS − 1) eHS − 1 ≤ 2δHS . (2.41) By assuming a maximally mixed state (i.e., eH0 = DS ), and replacing the γij with independent random variables γn , we obtain the following condition on a “sufficiently large” chunk: DS (DS −1) 2 |γn |2 ≤ δDS HS (2.42) n=1 The interaction of the 1 DS (DS − 1) independent γ coefficients makes a rigorous solution to equation 2 2.42 very difficult. A qubit system has only one off-diagonal γ , so we treat it first as a simple case. Since γ depends on m in a complicated fashion, we use the additive decoherence factor introduced previously, d = − log γ . For a qubit system, Eq. 2.42 simplifies to: 1 d ≥ dδ ≡ − log (2δHS ) 2 (2.43) Since d increases additively with m, we treat it as an m-step biased random walk in d, where each step has a mean d and a variance ∆d that are derived from the environment. After m environments are added to √ chunk, d is distributed according to a normal distribution pm (d) with mean md the and variance m∆d. Leaving aside the calculation of d and ∆d for the moment, we define psuff (m) as the probability that m environments are sufficient – that is, that equation 2.43 is satisfied by m environments: ∞ psuff (m) = dδ pm (d)dd (2.44) The probability that m environments are required is preq (m) = psuff (m) − psuff (m − 1) m ∂ = psuff (n)dn, m−1 ∂n (2.45) (2.46) 38 and the expected fragment size m is ∞ m= m=0 ∞ m preq (m) m = m=0 ∞ m m−1 ∂ psuff (n)dn ∂n = = = 1 ∂ psuff (m)dm 2 ∂m 0 ∞ 1 ∂ + m psuff (m)dm 2 ∂m 0 ∞ 1 (1 − psuff (m)) dm + 2 0 m+ 1 + 2 ∞ dδ dm 0 −∞ pm (d)dd (2.47) By interchanging the order of integration, and substituting the standard expression for the normal distribution pm (d) given above, we end up with m= dδ 1 ∆2 + 2+ 2 d 2d (2.48) For larger systems, computing m correctly is much more difficult. The general redundancy condition (Eq. 2.42) involves a sum of 1 DS (DS + 1) terms, where the condition for qubit systems 2 (Eq. 2.43) has just one |γ 2 | term. In theory, we could derive a probability distribution for this sum, and subject it to the same analysis just performed for pm (d). This is extremely complex, so we take a simpler route and replace the sum over terms with 1 DS (DS + 1) · γ 2 : 2 DS (DS − 1) 2 γ 2 γ2 d ≤ δDS HS 2δHS DS − 1 1 ≥ dδ ≡ − log 2 ≤ 2δHS DS − 1 . (2.49) The system’s size is incorporated into a redefinition of dδ . Equation 2.48 is still valid, so the general result for mean fragment size is: m= log(DS − 1) − log (2δHS ) ∆d2 1 + 2 +2 2d 2d (2.50) Finally, by applying Eq. 2.39, the estimated specific redundancy is: rδ = 2d (1 − δ ) ∆2 + d + d (log(DS − 1) − log(2δHS )) 2 2 (2.51) Calculating d and ∆d in terms of DE is somewhat involved, so we present the details in Appendix A.6. The result, however, is d ∆d 2 = = 1 (Ψ(DE ) + γ ) 2 2 π Ψ1 (DE ) − , 24 4 (2.52) (2.53) 39 DE d ∆d 2 1 2 1 2 3 3 4 √ 5 4 4 11 12 7 12 5 25 24 √ 205 24 6 137 120 √ 5269 120 √ 8 363 280 266681 840 Table 2.1: The table shows the first few values of d and ∆d, for environments of size DE ∈ [2, 3, 4, 5, 6, 8]. See Appendix A.6 for details on the calculation. in terms of the digamma (Ψ(n)) and trigamma (Ψ1 (n) functions [145, 146], and the Euler-Mascheroni constant γ = 0.577 . . .. As these functions are not necessarily familiar to all readers, we present the 1 π √ . In Figure 2.15, we first few values in Table 2.1. For large DE , d 2 (log(DE ) + γ ) and ∆d 24 compare numerical results to the approximation of Eq. 2.51. Finally, in order to get an intuitive feel for the dependence of rδ on its parameters (DS , DE , and δ ), we consider large systems and large environments, so that H0 1, d ∼ 1 log(DE ), and 2 2 1. We can then throw out most of the terms in Eq. 2.51, simplifying ∆d ∼ π , and assume that δ 24 to: log(DE ) (2.54) rδ = log(DS ) − log(δ ) We can distill from this a capsule summary of the amount of redundancy in typical branching states produced by simple decoherence models: 1. Redundancy is proportional to the number of independent environments, Nenv . More environments produce more redundancy. ¯ 2. Redundancy is proportional to the mean decoherence factor of a single environment, d, which grows as log DE . Larger environments produce more redundancy, in proportion to their information capacity. 3. Redundancy is (roughly) inversely proportional to the total amount of available information about the system, HS . Larger systems require more space in the environment. 4. The deficit δ appears as a logarithmic addition to HS . Thus, decreasing the amount of unrequired information is equivalent to having a bigger system. Redundancy is only logdependent on the deficit, δ . 2.4 Conclusions and extensions The partial information plots introduced in Sec. 2.1.2 (and previously in [20]) provide a useful intuitive picture how E stores information about S . Information is physical – it has to be stored somewhere, and to retrieve it we must measure the system on which it’s stored. To understand the properties of information, we naturally look at the properties of this retrieval process. The shape of the PIPs for branching states (see Fig. 2.16) indicates a well-defined division of the information about S that can be obtained from E (IS :E ), into three parts: IS :E = IR + INR + IQ . (2.55) The redundant information (IR ) is classical. It can be obtained easily11 , by many independent observers, through semi-local measurements on E . The quantum information (IQ ) represents information about observables that are incompatible with the pointer observable. This is the information 11 As shown in [106], IR is not only easy to obtain, but difficult to ignore. 40 Specific Redundancy (DS = 16) 0.7 r (specific redundancy) 0.6 0.5 0.4 0.3 0.2 0.1 0 0.001 0.01 δ (Information deficit) 0.1 r (specific redundancy) Dε = 2 Dε = 3 Dε = 4 Dε = 8 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.001 Specific Redundancy (Dε = 2) DS = 2 DS = 3 DS = 4 DS = 8 DS = 16 0.01 δ (Information deficit) 0.1 (a) Specific Redundancy vs. Dε 0.6 0.5 0.4 0.3 0.2 0.1 0 2 4 DS = 16 DS = 8 DS = 4 DS = 2 6 8 10 12 14 Sub-environment size (Dε) 16 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2 4 r1% (specific redundancy) r1% (specific redundancy) (b) Specific Redundancy vs. DS Dε = 2 Dε = 3 Dε = 4 Dε = 8 Dε = 16 6 8 10 12 System size (DS) 14 16 (c) (d) Figure 2.15: Specific redundancy (rδ ≡ Rδ /Nenv ): numerical data (symbols) compared with theory (Eq. 2.51, solid lines). Plots (a) and (b) show the dependence of rδ on δ . In plot (a), a DS = 16 system is coupled to environments with DE = 2, 3, 4, 8. Plot (b) shows the effect of coupling assorted systems (DS = 2, 3, 4, 8, 16) to an environment with DE = 2. In plots (c) and (d), we fix δ = 0.01 and examine dependence on DS and DE . Discussion: Theory predicts the overall behavior of redundancy well. It is nearly perfect for DS = 2, but overestimates r for larger systems. The main problem is poor modeling of the multiple off-diagonal terms in systems with DS > 2. Theory breaks down for large δ (e.g., plot (a), for DS > 2). For large δ , a single environment can provide sufficient information, and R saturates. R then declines with increasing δ , because of the (1 − δ ) prefactor in Eq. 2.4. that Quantum Darwinism selects against. It can be accessed only through a global measurement on all of E , and is easily destroyed if E undergoes decoherence12 . Finally, non-redundant information (INR ) exists when the classical plateau in I (m) has a nonzero slope. INR represents a grey area, information that is gained “slowly but surely.” It forces us to allow for an information deficit (δ ) when computing redundancy. When the PIP is a straight line (see Fig. 2.4a), IS :E = INR . All the information is nonredundant; Quantum Darwinism is not a factor, and neither classical nor purely quantum information can be identified. For generic arbitrary states, however, all 2HS bits of information are obtained essentially at the same time. This doesn’t mean, of course, that we can really get 2 bits of information about a single bit – it simply means that by capturing the entire environment of a system, we gain the 12 The quantum information is encoded amongst the environments in much the same way that a classical bit can be encoded in the parity of many ancilla bits. 41 Quantum Darwinism in Action 200 IS:E {m} 150 imperfect "classical plateau" 100 50 0 0 20 40 60 % of environment captured 80 100 }} } % of information obtained IQ (quantum) INR (borderline) IR (classical) Figure 2.16: Quantum Darwinism selects certain observable properties of the system and propagates information about them throughout the environment. The preferred observable[s] become redundant at the expense of incompatible observables. As shown here, PIPs illustrate the results of Quantum Darwinism. Information about S becomes divided into three parts: redundant information (IR ), quantum information (IQ ), and non-redundant information (INR ). Redundant information is objective, and therefore classical. It can be obtained with relative ease. Quantum information represents the non-preferred observables, marginalized by Quantum Darwinism, which can only be measured by capturing all of E . Non-redundant information (determined by the slope of I (m) at m = Nenv ) represents the ambiguous borderline, undifferentiated as yet into classical and quantum 2 fractions. ability to measure arbitrary observables of that system, including the ones traditionally considered “quantum”, like relative phases between pointer states. An observer gains information about every observable at the same time. For such states there is no such thing as “quantum” or “classical” information – just a lot of observables, none of which are privileged over the others. Our investigation indicates that the structure of the environment is important. Information storage is dramatically different between randomly selected arbitrary states of the model universe, and randomly selected singly-branching states. A casual observer might be forgiven for hypothesizing that the physical Universe would, in all its complexity, evolve into states that are uniformly distributed. After all, thermodynamic arguments generally proceed from an assumption of maximal entropy (consistent with constraints)13 . However, we’ve shown here that in order for objects to display the redundancy that is characteristic of our Universe, there must be a great deal of structure to their correlation with environments. Uniformly distributed states are just not consistent with observations. We have not proved – or attempted to prove – that singly branching states do describe the Universe. We examine them here because they form a plausible ensemble which does typically display redundancy. Real decoherence and measurement processes are infinitely more complex processes 13 For instance, if nothing at all is known about the system, then equal probabilities are assigned to every state. A typical constraint is to specify the system’s energy, which leads to a microcanonical ensemble. The canonical ensemble is the one with the greatest entropy subject to the constraint of being in equilibrium with a reservoir. 42 than anything we’ve considered here. By establishing the existence of redundancy, as well as tools for examining it, we hope to provide a path for the investigation of more complex dynamical models. These models will produce some ensemble of states, which may be more general than the branching states studied here. If, however, a model does not lead to redundancy (e.g., the “random-state” model we began with), then the model must be flawed to some degree as a model of our Universe. The branching state analysis also provides a baseline for comparing quantitative redundancy in physical models. The specific redundancies we compute may be taken as upper bounds for physical processes. Although there exist states, and even ensembles, which display greater redundancy than generic branching states – GHZ states, for instance, have maximal specific redundancy (r = 1) – these states cannot be stable end products of physical processes. For example, the canonical C-NOT model of a quantum measurement can produce GHZ states as an end product. However, the environment must be initialized in just the right state. More importantly, the dynamical process which induces information storage will (if allowed to continue) begin to reverse itself immediately after the GHZ state is achieved. At some random time, the state of the universe will not be a GHZ state, but some other branching state. We expect that many physical models will lead to some variant of the branching states. 2.4.1 The next steps We see two lines of investigation in the examination of redundancy. The first is analyzing actual dynamical models, in order to verify that they store information about the system redundantly, and to compare different environments by the amount of redundancy they support. Almost every model of decoherence that has been considered is in principle amenable to this analysis, but certain modes of analysis are not useful. For instance, the master equation paradigm is not suited to examining the environment. In theory, upper bounds on redundancy can be obtained from the system’s density matrix. If ρS is not completely diagonalized, then redundancy is bounded by the logarithms of off-diagonal elements. However, extremely precise calculation of off-diagonal elements is required. The difference between ρx,x = 10−12 and ρx,x = 10−6 is negligible in comparison to ρx,x – but it doubles the possible redundancy. In general, analyzing redundancy requires direct examination of the state for the full S ⊗ E universe. The second area of outstanding problems is in better understanding information itself. Our approach, using QMI, is excellent for demonstrating the redundancy of all the information about a system, but begins to fail when only part of the information about a system is redundant. Examining the PIP provides at least an indication of what is happening, but the root of the problem is an inability to identify what a bit of information is about. This problem has been examined in [106, 105], but not definitively solved. A better understanding of the contextual nature of information would not only facilitate both theoretical and numerical work on redundancy, but advance our understanding of information in general. 43 Chapter 3 Dynamical redundancy in spin systems Some twenty-five years ago, decoherence was introduced in order to explain the transition to classical behavior in fundamentally quantum systems. With the development of quantum information science as field of technology, decoherence gained a new application as the primary failure mechanism of quantum information processing devices. Its original role in understanding the quantum-classical transition has been expanded by the idea of objectivity as an important property of classical systems. The objectivity of a property is dependent on that property being redundantly stored throughout the environment, where multiple observers can independently access it. Redundancy (of environmentally-stored information) is thus a very practical signature of objectivity – which appears to be a more ethereal concept. In this chapter, we examine the redundancy-producing ability of some simple decoherence processes. The framework and methods used here were developed in Chapter 2 (and previously in [20]). We conjectured in Chapter 2 that the simplest models of decoherence and quantum measurement will lead to the ensemble of “singly-branching states” (see Eq. 3.1 below), so we begin 1 by testing this conjecture, with a model consisting spin- 2 particles. We proceed to investigate the behavior of this model in depth, identifying the parameters that influence the development of redundancy. We conclude with some remarks on the connections between redundancy and decoherence in general. 3.1 Overview Our earlier work on quantifying redundancy, in Chapter 2, sidestepped the dynamical evolution of redundancy by conjecturing that physical processes lead to states that are selected randomly and uniformly from a simple ensemble. The ensemble of singly-branching states (or just “branching states”) of the form DS Nenv |ψ = i=1 si |i ⊗ n=1 ( |Enn) . (3.1) was found to display redundant information storage. We used partial information plots, or PIPs, to identify the presence of redundancy. A theoretical model to predict the shape of these PIPs was obtained, using the ensemble-averaged decoherence factor (|γ |2 ). We also examined a quantitative measure of redundancy, Rδ , defined as the redundancy of a fraction 1 − δ of the available information. It scales with the number of environments (Nenv ), as 44 Rδ = rδ Nenv . We refer to rδ as specific redundancy. The average specific redundancy for branchingstate ensembles was accurately predicted using the additive decoherence factor, defined as d = − log(γ ) (3.2) The ensemble average (d) is the most important factor in determining typical redundancies, but the variance (∆d) adds a significant correction to the formula (Eq. 2.51 in Chapter 2): rδ = 2d (1 − δ ) ∆d2 + d + d (log(DS − 1) − log(2δHS )) 2 2 . (3.3) We now use these tools to examine the dynamical emergence of redundancy. We also introduce a new concept, pliability. Pliability measures the environment’s ability to respond to the system and produce decoherence. We use both γ and d as measures of pliability, and we use the formulae derived in Chapter 2 to predict specific redundancy. Partial information plots provide a “big-picture” view of the emergence of redundancy, but we turn to Rδ (and rδ ) for quantitative results. We showed in Chapter 2 that the exact value of δ is largely unimportant, so we fix δ = 10% throughout this chapter. The first half of this chapter focuses on the development of redundancy – prerequisites for its existence, the speed with which it emerges, and its asymptotic or time-averaged level. We use the “random-state average” computed in Chapter 2 as a baseline for evaluating dynamical redundancy, and find it to be a good (though not exact) model for redundancy in two classes of dynamical models1 . The second half of this chapter is devoted to determining when, why, and how redundancy is dynamically destroyed. We consider two important models where the random-state model breaks down. In these models, the redundancy that initially appears is gradually eroded, as information diffuses throughout the environment. We identify these models, explain why redundancy disappears, and attempt to understand the observed behavior. 3.1.1 Our model universe In order to study decoherence (or quantum open systems in general), the universe must be divided into a central system (S ) and its environment (E ). To study redundancy, we subdivide the environment into Nenv indivisible subenvironments (En ). The individual En may be combined into fragments (E{m} ) consisting of m subenvironments. The supersystem consisting of S and all the En is the universe.2 We continue the thread presented in Chapter 2, by considering a spin-j system coupled to a bath of spin-j particles. In particular, we focus on the j = j = 1 case. This model is 2 particularly relevant to quantum information theory, as spin- 1 particles are equivalent to qubits. 2 1 Where simulations are computationally intensive, use of spin- 2 systems allows the most Nenv . We also avoid the complications that arise when the system has two or more simultaneously measurable properties. When practical, we extend our analysis to arbitrary j and j . Initial product states – pure states with no correlation between S and any of the En , or between the En – evolve into correlated states, whose information-storage properties we analyze using the techniques presented in Chapter 2. This dynamical evolution is governed by a Hamiltonian (H). The Hamiltonian must respect (approximately) the tensor-product structure of the universe – otherwise that structure is largely irrelevant. Arbitrary Hamiltonians will generate arbitrary states of the universe, which do not record the system’s state redundantly (as we showed in Chapter 2). Requiring H to respect the observed tensor-product structure is physically well-motivated. Locality (e.g., tensor-product) structures reflect the way that external observers perceive our model interaction-only and quantum-measurement models. use of this terminology should not be taken to imply that this model universe is necessarily a good representation of the physical Universe in which we live. 2 Our 1 The 45 universe. Such observers perceive and interact with the system and its environments by exploiting the same Hamiltonians that govern the universe’s dynamics. The dynamics are bound to respect the structure of the environment – if they did not, then we, as observers, would ascribe a different structure to the environment! It is this argument which makes redundancy possible in the first place; of all the possible Hamiltonians, only a few very basic types actually exist. Our choices of dynamical models are strongly influenced by two principles derived from the locality considerations discussed above: env • We separate the Hamiltonian into a local part (Hsys + Henv . . . Henv ), and an interaction part (Hint ). The interaction Hamiltonian is more important, as it generates decoherence and information flow between subsystems. (1) (N ) • The system and the subenvironments are assumed to be qualitatively similar, so that their local Hamiltonians are of the same form. In this chapter, the local dynamics are generated by spin Hamiltonians, of the form H = v ·J. The interaction Hamiltonian is: (a) restricted to 2-body interactions, and (b) composed of terms similar to the local Hamiltonians. Specifically, the terms in Hint are products of local spin operators, ( S ) (E ) e.g. gn Jz Jx n . Finally, the subenvironments are assumed not to interact directly with each other. An excellent physical realization of this class of models is a localized electron spin, which interacts with a bath of nuclear spins. This includes quantum dot designs for quantum computers, where the “interesting” degree of freedom (in which information is stored) is the spin of a single electron in the dot.3 This model is reviewed in [31]. In that paper, the overall Hamiltonian is (in our notation) H = bJ ( S ) + J ( En ) + J ( S ) · kn J(En ) (3.4) z z n n Transformed into the rotating frame of the nuclear spins, this Hamiltonian becomes H = (b − )J(S ) + J(S ) z z n kn J(En ) + z 1 (S ) J 2x kn J(En ) + J(S ) x x n n kn J(En ) . y (3.5) The couplings to the magnetic field are: b = g ∗ µB Bz and = gI µN Bz , where g ∗ is the electron g -factor, gI is the nuclear g -factor, µB is the Bohr magneton, µN is the nuclear magneton, and Bz is the applied magnetic field. The kn couplings are hyperfine couplings from the Fermi contact hyperfine interaction model. The only tunable parameter in this model for an electron spin in a bath of nuclear spins is the applied magnetic field. When Bz is large, the Jx · Jx and Jy · Jy terms can be neglected (as in [31]); when Bz is turned off, they become significant. In quantum computing applications, the need to exert unitary control over the electron spin will require the addition of a controllable HS term to the model – possibly implemented by a highly localized magnetic field as is used in computer hard drives for data recording. Such a technology could also be used to control the nuclear spins’ Hamiltonians. We begin by considering all the possible Hamiltonians that could be applicable to such a spin-bath model. The locality constraints above reduce the infinite range of dynamical models to a manageable set, which we divide into five classes: • Interaction-only models: Neither S nor the En have their own dynamics. Hint has the form M(S ) ⊗ n kn R(n) , which we abbreviate as an “M ⊗ R interaction”. Each environment records information about M independently, and M is a perfect pointer observable. In the electron nuclear spin model mentioned above, this is the strong-applied-field limit. 3 Proposals to store information in the dot’s charge, or in motional degrees of freedom for the electron, have different dynamics. 46 • Generalized quantum-measurement models: Hint has the same form as in the interactiononly model, but the environments have independent (or unconditional ) dynamics generated ( n) by Henv = V(n) . This could be generated in the electron - nuclear spin model by quantum control of the nuclear bath (which might be desirable, e.g. to reduce decoherence). • Dynamical-system models: The system has its own dynamics, generated by Hsys . Hint is the same as in the interaction-only model. In the electron - nuclear spin model, this is the result of controlling the electron spin, in a strong applied field. • Multiple-measurement models: Neither S nor E have independent dynamics. Hint is a sum of distinct M ⊗ R terms. Each environment is effectively trying to record multiple noncommuting observables at the same time. In the electron - nuclear spin model, this represents the case when the applied field is weak. • Dissipative models: To either the dynamical-system or multiple-measurement models, we add internal subenvironment dynamics. The resulting Hamiltonian is complex enough to generate effectively ergodic dynamics (subject to energy conservation). In the electron - nuclear spin model, this case is the most realistic – it results when we admit that all the previous cases were really approximations to a complex multibody Hamiltonian. These five classes have a natural division into two categories. The interaction-only and quantum-measurement models generate singly-branching states, which were discussed in detail in Chapter 2. They can be efficiently simulated, and well-described by the theory developed in Chapter 2. Dynamical-system, multiple-measurement, and dissipative models do not stay within the branching-state ensemble. As a result, they are much more costly to simulate, and demonstrate behavior beyond the branching-state theory. One model that we do not examine here is the interacting-environments model. In such models, the individual subenvironments are allowed to interact with each other without the central system’s intervention. It is easy to see that interaction between subenvironments will destroy redundancy, by transferring locally-available information into entangled modes of many En . However, interacting-environment models might help to explain the behavior of other, more interesting models – e.g., ones where the environments interact indirectly, through the system. We discuss each of these models in detail in the appropriate section. For a schematic overview of all models, with major conclusions and interconnections, see Fig. 3.15. 3.2 Interaction-only models The simplest possible interaction model that respects the locality structure of the environment is one where the Hamiltonian consists solely of an interaction term. A fairly general form for such interaction terms is Hint = M(S ) ⊗ kn R(n) . (3.6) n The measurement operator (M) determines what property of S is measured, while the recording operators (R(n) ) specify how that information is recorded. When the system is in an eigenstate of M with eigenvalue µ, the nth subenvironment (En ) feels a Hamiltonian H(n) = µkn R(n) . Eigenstates of M do not decohere, because they generate coherent evolution of the environments. M is the system’s pointer observable, and its eigenstates are the pointer states. Superpositions of distinguishable pointer states (with different values of µ) will experience decoherence, as the environment evolves into distinct states conditional upon µ. This decoherence indicates that the environment has obtained information about M. 47 In the model that we implement here4 , M(S ) = R(n) = Jz . The coupling strengths (kn ) are Gaussian random variables. We choose the initial state of the system to be unbiased with respect 1 to Jz : |S0 = √2 (|↑ + |↓ ). This maximizes the amount of available information (about M). We assign a random, uniformly distributed initial state to each subenvironment. The model’s only 1 2 tunable parameter is the root-mean-square coupling strength, g0 ≡ kn 2 , which we set to 1. All time- and energy-scales are thus relative to g0 . Summary of the Interaction-Only model H = Hint |ψ0 = J (S ) ⊗ z n kn J(n) , z (3.7) (3.8) = |↑ + |↓ (1) (1) √ ⊗ |E0 ⊗ . . . |E0 2 ( with |E0n) uniformly distributed, 2 kn 2 P (kn ) ∝ e− . Nenv = 12 . . . 128 3.2.1 Results Previously (in Chapter 2), we used partial information plots or PIPs to examine the qualitative aspects of information storage. Fig. 3.1 shows typical PIPs (obtained from numerical simulations) for interaction-only models with Nenv = 12 and Nenv = 128. The dependent axes represent time (t) and the size (m) of a captured fragment of the environment. We normalize the average information content (I S :E{m} of an m-sized fragment, by subtracting the system’s entropy (HS ). A fragment E{m} for which IS :Em − HS ≈ 0 provides “sufficient” information – that is, virtually all the classical information about S . The short-time plots in Fig. 3.1 show the total information obtainable from E rising rapidly, accompanied (initially) by little or no redundancy. When HS has nearly reached its asymptotic value, the I (m) curve is still relatively straight. As more time elapses, it develops the curvature characteristic of redundancy. By t ∼ 1, substantial curvature has developed, and remains roughly constant thereafter, except for fluctuations. These fluctuations are much more evident for Nenv = 12 than for Nenv = 128. The larger (Nenv = 128) environment has much higher levels of redundancy, and a more distinct separation between the timescales for decoherence and redundancy (see below). We can condense the information content of the PIPs, by computing quantitative redundancy (Rδ ) of “all but a fraction δ ” of the available information. To simplify matters, we fix δ = 10%.5 Fig. 3.2a presents R10% for the same simulations shown in Fig. 3.1, along with other values of Nenv . R10% reaches its asymptotic value by t ∼ 1, but continues to fluctuate thereafter. We averaged over 10 independent simulation runs (each with different randomly chosen couplings kn ), to reduce the fluctuations. This smoothed data is plotted in Fig. 3.2b. The interaction-only model for Nenv = 12 is used later as a baseline (relative to other models), so for Nenv = 12 we have also averaged R10% over a set of 600 runs. The results indicate that all the time-dependence in Fig. 3.2, except the initial rise in R, stems from random fluctuation. Scaling of R with Nenv Fig. 3.3 confirms that Rδ scales with Nenv , as predicted in Chapter 2. Specific redundancy (rδ ) is the the amount of redundancy per subenvironment. We find r10% = 0.193 for the interactiononly model, which is substantially lower than the random-state average derived in Chapter 2. In 4 Note 5 e.g., that this form for the M ⊗ R interaction is fully general – any other interaction would be unitarily equivalent. when the system has 1 bit of entropy, we demand 0.9 bits from each fragment E{m} . 48 Interaction-only: Nenv=12, t < 1.5 IS:ε {m} Interaction-only: Nenv=12, t < 15 IS:ε {m} - HS - HS 12 1.5 9 m 6 3 0.5 1 time 12 15 9 m 6 3 5 10 time (a) Interaction-only: Nenv=128, t < 1 IS:ε {m} (b) Interaction-only: Nenv=128, t < 8 IS:ε {m} - HS - HS 128 96 m 0.9 64 32 0.3 0.6 time 128 96 m 64 32 3 6 time (c) (d) Figure 3.1: Partial information plots provide a qualitative view of how the environment stores information. These time-series PIPs represent single simulation runs for two environments in the interaction-only model. Plots (a)-(b) show short- and long-time behavior of a 12-spin environment, while Plots (c)-(d) show short- and long-time behavior of 128-spin environment. Important features of the short-time plots include the initial ramp-up of total entropy (which occurs on the decoherence timescale), followed by the development of curvature in I (m), which indicates redundancy. The long-time behavior shows a stable level of redundancy for t > 1, with fluctuations that are more obvious for the smaller (Nenv = 12) environment (plot (d)). Sec. 3.2.2, we present a theoretical model which predicts r10% correctly to within the numerical error. A comparison of theory and simulation for other values of δ is presented in Table 3.1. Timescales for decoherence and redundancy Redundancy develops on a slower timescale than decoherence. This is apparent in Fig. 3.4, where R10% (t) and HS (t) (for Nenv = 12 and Nenv = 128) are scaled to fit on the same plot. Changing the size of the environment reduces the decoherence time by a factor of 5, but the time required to develop maximum redundancy stays fixed at t ∼ 0.8. Decoherence can be produced by a small, quick-reacting fraction of the environment, but redundancy requires the entire environment (or at least a majority of it) to become involved. We noted in Chapter 2, when examining PIPs, that the central “classical plateau” region is exceedingly flat. Its slope decreases exponentially as m increases. The dynamical PIPs in Fig. 3.1 show that the same is true after a while, but this flattening does not occur immediately. A flat classical plateau is an important prerequisite for clear redundancy: if this region of the PIP is not 49 Redundancy (R10%) in the Interaction-Only model (single run) 35 30 25 20 15 10 5 0 0 5 10 15 20 25 Time (arbitrary units) 30 35 40 Nenv = 6 Nenv = 12 Nenv = 24 Nenv = 48 Nenv = 80 Nenv = 128 (a) 30 25 R10% 20 15 10 5 0 0 5 10 R10% Redundancy (R10%) in the Interaction-Only model (10-run avg) Nenv = 6 Nenv = 12 Nenv = 24 Nenv = 48 Nenv = 80 Nenv = 128 (b) 15 20 25 Time (arbitrary units) 30 35 40 Figure 3.2: By computing the redundancy (Rδ ) of “all but δ ” of the total I , we distill out the important features of PIPs. Here, we plot R10% versus time for several interaction-only environments. A 1 spin- 2 system is coupled to Nenv = 6, 12, 24, 48, 80, 128 environments. The most significant features of the data are a consistent initial rise on a timescale t ∼ 1 (see Fig. B.1 for more detail), and an asymptotic level of redundancy that scales proportional to Nenv (see Fig. 3.3 for more detail). The observed fluctuations are due to: (1) particular dynamical timescales of individual simulation runs; and (2) Monte Carlo error in the sampling process for computing redundancy. Plot (a) represents single simulations. In plot (b), 10 different runs are averaged to eliminate artifacts from any single simulation (and slightly smoothed to eliminate random noise). flat, then capturing more of E yields new information! We can use the slope of the PIP’s central region to define the amount of nonredundant entropy – that is, the portion of HS which cannot be eliminated by measuring a small fraction of E : 1 ∂ I (m) Nenv ∂m HNR ≡ (3.9) m= Nenv 2 We also plot HNR in Fig. 3.4. It rises with HS initially, then begins to decline toward zero. Redundancy begins to develop when HS declines, but does not reach its maximum level until much later (in the case of a large environment). 2 The timescale for redundancy is set by g0 = kn 2 = 1, whereas the decoherence timescale is determined by a few of the the largest |kn | (i.e., the subenvironments that respond most quickly). For sufficiently large environments, decoherence can occur arbitrarily quickly, relative to the fixed redundancy timescale. Further details can be found in Appendix B.2. 1 50 Redundancy (R10%) vs. Nenv (Interaction-Only model) 25 20 15 10 5 0 0 20 40 Simulation data Theory 60 80 100 # of subenvironments 120 140 Figure 3.3: This plot confirms the theory from Section 3.2.2. The model predicts that R = rNenv , with r10% = 0.193. The time-averaged simulation data agrees with the theory to within numerical error. A linear fit to the data yields precisely the predicted specific redundancy. Conclusions The interaction-only model provides a baseline for dynamical redundancy. Numerical simulations are relatively easy to perform, and the results show clear qualitative agreement with the theory in Chapter 2. The timescale for development of redundancy is a new result, but conceptually simple and unsurprising. The one major discrepancy between Chapter 2 and the numerical results presented above is the amount of redundancy, which disagrees with the random-state prediction by a substantial amount. The concise explanation is that dynamically generated conditional states do not explore the available Hilbert space optimally. In the next section, we explain the reduction in redundancy in terms of the environment’s pliability, and use this theory to accurately predict specific redundancy. 3.2.2 R10% Theory The interaction Hamiltonian (Eq. 3.6) produces branching states (see Chapter 2), with the following structure: DS |ψ = i=1 (si |i S (1) (2) ( ⊗ |Ei ⊗ |Ei ⊗ . . . |Ei Nenv ) ) . (3.10) Distinct pointer states of the system (e.g., |i S and |j S ) are correlated with conditional states of the environment. Each conditional state of the whole environment is a product of the conditional 51 Timescales (decoherence vs. redundancy): Nenv=12 1.2 Fraction of fiducial value 1 0.8 0.6 0.4 0.2 0 0 Entropy R10% * 1/1.67 Non-redundant Entropy 0.2 0.4 time 0.6 0.8 1 Fraction of fiducial value Timescales (decoherence vs. redundancy): Nenv=128 1.2 1 0.8 0.6 0.4 0.2 0 0 Entropy R10% * 1/25 Non-redundant Entropy 0.2 0.4 time 0.6 0.8 1 (a) (b) Figure 3.4: These plots illustrate one of our most basic and important results: decoherence and redundancy happen on different timescales. For baths of 12 (plot (a)) and 128 (plot (b)) spins, we plot the time dependence of: (1) HS , a measure of total decoherence; (2) R10% , a measure of total redundancy; and (3) HNR (non-redundant entropy ), a measure of information that cannot be obtained redundantly. Discussion: As Nenv increases, the decoherence timescale declines as 1 τD ∝ √N . The timescale for maximum redundancy remains τR ∼ 0.8. A few rapidly-responding env environments produce near-complete decoherence, while maximum redundancy requires participation by a majority of the environment. Initially, all entropy is non-redundant, but HNR peaks at 1 about 2 Hmax . Its decline marks the onset of redundancy development, but can occur well before R reaches its maximum. ( states for each subenvironment (|Ei n) ). Because of this structure, the subenvironments are never entangled with each other. Superpositions of pointer states (e.g., |i and |j ) decohere. The degree of decoherence is measured by a multiplicative decoherence factor γ : γij = n γij = n ( n) Ej (n) |Ei (n) . (3.11) Dynamically generated branching states differ from randomly selected branching states in two ways. First, time plays a role, because S and E are uncorrelated at t = 0. Only after sufficient time has passed can the environment register the state of the system, by evolving into distinct conditional (on the state of S ) states. Second, each subenvironment’s conditional states are restricted to one orbit of one Hamiltonian. They cannot explore the full extent of Hilbert space. One of these phenomena is transient, while the other is asymptotic. Both, however, reflect a reduction in the environment’s pliability – ability to be imprinted by information about the state of S . Pliability is a concept, not a quantity. A pliable environment can (and will) evolve into highly distinguishable6 states conditional upon the value of the pointer observable. The opposite of pliability is stiffness. Compared to the random-state model, dynamical models have: (1) reduced (and time-dependent) pliability at short times; and (2) moderately reduced pliability at long times. We emphasize that our concept of pliability is not inherently connected with time. The speed with which an environment reacts to the system does not determine its pliability. An environment which reacts quickly merely has a more rapid increase in its pliability. We define pliability in this fashion in order to apply it to models with no concept of time (e.g. the random-state model). 6 Distinguishability: States are distinguishable inasmuch as they can be reliably told apart by a measurement. If all measurements are allowed, then the distinguishability of |ψ and |ψ is just determined by their fidelity, F = ψ|ψ . When the conditional states of E corresponding to |i S and |j S are distinguishable, superpositions of |i and |j are decohered. Thus, “the environment’s conditional states are distinguishable” means “the system has decohered,” which in turn means “the environment has information about the system.” 52 A measure of pliability should characterize the tendency of the environment to evolve into distinguishable conditional states, producing decoherence. Consider any measure of instantaneous decohering power (such as the decoherence factor γ ). Averaging it over an ensemble of possible conditional states7 provides a measure of pliability. In Chapter 2, we found two distinct measures of instantaneous decohering power: |γ |2 , and d. The total decoherence produced by a collection of subenvironments depends on the average (over all subenvironments) of |γ 2 |. Redundancy, however, depends on the average (over subenvironments) of the additive decoherence factor, d = − log(γ ).8 We use both |γ |2 and d as measures of pliability. While d is essential to predicting redundancy, |γ |2 can be used to define a subenvironment’s dynamical capacity ( n) cd ≡ − log |γ (n) |2 , (3.12) in close analogy to its information capacity c(n) ≡ log DE ( n) . (3.13) We showed in Chapter 2 that the average of |γ |2 over random (uniformly distributed) conditional 1 states is |γ |2 = DE . Therefore, when dynamics produce uniformly distributed conditional states, the dynamical capacity is equal to the information capacity (cd = c = log DE ). A subenvironment’s dynamical capacity measures how much of its information capacity is actually accessible. We have investigated the pliability of spin-j environments in exhaustive detail. Precisely because the results are detailed and extensive, they are relegated to Appendix B.3. In this section, and in the section on quantum-measurement models, we summarize the relevant results for j = 1 1 2 . Our primary concern is an understanding of the time-averaged dynamical pliability of spin- 2 environments, which we use to predict (correctly) the specific redundancy observed in the previous section. We use the following formulae from Chapter 2: rδ Rδ mδ Rδ Nenv =∞ Nenv Nenv ≡ (1 − δ ) mδ log(DS − 1) − log (2δHS ) ∆d2 1 + = 2 + 2. 2d 2d ≡ lim (3.14) (3.15) (3.16) Computing decoherence factors We can rewrite the interaction Hamiltonian (Eq. 3.6) by decomposing the measurement operator as M = i µi |i i|: Hint = i µi |i i|S ⊗ n kn R(n) (3.17) (3.18) = i,n µi kn |i i| ⊗ R(n) . An initial product state, DS |ψ0 = i=1 si |i S (1) (2) ( ⊗ |E0 ⊗ |E0 ⊗ . . . |E0Nenv ) , (3.19) 7 Note that this ensemble could be a single state, if the environment’s initial state, and evolution, were fixed. In other words, averaging over parameters/states/whatever is an option, not a requirement. 8 More precisely, both the mean (d) and variance (∆d) are used to predict R . δ 53 δ rmodel rsimulation 1 2 1 3 1 4 1 5 1 10 .244 .229 .246 .238 .236 .232 .226 .224 .193 .193 Table 3.1: A comparison of the specific redundancy (r) predicted by our theoretical model to the simulation results, for assorted values of the deficit δ . The simulation data have about 0.5% accuracy at best, so uncertainties are approximately ±0.001. The model of Section 3.2.2 is correct to within numerical error for r10% . Even when δ = 0.5, far outside of its intended range of validity, the model is still correct to within 6%. will evolve under this Hamiltonian into a branching state, DS |ψt = i=1 (si |i S (1) (2) ( ⊗ |Ei (t) ⊗ |Ei (t) ⊗ . . . |Ei Nenv ) (t) ) . (3.20) ( ( The conditional state of a single subenvironment, |Ei n) (t) , is given in terms of its initial state (|E0n) ) ( n) and recording operator (R ) by (n) ( |Ei n) (t) = e−iµi kn R t ( |E0n) . (3.21) Finally, the contribution of En toward the decoherence between |i and |j is given by the decoherence ( n) factor γij : γij (t) = ( n) E0 (n) | eikn t(µj −µi )R (S ) ( n) (n) ( |E0n) . (3.22) In our interaction-only model, Hint = n kn Jz ⊗ Jz . The eigenvalues of the pointer observable are µ = ± 1 and the recording operators are R(n) = Jz , so 2 γ (n) (t) = Computing pliability ( The decoherence factor for a single environment depends on time (t), the initial state (|E0n) ) of En , and the coupling constant (kn ). In our model, the initial states are chosen randomly from the uniform ensemble, and the kn are univariate Gaussian random variables. We characterize the pliability of a single environment by averaging d over |ψ0 , k , and t: ψ0 (n) ( | eikn tJz |ψ0n) . (3.23) d ∆d 2 =− = dψ0 dψ0 P (k )dk P (k )dk dt log (γ (t)) dt log2 (γ (t)) − d 2 (3.24) (3.25) where P (k ) is a normalized univariate Gaussian distribution, and the integration measures dt and dψ0 are normalized to 1.9 9 Integration over initial states is, more properly, integration over the invariant Lebesgue measure on CP (D − 1) E induced by Haar measure on SU (DE ). For spin- 1 systems, this means integration over the surface of the Bloch sphere. 2 For larger environments, explicit parametrization of the measure is useless, as the integrals cannot be performed analytically. In these cases we integrate numerically by generating a lot of random states on the surface of a complex hypersphere. 54 In general, these integrals cannot be performed analytically. We turn to numerical averaging to obtain results, which are presented and discussed in Appendix B.3.10 d ∆d mδ rδ = = = 0.306 0.421 0.909 − 1.629 ln δ 1−δ = 0.909 − 1.629 ln δ Theoretical predictions of rδ are tabulated in Table 3.1 and compared with numerical results. The fit is extremely good – for r10% the predicted value and the numerical value agree to within numerical error (approximately 0.5%). As the deficit (δ ) is increased, we expect the model that we adopted in Chapter 2 to become less accurate. Nonetheless, even at δ = 50%, theory and numerics agree surprisingly well. 3.2.3 Discussion The basic theory behind the calculations above was derived in Chapter 2, where we modelled the sequential capture of subenvironments as a random walk in d (the additive decoherence factor). We have extended it here simply by using an ensemble of dynamically generated branching states instead of averaging over all branching states. The excellent agreement with numerical simulation validates the model described in Chapter 2. The random-state model succesfully describes the mechanism that produces redundancy in the interaction-only model. However, the information is stored with less redundancy in the dynamical model. In the interaction-only model, each spin- 1 environment contributes approximately 2 r10% = 0.193, compared with the random-state average of r10% = 0.302. The immediate cause is the relatively lower pliability of dynamical environments: ddynamical = 0.306, whereas drandom = 0.5.11 We can push the question a bit further by asking why the environments’ pliability is less in dynamical models. The explanation lies in the effect that conditional dynamics (generated by R), have on the subenvironments’ random initial states. Letting12 R = Jz , we can trace out its orbits on the Bloch sphere. As states evolve, they rotate around the z -axis, tracing out “lines of latitude.” An equatorial state (e.g., an eigenstate of Jx or Jy ) will trace out the entire equator, eventually evolving into an orthogonal state before returning to the initial point. A polar state (e.g., an eigenstate of Jz ) will, in contrast, remain fixed. In short, initial states near the equator can evolve into conditional states which are nearly orthogonal, while initial states near the poles are largely unaffected by the system. An environment whose initial state is equatorial is highly pliable, while one with an initial polar state is unpliable. Now consider randomly assigned conditional states. There is no initial state to average over – instead, the conditional states automatically explore the entire Bloch sphere. They are uniformly distributed. The distribution of decoherence factors is equivalent to the one we would obtain dynamically if every initial state were equatorial, and thus maximally pliable! The reduced pliability of dynamical models stems from the possibility of highly un pliable initial states. a curiosity, it turns out to be possible to compute d analytically for the particular case of a spin- 1 interaction2 only model. The result is d = 1 − ln(2), which agrees precisely with the numerical result. Since this is the only case we have succeeded in computing analytically (even ∆d for the same case is intractible), we do not consider the result sufficiently useful to mention outside of this footnote. 11 The contribution of ∆d toward specific redundancy is rather small (though not numerically negligible) in comparison with the effect of changes in d. As a result, we usually ignore ∆d in the discussion. 12 This implies no loss of generality for spin- 1 systems. Every normalized traceless operator on a 2-dimensional 2 Hilbert space is unitarily equivalent to Jz . For j > 1 , there are more possible forms for R. 2 10 As 55 Dynamical capacity (cd = − log |γ |2 ) is a useful measure of the reduction in pliability. For arbitrary j , we find (see Appendix B.3) c ≡ crandom states cd ≡ cdynamical = = log(2j + 1) log(j + 1), (3.26) (3.27) where c is information capacity. For large j , cd ≈ c − log(2); the dynamical capacity is 1 bit less than the total information capacity. For j = 1 , cd = log 3 whereas c = log(2). Recall that |γ |2 2 2 determines the environment’s decohering power (i.e., the average reduction in off-diagonal elements 1 of ρS ). Thus, a single spin- 2 environment which “measures” the system dynamically will decohere the system by less (on average) than an identical environment whose conditional states somehow explore the entire available Hilbert space. This difference is not only apparent in reduced dynamical capacity, but also in reduced specific redundancy. We shall see in the next section that the internal dynamics of an environment can augment its pliability, by allowing the conditional states to explore more of Hilbert space. 3.3 Quantum-measurement models In the interaction-only model, the environment has no internal dynamics. The subenvironments simply sit there and get pushed around by the system. In real systems, the individual subenvironments13 have internal Hamiltonians. This generates a class of generalized quantum-measurement models, which represent the interaction process for quantum measurements (thus the name). The environment plays the role of the measurement apparatus. Note that no collapse of the wavepacket occurs. “Measurement,” refers to the pre-measurement process, whereby the system and the apparatus become entangled. This is sufficient to produce decoherence and inscribe information about S in the environment, so collapse is unnecessary for (and irrelevant to) our purposes. Quantum measurement models are a refinement of the interaction-only model. Each subenvironment still feels a Hamiltonian which is conditional upon the value of the pointer observable. The difference is that this conditional Hamiltonian consists of a conditional part (R, the familiar recording operator), and an unconditional part (V, the internal Hamiltonian). Roughly speaking, the environment feels either Heff = V + R or Heff = V − R. More precisely, the environment evolves (conditional upon the system’s state) according to Heff = V + µR, where µ depends on the state of S. The universe still evolves into branching states, which means that 1. efficient simulation is possible, even for very large environments; and 2. the theory that we developed in Chapter 2 and expanded in the previous section is still valid, and needs only to be expanded further. We will find that the environment’s pliability is altered by its internal dynamics, in both the transient and asymptotic regimes. We continue the practice of choosing uniform random initial states for the subenvironments, and univariate Gaussian couplings kn . We fix all the environment Hamiltonians identically: V(n) = V. The Hamiltonian for the model is H= n ( 1l(S ) ⊗ V(n) + J(S ) ⊗ kn Jzn) , z (3.28) and the conditional Hamiltonian felt by En is: Heff = V ± kn Jz ( n) (3.29) 13 We assume that the environment’s internal dynamics respect its tensor product structure, and therefore that the subenvironments do not interact directly with each other. 56 The two parameters governing the model (in addition to Nenv ) are (1) the strength of the environment’s internal dynamics, v ≡ ||V||, and (2) the relative orientation of R and V.14 In these simulations, we set V = vz Jz + vx Jx , so that vz and vx completely parametrize the environment’s dynamics relative to the recording operator. We consider three different models: dynamical decoupling models, where vx = 0 and vz = 0; (b) super-pliable models, where vx ∼ vz ; and (c) transient decoupling models, where vx vz . A fourth regime, where vx = 0, is identical to the interaction-only model. Summary of the Quantum-Measurement model ( = JzS ) ⊗ n H = Hint + Henv |ψ0 kn J(n) + z n vx J(n) + vz J(n) , x z (3.30) (3.31) = |↑ + |↓ (1) (1) √ ⊗ |E0 ⊗ . . . |E0 2 ( with |E0n) uniformly distributed, 2 kn 2 P (kn ) ∝ e− . Nenv = 12 . . . 128 3.3.1 Results We begin by examining PIPs, and their evolution over time. Figure 3.5 illustrates two dynamically decoupled environments; a small one with Nenv = 12 and V = 0.8Jx , and a large one with Nenv = 128 and V = 3.2Jx . Figure 3.6 illustrates a super-pliable environment with 4 Nenv = 12 and V = 0.8 5 Jx + 3 Jz , and also a transient-decoupling environment with Nenv = 128 5 199 1 and V = 3.2 200 Jx + 10 Jz . V was chosen so that v = 0.8 for the Nenv = 12 environments, and v = 3.2 for the Nenv = 128 environments. Dynamically decoupled enviroments When we add internal dynamics, generated by environment Hamiltonians V = vx Jx , to the Jz ⊗ Jz interaction model, the result is a reduction in the amount of redundancy. The PIPs for vx = 0.8 in Figs. 3.5a-b, when compared with the equivalent interaction-only plots (Figs. 3.1a-b), are less sharply curved. The classical plateau around m = Nenv is less distinct, and fluctuations 2 in the shape of the curve (over time) are more pronounced. In Figs. 3.5c-d, we increase the size of the environment to Nenv = 128 and set vx = 3.2. The same reduction in curvature (compared with Figs. 3.1c-d) is apparent, along with a pronounced oscillation. At regular intervals (the first is visible around t = 0.85 in plot (d)), the PIP is distorted by a recurrence. Both effects are due to dynamical decoupling, which we discuss in Sec. 3.3.2. Super-pliable environments When the internal Hamiltonian is V = vx Jx + vz Jz (i.e., not orthogonal to R), redundancy is enhanced rather than diminished. The PIPs in Figs. 3.6a-b show the effect of a Hamiltonian com3 4 posed of roughly equal commuting and orthogonal (w/respect to R) parts, V = 0.8 5 Jx + 5 Jz . Careful comparison with the corresponding interaction-only plots (Figs. 3.1a-b) shows slightly enhanced curvature, with a flatter and broader classical plateau in the center. Internal dynamics that is not orthogonal to the measurement’s recording operator enhances pliability, increasing the environment’s capacity to record information. much more detailed treatment of these parameters and their significance for spin- 1 and spin-j environments is 2 presented in Appendix B.3. 14 A 57 Quantum-measurement (V = 0.8Jx): Nenv=12, t < 1.5 IS:ε {m} Quantum-measurement (V = 0.8Jx): Nenv=12, t < 15 IS:ε {m} - HS - HS 12 1.5 9 m 6 3 0.5 1 time 12 15 9 m 6 3 5 10 time (a) Quantum-measurement (V = 3.2Jx): Nenv=128, t < 1 IS:ε {m} (b) Quantum-measurement (V = 3.2Jx): Nenv=128, t < 8 IS:ε {m} - HS - HS 128 96 m 0.9 64 32 0.3 0.6 time 128 96 m 64 32 3 6 time (c) (d) Figure 3.5: We begin studying quantum-measurement models by examining representative PIPs for dynamical decoupling models. These plots correspond directly to those in Fig. 3.1; plots (a)-(b) are for 12-spin baths, whereas Nenv = 128 in plots (c)-(d). The environment spins are given internal dynamics by a Hamiltonian V = vx Jx , which is orthogonal to the measurement interaction’s recording operator. In plots (a)-(b), vx = 0.8. Curvature in these PIPs (compared with the interaction-only PIPs) is less sharp, indicating less redundancy and a stiffer (less pliable) environment. Periodic recurrences cause redundancy to drop drastically. The same phenomena are apparent in plots (c)(d), where vx = 3.2. Recurrences with a period of T = 2π/vx are even more pronounced, as is the sharply reduced curvature of I (m). The recurrences are a result of choosing all the V(n) identically, and would be washed out if the subenvironments had different internal Hamiltonians. Transient decoupling Plots (c)-(d) do not represent the Nenv = 128 results for this super-pliable model, because it’s almost impossible to see any difference (see Fig. 3.7, however). Instead, plots (c)-(d) show transient decoupling. When the internal Hamiltonian is almost orthogonal to R (e.g., as in Figs. 3.6c-d, V = 199 1 3.2 200 Jx + 10 Jz ), we see a mixture of dynamical decoupling and superpliability. At short times (Fig. 3.6c) the PIP evolves very much as in the dynamical decoupling model (Fig. 3.5c). As time progresses (Fig. 3.6d), two things happen: the oscillatory behavior damps out, and the curvature becomes much sharper (indicating greater redundancy). Comparing Fig. 3.6d to the equivalent interaction-only data (Fig. 3.1d), we see that the asymptotic level of redundancy takes much longer – by an order of magnitude – to develop. The timescale on which this transient decoupling gives way to asymptotic superpliability is determined 58 V = 0.48Jx + 0.64Jz: Nenv=12, t < 1.5 IS:ε {m} V = 0.48Jx + 0.64Jz: Nenv=12, t < 15 IS:ε {m} - HS - HS 12 1.5 9 m 6 3 0.5 1 time 12 15 9 m 6 3 5 10 time (a) V = 3.184Jx + 0.32Jz: Nenv=128, t < 1 IS:ε {m} (b) V = 3.184Jx + 0.32Jz: Nenv=128, t < 8 IS:ε {m} - HS - HS 128 96 m 0.9 64 32 0.3 0.6 time 128 96 m 64 32 3 6 time (c) (d) Figure 3.6: We continue investigating quantum-measurement models by examining a super-pliable 4 environment with Nenv = 12 and V = 0.8 3 Jx + 5 Jz (plots (a)-(b)), and the effects of transient 5 1 199 decoupling for Nenv = 128 and V = 3.2 200 Jx + 10 Jz (plots (c)-(d)). By comparing plots (a)(b) with the corresponding interaction-only results in Fig. 3.1, an increase in curvature (indicating enhanced redundancy, and greater pliability of the environment) can be seen. In plots (c)-(d), short-time behavior mirrors the dynamical decoupling in Fig. 3.5c-d, but a transition to super− pliable behavior occurs around t = vz 1 . by the magnitude of the vz Jz term in V – or, more generally, by the portion of V which commutes with R. Again, we discuss this behavior in detail in Sec. 3.3.2. Quantitative redundancy We turn to plots of quantitative redundancy, to distill out the most important features of the data. Figure 3.7 compares R10% (t) for selected quantum-measurement environments with Nenv = 12 and 128. The plots in Figs. 3.7a and 3.7b look very much alike, although R is consistently much higher for Nenv = 128. This illustrates that quantum-measurement models display the same scaling of Rδ with Nenv that we found in the interaction-only model. Consistent with Fig. 3.5, we see that the presence of V induces pronounced oscillation in Rδ . In the interaction-only model, the Gaussian distribution of coupling constants smears out the environments’ frequency spectrum (except for an initial transient). The fixed environment Hamiltonian, however, creates a peak around ω = v . As time advances, the environments lose coherence with one another because their frequencies are perturbed by the interaction Hamiltonian. 59 Quantum Measurement Models: R10% (avg.) versus time (N=12) 2.5 2 R10% 1.5 1 0.5 0 0 5 10 V= 0 V = 1.6Jx V = 1.6(0.6Jx + 0.8Jz V = 1.6(0.995Jx + 0.1Jz 15 20 25 Time (arbitrary units) 30 35 40 (a) 40 35 30 25 R10% 20 15 10 5 0 0 5 10 Quantum Measurement Models: R10% (avg.) versus time (N=128) V= 0 V = 1.6Jx V = 1.6(0.6Jx + 0.8Jz V = 1.6(0.995Jx + 0.1Jz 15 20 25 Time (arbitrary units) 30 35 40 (b) Figure 3.7: We summarize the trends evident in Figs. 3.1, 3.5, and 3.6 by plotting R10% vs. time for selected models. Plots (a) and (b) present data from 12- and 128-spin baths (respectively). One important feature of the plots is that they illustrate the scaling of Rδ with Nenv ; the plots for Nenv = 128 look like scaled versions of those for Nenv = 12. Each set of plots includes (1) interaction-only, (2) dynamical-decoupling, (3) super-pliable, and (4) transient decoupling models. In all cases (except the reference, interaction-only, model), the environment’s Hamiltonian has a magnitude v = 1.6; the different models only change the relative orientation of V and R. This damps the oscillations in R10% 15 , and redundancy eventually settles down to an asymptotic level. A comparison of asymptotic R10% for v = 1.6 should clarify the motivation for the models’ names. The “super-pliable” model displays ∼ 20% more redundancy than the interaction-only model16 , while the “dynamically decoupled” model displays ∼ 50% less. This effect is best expressed in terms of specific redundancy.17 In Fig. 3.8, we compare the specific redundancy of the three quantum-measurement variants, for a wide range of v , to the interaction-only value. In all three variants, weak (v 1) environment dynamics enhance redundancy slightly. In dynamically15 Except in the super-pliable model, where they persist at a lower amplitude. This is due to the v J part of V, zz which commutes with the interaction. 16 But still less than in the random-state model we considered in Chapter 2. 17 The plots in Appendix B.4 confirm that the general behavior seen in Fig. 3.7 scales well with N env , as well as showing the effect of other values of v for all three variants of the model. 60 Specific Redundancy of Quantum-Measurement Models 0.3 r10% (specific redundancy) 0.25 0.2 0.15 0.1 0.05 0 Random-state model Interaction-only (V = 0) Venv = Jx Venv = .6Jx + .8Jz Venv = .995Jx + .1Jz 0.125 0.25 0.5 1 2 v (strength of Henv) 4 8 Figure 3.8: The dependence of asymptotic (long-time) specific redundancy in quantum-measurement models is a concise measure of their overall information-storing ability. The specific redundancy 1 (r10% ) of spin- 2 baths is 0.193 for the interaction-only model (shown as the dashed line), which provides a baseline for the various quantum-measurement models. Weak internal dynamics enhance the environment’s pliability slightly, regardless of R’s orientation relative to V. As v increases, the models with V ∝ Jx exhibit increasingly severe dynamical decoupling. When R has a component that commutes with V, pliability is enhanced most around v = 1, but declines toward the interactiononly level for v 1 and v 1. 61 decoupled variants (where V = vx Jx ), redundancy begins to decline around v ∼ 1. For super-pliable models, redundancy is enhanced by almost 25% at v ∼ 1. As v increases, the dynamically decoupled environment becomes less and less effective at recording the system, while the super-pliable models converge back to interaction-only behavior. We explain both effects using pliability, in Sec. 3.3.2 below. Before moving on to theory, we note one more important effect. When the internal dynamics of the environment are almost, but not quite, orthogonal to the recording operator (e.g., V = 0.995Jx + 0.1Jz ), redundancy is initially suppressed (Fig. 3.7). After some time, redundancy rises, approaching the same level as in super-pliable models. This is transient decoupling. Dynamical decoupling is unstable, because it depends on vz being exactly zero. A component of R in V will destroy the dynamical decoupling on a timescale tcoupling ∼ 1 vz (3.32) In experimental setups, tiny perturbations will eventually destroy the dynamical decoupling effect. 3.3.2 Theory and discussion The results of the previous section can be summarized as follows: • The internal dynamics of a subenvironment can either increase or decrease its average pliability. • When V is orthogonal to R, V makes the environment less pliable (e.g., more stiff). • When V commutes with R, there is no change (w/respect to the interaction-only model). • When V and R are relatively randomly oriented (neither orthogonal nor commuting), V makes the environment more pliable. • If V is almost orthogonal to R, it induces transient stiffness, which evolves into super-pliability. In Appendix B.4, we present a detailed exploration of these phenomena in individual subenvironments, extend the analysis to spin-j baths, and generally provide a more rigorous (if mathematically gruesome) foundation for our pliability analysis. In this section, we skip over some of the gory details to focus on the summarized points above. A subenvironment that starts in state |ψ0 , and experiences a Hamiltonian Hi conditional upon the value µi of the pointer observable, will evolve into a conditional state |ψi = e−iHi t |ψ0 . (3.33) (S ) The resulting decoherence factor, which describes the effect that this subenvironment has on ρij , is γij = ψ0 | eiHj t e−iHi t |ψ0 (3.34) When Hj and Hi commute (as in the interaction-only model), this reduces to γij = ei(Hj −Hi )t . (3.35) Both conditional states evolve along the same orbit, determined by a single operator (Hj − Hi ). Pliability depends on: (1) the norm of Hj − Hi , which determines the duration of initial stiffness; and (2) the orientation of |ψ0 relative to Hj − Hi . Eigenstates of Hj − Hi are very stiff, while unbiased states are very pliable. This accounts for the reduced (relative to random-state averages) pliability of the interaction-only model. 62 Superpliability When Hj and Hi do not commute, the expression in Eq. 3.34 does not reduce to dependence on a single operator (e.g., Hj − Hi ). Instead, the conditional states evolve along different orbits, determined respectively by Hi and Hj . In order for a given |ψ0 to be invariant under the tparameterized family of unitary transformations in Eq. 3.34, it must be an eigenstate both of Hj and Hi – not merely an eigenstate of their difference. Since [Hi , Hj ] = 0, no state can be an eigenstate of both. As a result, there are no maximally stiff initial states when the conditional Hamiltonians don’t commute. This explains the enhanced pliability that we see for V = vx Jx + vz Jz . Fewer of the environment’s possible configurations are stiff, and so the average over initial states yields greater pliability. The reduction in stiffness (i.e., enhanced pliability) is strongest when vx = vz . In Sec. 3.2.3, using dynamical capacity (cd ) as a pliability measure, we found crandom = +J 3 log(2), versus cdynamical = log 2 for V = 0. For a super-pliable model (with V = Jx√2 z ) we obtain cdynamical = log(2). The appropriate V can increase a subenvironment’s pliability all the way to random-state level. Numerical computation of d (see Appendix B.4) yields a maximum value 1 (at vx = vz ) that actually exceeds the random-state average slightly (0.529 vs. 2 ). Our numerical results (e.g., Fig. 3.8) never show such a high value of r because the effective (observed) pliability is an average over all subenvironments. The kn are selected randomly, so the ratio of ||V|| to ||R|| varies from one subenvironment to the next. Under most circumstances, we can view V as a mixing influence. By “stirring” the state of En around, internal dynamics provide more available Hilbert space in which information can be stored. Dynamical decoupling The reduction of redundancy in dynamical decoupling models is a special exception to the mixing intuition. Dynamical decoupling occurs when V and R have a very special relative orientation, which prevents ergodic mixing. This effect has been recognized for some time in NMR systems[136]. A spin which rotates sufficiently rapidly on its own is largely immune to outside influences – that is, it is decoupled from systems which would otherwise alter its state. Broadly speaking, dynamical decoupling reflects the principle that systems with high internal energies are usually unaffected by low-energy interactions. In our model, dynamical decoupling occurs only when vz = 0, so that V is orthogonal to R. In this case, the two conditional Hamiltonians H± = V ± R have identical spectra. The conditional states |ψ+ and |ψ− evolve along different orbits, but at the same rate. When v > 1, the conditional Hamiltonians are dominated by V, so |ψ+ and |ψ− evolve at the same rate along closely separated orbits. The conditional states’ relative motion is small, they never diverge substantially, and γ remains close to 1. The result is a (potentially drastic) reduction in pliability for all initial states, because R is overwhelmed by V. Transient decoupling Dynamical decoupling seems to have great promise as a means of preventing decoherence. However, it depends crucially on the eigenvalues of H± being identical. This occurs only when V and R are orthogonal. By dividing R into a “diagonal” component Rdiag that commutes with V, and an “off-diagonal” component Roffdiag that is orthogonal to V, we can separate the eigenvalue shifts in H± from the eigenvector shifts.18 The two components together induce conditional states to explore their full Hilbert space (the super-pliability effect). 18 Equivalently, we can divide V into parts which commute and don’t commute with R. This is the approach we take in generating simulation data, since it’s easier to keep the core of Hint fixed throughout all simulations. 63 Rdiag alone has no effect, since it commutes with V, while Roffdiag alone causes dynamical decoupling. In practice, however, dynamical decoupling will be a transient effect, because even a small amount of Roffdiag (e.g., setting vz > 0 in our model) will eventually cause a transition from dynamical decoupling to super-pliability. The timescale for the transition is set by v ||Rdiag ||.19 3.4 The dynamical-system model Realistic systems evolve, according to their own internal dynamics, at the same time that they get decohered. So far, we have ignored the system’s internal dynamics in order to focus on the measurement process.20 In this section, we expand our model to consider the effects of a system Hamiltonian (Hsys ). If the system’s internal dynamics commute with M (the measurement operator), then they are irrelevant to information transfer. The environment records only information about M, which is unaffected by the internal dynamics. We restrict our study to Hamiltonians that don’t commute with M. Since the pointer observable for our models is M = Jz , we consider Hsys = E0 Jy , (3.36) which generates a rotation around the y -axis on the Bloch sphere. This is not intended to be a fully general model. We’re interested in how the presence of a nontrivial Hsys affects redundancy, not the quantitative details of the process. An important feature of this model is that it does not respect the branching-state structure. In branching states, each pointer state is correlated with a single branch (e.g., product state) of the environment. Because of Hsys , each pointer state evolves into a superposition of pointer states, which then decoheres. The environment ends up attempting to measure a history, not an observable. One consequence is that a much more general simulation algorithm is required, which is not efficient.21 This severely limits the size of the models we can simulate, since the time required scales exponentially with Nenv . We have studied Nenv = 12, because the environment is large enough to show redundant information storage, yet small enough that simulation runs take hours instead of days or weeks. Summary of the Dynamical-System model H = Hint + Hsys |ψ0 = J(S ) ⊗ z n kn J(n) + E0 Jy , z (3.37) (3.38) = |↑ + |↓ (1) (1) √ ⊗ |E0 ⊗ . . . |E0 2 ( with |E0n) uniformly distributed, 2 kn 2 P (kn ) ∝ e− . Nenv = 12 3.4.1 Results The dynamical-system model is generated by adding an Hsys term to the interaction-only Hamiltonian: H = E0 J (S ) + J (S ) ⊗ kn J(n) . (3.39) y z z n if we divide V, by vdiagonal ||R||. noted previously, “measurement” in this context refers only to the unitary process by which the environment gains information about the system. No collapse or other nonunitarity is implied. 21 Branching-state models can be simulated in O (N 3 env DS DE ), but evolution of a general quantum state scales as 2 D 2Nenv ). O(DS E 20 As 19 Or, 64 System-Dynamics (Hsys = 0.1Jy): Nenv=12 IS:ε {m} System-Dynamics (Hsys = 0.1Jy): Nenv=12 IS:ε {m} - HS - HS 12 9 m 6 3 20 60 40 time 80 100 12 9 m 6 3 100 300 200 time 400 500 (a) (b) Figure 3.9: The dynamical-system model adds a term Hsys = E0 Jy to the interaction (Jz ⊗ Jz ) Hamiltonian. In this figure we show two series of partial information plots for Hsys = 0.1Jy . Plot (a) shows t = 0 . . . 100; plot (b) shows t = 0 . . . 500. The initial development of redundancy (not 1 pictured) is identical to the interaction-only model. Over long timescales (t ≥ E0 ), the initial redundancy is eroded. By t = 100 the PIP is essentially flat. On even longer timescales, antiredundancy develops as the information about S becomes encoded in entangled modes of the En . We vary the strength of Hsys from E0 = 0.01 . . . 1.0. The most interesting behavior occurs for E0 g0 , where g0 is the mean 2-body interaction energy: g0 = 2 kn . (3.40) In this regime, the initial effect of system dynamics is negligible; decoherence and redundancy develop before Hsys causes the system to evolve. We focus on the long-time behavior of information. Partial information plots A representative PIP (for E0 = 0.1) is shown in Figure 3.9, over the time interval t = 0 . . . 100.22 Initially, I (m) shows pronounced curvature, identical to that seen in the interactiononly model. As time progresses, the curve flattens out. For long times, the PIP begins to display anti-redundancy, which we found in Chapter 2 to be characteristic of random (non-branching) states. Quantitative redundancy To gain a quantitative view of Hsys ’s effect, and its dependence on E0 , we turn to redundancy. In Fig. 3.10 we plot R10% versus time for several values of E0 . All data series represent the average of 10 simulation runs, and are smoothed slightly to minimize Monte Carlo noise. We examine two time intervals: t = 0 . . . 50, in order to see initial behavior; and t = 0 . . . 500, to examine asymptotics. Hsys consistently produces a gradual decline in redundancy, down to R = 1 and below23 , which is not accompanied by reduced decoherence levels (i.e., HS ). The timescale for this decline is roughly proportional to E0 . For sufficiently large E0 , R10% never even reaches the interaction-only level. We have never observed an example of redundancy recurring at a later time, nor of Hsys increasing redundancy (even transiently). data points are expensive, we only compute a PIP every 1.0 time units. R = 1 indicates no redundancy – just one copy of the information is present in the environment. However, our R is a lower bound, so even when the PIP is a straight line (indicating no redundancy), R may be less than 1. Antiredundancy can also reduce R a bit. In general, R ≤ 1 should be taken to mean “there is no substantial redundancy,” with no more precise implications. 23 Technically, 22 Because 65 Smoothed Averaged (10 runs) R10%(t); H = E0Jy(S) + Hint 2.5 2 1.5 1 0.5 0 5 10 E0 = 0 E0 = 0.03 E0 = 0.06 E0 = 0.1 E0 = 0.15 E0 = 0.25 E0 = 1 E0 = 10 R10% (a) 2.5 2 1.5 1 0.5 0 50 100 150 15 20 25 Time (arbitrary units) 30 35 40 Averaged (10 runs) R10%(t); H = E0Jy(S) + Hint E0 = 0.01 E0 = 0.02 E0 = 0.03 E0 = 0.06 E0 = 0.1 E0 = 0.15 R10% (b) 200 250 300 Time (arbitrary units) 350 400 450 500 Figure 3.10: Adding a system Hamiltonian Hsys ∝ E0 to the model causes time-dependent destruction of redundancy. These plots show data averaged over 10 simulation runs, for values of E0 from 0 to 10 (relative to an RMS interaction strength g0 = 1). Plot (a) has been smoothed to minimize Monte Carlo error, and compares the stably fluctuating behavior of R10% for the interaction-only model to the decline produced by Hsys , on relatively short timescales. Plot (b) extends to much longer timescales (at the cost of fine resolution in time), and illustrates qualitatively similar behavior for very small E0 . 3.4.2 Theory Introducing system dynamics changes the structure of the decoherence model. The universe cannot be described by branching states, which makes simulation much more arduous. More importantly, the theoretical model that we developed in Chapter 2, that helped us to understand quantum-measurement models, is not valid. A full and quantitative understanding of how Hsys affects branching states probably demands a currently nonexistent theory for multipartite entanglement. To understand the impact of Hsys , we need to consider entanglement. Entanglement is a form of correlation, but not all correlation between quantum systems is entanglement. The short, intuitive rule for identifying entanglement is that superpositions of product states (e.g., ψ = √ (|00 + |11 ) / 2) are entangled, whereas mixtures of product states (e.g., ρ = 1 (|00 00| + |11 11|)) 2 are not entangled. The rigorous definition is: ρ is entangled if it is not separable; ρ is separable 66 if it can be written as a mixture of product states. If two quantum systems are correlated, and their joint state is pure, then they are entangled.24 Decoherence results from entanglement between the system and the rest of the universe25 , but the correlation between S and an small fragment of the environment is typically classical (i.e., ρSE{m} is separable, and the discord [107] between S and E{m} vanishes). The existence of redundancy relies upon this structure, because while perfect entanglement is monogamous, classical correlation need not be26 . Branching states do not allow entanglement between the subenvironments. If we take Eq. 3.1 and trace out the system, the density matrix ρE is completely separable. In the interaction-only and quantum-measurement models, the subenvironments never interact with one another. Information about En cannot be transferred to Em , because there is no opportunity for the state of En to influence the evolution of Em . As a result, En is correlated with Em only because they are both correlated with S . Hsys breaks this symmetry. The subenvironments do not couple directly to each other, but they can interact through the system. This occurs because quantum information flow is always (S ) bidirectional. A Jz ⊗ Jz interaction between S and E1 will transcribe information about Jz into ( E1 ) E1 – but at the same time, information about Jz is transcribed into S . If no other process disturbs the interaction, the two transcriptions proceed independently. (S ) (E ) (E ) (E ) (S ) Information about Jz is recorded in Jx 1 and Jy 1 , and information about Jz 1 is recorded in Jx (S ) and Jy . However, in the interaction picture of quantum mechanics, Hsys causes the measurement (S ) operator to evolve. Initially, the En measure Jz . After a time t they measure M = e− i Hsys t Jz e i Hsys t (3.41) instead. If, e.g., M = Jx , then the operator recorded by En at time t is the same one into which some property of Em was transcribed at t = 0. The subenvironments become entangled through an effective interaction between the En , mediated by S . This is significant for redundancy because product states of the environment are dynamically transformed into entangled states. Decoherence initially correlates each pointer state with a product state of the environment. These product states are transformed, by interactions between the En , into entangled states. The resulting entangled conditional states cannot be distinguished by local measurements on individual environments. An observer is forced to capture larger fragments of the environment in order to identify which pointer state the system is in. Redundancy disappears as entanglement (between the subenvironments) develops. We can attempt to estimate the rate of redundancy loss by identifying the portion of e−iHt that generates entanglement between En and Em . We write H as a sum of Hsys and Hint , and expand the evolution operator in a Taylor series: e−iHt + i 6 1 H2 + Hsys Hint + Hint Hsys + H2 t2 sys int 2 H3 + H2 Hint + Hsys Hint Hsys + Hsys H2 sys sys int t3 (3.42) +Hint H2 + Hint Hsys Hint + H2 Hsys + H3 sys int int 1l − i (Hsys + Hint ) t − Of all the terms in the expansion up to 3rd order, the only one that induces entanglement between the i environments is 6 Hint Hsys Hint t3 . This yields a rough estimate of the timescale on which noticeable 24 Proof: the joint state is pure, so it is not equal to any mixture. It is itself not a product state; therefore it cannot be a mixture of product states, so the joint state is not separable. 25 We assume, as an axiom, that the state of the entire universe is pure. Systems in mixed states are assumed to be subsystems of a larger (pure) system, whose entropy results from entanglement with other subsystems. 26 We refer here to the monogamy of quantum entanglement; basically, if system A is fully entangled with system B, then it cannot also be entangled with a third system E . See [78] for more detail. 67 effects should begin to appear:27 te ∼ 6 E0 1 /3 . (3.43) Unfortunately (as we discuss below), Eq. 3.43 does not describe the observed behavior well. 3.4.3 Discussion There are several problems with the derivation of Eq. 3.43. Most importantly, we have no quantitative model for how entanglement among the environments reduces redundancy. This exacer2 bates the second problem: there are Nenv different interaction terms of the form Hint Hsys Hint , each of which entangles a different pair of subenvironments. Finally, the simulation data aren’t sufficient to elucidate what is happening. They tend, however, to indicate a linear or superlinear relationship − between E0 1 and the entanglement timescale (te ), not the sublinear relationship indicated by Eq. 3.43. System dynamics clearly destroy redundancy (but not decoherence), on a timescale that decreases as E0 rises. The general picture that we outline above, wherein the primary cause of redundancy decay is the emergence of entanglement within the environment, seems likely to hold up under scrutiny. However, in order to go further, we need a better understanding of many-body entanglement. 3rd-order terms of the form Hsys Hint Hsys generate two-body entanglement, but this is not the end of the story. Higher-order terms can generate entanglement between larger and larger clusters of subenvironments. This is particularly awkward because there are a lot of higher-order terms, corresponding to the exponentially large (in N ) number of N -body clusters. Ideally, we would like to synthesize all of these entanglement-generating terms into a single quantity which would characterize how much they affect redundancy. This is impractical without understanding (1) how to measure multiparty entanglement, particularly the irreducible sort that does not resolve into bipartite entanglement, and (2) how different kinds of entanglement impinge upon one another. At this writing, neither of these questions is well understood. Better numerical data might motivate a better theory. However, numerical experiments are constrained by the computational complexity of simulation. Consider: with current workstation technology, it might be possible to simulate a quantum system with 108 Hilbert space dimensions 1 (considering only memory constraints), which corresponds to about 26 spin- 2 environments. A single data point would require approximately 300, 000 years to compute. Obviously, even improving efficiency by a few orders of magnitude will not make this feasible. The point is, even a modest increase in Nenv imposes substantial computational demands, for limited improvement in the data. 3.5 Multiple-measurement interactions In all the models we have studied so far, the system interacts with the environments through a M ⊗ R interaction, which unambiguously measures a single observable. In the absence of system dynamics, the pointer states are perfectly preserved. Real interactions are not so simple. A general interaction between S and E can always be written as a sum of terms: Hint = M1 ⊗ R1 + M2 ⊗ R2 . . . (3.44) We refer to models with more than one M ⊗ R term in Hint as multiple-measurement models. The environment is effectively attempting to measure several noncommuting observables simultaneously, which generates complicated dynamics. 27 If the units in Eq. 3.43 do not appear correct, it is because the denominator is actually (g 2 E ). We have previously 00 defined the unit of time by setting g0 = 1, however. 68 Given the tremendous range of possible multiple-measurement interactions, we follow the same approach as when considering system dynamics. By picking a particular form for Hint , we explore the qualitative effect of multiple measurements, instead of seeking a general theory to predict information storage properties from the specific form of Hint . We consider (and simulate) two simple interactions. The Z-Y model adds a weak measurement of Jy to the standard Jz measurement: Hint = n kn J(S ) ⊗ J(n) + gy J(S ) ⊗ J(n) . z z y y (3.45) This is perhaps the simplest possible way to “pollute” the original measurement with another interaction, but it lacks symmetry and is not particularly realistic. A more realistic interaction is the dipole model, based on the truncated dipolar interaction between spins: Hint = n kn J(S ) ⊗ J(n) − gd J(S ) ⊗ J(n) + J(S ) ⊗ J(n) z z y y x x . (3.46) This represents the weak-applied-field limit of the electron spin / nuclear bath model in [31]. For gd = 0.5, the standard dipolar Hamiltonian is recovered; for gd = 1.0 we would obtain a Heisenberg (J · J) model. By varying gd we can transition smoothly from a simple measurement of Jz to a √ dipolar interaction. Note that the interaction energy for the dipole model is proportional to 2gd , whereas for the Z − Y model it is proportional to gy . Summary of the Multiple-Measurement models HZY Hdipole |ψ0 = J (S ) ⊗ z n kn J(n) + gy J(S ) ⊗ z y n kn J(n) , y kn J(n) + J(S ) ⊗ y y n n (3.47) kn J(n) y = J (S ) ⊗ z n kn J(n) − gd z J(S ) ⊗ y , (3.48) (3.49) = |↑ + |↓ (1) (1) √ ⊗ |E0 ⊗ . . . |E0 2 ( with |E0n) uniformly distributed, 2 kn 2 P (kn ) ∝ e− . Nenv = 12 3.5.1 Results Partial information plots, for both the multiple-measurement models we considered, are virtually indistinguishable from those for the dynamical-system model treated previously. We present one PIP in Fig. 3.11, primarily to confirm this general behavior. The one noteworthy difference √ is that in similar plots, the energy of the redundancy-destroying term (gy or 2gd ) is much less than the corresponding system self-energy (E0 ). PIPs provide only a qualitative picture, and this quantitative effect is more evident in plots of Rδ . Plotting the evolution of redundancy (R10% ) for the Z-Y (Fig. 3.12) and dipole (Fig. 3.13) models confirms several hypotheses. First of all, the two interactions produce virtually identical √ results when the energies are scaled appropriately (i.e., by letting gy = 2gd ). Second, the existence of multiple measurement interactions causes gradual redundancy decay (confirming the indication of Fig. 3.11). Finally, the decay is almost indistinguishable from that produced by a dynamical system. This indicates that there may be a connection between the two models. In the discussion below, we confirm that suspicion. 69 Multiple-Measurement (Hint = Jz⊗Jz - 0.02(Jx⊗Jx + Jy⊗Jy)) IS:ε {m} Multiple-Measurement (Hint = Jz⊗Jz - 0.02(Jx⊗Jx + Jy⊗Jy)) IS:ε {m} - HS - HS 12 9 m 6 3 20 60 40 time 80 100 12 9 m 6 3 100 300 200 time 400 500 (a) (b) Figure 3.11: One variant of the multiple-measurement model implements a dipole interaction, of the form Hint ∝ Jz ⊗ Jz − gd (Jx ⊗ Jx + Jy ⊗ Jy ). Here, we illustrate two series of PIPs for gd = 0.02. Plot (a) shows t = 0 . . . 100; plot (b) shows t = 0 . . . 500. The general behavior is virtually identical to that seen in Fig. 3.9, although the interaction energy is 5 times smaller. Redundancy develops initially according to the interaction-only model, then gradually drains away as the environments become entangled with each other. Eventually, information becomes encoded in entangled modes. 3.5.2 Discussion Multiple measurements are closely akin, in the effects they produce, to system dynamics. In the latter case, the system’s evolution causes the environment to measure different observables at different times. In multiple-measurement models, the environment is also measuring multiple noncommuting observables – but at the same time. It is unsurprising that the two models produce such similar results. The main difference is that multiple measurements seem much more effective at destroying redundancy. Comparable behavior is obtained when g is just 1/3 as large as E0 – for instance, compare gy = 0.02 in Fig. 3.12a to E0 = 0.06 in Fig. 3.10a. We conjecture that this occurs because multiple-measurement models induce entanglement within the environment at 2nd order instead of 3rd order. Consider the Z-Y interaction Hamiltonian, Hint = n ( kn J(S ) ⊗ J(n) + gy kn J(S ) ⊗ Jyn) . z z y (3.50) When we expand e−iHt in a power series, the 2nd order terms include e−iHt = . . . − n,m (S ) ( gy kn km J(S ) ⊗ Jxn) + J(m) x x (3.51) Information about Jx is recorded into joint modes of the environment at 2nd order. The states |+ and |− become correlated with entangled states of En and Em (for all n and m). The actual effect is complicated and messy because of competing influences. For small gy , the single-environment couplings still dominate. Over time, however, there is nothing to prevent multiparty entanglement among the environments from building up. Information about S is still present in the environment, but not in fragments of the environment. This destroys redundancy, without affecting decoherence. When we studied the effects of system dynamics, power-series expansion of H failed to predict the rate of redundancy decay correctly. One reason for this is poor understanding of multiparty entanglement. The same problem arises in multiple-measurement models. We can identify 70 Smoothed Averaged R10%(t); Hint = Jz⊗Jz + gyJy⊗Jy 2.5 2 1.5 1 0.5 0 5 10 gy = 0 gy = 0.01 gy = 0.02 gy = 0.04 gy = 0.05 gy = 0.1 gy = 0.5 gy = 1 R10% (a) 2.5 2 1.5 1 0.5 0 50 100 150 15 20 25 Time (arbitrary units) 30 35 40 Averaged R10%(t); Hint = Jz⊗Jz + gyJy⊗Jy gy = 0.01 gy = 0.02 gy = 0.04 gy = 0.05 R10% (b) 200 250 300 Time (arbitrary units) 350 400 450 500 Figure 3.12: These plots, like those in Fig. 3.10, present the short-time (plot (a)) and long-time (plot (b)) behavior of R10% . Here, we add a second interaction Hamiltonian, gy Jy ⊗ Jy , to the original Jz Jz interaction, producing a multiple-measurement model. The effect appears identical to that of adding system dynamics; redundancy is damped at a rate that increases with gy . Multiple measurements appear to be substantially more effective than system dynamics at destroying redundancy. For example, compare the gy = 0.1 curve in plot (a) to the E0 = 0.25 curve in Fig. 3.10a. the terms that lead to entangled states of E , but not the rate at which they produce entanglement, nor the effect of that entanglement on Rδ (t). It may nonetheless be possible to compare two similar models – e.g., dynamical-system and multiple-measurement – by comparing the terms that produce entanglement. This is still not an easy problem: the environment-entangling terms in the respective evolution operators are not exactly the same, and the number of entangling terms is quite large. Encouragingly, multiple measurement terms destroy redundancy more efficiently than Hsys terms. This observation makes sense: entangling terms appear at 2nd order in the Z − Y model, but only at 3rd order in the Hsys model. One approach to this problem is to examine the interacting-environment models mentioned briefly in Section 3.1.1. Entangling terms would appear at 1st order, and might be easier to understand. 71 Smoothed Averaged R10%(t); Hint = Jz⊗Jz - gd(Jx⊗Jx + Jy⊗Jy) 2.5 2 1.5 1 0.5 0 5 10 gd = 0 gd = 0.01 gd = 0.02 gd = 0.05 gd = 0.1 gd = 0.5 R10% (a) 2.5 2 1.5 1 0.5 0 50 100 150 15 20 25 Time (arbitrary units) 30 35 40 Averaged R10%(t); Hint = Jz⊗Jz - gd(Jx⊗Jx + Jy⊗Jy) gd = 0.01 gd = 0.02 gd = 0.05 R10% (b) 200 250 300 Time (arbitrary units) 350 400 450 500 Figure 3.13: These plots duplicate Figure 3.12, but show the results of simulating a dipole interaction H = Jz ⊗ Jz − gd (Jx ⊗ Jx + Jy ⊗ Jy ). The results appear equivalent to those obtained from the Z − Y model and shown in Fig. 3.12. 3.6 Dissipative models One more type of model needs to be mentioned, albeit briefly. By adding either (1) a system Hamiltonian, or (2) multiple measurements, to the quantum-measurement model, a dissipative model is created. The distinguishing feature of dissipative models is that because both S and E have nontrivial Hamiltonians, energy can be exchanged between them. Thermal equilibrium is a valid concept. Furthermore, the interaction between S and E drives the universe toward thermal equilibrium. From a dynamical (as opposed to thermo dynamical) perspective, dissipative models have virtually no constants of the motion. There are no remaining symmetries to prevent ergodicity. The end result, whether we explain it using thermodynamics or analytic mechanics, is a more rapid approach to equilibrium. In thermal equilibrium, the conditional states (of E ) corresponding to distinguishable states of S are randomly selected, or appear to be. This means no redundancy – in fact, as shown in Chapter 2, information is stored with anti -redundancy. That is, it is recorded in highly entangled modes of the environment. Virtually no information can be obtained about S unless a majority of the environment is captured. 72 Dissipative (H = Hsys + Henv + Hint): Nenv=12) IS:ε {m} Dissipative (H = Hsys + Henv + Hint): Nenv=12) IS:ε {m} - HS - HS 12 9 m 6 3 20 60 40 time 80 100 12 9 m 6 3 100 300 200 time 400 500 (a) Dissipative (H = Henv + Hint + Hdipole): Nenv=12 IS:ε {m} (b) Dissipative (Henv + Hint + Hdipole): Nenv=12 IS:ε {m} - HS - HS 12 9 m 6 3 20 60 40 time 80 100 12 9 m 6 3 100 300 200 time 400 500 (c) (d) Figure 3.14: Adding environment dynamics to the dynamical-system (plots (a)-(b)) or multiplemeasurement (plots (c)-(d)) models creates a dissipative model. Adding subenvironment Hamiltonians, V(n) = 0.6Jx + 0.8Jz , enhances the decay of redundancy. These plots correspond exactly to those in Fig. 3.9 and Fig. 3.11. Comparing plots, we see that redundancy decays much faster in the dissipative versions. Information becomes encoded in entangled states, as indicated by the reverse curvature of PIPs. The universe relaxes toward equilibrium, conditional states are randomly distributed, and nonredundant information storage (typical of the uniform ensemble; see Chapter 2) emerges. Figure 3.14 shows PIPs for both types (dynamical-system and multiple-measurement) of dissipative model. The data show clearly that redundancy is rapidly destroyed, and that “encoding” of information emerges shortly thereafter. Not only is redundancy eliminated more rapidly than in the non-dissipative models, but the encoding afterward is more severe. 3.7 Conclusions, discussion, and future directions In this work, we have taken the first steps toward an understanding of the dynamical development of redundancy in decoherence processes. We have established a substantial foundation for understanding how redundancy emerges in quantum measurement models, and the properties of the environment which affect its storage of information. We study redundant information storage in order to understand how classical objectivity emerges from the quantum substrate. This is a large question. We try to answer three main questions in this particular work. 73 Models for Spin Bath Environments System Dynamics Measurement Interaction Hsys=E J S 0Y Hint = J ° ∑n k J S Z Extra “Y­Y” Interaction n nZ H Y-Y =g y J Y ° ∑n k n JY S n Environment Dynamics Extra “Dipole” Interaction Henv =∑n v J v J n xX n zZ Hdipole =gd ∑n k n JY ° JY JX ° JX S n S n Interaction-Only Model H= Jz ° ∑n k n Jz S n - yields branching states - stable redundancy - well understood Dynamical-System Model H=Hint Hsys - no branching states - redundancy decays - partial understanding Quantum-measurement models Multiple-Measurement Models H=Hint Henv - yield branching states - stable redundancy - fairly well understood Variants: - dynamically decoupled - superpliable - transiently decoupled H=Hint HY-Y H=Hint Hdipole - no branching states - redundancy decays - partial understanding or Dynamically Decoupled Dissipative Model (1) Superpliable Transiently Decoupled Dissipative Model (2) H=Hint HsysHenv - no branching states - redundancy decays rapidly - thermal equilibrium - partial understanding H=HintHdipoleHenv - no branching states - redundancy decays rapidly - thermal equilibrium - partial understanding Figure 3.15: A schematic diagram of all the spin bath models we have considered in this chapter. Hamiltonians, at the top of the figure, are connected to the models which depend upon them. Basic models are found in the center of the figure. Dissipative models, connected to the models from which they are built, are at the bottom. 74 • Does redundancy emerge in realistic decoherence processes? It does, in certain models. The quantum-measurement models28 examined in Sections 3.2 and 3.3 store information about the system with a redundancy that is limited only by their size. We have also shown that more complex models store information redundantly for a limited time. • What determines the degree of redundancy? For quantum-measurement models, the degree of redundancy is determined by the subenvironments’ pliability. Pliability, in turn, is determined by the internal dynamics of the subenvironments, and its relationship with the recording portion of Hint . Dynamical decoupling can reduce redundancy dramatically, but is almost certainly transient in most physical models. The superpliability of spin- 1 environments, 2 which only occurs in the resonant regime ||R|| ∼ ||V||, is shown in Appendix B.3 to be become generic (i.e., not strongly dependent on a resonance) for environments composed of large-j spins. Sufficiently large and complex subenvironments are a much more complicated problem. • Is redundancy a generic feature of [most] decoherence models? There are multiple answers: yes, no, and maybe. For (1) small perturbations around quantum-measurement models, and (2) short timescales (relative to the perturbation), substantial redundancy appears. Our [limited] results indicate, however, that almost any such perturbation will eventually destroy redundancy. Dissipation destroys it even faster. Most interactions produce non-singly-branching states, and we showed in Chapter 2 that redundancy is scarce outside the singly-branching states. Finally, there are many, many models that we have not considered here. We cannot yet make any substantive statements about all decoherence models, except that while decoherence may occur without redundancy, redundancy cannot occur without decoherence. Our results indicate a host of further questions. We have already considered a few of them; these may be found in the Appendices. Some general features merit discussion here. Pliability plays an important role in both redundancy and decoherence, since a stiff environment cannot “measure” the system. Redundancy is far more sensitive to the pliability of individual subenvironments or fragments than is decoherence. A sufficiently large collection of stiff subenviron1 ments can still produce strong decoherence. We showed, in Sec. 3.3.2, that baths of spin- 2 particles have fairly robust pliability, except in the [unlikely] case of perfect dynamical decoupling. A thorough investigation of pliability for other environments would be very useful, and we have begun this process in Appendix B.3. Small variations in pliability are relatively unimportant – it doesn’t really matter whether a given bit of information is 500-fold redundant or 1000-fold redundant – but effects that dramatically increase or reduce pliability (dynamical decoupling, for example) are relevant to quantum information processing. The relationship between decoherence and redundancy is also of great interest. Decoherence is obviously necessary for redundancy, but not sufficient. In Appendix B.5, we consider the relationship between decoherence (as measured by the system’s entropy) and redundancy. In quantum-measurement models, we show that in order to have redundancy, decoherence must not only exist, but be very nearly complete. Off-diagonal elements of ρS must be vanishingly small, because each additional fragment that has “pretty good” information suppresses off-diagonal terms anew. More interestingly, redundancy decay (when it appears) is not reflected in the strength of decoherence. This supports our theory that redundancy is lost when conditional states of E evolve into entangled states, and fragments of E can no longer supply useful information. The more general models (dynamical-system and multiple-measurement) are the most relevant to understanding classicality and the Universe in which we exist. Our simulation results indicate that redundancy is (unlike decoherence) fragile, subject to destruction by even very weak influences. It’s important to emphasize, however, that we have examined one extremely small “universe,” with a very specific Hamiltonian. Physical systems similar to the spin bath we examine here have further 28 The interaction-only model is a special case of the quantum-measurement class. 75 features which we have not considered. For instance, although a nuclear spin may interact strongly with fewer than 12 other spins, it has weak couplings to many more distant spins, which may have unforeseen effects. The environments are, in turn, coupled to other spins, which are coupled to even more spins. This “onion” model of the environment, where a local layer of subenvironments interacts with a larger bath, deserves much closer examination. An additional effect which we cannot treat in the spin bath model is the transient nature of many interactions. Observers gain information about most systems by monitoring the electromagnetic field, and its component photons. A single photon does not interact continuously with the central system; rather, it scatters off the system, records information about it, and flies off to be intercepted (or escape to infinity). This is a very important paradigm for understanding the Universe around us, but is not accessible through the sort of finite model we have considered here. In conclusion, while we feel that a foundation for understanding the dynamics of information storage has been laid in this work, the field of interesting problems remains barely scratched. Another, more oblique, consequence of our investigation is perhaps the most urgent and exciting of all the future directions. By examining redundancy, we have pushed the boundaries of open systems research outward. The traditional paradigm has been to focus on the central system, and to study decoherence through reduced density matrices, master equations, and entropy. These tools measure the amount of correlation between S and E . Redundancy shows that the nature of those correlations can be equally important. For ρS , it is irrelevant whether pointer states are correlated with product states of the En or with entangled states. For an observer seeking to learn about S , however, the difference is of paramount importance. It determines whether the information stored in E is easy, or absurdly difficult, to access. Eventually, this program of research will provide a better understanding of what it means for one object to “have information about” another object. This includes quantitative questions such as “How much information?” and more complex questions such as “Information about what property?” Such a theory not only will clarify many of the currently unsolved questions about redundancy (e.g., how to characterize the effect of emergent entanglement among the subenvironments), but has the potential to contribute to many areas of quantum information science. 76 Chapter 4 Redundancy in quantum Brownian motion In previous chapters, we have examined the concept and development of redundancy, both abstractly and in simple spin bath models. In this chapter, we consider a much more realistic model of decoherence, quantum Brownian motion (QBM). QBM environments are one of two canonical models for decoherence (spin baths are the other), and have been examined by over 100 papers in the literature, dating back to the 1940s. Exact and approximate master equations have been derived analytically for QBM, but the master equation paradigm explicitly discards the environment. Here, we apply the “Environment as a Witness” paradigm to QBM. By explicit evolution of states, we analyze how decoherence records information about an oscillator (S ) in a large but finite bath of oscillators (E1 . . . ENenv , where Nenv ∼ 64−2048). We focus on two aspects of information storage: (1) how redundantly is the information stored? and (2) where, within the environment, is information stored? The QBM model of decoherence has some outstanding advantages. It describes several physically realistic environments very well, and its linearity allows us to simulate exactly the dynamics of thousands of subenvironments. It introduces effects that are not seen in spin baths. The central system (a harmonic oscillator) has many independent properties that can be separately measured by the environment. Unlike spins, the oscillators that make up a QBM environment are not identical – each oscillator mode is distinguishable from the others by its frequency. We neither expect nor find information to be partitioned evenly among the various modes. On the one hand, these complications hamper the construction of a quantitative theory. On the other, they highlight the shortcomings in our present understanding of information/correlation, and illustrate what new developments are needed. 4.1 Introduction and background Quantum Brownian motion’s long and illustrious research history, stands out in a field (quantum information theory) in which very little predates the 1980s. QBM models arose naturally from attempts to model friction and dissipation in quantum systems. The canonical “original” references are Feynman and Vernon [45], and Caldeira and Leggett [26]. As the study of decoherence emerged in the 1980s, it began to be appreciated that decoherence is naturally associated with dissipation, and that it generally occurs on a much faster timescale [140]. In the 1990s, exact and approximate master equations for the central system’s density matrix were derived [66, 61]. These master equations provide (in principle) a complete understanding of certain systems’ open dynamics. Classical Brownian motion describes a particle (S ) constantly buffeted by the molecules of 77 a surrounding fluid (E ). Interaction with the environment produces dissipation (a particle initially in motion slows down) and fluctuation (once at “rest”, the particle continues to move diffusively). Brownian motion is quantized by replacing the bath of classical particles with a quantum field Φ(x), which interacts with S through a position-amplitude coupling: Hint ∝ xS Φ(0). ˆˆ (4.1) A wide variety of quantum open systems have been treated using this model. Its generality comes from the fact that low-energy interactions with almost any continuous system are well-approximated by linear couplings to a harmonic environment. QBM has been used to describe decoherence in cosmology [65, 69, 60], interaction of an electronic or photonic signal with a noisy transmission line [150, 151, 152, 84], dissipation of electromagnetic field modes in a cavity [50], nuclear fission [154], and a two-level particle interacting with a generic bath [99, 5, 26, 10]. We begin by briefly reviewing the properties of QBM models, and the results of prior work on master equations and decoherence. This work is only partially relevant to our project; we primarily use these master equation results to constrain and verify our numerical simulations. To discover how the information lost through decoherence is stored in the environment, we need to know the environment’s state. We thus proceed to review briefly the techniques we use to simulate dynamics, and the constraints placed on the model. We then present the results of our study, confirming in particular that information is stored redundantly in ohmic QBM models. After discussion and some theoretical explanation of the results, we conclude with a survey of the most pressing unanswered questions, and the next steps toward answering them. 4.1.1 General properties of the QBM model The bath can be treated in two ways: as a quantum field Φ(x), or as a collection of [infinitely many] independent harmonic oscillators. The two pictures are entirely equivalent; each oscillator in the latter model is simply a normal mode of the field Φ. We find the independent-oscillators model far more convenient for our purposes, so we will adopt it for the remainder of this chapter. The alternate picture should be kept in mind, however, as it motivates both our methods and some of our further questions at the end. In particular, if the field is constrained to a fixed interval (that is, a “box” in space), then the possible frequencies (ω ) of the oscillators are no longer continuous, but integer multiples of a base ω0 . Our numerical models have exactly this property, because an infinite collection of modes cannot be simulated. Our results thus represent both an approximation to the continuous-spectrum case and an exact treatment for a field in a box. The central system is a harmonic oscillator, with mass mS and bare frequency Ω0 . The environment consists of Nenv harmonic oscillators, each with mass mn and frequency ωn , and coupled to the system through a position-position interaction with coupling strength Cn . The Hamiltonian for such a “universe” is1 H= mS Ω2 x2 p2 0S S + + 2mS 2 Nenv n=1 2 22 qn mn ωn yn + 2mn 2 Nenv + xS n=1 Cn , (4.2) where xS and pS are the position and momentum of the system, and yn and qn are the position and momentum for the nth environment. Choosing a harmonic oscillator as the central system is not strictly necessary, but it ensures linearity of the total Hamiltonian. This model can be solved exactly (for the master equation of the system), or simulated efficiently (to obtain the global state). A nonlinear Hsys would break overall linearity, permitting only approximate solutions for the master equation (and increasing the complexity of simulation). 1 We attempt as much as possible to follow the notation of [66] throughout this chapter, although we extend their notation substantially in describing the individual oscillators of the environment. In this area, we follow the notation of [19]. 78 Each oscillator in the environment has a unique frequency ωn which “labels” it2 , so we will frequently refer to an individual oscillator by its frequency. In previous analytical studies of QBM, the ωn are continuous in the range 0 . . . ∞. The dynamical properties of E are then determined entirely by its spectral density I (ω ), defined as I (ω ) = n δ (ω − ωn ) 2 Cn 2mn ωn (4.3) Spectral density measures the coupling between S and a given frequency band of E . The integral of I (ω ), over some interval ∆ω , determines how strongly the system interacts with that part of the environment that resonates at frequencies within ∆ω . As such, the spectral density depends both on the number of oscillators present at a given frequency, and on the strength with which they couple to the system. Hu et al point out, “In fact, two different environments with the same I (ω ) are effectively equivalent in so far as their influence on the dynamics of the system is concerned.”[66] QBM environments are, in theory, infinitely diverse. Every distinct I (ω ) represents a different environment. In practice, two principles constrain the possible models. First, physical intuition indicates that for very high frequencies the spectral density should vanish. A cutoff frequency Λ is therefore always applied, and I (ω ) is made to disappear rapidly for ω > Λ. Analytical treatments 2 2 generally use a smooth rolloff such as e−ω /Λ ([66]) to simplify integrations; in numerical work the simple solution is to only include ω < Λ. Second, I (ω ) is assumed to have a simple power-law dependence on ω , so I (ω ) ∝ ω p for ω < Λ. The most important and commonly studied power law is the ohmic spectrum, where p = 1. Other power-law dependences are classified as sub-ohmic (p < 1) or superohmic (p > 1), and induce dissipative forces that are nonlinear in the velocity. In this chapter, we consider only ohmic models. An ohmic QBM model has only one free parameter, the strength of the coupling constant. Following Hu et al [66], we parametrize the spectral density as I (ω ) = 2mS γ0 ω. π (4.4) The coupling constant (γ0 ) has units of inverse time, and manifests itself as a frictional coefficient. 4.1.2 Practical issues in simulation of QBM We study the dynamics of information by computing (numerically) the evolution of a central oscillator and the QBM environment with which it interacts. In order to do so, we must modify the model in some ways, and use some existing results. Spectral bands The continuous spectrum ω ∈ [0 . . . Λ] is replaced by a finite collection of oscillators with frequencies distributed over the interval [0 . . . Λ]. Each oscillator (ωn ) represents a band of frequencies, whose width is ∆ω . This collection represents a continuous spectrum faithfully until a time 2 tc ∼ ∆π . For t < tc , all the distinct oscillators represented by a single band evolve and respond ω to the system in unison, but for t > tc , nontrivial dynamics within a single band are possible. We choose the ωn to be linearly spaced: 1 ωn = (n − )ω0 . (4.5) 2 is unambiguous in the case of master equation studies. If two oscillators i and j have identical ω , they can q 2 2 always be replaced by a single oscillator k with a higher coupling (Ck = Ci + Cj ), which influences S in the same way. In our study this is not necessarily true; if a single oscillator has extremely good information about S , then by separating it into two identical oscillators which each have fairly good information, we can increase the redundancy of that information. In practice, this is not an issue. Any time an oscillator is divided this way, we assign distinct frequencies to the resulting oscillators. 2 This 79 This ensures that tc is impossible to ignore, since a massive Poincar´ recurrence occurs at t = tc . e We could eliminated the Poincar´ recurrence by choosing a more random distribution of frequencies, e but the linear spacing ensures that the data are valid for all t < tc . Gaussian states We constrain the state of the universe to be Gaussian. This permits efficient simulation of the dynamics for arbitrary times, because the dynamics of Gaussian states mirror the dynamics of classical probability distributions. A Gaussian state for a single degree of freedom x with associated ˆ 2 momentum p can be fully described by five parameters: x , p , ∆x2 = x2 − x , ∆p2 = ˆ ˆ ˆ ˆ ˆ 2 1 ˆ ˆ ˆˆ ˆˆ p2 − p , and ∆(xp) = 2 xp + px − x p . These quantities have a geometric representation on ˆ ˆ phase space, as (1) a symplectic vector identifying the center of the Gaussian, z= x ˆ p ˆ , (4.6) and (2) a rank-2 symmetric variance tensor (also known as a covariance matrix) containing the uncertainties, ∆x2 ∆(xp) V= . (4.7) ∆(xp) ∆p2 A Gaussian state with N degrees of freedom is described in the same way, except that z has 2N components, and V is 2N × 2N . A convenient feature of linear Hamiltonians (any combination of linearly coupled harmonic oscillators, free particles, and inverted oscillators) is that the rank-1 and rank-2 components of the state (z and V ) never interact. They transform via different irreducible representations of the symplectic group. If we choose, we can ignore either one or the other. Decoherence, information, and the correlations between subsystems are all obtained from the rank-2 portion of the state (V ). For the remainder of this chapter, we will refer to V as “the state”, and ignore z . This representation of the state (in terms of V ), describes a Wigner function (W (z )), which is in turn a representation of the density matrix ρ. An efficient algorithm for evolving states Because the dynamics of the universe are linear, the symplectic vector z transforms as z (t) = T z0 , where T is a 2N × 2N symplectic matrix. The variance tensor V is a symmetric tensor, so it transforms like zz T : V (t) = T V0 T T . (4.8) The method that we use for computing T is discussed in detail in Chapter 6. We present only a summary here. The Hamiltonian of the entire universe (S ⊕ E ) describes a collection of coupled harmonic oscillators, which has a set of N = Nenv + 1 normal modes. We find the normal modes (and their frequencies) by first eliminating the various masses by applying a scaling transformation M, then finding an orthogonal matrix R that diagonalizes the potential energy matrix. R (which transforms to normal modes) is a point transformation. It acts only on the coordinates, because the couplings are only between position coordinates (i.e., there are no pi pj or xi pj couplings). Each normal mode ˆˆ ˆˆ has a frequency ωn , where (ωn )2 is an eigenvalue of the potential energy matrix. The time evolution ( n) matrix (T0 ) is block-diagonal in the normal-mode basis, with symplectic blocks T0 that implement 2 oscillatory motion when (ωn ) > 0, T0 ( n) = cos ω t −ω sin ω t sin ω t ω cos ω t , (4.9) 80 or exponential divergence when (ωn )2 < 0, Tn = cosh ω t ω sinh ω t sinh ω t ω cosh ω t . (4.10) We transform (T0 ) back to the original basis of observable modes to obtain: T (t) = M−1 RT T0 (t)RM−1 . (4.11) Master equation coefficients (Eq. 4.19) can also be obtained through manipulation of T . In this chapter, however, we use T only to evolve states. We refer the interested reader to [19]. Entropy of Gaussian states Our primary concern is with information, which requires computing the entropy of arbitrary subsystems. The entropy of a Gaussian state on a single degree-of-freedom phase space can be expressed conveniently in terms of its scaled symplectic area A, where A = Trρ2 −1 = a/a0 ∆x2 ∆p2 − (∆(xp)) 2 , 2 (4.12) (4.13) (4.14) a≡ a0 using a formula derived in [169]: H (A) = ≡ (A + 1) ln(A + 1) − (A − 1) ln(A − 1) − ln(2). 2 (4.15) A convenient approximation which is exact for H = 0 and as H → ∞, and always accurate to within 1 − ln 2 0.31, is H ln(A) + 1 − ln(2), (4.16) The astute reader may note that the symplectic area (a in Eq. 4.13) is the square root of the determinant of the state’s variance tensor (Eq. 4.7): a2 = det(V ). (4.17) This suggests a generalization to multiple degrees of freedom, by replacing a2 with the determinant N of the 2N × 2N variance tensor, and a0 with 2 . Further consideration, however, reveals that this method is incorrect. The correct method is to symplectically diagonalize the variance tensor V – that is, reduce it to 2 × 2-block-diagonal form using symplectic transformations instead of orthogonal rotations. This yields N symplectic areas: a1 , a2 , . . . aN . Entropy is given by a sum: N Htotal = i=1 H (Ai ), (4.18) where H (Ai ) is defined in Eq. 4.15. 4.1.3 Relevant known properties of QBM models The environment’s dynamics are determined by three important inverse timescales: ω0 , Λ, and γ0 . ω0 and Λ represent infrared and ultraviolet cutoffs (respectively), and γ0 determines how strongly the environment couples to the system. A fourth timescale, which interacts with the environment timescales to determine the overall dynamics, is the frequency of the system (ωS ). The relationship between ωS and Ω0 is discussed below. 81 The Master Equation Previous research into QBM has focused on deriving master equations for ρS . We present a brief review of the master equation here; for details the reader is referred to [26, 140, 66]. The system’s state evolves as mωeff 1 ∂ x2 , ρ + 21 p2 , ρ + γeff [ˆ, {p, ρ}] − F (t) [ˆ, ρ] ˆˆ xˆ 2 mˆˆ 2 x ˆˆ ρ= i ˆ ∂t −f1 (t) [ˆ, [ˆ, ρ]] + f2 (t) [ˆ, [ˆ, ρ]] x xˆ x pˆ 2 . (4.19) The terms in the master equation fall naturally into two groups. The first group, with coefficient (i )−1 in Eq. 4.19, represent classically relevant processes. These include (a) renormalized unitary 2 evolution with frequency ωeff , (b) dissipation with frictional coefficient γeff , and (c) a driving force with amplitude F (t). Each of these terms impacts the evolution of x and p . The second group of terms, proportional to f1 and f2 , have purely “quantum”3 effects. They affect the variances ∆x2 , ∆p2 , and ∆(xp), but do not change the linear expectation values. The effects of each term, and the behavior of their coefficients, are discussed at great length in [26, 140, 66], and in [19] in a slightly different context. Essentially, the first group of terms act as classical intuition suggests. The system’s state rotates in phase space, according to the renormalized mω 2 p2 Hamiltonian Heff = 2m + 2eff . Energy dissipation due to γeff drives the system toward the origin of its coordinate system, while F (t) acts exactly as a classical driving force.4 The second (“quantum”) group of terms cause diffusion in phase space. The term proportional to f1 , known as the normal diffusion term, simply induces diffusion in p. Positive values of f1 cause ∆p2 to grow linearly as a function of time. This diffusion smears out the interference fringes in the Wigner function of a Schr¨dinger cat state, effectively measuring x. o The other diffusion term produces anomalous diffusion. Its effects are similar to those of the normal diffusion term, except that it affects only the skew-variance term (∆(xp)). Unlike normal diffusion, anomalous diffusion is not guaranteed to increase entropy (HS ), even when f2 is positive. Instead, the entropic effect of the f2 term depends on the value of ∆(xp). All of the coefficients in the master equation vary with time. A great deal of work, culminating in [66], has been devoted to computing their precise time-dependence. A complete review of this work is beyond our current scope, but we can make a few general comments. At t = 0, all the coefficients are equal to their bare values: ωeff = Ω0 , and γeff = f1 = f2 = 0. For ohmic environments, both ωeff (t) and γeff (t) undergo a rapid (on a timescale t ∼ Λ−1 ) transition to equilibrium values which are subsequently stable. Unlike ωeff (t) and γeff (t), the diffusion coefficients depend on the initial state of the environment. For the low-temperature case we study here, f1 rises rapidly, then drops (again, on a timescale t ∼ Λ−1 ) to an equilibrium value. Frequency renormalization The preceding discussion of the master equation is relevant because a basic understanding of the system’s dynamics is required to choose sensible input parameters for the numerical simulations. Dissipation, the most obvious of these effects, is governed by γeff (t). Prior studies [140, 66] have shown that for ohmic spectral densities (Eq. 4.4), γeff (t) equilibrates to the coupling strength γ0 after an initial delay. To control the rate of dissipation, we vary γ0 . A more subtle effect which is crucial to an understanding of the dynamics, is frequency 2 renormalization. After an initial delay, ωeff equilibrates (except for high-frequency oscillations) to a 3 Actually, every feature of the master equation and of Gaussian-state dynamics can be considered “classical.” Only the introduction of a minimum phase-space area ( 2 ) is truly quantum mechanical. What we mean by calling these terms “quantum” is that their effects do not appear for classical pure states – points in phase space. For classical distributions, these terms produce diffusion exactly as they do for quantum states. 4 F (t) vanishes for a balanced environment, where y n = qn = 0 for all n. By ensuring that the environment is balanced, we are free to disregard F (t) henceforth. 82 2 renormalized value ωS , 2 ω S = Ω 2 − δ Ω2 0 (4.20) where we have calculated the equilibrium value of the frequency shift (for an ohmic environment) as δ Ω2 = 4mS γ0 (Λ − ωmin ) π (4.21) This frequency shift cannot be ignored. As Λ → ∞, δ Ω2 → ∞. For finite Λ, the frequency shift can 2 still be much larger than the bare frequency Ω0 . It is ωeff which determines the dynamics of the coupled system, however. The bare frequency must be considered nonphysical, so we adjust Ω0 in order to produce the desired physical frequency: Ω0 = 2 ω S + δ Ω2 (4.22) Throughout this chapter, we will use ωS to indicate the renormalized frequency. The bare frequency Ω0 is largely irrelevant, as can be seen in the following argument. We consider the eigenvalues of the whole universe’s N × N potential-energy matrix. Its diagonal 2 elements are the individual oscillators’ bare frequencies (Ω2 , {ωn }), and its off-diagonal elements are 0 the coefficients (Cn ) of the xS · yn coupling terms. Its eigenvalues are the squared normal-mode 2 frequencies of the coupled universe. If Ω0 is chosen so that the renormalized ωS is less than 0, then the lowest eigenvalue of the coupling matrix also becomes negative. The result is a normal mode with an imaginary frequency – an inverted oscillator. The position and momentum of this imaginary-frequency mode rapidly diverge, dominating the dynamics of the universe and leading to radically nonphysical behavior. By choosing Ω0 so that the renormalized frequency is positive, we eliminate this nonphysical behavior. We conclude that the renormalized frequency really is the physically relevant quantity. 4.2 Simulation results: redundancy of stored information In this section, we focus on the development of redundancy. Numerical data from simulations of QBM universes are the heart of our analysis. We compute Ψ(t), represented as a variance tensor (V (t)), and analyze the correlations between subsystems in order to track information about the central system. We focus on analysis of redundancy in this section. In Sec. 4.5, we will identify which bands of the environment have the most information about S . The analysis in this section, however, is concerned with how the environment stores information. We begin with a brief discussion of our simulation methods. An examination of PIPs leads to a relatively new concept, non-redundant information. Finally, we look at quantitative measures for redundancy, and the effects of varying parameters of the model. Many of the figures in this section serve two distinct purposes: they present numerical data, and they also compare that numerical data to a theoretical model. This model is mathematically dense and only partially successful, so we delay its presentation until Sec. 4.3. The theoretical predictions are displayed on the plots in this section, in order to keep to the total number of figures to a minimum. Readers are advised to ignore these dashed lines on the first reading, then refer back to them while perusing Sec. 4.3. 4.2.1 Methods and parameters We use the algorithm discussed in Section 4.1.2 to evolve a universe consisting of N = Nenv + 1 coupled oscillators. Virtually every parameter of the model can be tuned and varied using more than 40 command-line options. The simulation code generates the full evolved state of the universe (V (t)), as well as master equation coefficients (Eq. 4.19), which do not concern us here. 83 We can compute every interesting property of the system and environment using V (t). In this work, we will consider only information-based properties, such as: the system’s entropy (HS ); the mutual information between S and individual bands of the environment (IS :Eω ); and the redundancy of the stored information (Rδ ). The simulation’s properties are governed by: (1) global parameters, (2) parameters governing the environment, and (3) parameters governing the system. Global parameters, which we do not vary at all, include: 2 = 1, mS = 1, and mn = 1. Primary parameters for the environment are: 1. the number of bands (Nenv ), 2. the cutoff frequency (Λ), 3. the coupling strength (γ0 ). Nenv and Λ jointly determine the frequency spacing, ω0 = Λ/Nenv . This ω0 is also the infrared − cutoff frequency. Since a massive Poincar´ recurrence disrupts the dynamics at trec = 2πω0 1 , we e must choose Nenv and Λ so that the interesting timescales fall within the recurrence time. Our base environment has the properties: Nenv = 256 Λ = 16 γ= 1 10 ω0 = 1 16 trec = 32π 100 (4.23) The environments are all initialized in their ground states, which means that the initial state of the entire environment is a thermal state with T = 0. Coupling to the system injects some energy, however, so when equilibrium is reached, it is not at zero temperature. The system’s initial configuration consists of its frequency (ωS ), and its initial state (ρS ). Setting ωS requires a bit of care, primarily because of the frequency renormalization discussed in 2 Section 4.1.3. The bare frequency of the system is chosen as Ω2 = ωS + δ Ω2 , where δ Ω2 is given by 0 Eq. 4.21. We initialize the system in a squeezed coherent state with x = p = 0. The squeezing can be accomplished either by fixing ∆x2 or by squeezing the ground state via a parameter r: ∆x2 ∆ p2 = 1 r 2mS ωS mS ωS 2 2 (4.24) (4.25) =r Note that for a pure Gaussian (i.e., squeezed coherent) state, ∆x2 ∆p2 = 4 ≡ 1. We use squeezed states in the same way that Schr¨dinger cat states have previously been o used – i.e., as highly “measurable” states. A Gaussian with large ∆x (and correspondingly small ∆p) can be thought of as a superposition of many position eigenstates, which decoheres into a mixture of localized coherent states. Future work will consider arbitrary superpositions of Gaussian states, including cat states, as a natural extension. 4.2.2 Partial information plots As in previous analyses of information storage (Chapters 2 and 3), we begin by considering partial information plots (PIPs). These PIPs have the same basic structure as those considered in the previous chapters (see, e.g., Fig. 2.16): we plot the average information (I ) obtained from a fragment of the environment, against the fragment’s size. As before, we average I over many fragments of the same size in order to get I . The main difference in this chapter is that the number of subenvironments (m) is no longer a good measure of a fragment’s size. This is because the environment has, in principle, a continuous frequency spectrum. When we subdivide each oscillator in two, we also reduce the coupling constant to each of the resulting oscillators. To accommodate this 84 difference, we keep track of the fraction (f ) of the environment’s bandwidth that a given fragment contains. This bandwidth is not generally contiguous – a typical fragment containing 25% of a 256oscillator environment will be composed of 64 oscillators selected randomly from the entire available spectrum. In all other respects, PIPs which plot I (f ) from f = 0 . . . 1 are equivalent to those which plot I (m) from m = 0 . . . Nenv . We generally fix the system’s initial state: a squeezed coherent state with ∆x = 40. The central system’s frequency (ωS ) is then the most important parameter that determines the dynamics of information storage. Figure 4.1 shows PIPs for an underdamped oscillator with ωS = 4, on short and long timescales. The behavior shown in Fig. 4.1 is typical for underdamped oscillator systems that resonate within the coupling range of the environment – i.e., ω0 < ωS < Λ. We begin by noting similarities and differences between these PIPs and the ones obtained for spin baths in Chapter 3. A basic difference is that the QBM environment has a scalable division into subenvironments. The collection of Nenv environments represents an infinitely subdivisible spectrum, and as Nenv is increased, the coupling to each subenvironment is made weaker. The appropriate measure of “how much environment has been captured” is not the number of captured bands (m), but rather the fraction of the environment’s total bandwidth that has been captured, f ≡ Nm . We have verified that the shape of a PIP is largely independent of Nenv . env A representative case (ωS = 4) The curvature of the short-time PIPs in Figs. 4.1a-b indicates redundancy, just as in Chapter 3. We note three substantial differences: 1. The total information (IS :E ) between S and E , which is represented in Fig. 4.1 by the total relief between I (1) and I (0), is not bounded. The system’s Hilbert space is infinite-dimensional, and since the environment is coupled to xS , it measures the system’s position. This can be resolved, in theory, with arbitrary precision. The system’s initial state thus contains theoretically infinite amounts of information. IS :E is limited only by the precision with which the environment can measure the system’s initial state. 2. Relatively large (e.g., effectively unlimited in the same sense as IS :E ) amounts of information can be gained from very small fragments of the environment. The initial (at f = 0) rate of information gain is extremely high. In spin bath models, by contrast, the information capacity of a single subenvironment limits the rate of information gain (see Chapter 2). For QBM, the information capacity of a single subenvironment is (like IS :E ) bounded only by the amount of information available. 3. At relatively short times (i.e., after the measurement interaction has occurred and IS :E has peaked, but before dissipation has really set in), the midpoint slope of the PIP ( ∂ I ) is ∂f 1 f= 2 not almost zero, as is typically the case for spin baths (Chapters 2 and 3). The flat “classical plateau” that we saw for spin bath models has been replaced by a slanted region, where ∂I 2. The slope of this plateau is a measure of non-redundant information (discussed ∂f 1 f= 2 in Section 4.2.3). After a few dissipation timescales, however (e.g., t = 25), it becomes flat. The shape of the PIPs in Figs. 4.1a-b stabilizes by t ∼ 0.3, although the total amount of information continues to rise until t ∼ 1. At longer times, however, Figs. 4.1c-d show new behavior. The total information declines (roughly linearly) to nearly zero (by t ∼ 60). On a much shorter timescale, the slope of the central plateau declines as well. A classical – i.e., flat – plateau emerges well before the total information declines. As previous studies (e.g., [140, 66]) have noted, decoherence and dissipation occur on 1 different timescales in Brownian motion.5 In our model, τdecoherence is roughly determined by Λ ≈ 5 For further details on relative timescales in QBM, see Section 4.4. 85 Partial Information Plots: ωs = 4 2 IS:ε - HS 1 0 -1 -2 0 0.2 0.4 0.6 f (dimensionless) 0.8 1 t= 0 t = 0.02 t = 0.04 t = 0.08 Theory IS:ε - HS f Evolution of a PIP: ωs = 4 (t=0..1) 4 2 0 -2 -41 0.8 f 0.6 0.6 0.4 Time 0.8 1 f 0.4 0.2 0.2 (a) Partial Information Plots: ωs = 4 2 IS:ε - HS 1 0 -1 -2 0 0.2 0.4 0.6 f (dimensionless) 0.8 1 t= 1 t= 5 t = 15 t = 25 Theory IS:ε - HS f (b) Evolution of a PIP: ωs = 4 (t=0..75) 4 2 0 -2 -41 0.8 f 0.6 60 40 50 20 30 Time 70 f 0.4 0.2 10 (c) (d) Figure 4.1: PIPs for an underdamped oscillator with ωS = 4, over two different timescales. These PIPs plot the mean information (I ) that can be obtained from a fragment E{f } of the environment, against the fragment’s size. Fragment size is measured as a fraction of the whole environment’s bandwidth. In this simulation run, S is coupled to the standard environment of Eq. 4.23. We show time-series PIPs on the right (plots (b),(d)) for two different time intervals. The total amount of information that E has about S is shown by the f = 1 cross-section of the PIP. On the left (plots (a),(c)), single PIPs from representative times are shown. In these plots, the dashed lines are theoretical predictions derived in Sec. 4.3. Discussion: Plot (b) shows that IS :E rises rapidly from t = 0 to t = 0.2. The PIP becomes curved (indicating redundancy) on the same timescale. However, the “classical plateau” around f = 1 2 is slanted, which indicates that some information is non-redundant. The dynamics on longer time scales is shown in plot (d). For t > 1, dissipative effects cause IS :E to declines gradually. However, the central plateau of the PIP becomes flat. By t ∼ 20, the PIP displays a true “classical plateau,” which indicates the presence of redundancy that does not depend on the deficit (δ ) that we allow. 1 0.1, whereas τdissipation is set by γ0 ≈ 10. We therefore designate the short- and long-time regimes pictured in Fig. 4.1 (separated by t = 1) as the measurement regime and dissipation regime, respectively. The measurement (0 ≤ t ≤ 1) regime: Quite rapidly, the environment gains all the information about S that it will ever obtain. Correlation between S and E (as measured by IS :E ) rises quickly as the high-frequency bands of E respond, then continue to rise slowly as more bands become involved. The PIPs for these times are definitely curved – indicating redundancy of some sort – but the amount of redundancy is ambiguous. The reason for this is that the PIP is not flat 86 around f = 1 . As a result, the amount of redundancy depends heavily on δ , which measures how 2 “picky” we are about getting all the possible information. Objectivity and redundancy are therefore ambiguous. For different standards of what “sufficient” information is, the stored information can be redundant or not. The dissipative (t > 1) regime: Dissipation begins to be noticeable around t ∼ 1, as the total available information begins to decline. The decline in total information is accompanied ∂I by a decline in the non-redundant information (INR ), represented by ∂m at f = 1 . The elimination 2 of the non-redundant information makes the remaining information unambiguously redundant. The system’s state becomes unambiguously objective – i.e., everything that can be discovered about it can be determined by measuring a small fragment of E – as dissipation destroys non-redundant information preferentially. By t = 25 (see Fig. 4.1c), a classical plateau is very much in evidence, yet IS :E remains large. As dissipation continues, IS :E eventually declines nearly to zero. At this point, nothing can be found out about the system by measuring the environment. Redundancy is essentially a moot point, for there is no information (about S ) left to be redundant. This point has led to some confusion, so it is worth considering it further. One might easily say, “How can there be no information in the system’s state? There used to be information, and you can’t destroy information!” The key point is that we are concerned with the possibility of obtaining information that we do not already have. When the system’s state is pure – i.e., the universe is in a product state of S and E – we already have perfect knowledge of the system’s state. There is no entropy to be reduced. For instance, if S and E are in a product state at t = 0, and evolve into an entangled state at t = 1, then information has been “created.” More precisely, it has been stored in the environment. If at t = 2, they have evolved back into a product state, then information has been “destroyed” – or, rather, erased from the environment and put back into the system. So, after dissipation has proceeded for a long time, the system has lost almost all its energy to E , and is therefore almost certainly in a coherent state at the origin (i.e., a T ≈ 0 thermal state). Since we can predict its state with good accuracy, there is little uncertainty about S that can be resolved by measuring E . Exceptions and special cases The discussion in the previous section (of a system with ωS = 4) accurately describes all systems for which: (1) ωS γ0 , and (2) ωS < Λ. In other words, underdamped oscillators with frequencies well below the ultraviolet cutoff. In this section, we briefly consider what happens when these conditions are not met – i.e., the failure modes of the heuristic, descriptive model just outlined. Figure 4.2 illustrates the behavior of: 1. plots (a-b) – a free particle6 with ωS = 0. 2. plots (c-d) – an underdamped oscillator with ωS = 1. This case is presented purely to confirm that all systems which meet the criteria above display qualitatively identical behavior; no more need be said. 3. plots (e-f ) – an ultra-high-frequency oscillator with ωS = Λ = 16. Both ωS = 0 and ωS = 16 are marginal cases. The frequency of the central system is right at the margin of the environment’s bandwidth. However, while the free particle is a physically relevant example, the ultra-high-frequency oscillator is difficult to justify on physical grounds.7 We examine it nonetheless, to gain insight into the model. 6 That is, the renormalized Hamiltonian is that of a free particle. This involves a very careful adjustment of the bare frequency (Ω0 ) in order to avoid having an unstable mode in the joint system. 7 The justification for including a high-frequency cutoff in the first place is the assumption that Λ is “much larger than any relevant frequency.” For an environment described by a continuous field, we have thus violated a fundamental assumption. In order to justify this model physically, the environment must be a lattice system with limited bandwidth – e.g., solid-state systems described by phonon modes. 87 Partial Information Plots: ωs = 0 2 1 IS:ε - HS 0 -1 -2 0 0.2 0.4 0.6 f (dimensionless) 0.8 1 f t= 1 t= 5 t = 15 t = 25 t = 95 Theory IS:ε - HS f Evolution of a PIP: ωs = 0 (t=0..75) 4 2 0 -2 -41 0.8 f 0.6 40 50 20 30 Time 60 70 0.4 0.2 10 (a) Partial Information Plots: ωs = 1 2 1 IS:ε - HS f 0 -1 -2 0 0.2 0.4 0.6 f (dimensionless) 0.8 1 t= 1 t= 5 t = 15 t = 25 t = 95 Theory IS:ε - HS f (b) Evolution of a PIP: ωs = 1 (t=0..75) 4 2 0 -2 -41 0.8 f 0.6 60 40 50 20 30 Time 70 0.4 0.2 10 (c) Partial Information Plots: ωs = 16 3 2 IS:ε - HS f 1 0 -1 -2 -3 0 0.2 0.4 0.6 f (dimensionless) 0.8 1 t= 1 t= 5 t = 15 t = 25 t = 95 Theory IS:ε - HS f (d) Evolution of a PIP: ωs = 16 (t=0..75) 4 2 0 -2 -41 0.8 f 0.6 40 50 20 30 Time 60 70 0.4 0.2 10 (e) (f ) Figure 4.2: Long-time PIPs for free-particle (ωS = 0) and underdamped (ωS = 1, 16) systems. The short-time behavior for these systems is virtually identical to that shown in Fig. 4.1a-b for ωS = 4. As in Fig. 4.1, the systems are coupled to the standard environment (Eq. 4.23). Plots (a),(c),(e) are representative slices from plots (b),(d),(f ); dashed lines represent predictions of the theory in Sec. 4.3. Discussion: For ωS = 1 (plots (c,d)), information storage evolves very similarly to ωS = 4 (Fig. 4.1c-d). The free particle (plots (a-b)) and ultra-high-frequency oscillator (plots (e-f )) do not appear to undergo dissipation in the same way. Neither the total information nor the slope of the central plateau converge to zero. 88 The free particle’s Hamiltonian, H= p2 2mS (4.26) is independent of position. As a result, dissipation contracts only ∆p, not ∆x. Rapid decoherence at t = 0 induces a large spreading in momentum. In the dissipative regime, total information declines as ∆p shrinks back to a steady-state level by t ∼ 20. Thereafter, however, ∆x and IS :E continue to increase slowly due to quantum diffusion. The central plateau never becomes flat – after decreasing ∂I slightly, the plateau’s slope ∂m equilibrates around t ∼ 20, then begins to rise very slowly. Similar effects are apparent in the ultra-high-frequency oscillator. Here, no dissipation is observed. Decoherence occurs in the measurement regime, but after t ∼ 1, the shape of the PIP remains constant except for fluctuations. Neither IS :E nor the slope of the central plateau decline noticeably. Both of these cases are exceptions to the much more general behavior described in the previous section. They are valuable because they indicate that resonance between S and E is a key factor in the development of unambiguous classicality. Well-behaved systems, with γ0 ωS Λ, experience dissipation. Dissipation appears to preferentially destroy non-redundant information, leaving only redundant (i.e., objective) information. The marginal systems just discussed do not resonate with the environment, and do not appear to feel the same dissipative effect that normal underdamped oscillators do. We will return to this line of analysis later. 4.2.3 Total and non-redundant information The partial information plots of Sec. 4.2.2 provide a detailed view of how information is stored in the environment. In this section, we focus on two particularly important quantities derived from the PIPs. The first is the total information available from the environment, or IS :E . Because the state of the universe is pure, IS :E = 2HS ; total information measures how much entropy the environment has induced in the system. The second quantity is non-redundant information (INR ), which measures how “classical” the central plateau is. Information is measured here in nits, the natural units of entropy (one bit = ln(2) nits). Overview of INR We introduced the concept of non-redundant information in Chapter 2 (see Fig. 2.16). The total information is divided into redundant information (easily obtained by multiple observers, and therefore classical) and quantum information, which cannot be obtained without capturing the whole environment. In between lies a sort of gray zone of undifferentiated information, which is neither easy nor particularly difficult to obtain. The signature of non-redundant information is a sloping central plateau in the PIP. We define INR in terms of that slope: INR ≡ Nenv = ∂I ∂f ∂I ∂m . f= 1 2 (4.27) m= Nenv 2 (4.28) The existence of nonzero INR means that the objectivity and classicality of recorded information is ambiguous. Although we accepted in Chapter 2 that not all the available information can be redundant, it was also assumed that the precise magnitude of the permitted deficit (δ ) would be largely unimportant. Non-redundant information violates this assumption: if δ IS :E ≥ INR , then small changes in δ may change Rδ substantially. In spin bath models (see Chapters 2 and 3), INR is not significant. 89 Ohmic QBM model (ω ≤ 1): IS:ε vs. Time (initial ∆x = 40) 9 8 7 6 5 4 3 2 1 0 0 20 40 60 Time (arbitrary units) 80 100 ωS = 0 ωS = 1/8 ωS = 1/4 ωS = 1/2 ωS = 1 Theory (a) 10 9 8 7 6 5 4 3 2 1 0 0 10 20 30 IS:ε (nits) Ohmic QBM model (ω ≥ 1): IS:ε vs. Time (initial ∆x = 40) ωS = 1 ωS = 2 ωS = 4 ωS = 8 ωS = 16 ωS = 18 Theory IS:ε (nits) (b) 40 50 60 Time (arbitrary units) 70 80 90 Figure 4.3: The total mutual information between S and the QBM environment (E ) is plotted, versus time, for an array of systems. Plot (a) depicts behavior for a free particle (ωS = 0) and low-frequency oscillators. Plot (b) shows high-frequency (ωS ≥ 1) oscillators, as well as two ultra-high-frequency oscillators (ωS = 16, 18). Theoretical predictions from the quantum-measurement (Hsys = 0) theory in Section 4.3 are shown as dashed lines. Discussion: In all cases, IS :E increases rapidly in the t < 1 regime, as the environment initially measures the system. After t > 1, dissipation affects the underdamped, dissipatively coupled oscillators (γ0 < ωS < Λ) predictably. Total information ˙ decreases linearly as I = −2γ0 until it reaches an asymptotic “thermal” level. Marginal cases (ωS = 0, 16, 18) are not damped in the same way. In particular, ultra-high-frequency oscillators are not affected by dissipation. Results Figure 4.3 shows total information over the full timescale of the simulation. The base environment (Eq. 4.23) is used to decohere a wide range of systems, ranging from a free particle to ultra-high-frequency oscillators. Except for the marginal cases discussed previously (e.g., ωS = 0, ωS ≥ 16), a consistent pattern is visible. The environment very rapidly gains a great deal of information about S in the measurement (t < 1) regime (see also Fig. 4.10 for a high-resolution examination of this interval). Then, that information slowly erodes away in the dissipation regime. The rate of decay is precisely determined by the dissipation constant: ∂HS ∂ I S :E =2 = −2γ0 . ∂t ∂t (4.29) 90 Ohmic QBM model (ω ≤ 1): INR vs. Time (initial ∆x = 40) 2.5 2 INR (nits) 1.5 1 0.5 0 0 10 20 30 ωS = 0 ωS = 1/8 ωS = 1/4 ωS = 1/2 ωS = 1 Theory (a) 2.5 2 INR (nits) 1.5 1 0.5 0 0 10 40 50 60 Time (arbitrary units) 70 80 90 Ohmic QBM model (ω ≥ 1): INR vs. Time ωS = 1 ωS = 2 ωS = 4 ωS = 8 ωS = 16 ωS = 18 (b) 20 30 Time (arbitrary units) 40 50 Figure 4.4: Nonredundant information (derived from the central slope of the PIP) is plotted, versus time, for an array of systems. Behavior of INR corresponds well with the behavior of IS :E in Fig. 4.3. For dissipatively coupled systems (γ0 < ωS < Λ), non-redundant information climbs to INR = 2 during the measurement process, then declines to a thermal level through dissipation. The decline thermal is initially linear, but becomes an exponential decay as INR approaches INR . When the system’s frequency is too high to resonate with E , dissipation is eliminated. For ωS = 16, 18, INR appears not to dissipate at all. Substantial fluctuations characterize the case with ωS = Λ = 16. During the decay period, IS :E oscillates slightly at the system frequency (ωS ), as the system’s internal dynamics alternately facilitate and interfere with dissipation. Eventually, IS :E stabilizes as the system reaches thermal equilibrium with its environment. These data also confirms that the marginal cases ωS = 0 and ωS ≥ 16 are not subject to dissipation in the same way. The free particle (Fig. 4.3a) does undergo dissipation, but does not end in a stable equilibrium state. Rather, IS :E reaches a minimum value, then commences to grow slowly and without bound, as ∆x is unconstrained by dissipation. The two ultra-high-frequency oscillators shown (ωS = 16, 18, in Fig. 4.3b) appear to be completely immune to dissipation. Information is gained by E in the measurement regime, but IS :E remains constant through the dissipative regime. Further investigation reveals that as ωS becomes higher, the initial measurement by E becomes weaker. If ωS is sufficiently high, the system is almost completely decoupled from the environment. Thus, there is only a small regime around ωS ∼ Λ in which decoherence occurs unaccompanied by dissipation. Figure 4.4 shows the corresponding behavior of INR . Immediately after t = 0, we find 91 INR = 2 in every single case! This is not a coincidence; in Sec. 4.3, we derive INR = 2 in the zero-dissipation limit. For well-behaved oscillator systems, INR begins immediately after t = 0 to decline, at the same rate as IS :E : ∂ INR ≈ −2γ0 (4.30) ∂t Eventually, INR equilibrates. The equilibrium level is precisely the same as the equilibrium IS :E – at equilibrium, all information is non-redundant. As INR approaches its equilibrium level, its rate of decline slows and becomes an asymptotic approach. The same phenomenon occurs for IS :E (Fig. 4.3), but it is more obvious for INR . We discuss the underlying dynamics in more detail below. Once again, the marginal cases show a clear divergence from typical behavior. For the free particle, INR drops slightly at first, then begins to rise slowly again. Further investigation indicates that if the initial wavepacket spread (∆x) is large enough, INR ≈ 2 at all times. The results for ωS = 16, 18 are even more sharply different; INR does not drop at all from its initial value, except for fluctuations that appear to be numerical artifacts. Discussion The behavior of INR observed in this section is probably the most remarkable thing about the entire oscillator-bath model. First of all, at the conclusion of the measurement process (i.e., when IS :E reaches its maximum value) INR ≈ 2 for virtually every model we examined. This remarkable constancy, independent of system and environment parameters alike, is confirmed in Sec. 4.3. It contrasts sharply with everything observed in Chapters 2 and 3. For the spin baths investigated there, the central plateau of a PIP is truly classical – INR is negligible in the presence of strong decoherence, and decreases exponentially with Nenv . We infer that the measurement process in QBM is different from the measurement process in spin baths. In QBM, some information is left undifferentiated – neither redundant nor “quantum.” Even more remarkable is the fact that dissipation rapidly destroys INR . Since INR and IS :E decline at the same rate, the initial decline in IS :E represents the loss only of non-redundant information. Redundant information (IR ≡ IS :E − INR ) is almost undiminished. Instead (see Sec. 4.2.4) it becomes more redundant. Again, this situation is in sharp contrast to spin baths, where any effect beyond a simple measurement process (e.g., dissipation) actually increases INR , destroying redundancy. Our theoretical model (Sec. 4.3) does not include dissipation, and therefore cannot explain or model this effect. We can, however, shed some light on the function form of the decay for IS :E and INR . Mutual information is derived from entropy, which for Gaussian states is approximately the logarithm of symplectic area. Dissipation is caused by the master equation’s frictional term (proportional to γeff ≈ γ0 ). For a harmonic oscillator, friction causes the symplectic area of an extended state to decay as A(t) = e−γ0 t (A0 − Athermal ) + Athermal , (4.31) where Athermal is the area of the oscillator’s equilibrium state.8 Since entropy (and therefore information) scales approximately as log(A), I decreases linearly when A(t) Athermal , but approaches exponentially toward its equilibrium value when A(t) ∼ Athermal . 4.2.4 Redundancy The relative behavior of total (IS :E ) and non-redundant (INR ) information has far-reaching consequences for redundancy. QBM environments do not achieve unambiguous redundancy during 8 Viewed in terms of the master equation, A thermal results from equilibrium between the frictional term (γeff ) and the normal diffusion term (f1 ), which tends to increase A. 92 Ohmic QBM model (ω ≤ 1): R10% vs. Time (initial ∆x = 40) 7 6 5 R10% 4 3 2 1 0 20 40 60 Time (arbitrary units) 80 100 ωS = 0 ωS = 1/8 ωS = 1/4 ωS = 1/2 ωS = 1 Theory (a) 140 120 100 R10% 80 60 40 20 0 0 10 20 30 Ohmic QBM model (ω ≥ 1): R10% vs. Time (initial ∆x = 40) ωS = 1 ωS = 2 ωS = 4 ωS = 8 ωS = 16 ωS = 18 (b) 40 50 60 Time (arbitrary units) 70 80 90 Figure 4.5: The redundancy of a fixed fraction (δ = 10%) of the total information is plotted against time. The system’s frequency is varied from ωS = 0 (free particle), up to ωS = 18, at which point the system is no longer dissipatively coupled to the environment. The environment’s parameters are given by Eq. 4.23, and the system’s initial squeezing is ∆x = 40. These plots show clearly that the dominant role in producing redundancy belongs to dissipation, not the initial measurement process. the initial measurement process – unlike spin baths, where strong decoherence automatically eliminates INR . For example, in the models examined so far, IS :E rapidly rises to approximately 9 nits. At the same time, INR ∼ 2 nits. Since 2 of the total information is non -redundant, we would expect 9 Rδ to be relatively small for any δ < 0.22. However, examination of the PIPs indicates that Rδ is large for sufficiently large δ . The amount of redundancy may be strongly dependent on δ , and objectivity is thus somewhat ambiguous. We consider two measures of redundancy in this section. First, we consider Rδ , the redundancy of a fixed fraction (1 − δ ) of IS :E . For continuity with previous work, we set δ = 10%. Because some information is non-redundant, we are also motivated to examine the redundancy of the redundant fraction of IS :E . This “redundancy of the redundant information” is denoted RR . We compute it by substituting δ = IIRE into Rδ , where IR ≡ IS :E − INR . S: Redundancy of a fixed fraction (Rδ , or R10% ) Fig. 4.5 shows R10% (t) for a wide range of low-frequency (Fig. 4.5a) and high-frequency (Fig. 4.5b) systems. The initial conditions (other than ωS ) for each simulation run are identical; in 93 particular, the system’s initial state is squeezed so that ∆x = 40.9 The dynamics of redundancy are similar for all well-behaved systems.10 The initial measurement process, during which the environment obtains all its information, produces very little redundancy. At t = 1, R10% 2. As dissipation proceeds, however, R10% begins to rise almost exponentially.11 Redundancy continues to rise, until it reaches a maximum at a time that increases logarithmically with ωS . At this point, R10% drops rapidly, down to approximately 0.9 (indicating a straight-line PIP, as in Fig. 2.4a). By careful examination of PIPs, we can stitch together a picture of redundancy dynamics. The initial measurement process (t < 1) records information in the environment. Because some of the information is non-redundant, initial redundancy is low. As dissipation destroys non-redundant information (both INR and IS :E decline linearly at first), the redundancy of the remaining informa− tion climbs. Around t ∼ γ0 1 , INR approaches its asymptotic value. Redundancy continues to rise, albeit at a reduced rate, as the PIP continues to flatten out.12 Dissipation continues to decrease the total information (IS :E ), and at some point IS :E drops below some critical multiple of INR . Redundancy drops rapidly as the last shreds of redundant information are destroyed by dissipation. When Rδ equilibrates to R ∼ 0.9, all redundant information has been eliminated. Redundancy of the redundant information (RR ) Figure 4.6 shows RR (t) for the same models considered in Fig. 4.5. RR is computed by first calculating INR (using the slope of I (f ) at f = 1 ), then computing the redundancy of the 2 remaining information (IR ≡ IS :E − INR ). The general behavior of RR (Fig. 4.6) agrees with that of R10% (Fig. 4.5). Redundancy is initially low, rises as dissipation proceeds, then declines and vanishes after reaching a maximum level. Three differences are notable: (1) the initial value of RR is higher than that for R10% ; (2) the maximum value is substantially lower; and (3) the timescale for maximum R is approximately doubled. None of these is particularly remarkable, given the definition of RR . Its initial value is higher, for instance, because the large initial value of INR means that IR is less than 90% of the total information; thus the redundancy of IR is greater. The maximum value is lower because after dissipation has reduced INR , IR is similarly greater than 90% of IS :E . The change in timescale results from the same phenomenon. Unfortunately, the plots of RR are much noisier than those for R10% . This is partly due to numerical error in INR , which is difficult to compute accurately. The root of the problem is that in computing the derivative of I (f ), the average values of I 1 − and I 1 + must be subtracted 2 2 from each other. This is a classic example of an unstable problem, since (a) must be small in order to get ∂ I accurately, but (b) any errors or fluctuations in the averaging of I are magnified by 1 . ∂f When IS :E INR , this problem is manageable – but when IS :E < 2INR , small fluctuations in INR can be disastrous. This explains the increased noise at late times in Fig. 4.6. Errors in RR are also enhanced by the shape of PIPs. Each PIP (see Fig. 4.1) has a region 1 where I increases rapidly (f 1), a region where I increases slowly (f ∼ 2 ), and a “knee” where the two regions join. By fixing δ so as to compute the redundancy of IR only, we ensure that the intersection of the line Irequired = 1−δ IS :E with each PIP occurs right around the “knee.” This is 2 exactly where small fluctuations in Irequired have the greatest effect. Computing RR accurately is thus difficult for several reasons. 1 ∆p = 40 , since ∆x∆p = 2 ≡ 1. should be apparent by now that the marginal (ωS = 0, 16, 18) cases are “failure modes” of the model. They are included in Fig. 4.5 for completeness. It is worth noting that R10% for the marginal, non-dissipative cases is predicted quite well by the zero-dissipation theory in Sec. 4.3. 11 Careful examination of the data shows that the rise is super-polynomial, but not quite exponential. Its exact form is still unclear. 12 As pointed out previously, I S :E and INR are important characteristic features of the PIP. They are not, however, sufficient to completely describe it. Even after INR equilibrates, the PIP continues to change shape as the classical plateau broadens, increasing redundancy. See Figs. 4.1c and 4.2c. 9 and 10 It 94 Ohmic QBM model (ω ≤ 1): RR (smoothed) vs. Time (initial ∆x = 40) 30 25 20 RR 15 10 5 0 0 10 20 30 40 Time (arbitrary units) 50 60 70 ωS = 0 ωS = 1/8 ωS = 1/4 ωS = 1/2 ωS = 1 Theory (a) 60 50 40 RR 30 20 10 0 0 10 20 30 Ohmic QBM model (ω ≥ 1): RR (smoothed) vs. Time ωS = 1 ωS = 2 ωS = 4 ωS = 8 ωS = 16 ωS = 18 (b) 40 50 60 Time (arbitrary units) 70 80 90 Figure 4.6: The redundancy of the redundant information (RR ) is plotted against time. The system’s frequency is varied from ωS = 0 (free particle), up to ωS = 18, at which point the system is no longer dissipatively coupled to the environment. The environment’s parameters are given by Eq. 4.23, and the system’s initial squeezing is ∆x = 40. While these plots confirm the overall dynamics of redundancy inferred from Fig. 4.5, the difficulty of computing RR with good precision is also clear. Discussion The results presented in this section leave no doubt that information is stored redundantly in QBM environments. Since Brownian motion is such a widely applicable model, this adds weight to the conjecture that physical environments generally produce redundancy, and therefore objectivity. Figure 4.7 adds another conclusion: redundancy is proportional to the “measurability” of the initial state. By varying the initial ∆x of the system, we provide more or less information for the environment to measure. Note that maximum redundancy appears to increase linearly with ∆x, whereas total information increases only logarithmically. For “classical” superpositions, the redundancy of the stored information will be truly huge. Besides the fundamental result that information is stored redundantly (and massively so for highly measurable states), we identify two particularly interesting aspects of information storage from the preceding results. First, measurement alone does not produce maximum redundancy.13 13 If I S :E is sufficiently large, then it will dwarf INR . Redundancy will be much less ambiguous (and more massive) if IS :E is very large. Nonetheless, the destruction of IS :E will still increase redundancy. 95 R10%(t): Varying ∆x 1000 ∆x = 10 ∆x = 20 ∆x = 40 ∆x = 80 ∆x = 160 ∆x = 320 ∆x = 640 100 R10% 10 1 0 10 20 30 Time 40 50 60 Figure 4.7: Redundancy (R10% ) is plotted on a logarithmic scale for different initial states of the system. The initial states are parameterized by their initial uncertainty in position, ∆x. Total redundancy increases proportional to ∆x. For “classically large” superpositions, information will be stored with massive redundancy – the value of x will be objective. The time over which redundancy ˆ is preserved also increases with ∆x, because it takes longer for the stored information to dissipate. Information remains redundant until it dissipates completely. Beyond ∆x = 640, the capacity of the environment is essentially saturated (that is, less than 2 individual bands of the environment are required to obtain nearly-complete information). Second, truly unambiguous redundancy is induced by dissipation. The former is surprising in light of the results for spin baths, but the role of dissipation is truly remarkable. It causes the eventual vanishing of redundancy (as the state of S becomes uncorrelated with everything, including its own initial state) – but first, dissipation enhances redundancy, by first eliminating the non -redundant information. This seems to have great significance for large, classical objects. The total, theoretical information content of a classical object is typically stupendous; a cubic meter of air contains over 102 5 bits of information just in the molecules’ internal states! The amount of information that we can practically gain about such a macroscopic system by measuring its environment is many orders of magnitude less. When dissipation destroys most of the information about a system – e.g., all the information that is not massively redundant – we can confidently assume that the only available information is that which is easily available. Yet even that information has a sort of expiration date, beyond which dissipation destroys its correlation with the central system. In the next section, we aim to understand why measurement does not produce unambiguous redundancy, by examining an analytically solvable model. Our model will be a quantummeasurement type (using terminology from Chapter 3), whereas the preceding simulations are more complex because the central system evolves. As a result, our model does not account for dissipation at all. However, it predicts decoherence and PIPs in the measurement regime (t < 1) very well. By modeling the measurement regime, we (1) elucidate why information is stored differently in QBM and spin bath models, and (2) identify features of the data that must result from dissipation. 96 4.3 Theoretical analysis of a quantum-measurement model for QBM Brownian motion is appealing because it provides a complex and realistic model of how physical systems decohere. QBM allows us to investigate redundancy in a familiar context. The simulation data presented in Section 4.2, however, are difficult to understand analytically because of the model’s complexity. In this section we solve a simplified model, based on the quantummeasurement models of Chapter 3, to illuminate the physics underlying the measurement process. Throughout this section, we will take advantage of the fact that coupled oscillator Hamiltonians are isomorphic to their classical versions, and do our calculations in phase space. Thus, for instance, x represents a classical phase space variable, and does not need to be treated as an operator. Density matrices (e.g., ρ(x, x )) are given as functionals of two classical variables. Several categories of decoherence models were identified (with reference to spin baths) in Chapter 3. The QBM models analyzed in Sec. 4.2 combine features of the dynamical-system and quantum-measurement models.14 By eliminating the system dynamics (e.g., setting Hsys = 0), we obtain a quantum-measurement model, which combines a single interaction term with environment dynamics: Nenv Nenv 22 2 mn ω n y n qn +x Cn yn (4.32) + H= 2mn 2 n=1 n=1 This quantum-measurement model does not actually fit into the framework of the numerical models discussed in Section 4.1.2. If we write H in the symplectic matrix form discussed previously, the entire 2 × 2 block corresponding to S is filled with zeros. No scaling of the masses mS and mn can 1 2 equalize the terms 0 · p2 and 2mn qn , which means that the normal modes of Eq. 4.32 cannot be obtained by a point transformation (e.g., one which acts only on the coordinates). We must either resort to canonical transformations which mix coordinates and momenta, or take a coordinated limit 2 [mS , ωS ] → [∞, 0] so that mS ωS → 0. Thus, while Eq. 4.32 is a perfectly valid model, it lies outside the set of models that we simulate in Section 4.2. We recall that, in Chapter 2, we approached redundancy by defining an additive measure of decoherence, d. We proceed here in the same fashion, although the d-factor for Gaussian states appears in a different (and more natural) fashion. The general approach will mirror that of Chapter 2: we identify the manner in which the action of multiple environments is additive; then we write the mutual information between S and a fragment E{m} in terms of the additive parameter, and estimate redundancy. The parallels with spin bath models begin to break down at this point, although our final result is similar. 4.3.1 Initial conditions We will assume that each subenvironment is initialized in its ground state, uncorrelated with S or other subenvironments: ρ(En ) (y, y , 0) = mn ωn − mωn (y2 +y 2 ) e2 . π (4.33) The possible alternatives include mixed, squeezed, and displaced states, each of which is briefly mentioned in the following discussion. The initial states could be mixed. A unique feature of QBM models is that thermal states are just as easy to consider as pure states, since the Gaussian framework encompasses both. Our investigations of thermal and mixed environments are as yet inconclusive; the difficulty of considering 14 QBM models do not have multiple-measurement terms of the form M ⊗ R + M ⊗ R ; the interaction between 1 1 2 2 system and bath unambiguously selects xS as the measurement operator, and the environment coordinates yE as the ˆ ˆ recording operators. 97 mixed states in the spin-bath framework makes it more difficult to understand the basic principles. We therefore restrict our scope to initial pure states of the environment. The initial states could also be squeezed or displaced. The primary effect of squeezing is to induce oscillations around the coherent-state behavior, which further complicates an alreadycomplex model. Displacing the state affects only z , not V . As pointed out in 4.1.2, the rank-1 and rank-2 components of the state never interact with each other. The only effect of using displaced coherent states would be to induce a time-dependent driving term proportional to F (t) into the master equation (Eq. 4.19). Since this does not affect the dynamics of information and correlation at all, we will always assume that the Ei are initialized in their ground states. This does not imply that the initial joint state of the whole environment is a ground state. The coupling term, turned on at t = 0, couples the environments indirectly. The true “ground state” of E , in the presence of S , is dependent on the value of xS . ˆ The state of the central system determines its measurability. The coupling between S and E is through position, and therefore the environment measures x. Larger values of ∆x lead to more information that can be recorded in the environment. We initialize the system in a squeezed (but not displaced) Gaussian pure state, ρS (x, x , 0) = √ in order to keep the mathematics simple. x2 +x 2 1 e− 4∆x2 , 2π ∆ x (4.34) 4.3.2 xS : ˆ Derivation of Gaussian decoherence factors Each subenvironment En feels a Hamiltonian Hx , that is conditional upon the value of 2 22 qn mn ω n y n + + Cn xyn 2mn 2 2 2 qn mn ω n + 2mn 2 ( n) H(n) x = = (4.35) 2 yn + Cn x 2 mn ωn − 2 Cn x2 2 2mn ωn (4.36) Coupling to S displaces the origin of the coordinate system from yn = 0 to yn = −δyn , where δyn ≡ Cn x . 2 mn ωn (4.37) En evolves along different orbits, conditional on the distinct eigenstates of x. As illustrated in Fig. ˆ 4.8, the ground state becomes a coherent state, which follows a circular path whose radius is δy . The overlap between the two coherent states produced by Hx and Hx , and centered respectively at (y, q ) and (y , q ), is given by |γx,x |2 = exp − (q − q )2 (y − y )2 −− 2 4∆y 4∆q 2 (4.38) Solving the harmonic oscillator equation of motion yields y − y = δy (1 − cos ωn t) and q − q = ∆q δy ∆y (sin ωn t), so γx,x ( n) 2 = = δy 2 (1 − cos ωn t) 2∆y 2 2 Cn exp − (x − x )2 (1 − cos ωn t) , 3 mn ω n exp − (4.39) (4.40) 98 Figure 4.8: Illustration of how initial coherent states evolve under x-conditional Hamiltonians. In this diagram, the crosshatched disc at the top of the figure represents the initial ground state of an environment. When xS = 0, the environment feels no perturbation, and the initial state remains in place. For xS = 1, the environment’s equilibrium point is displaced by δy , and the initial state rotates (red discs) around the new equilibrium. A stronger perturbation from xS = 2 increases δy , and the environment’s state oscillates with a larger amplitude. After a full period t = 2π/ωn , however, the state returns to its initial position. 2 where we have used the identity ∆yn = /2mn ωn (for the ground state of a harmonic oscillator) in the last line. Each subenvironment En thus contributes a Gaussian damping factor to the off-diagonal terms in ρS (x, x ). Defining d ( n) (t) ≡ − log γx,x ( n) (x − x )2 Eq. 4.40 indicates that the effect on ρS of a single subenvironment En is: ρS (x, x ) −→ exp −d(n) (x − x )2 ρS (x, x ). (4.42) Since the effects of multiple environments combine multiplicatively, the factor d(n) in the exponent is an additive decoherence factor. The effect on ρS of many subenvironments is: ρS (x, x ) −→ exp − n The additive decoherence factor (d) appears more naturally here than in Chapter 2. Additionally, the dependence of d on the pointer-state index (x) has been absorbed into the definition, since log(γ ) is always proportional to (x − x )2 .  ¡¡ ¡¡ ¡¡  ¡ ¡¡ ¡¡ ¡¡ ¡¡ ¡¡  ¡¡ ¢ ¢ ¡ ¡¡ ¡ ¡ ¡¡ ¢¡¢¡ ¢   © ¡ ¡ ©¡¡ ¡¡ ¢ ¡¡ ¢ ¢ ©¡¡ ¡ ¡ ¡¡©© ¡¡ ¢ ¡© ¡ §§    ¡© §¡ ¡¡¡¡ ¡¡©¡¡© § ¡¡ ¡¡§¨  §¨ ¨ ¡¡ ¡¡ ¡¡ ¨§ § ¡¡ ¡¡ ¡¨¡§ ¡¡   ¨ ¡¡ ¡¡ ¡¨¡¨¨ ¡¡ ¡¡ ¡¡ ¡¡ ¡¡ ¡¡ x=1 x=2 = 2 Cn (1 − cos ωn t) , 3 2mn ωn x=0 ¡¡ ¡¡ ¡¡  ¡¡ ¡¡ ¡¡ ¡¥¡ ¦ ¥¦¡¦¡ ¦¡¥¡¦¥¦ ¦¥¡¦¡¥ ¥¡¥¡ ¡¡¥¦ ¤¡£¡!"¡"¡!" ¤ ¡!¡ £¤ " ¤£¡¤¡¡!¡ £¡£¡!"¡¡!" ¤¡¤¡£¤£¤ " £¡£¡¡!¡ ¡¡!"¡¡!" (4.41) d(n) (x − x )2 ρS (x, x ). (4.43) 99 4.3.3 form Structure and entropy of decohered states The Hamiltonian in 4.32 generates Gaussian singly-branching states (see Chapter 2) of the |ψ universe = (1) (2) ( ψS (x) |x ⊗ |Ex |Ex . . . |ExNenv ) dx, (4.44) ( where the |Exn) are Gaussian states. This structure is identical to the branching-state structure discussed in Chapter 2. The three density matrices needed to compute IS :E{m} (the mutual information between S and a fragment E{m} of the environment) are: ρS (x, x , t) = exp − n∈E d(n) (t)(x − x )2 ρS (x, x , 0) (4.45) ρE (x, x , t) = exp − n∈E{m} d(n) (t)(x − x )2 ρS (x, x , 0) d(n) (t)(x − x )2 ρS (x, x , 0) n∈E{m} (4.46) ρSE (x, x , t) = exp − (4.47) We use Eqs. 4.12-4.16 compute the entropy of a state decohered by a factor d. An initial squeezed pure state x2 +x 2 1 (4.48) ρS (x, x , 0) = √ e− 4∆x2 , 2π ∆ x decoheres into ρS (x, x , t) = √ 2 x2 +x 2 1 e− 4∆x2 −d(t)(x−x ) . 2π ∆ x (4.49) Suppression of off-diagonal coherences causes diffusion in momentum; ∆p increases, while ∆x remains constant. The scaled phase space area (Eq. 4.12) evolves as A(t) = As long as A 1 + 8∆x2 d(t). (4.50) 1, entropy is well-approximated by Eq. 4.16: H 1+ 1 log(2d∆x2 ). 2 (4.51) This formula is self-validating. If ∆x2 d ≥ 1 then (by Eq. 4.50) A ≥ 3, and the error in Eq. 4.51 is less than 1%. Substituting ∆x2 d ≥ 1 in Eq. 4.51 gives H ≥ 1 + 1 ln 2 1.35. This provides an ex 2 post facto test of the approximation’s validity: if H ≥ 1.35, then Eq. 4.51 is valid. We conclude with a few words on the general decohering properties of this model. Let us expand Eq. 4.51 as 1 H 1 + log (∆x) + log(2d). (4.52) 2 If ∆x2 is large enough, the dependence of H on d is quite weak. Of course, if d = 0 then there is no decoherence at all, so this analysis is somewhat na¨ ıve. In order for Eq. 4.51 to be valid, d must be at least ∆1 2 . Merely achieving this condition, however, is sufficient to produce the x bulk of the possible entropy. Further decoherence (i.e., greater d) increases H only logarithmically. Thus, an environment’s ability to induce nearly-complete decoherence (i.e., obtain nearly-complete information about S ) is almost entirely dependent on its ability to achieve the threshold d ≥ ∆1 2 . x 100 4.3.4 The ohmic quantum-measurement model We now use the following results of Section 4.3.2 to model an ohmic environment. • Equation 4.41 determines the decoherence factor d for single subenvironments. • Equations 4.45-4.47 provide a rule for computing d-factors for S , SE{m} , and E{m} . • Equation 4.51 gives the entropy of a subsystem (e.g., S , SE{m} , or E{m} ) in terms of its d-factor. Decoherence For a subenvironment representing a frequency band ωn − 2 constant (Cn ) is given in terms of the spectral density (Eq. 4.3) as ωn + ∆ω 2 2 Cn = ωn − ∆ω 2 ∆ω 2 . . . ωn + ∆ω 2 , the coupling 2mn ωn I (ω )dω (4.53) For an ohmic spectral density (Eq. 4.4), the coupling is 2 Cn = 2 4mS mn γ0 ωn ∆ω. π (4.54) Equation 4.41 yields 2mS γ0 (1 − cos ωn t) ∆ω. (4.55) π ωn When ∆ω is sufficiently small, it makes more sense to treat the environment as having a continuous spectrum, parametrized by ω , than as a discrete collection of ωn . In this limit, the bandwidth ∆ω becomes a differential element, and we have a differential decoherence factor d(n) (t) = dd(ω) 2mS γ0 (1 − cos ωn t) (t) = , dω π ωn Λ (4.56) which we integrate over the relevant spectrum ω ∈ [0 . . . Λ] to obtain the total amount of decoherence: dS (t) = = dd(ω) (t)dω dω 0 mS γ0 [log (Λt) + γEM − Ci (Λt)] . π (4.57) In Eq. 4.57, Ci indicates the cosine-integral function [144], which has a logarithmic divergence at the origin, but rapidly becomes negligible as its argument becomes greater than 1. γEM is the EulerMascheroni constant, not the γ we use to denote a coupling strength. We note immediately that d diverges logarithmically as t −→ ∞. This reflects the fact that low-frequency environments, while slow to respond, have potentially unlimited sensitivity. As extremely low frequency modes join in the measurement process, d continues to rise without bound. This may be physically relevant in certain models, but the numerical models we study next have an infrared cutoff ω0 . By cutting off the integral at ω = ω0 , we obtain an alternative result: dS = mS γ0 log π Λ ω0 + Ci (ω0 t) − Ci (Λt) . (4.58) − − The low-frequency cutoff terminates the logarithmic rise in dS (t) around t ∼ ω0 1 . After t ∼ ω0 1 , dS undergoes slow, damped oscillations around an asymptotic value of mS γ0 dt=∞ = log (Λ/ω0 ) . (4.59) π 101 Redundancy From the perspective of redundancy analysis, the QBM environment is fundamentally different from the spin-bath environments we considered in Chapter 2 and Chapter 3. Whereas spins are indivisible, the frequency spectrum of a QBM environment can be infinitely subdivided into smaller and smaller bands. The parameter Nenv indicates only how faithfully a discrete model reproduces this property. Since the individual subenvironments contribute infinitesimal d(ω) to the total decoherence, the size (m) of any fragment is effectively infinite. This eliminates the effect of ∆d, which means that the two varieties of PIP discussed in Chapter 2– I (m) and m(I ) – are identical. Instead of mucking about with m at all, we simply identify: 1. dS , the total amount of decoherence produced by the environment; 2. dδ , the amount of decoherence required to achieve nearly-complete information; 3. the fraction f = m/Nenv of the environment required to achieve dδ . The captured fraction (f ) appeared in Chapter 2. In this analysis we use it exclusively in place of m. The mutual information between the central system and a subenvironment Ef containing a fraction f of the environment’s bandwidth is I (f ) = H (dS ) + H (f dS ) − H ((1 − f ) dS ) . Using Eq. 4.51 for H (d), we obtain I (f ) ≈ HS + 1 log 2 f 1−f . (4.61) (4.60) This result is remarkably simple. The shape of the PIP is nearly independent of almost everything. The complicated functional dependence seen in Chapter 2 is replaced by a simple function of f alone, plus a constant HS . We note that Eq. 4.61 is derived using an approximation formula (Eq. 4.51) for H (d). Using the exact formula, (e.g., Eq. 4.15 instead of Eq. 4.16), gives the correct behavior near f = 0 and f = 1 (see Fig. 4.9). We can compute redundancy (Rδ ) using Eq. 4.61 for I (m). Rδ was defined in Chapter 2 as the redundancy of a fraction 1 − δ of the classical information (HS ): Rδ = (1 − δ )Nδ − 1, (4.62) where Nδ is the number of distinct fragments of the environment which provide the required information. We let Nδ = f1 , where fδ is the fraction of E required to obtain sufficient information. δ Setting I (f ) = (1 − δ )HS and using Eq. 4.61 yields Rδ = (1 − δ )e2δHS . Non-redundant information For spin systems, we showed in Chapter 2 that Rδ depends on δ only weakly (at least for a wide range of δ ). Rδ ’s independence of δ results from the way that I (m) − HS approaches zero as 1 e−m . This means that the slope of I (m) at m = 2 N is exponentially small, and the intermediate region of the PIP forms a “classical plateau.” For the QBM model studied here, Eq. 4.63 indicates that Rδ depends strongly on δ , raising 1 the question of whether a classical plateau exists at all. Using Eq. 4.61 the slope of I (f ) at f = 2 (4.63) 102 Partial Information Plots: Theory 3 5 Information (nits) 4 3 2 1 0 0 1 2 3 d∆x2 4 5 6 INR Itotal IS:ε and INR: Theory 2 1 IS:ε - HS 0 f -1 -2 -3 0 0.2 0.4 0.6 f (dimensionless) 0.8 d∆x2 = 1/64 2 d∆x2 = 1/16 d∆x = 1/4 d∆x2 = 1 d∆x2 = 2 d∆x2 = 8 d∆x2 = 100 approx. 1 Figure 4.9: Theoretical predictions for partial information plots, non-redundant information (inset), and total information (inset). In the quantum-measurement (Hsys = 0) model for QBM, information and redundancy are functions of d∆x2 , where d is the additive decohering power of the environment and ∆x2 is the width of the system’s initial state. Equation 4.61 indicates that the shape of the PIP is, for d∆x2 1, independent of all parameters. The system entropy HS = 1 IS :E and 2 nonredundant information INR are given, respectively, by the value and the slope of the PIP at f = 1/2. We normalize the PIPs (for easier comparison) by subtracting HS , which is plotted in the inset. 103 can be computed as ∂I ∂f = f= 1 2 4dS ∆x2 S 4dS ∆x2 + 1 S log 4dS ∆x2 + 1 + 1 S 4dS ∆x2 − 1 S (4.64) (4.65) ≈ 2. The classical plateau which characterized redundant information storage in the spin-bath model is replaced by a slanted region. This makes objectivity more ambiguous – when δ is important, how much information is “nearly complete information?” The explanation is fairly simple. Gaussian states contain an effectively infinite amount of information, so the total amount of information obtained by the environment is not bounded above. Each environment provides further refinement of an observer’s knowledge about S . Thus, we can never get arbitrarily close to complete knowledge of the system’s state without capturing a large fraction of the environment. If we are willing to settle for less-than-complete knowledge, then that information can be extremely redundant, as Eq. 4.63 shows. Depending on how much information (I ) we demand, I can be either unique or highly redundant. Non-redundant information (INR ) determines the dividing line. The slope of the PIP ( ∂ I ) represents the cost of gaining more information. If ∂ I is large, then additional information can ∂f ∂f be gained rapidly by capturing a little bit more of E . Conversely, small ∂ I means that additional ∂f 1 information is hard to get. The minimum value of ∂ I , which occurs at f = 2 , thus indicates the ∂f most difficult-to-obtain information of all – irreducibly non-redundant information: INR ≡ ∂I ∂f (4.66) 1 f= 2 For spin-bath models with sufficiently strong decoherence, INR is very small. For the model studied here, INR ≈ 2 as long as HS ≥ 1.35. For weaker decoherence or less-measurable ρS (i.e., HS < 1.35), there is necessarily less non-redundant information – the total information is only IS :E = 2HS . As HS → 0, nearly all information becomes nonredundant (see inset plot in Fig. 4.9). Since INR is non -redundant, we expect the remaining information, IS :E − INR , to be 1 redundant. More precisely, half of it (IR ≡ 2 (IS :E − INR )) is redundant; the other half is “quantum” information (IQ ), which can only be obtained by capturing a majority of the environment (see Chapter 2, particularly Fig. 2.16). If we compute the redundancy of IR , by solving for I (fR ) = HS − 1, we find15 INR RR = 1 − e2 . (4.67) 2H S The prefactor arises from the factor (1 − δ ) in the definition of Rδ . Aside from this correction, IR has a constant level (approximately 7.5) of redundancy. This provides a baseline for distinguishing whether a given amount of information will be redundant or not. More information than IR cannot be easily obtained by multiple observers, while less information will probably be massively redundant. 4.3.5 Discussion The theory derived above is accurate, but with limited applicability. Most importantly, it cannot model dissipation. Dissipation is a thermodynamic process; it requires equilibrating interactions between all subsystems. In the quantum-measurement model, the system cannot mediate interaction between subenvironments, because it has no Hamiltonian. Without any meaningful energy of its own, it cannot serve as a thermal conductor between the subenvironments. 15 At first glance, “the redundancy of the redundant information” seems a strange construction. Upon further reflection, however, it makes sense – we know that IR is redundant, but we don’t know how redundant it is. 104 What the theory does explain is the measurement regime. In particular, it predicts not only decoherence (e.g., the behavior of IS :E for t < 1; see Fig. 4.10 in Section 4.4) but also the manner in which information is stored (e.g., the shape of PIPs for t < 1). No previous theory can do this. The master equation theory developed in [26, 140, 66] predicts decoherence behavior over short and long timescales – but cannot predict redundancy or PIP behavior.16 First and foremost, we now understand why decoherence in QBM produces PIPs with the characteristic shape seen in Figs. 4.1, 4.2, and 4.9. Non-redundant information is an inescapable feature of the model, and the observation that INR = 2 in so many cases can be explained. Finally, our intuitive definitions of INR and IR in Chapter 2 are supported by the remarkable result for RR in Eq. 4.67. The “redundant information” (IR ) has a simple interpretation (in this model) as the amount of information which is about e2 -fold redundant, no matter what. Equally important is a clear picture of which processes are not explained by the quantummeasurement theory. This model describes the initial measurement process well (it fits the numerical data), but it allows persistent non-redundant information. The redundancy of the stored information is therefore ambiguous – there may or may not be redundancy, depending on how much of a deficit is permitted. The elimination of the non-redundant information – and therefore the emergence of unambiguous redundancy – must be due to dissipation. We have conjectured and assumed this all along, but now we can conclusively state that the measurement process does not eliminate INR or enhance redundancy beyond the initially-observed levels. An interesting future challenge is to construct a theory which does model dissipation. Such a theory should explain the shape of PIPs in the dissipative regime, the decay of INR , and the rise and fall of redundancy. 4.4 Timescales Our numerical analysis so far (Section 4.2) has focused on the dissipative regime (t > 1), where redundancy emerges. The initial measurement process produces decoherence and correlation between S and E on a much more rapid timescale. The measurement process is described well by the master equation theory developed in [26, 140, 66], and also by the theory developed in the previous section17 . In this section, we consider the measurement process in more detail, and show that decoherence and redundancy occur on different timescales. By estimating (in separate calculations) both τR and τd , we show that redundancy develops much more slowly than decoherence. The two are qualitatively different phenomena. 4.4.1 The redundancy timescale The entropy of a Gaussian state is approximately equal to the logarithm of its scaled symplectic area. We model the rapid decoherence of S by assuming that a maximum entropy H0 is achieved very rapidly. The system’s initial symplectic area is thus A0 = exp(H0 ). For t Λ−1 , dissipation dominates the evolution of ρS . At declines as At ≈ A0 e−2γ0 t , toward an equilibrium level Athermal = exp (Hthermal ). 16 The master equation formalism is quite powerful – it predicts the behavior of the central system very accurately. However, it has not been successfully applied to the “Environment as a witness” paradigm, although an attempt has been made in [42]. In the future, we hope to combine master equation theory with the approach presented here in order to model PIPs and redundancy in the dissipative regime. 17 The master equation theory has the advantage of being well known, and also of great generality. The quantummeasurement model of the previous section is less general, but describes how information is recorded in the environment. (4.68) 105 Total information (IS :E ) is thus: IS :E (t) = 2HS (t) 2 log [exp (H0 − 2γ0 t) + exp (Hthermal )] . (4.69) (4.70) We have observed that INR decays in the same fashion, so we conjecture the following form: INR (t) 2 log exp (1 − 2γ0 t) + exp 1 thermal I 2 NR , (4.71) which agrees well with simulation data. Without further details of how redundancy depends on IS :E and INR , we cannot predict the actual amount of redundancy from these expressions. However, we can safely conjecture that Rδ will be maximized approximately when the ratio IS :E is highest. INR Solving for ∂ ∂t INR IS :E = 0 yields t=tmax thermal η H0 , Hthermal , INR γ0 tmax = (4.72) where the dimensionless variable η is the solution to an equation which appears to have no analytic solution. Numerical experimentation, however, reveals that η ≈ 2 ± 1, and that η varies only thermal logarithmically with the parameters H0 , Hthermal , and INR . This ad hoc calculation agrees well −1 with a measured value of tmax ≈ 2γ0 for many simulations. 4.4.2 The decoherence timescale The timescale for decoherence is set by the initial rate of entropy increase. We combine Eqs. 4.16, 4.50, and 4.57 to obtain an expression for H (t), and expand it in a series around t = 0 to obtain ∆x2 mγ0 Λ2 2 ∆x2 mγ0 Λ2 2 H (t) ≈ − t ln t −1 . (4.73) 2π 2π The timescale is immediately evident from rewriting this as H (t) ≈ −2λ2 t2 (ln λt − 1). The rate of entropy increase is O(λ), so the decoherence timescale is td ≈ λ−1 = 1 Λ∆x 2π mS γ0 (4.74) 4.4.3 Discussion Comparing Eqs. 4.72 and 4.74 confirms that decoherence and redundancy “happen” on very different timescales. The most important parameters in τD are the cutoff frequency (Λ) and the initial width (∆x) of the system’s state. The coupling constant (γ0 ) appears only as a square root. Decoherence is driven by the portion of the environment that responds the fastest, even if the rapid response comes only from a few environments, or from weakly coupled environments. Redundancy, on the other hand, develops through dissipation – on a timescale determined almost entirely by the coupling constant (γ0 ). Moreover, as discussed in Section 4.2.3, redundancy seems not to be a direct result of the S − E coupling, but of the information sharing among the subenvironments, mediated through the system. In Fig. 4.10, we show the initial rate of decoherence, and its dependence on the major timescales in the problem (except ω0 , which only becomes relevant at relatively long times). In addition to showing that the quantum-measurement model is accurate for short times, these plots confirm the timescale dependence indicated in Eq. 4.74. Figure 4.10d shows a typical timescale for 106 Decoherence time vs. ωsys 9 8 7 6 5 4 3 2 1 0 0 0.2 0.4 0.6 Time (arbitrary units) 9 8 7 6 5 4 3 2 1 0 0 0.2 Decoherence time vs. γ IS:ε (nits) ωS = 0 ωS = 1 ωS = 4 ωS = 16 ωS = 18 Theory 0.8 1 IS:ε (nits) γ = 0.05 Theory γ = 0.1 Theory γ = 0.2 Theory 0.4 0.6 Time (arbitrary units) 0.8 1 (a) Decoherence time vs. Λ 9 8 7 6 5 4 3 2 1 0 0 0.2 0.4 0.6 Time (arbitrary units) 25 20 R10% 15 10 5 0 1 0 10 20 Λ= 4 Λ= 8 Λ = 16 (b) R10%(t): Varying Λ IS:ε (nits) Λ= 4 Theory Λ= 8 Theory Λ = 16 Theory 0.8 30 Time 40 50 60 (c) (d) Figure 4.10: The initial rate of decoherence is examined, and its dependence on the major timescales of the model (γ0 , Λ, and ωS ) is illustrated in plots (a)-(c). Plot (d) shows the timescale for redundancy development, and its independence of Λ, the most important timescale for decoherence. Of particular interest is the fact that the two timescales (τd and τR ) differ by almost 2 orders of magnitude, which confirms that decoherence and the emergence of redundancy are driven by different physics. We also note that the theoretical model derived in Section 4.3 predicts short-time behavior excellently, except in the cases where ωS ≥ Λ. redundancy development, and its independence of the cutoff frequency, which largely determines τD . The timescales differ by two orders of magnitude. Further examination of the relationship between Rδ and the various parameters of the environment is presented in Appendix C.1. We can understand the unexpected appearance of redundancy, and the timescale of its appearance, in terms of INR ’s rapid disappearance. However, we have only pushed the inexplicability onto another phenomenon, for we have not yet understood the mechanism by which INR vanishes. With an eye to understanding the physics of information flow, we consider one more aspect of the simulation results, the information provided by individual frequency bands of the environment. 4.5 Local information Redundancy and decoherence are global properties or effects of the environment, emerging from the collective action of all the Ei . We turn now to the local 18 properties of the individual 18 By “local” we mean local in frequency space. We elected to use frequency space to generate a tensor-product decomposition of the environment, when we adopted the independent-oscillators view of a quantum field environment. 107 IS:ε 2 1 ω Local Information vs. Time: ωS = 0 IS:ε 2 1 0 0 10 ω Local Information vs. Time: ωS = 1 0 0 20 40 Time 60 80 4 8 12 ωenv 16 20 Time30 40 50 4 8 12 ωenv 16 (a) IS:ε 3 2 1 0 0 10 20 Time30 40 50 4 8 12 ωenv 16 ω (b) IS:ε 5 4 3 2 1 0 0 20 40 Time 60 80 4 8 12 ωenv 16 ω Local Information vs. Time: ωS = 4 Local Information vs. Time: ωS = 16 (c) (d) Figure 4.11: The information provided by individual bands, IS :Eω , is plotted versus ω and t. Four different systems, including a free particle (ω = 0) and an ultra-high-frequency (ω = 16) oscillator, are illustrated. Each plot demonstrates that the resonant bands of the environment (ω ∼ ωS ) record the most information about the S , while low-frequency (ω < ωS ) bands are essentially decoupled from the system. High frequencies initially measure the system, but give up their information to thermalization more rapidly than resonant oscillators. When the system’s frequency lies on the borderline of the environment’s spectrum (plot (d)), thermalization does not occur. environments: specifically, how much information does each environment have about the system? Over the course of each simulation run, we compute the mutual information IS :Eω between the system and the oscillator with frequency ω . The perspective that this data provides is far from complete, for it completely ignores the relationship between IS :Eω and IS :Eω , when ω = ω . Together, however, redundancy and local information sketch out the most important properties of information storage. Redundancy treats all environments as indistinguishable, revealing only collective properties, whereas the local information discards collective properties to focus on individual bands. Figure 4.11 presents an overview of frequency-localized information storage for four different systems (ωS = 0, 1, 4, 16). We illustrate the predictions of the quantum-measurement theory in Fig. 4.12, and compare the theory to more detailed analyses of the numerics in Figs. 4.13-4.15 and the following discussion. Future work will consider the effects of a spatial locality structure. 108 Local information: Theory 3 2.5 2 IS:E ω t = 0.24 t= 1 t= 3 t= 8 1.5 1 0.5 0 0 2 4 6 8 ω (frequency) 10 12 14 16 (a) 4 3.5 3 IS:E ω Local Information: Theory ωε = 1/16 ωε = 1/8 ωε = 3/16 ωε = 1 ωε = 3 ωε = 16 2.5 2 1.5 1 0.5 0 0 10 20 30 40 time 50 60 70 80 90 (b) Figure 4.12: The distribution of local information predicted by the theoretical quantum-measurement model of Section 4.3. Plots (a) and (b) present complementary views of the same underlying function, I (ω, t). Plot (a) shows I (ω ) at selected times, while plot (b) shows Iω (t) for selected frequency bands of the environment. Summary of behavior: Each band participates roughly equally in the measurement process. The information provided by a given band (ω ) oscillates with a constant amplitude h(ω ), which decreases slowly with ω . Thus, low-frequency bands are slightly more “involved” in decoherence than high-frequency bands. Local information in the quantum-measurement theory Using the theory from Section 4.3, we can calculate IS :Eω for the HS = 0 model. The actual functional form of IS :Eω is quite complicated, but a fair approximation is IS :Eω ≈ hω (1 − cos ωt), where the amplitude hω declines slowly with ω , roughly as hω ≈ log(ω )/ω . Figure 4.12 shows IS :Eω (t) for representative values of ω and t. The key features are: (1) oscillation of IS :Eω between 0 and a fixed hω ; and (2) the monotonic decrease of hω with ω . Local information for a free particle When we examine the simulation data for ωS = 0, shown in Fig. 4.13, a different pattern − emerges. The Hsys = 0 theory seems to describe the initial (t γ0 1 ) behavior well, although the observed values of IS :Eω are consistently somewhat lower than the theory predicts. As t advances toward the dissipation timescale γ −1 , however, the oscillations in IS :Eω begin to damp out. Where the theory predicts that, after a full period, an environment will “forget” about the system (i.e., dω → 0), we see that the environments (particular at low frequencies) retain some information. By t ∼ γ −1 , the environments retain a great deal of information through their entire cycle, and by 109 Information localization: ωS = 0 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 ω (frequency) 10 12 14 16 t = 0.24 t= 1 t= 3 t= 8 t = 50 (a) IS:E ω Information localization: ωS = 0 2 1.5 IS:E ω 1 0.5 0 0 10 20 30 40 time 50 60 70 80 90 ωε = 1/16 ωε = 1/8 ωε = 3/16 ωε = 1 ωε = 3 ωε = 8 ωε = 16 (b) Figure 4.13: Local information for a free particle system. Initially the theoretical predictions (Fig. 4.12) are largely accurate, but as time progresses, the oscillations in IS :Eω decay. By t ∼ 20, the information provided by a given environment has equilibrated to the time-average of its theoretical value. The Hsys = 0 model still describes the relative amount of information provided by the various environments fairly well – i.e., low-frequency bands are more involved than high-frequency bands, but the high-frequency bands are still relevant – until late in the process, when the high-frequency bands become largely irrelevant. t = 50, the oscillations in IS :Eω have essentially damped out completely. After a few dissipation timescales, each environment holds its information statically, although the amount of information that each environment retains is still well-described by the Hsys = 0 theory. Irreversibility, indicated by the persistence of information storage in each band, emerges from the collective behavior of the bath – even though each oscillator, on its own, displays reversible behavior. The irreversibility results from interactions between the environments, mediated by Hsys . In Chapter 3, we conjectured that the same form of indirect interaction was responsible for the slow decay of redundancy. For the free particle in QBM, there is little redundancy to be lost. We turn instead to one of the underdamped oscillator systems whose state is redundantly recorded. Local information for an underdamped oscillator Simulation data for a system with ωS = 1 (Fig. 4.14) shows the same short-time qualitative agreement with the theory as for ω = 0. The symptoms of irreversibility – in particular, damping of the oscillations in IS :Eω – appear more rapidly. Even more interesting is the development of a substantial peak in IS :Eω around ω = ωS . The peak is visible by t = 8, and when redundancy reaches 110 Information localization: ωS = 1 2.5 2 ω 1.5 1 0.5 0 0 2 4 6 IS:E t = 0.24 t= 1 t= 3 t= 8 t = 20 t = 50 (a) 2.5 2 ω 8 ω (frequency) 10 12 14 16 Information localization: ωS = 1 ωε = 1/16 ωε = 1/8 ωε = 3/16 ωε = 1 ωε = 3 ωε = 8 1.5 1 0.5 0 0 20 40 time 60 80 100 (b) Figure 4.14: Local information for an underdamped oscillator, with ωS = 1. The quantummeasurement theory is no longer particularly relevant. Low-frequency bands (ω < ωS ) are essentially irrelevant, while the resonant bands (ω ≈ ωS ) contain an overwhelming fraction of the information. At sufficiently long times, dissipation eliminates virtually all information. its maximum at t = 20, appears fully developed. The low-frequency portions of the environment become largely decoupled from S at late times, while the high-frequency environments appear to behave in much the same fashion as for the free particle. Local information for an ultra-high-frequency oscillator A contrasting case is seen for ωS = Λ = 16 (Fig. 4.15). The system oscillates at the very highest frequency available in the environment, and only a fraction of the resonant peak that appears for ωS < Λ is apparent. If we ignore the pronounced resonance curve, we see that the interaction of this ultra-high-frequency oscillator with the environment fits the theoretical model for Hsys = 0 almost perfectly. The system is rapidly decohered, but without resonant environment modes to mediate the interaction between environments, dissipation cannot ensue. The plots of IS :Eω (t) make this extremely clear, as the mutual information of each relevant environment oscillates with no irreversibility. Finally, we note that the nonredundant information also does not decay, but remains precisely at INR = 2 as the theory predicts. Discussion Figure 4.11 presents a holistic view of the way that information is frequency-localized in the environment over time, for ωS = 0, 1, 4, 16. For very short times, information is distributed equally IS:E 111 Information localization: ωS = 16 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 2 4 6 8 ω (frequency) 10 12 14 16 t = 50 t= 8 t= 3 t= 1 t = 0.24 (a) 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 20 IS:E ω Information localization: ωS = 16 ωε = 1/16 ωε = 1 ωε = 8 ωε = 12 ωε = 14 ωε = 15 ωε = 16 IS:E ω 40 time 60 80 100 (b) Figure 4.15: Local information for an ultra-high-frequency oscillator, with ωS = Λ = 16. As ωS rises to – and above – Λ, dissipation vanishes. As a result, local information for ωS = 16 behaves in some ways like the quantum-measurement theory. Specifically, Iω (t) oscillates periodically instead of dissipating. However, the relative importance of high- and low-frequency bands is reversed. High-frequency bands contain the bulk of E ’s information, while low-frequency bands are almost irrelevant. over the ω > ωS portion of the environment, as predicted by the theory. Low frequency modes (ω ωS ) do not participate. As time progresses, portions of the environment with ω ∼ ωS provide more information than the rest. For ω = 0 and ω = 16, the system lies on the very edge of the environment (in frequency space), and dissipation is not particularly evident. The most interesting cases are ωS = 1 and ωS = 4, where the system lies deep within the environment. This is the − physically relevant case. Information is concentrated, on a timescale t ∼ γ0 1 , into resonant modes of the environment. The environment for an underdamped oscillator can be split roughly into three segments: irrelevant modes with ω < ωS ; resonant modes with ω ∼ ωS ; and independent modes with ω ωS . The irrelevant modes, unable to respond to the system’s [relatively] rapid motion, do not contribute substantially to decoherence. The independent modes, in contrast, do measure the system, but their ability to extract large amounts of information is limited by their own rapid oscillation. The information stored in the independent modes appears to be largely nonredundant, as each highfrequency band measures a slightly different property of the system weakly. The resonant modes provide the most important contribution to decoherence, and are crucial to the development of redundancy. We do not fully understand, as yet, the role that resonant modes play. It seems clear from the data, however, that the resonant modes produce irreversibility, by 112 mediating the interactions between distinct bands of the environment. In order for information to be redundant, it must be possible to get the same information from distinct portions of the environment. We conjecture that bands with widely disparate frequencies measure sufficiently different properties of the system that their information is not redundant. Very narrow bands in the resonant regime have copious information about S due to their strong coupling. Because these narrow bands have similar frequencies, the information they yield is redundant. When the resonant bands are removed, as in our simulation of ωS = Λ, redundancy vanishes. We commented previously that local information is an incomplete characterization of the way that E stores information about S . Looking at IS :Eω , we see only how “useful” a given band of the environment is when we have no prior information – if the ω band is the rst one we capture, that is. The existence of prior information, obtained from some other fragment of E , will reduce the marginal utility of the remaining bands. Partial information plots illustrate an orthogonal perspective, by ignoring the differences between bands and considering only the marginal utility of the mth captured band. A full characterization of information storage would measure the marginal utility of each band ω , dependent on the collection {ω1 , ω2 . . .} of bands already captured. The size of such a description scales exponentially with the size of the environment. However, we can obtain a similar perspective by considering just two parameters: (1) the label ω of the band; and (2) the fraction f of the environment that has already been captured. We track the differential information, which is the amount ∆I of information that environment ω contributes (on average) when a random collection of f Nenv bands has already been captured. Differential information, of which we show plots in Appendix C.2, provides an excellent intuitive picture of how information is stored in the environment – although it does not seem be useful for quantitative computation at this point, which is why we have relegated it to an appendix. 4.6 Concluding remarks, future directions We have pursued two related questions in this chapter: • Does decoherence in quantum Brownian motion store information about S redundantly? How does this redundancy vary with time and parameters? • Viewed as a witness, how can the environment’s stored information about S be accessed? In Section 4.2, we presented convincing numerical evidence that information is stored redundantly. We also showed that information storage varies over three regimes with the passage of time: t γ −1 Immediately after S and E are coupled, their interaction with each other records information about S in the environment. A fixed amount of information (about 3 bits) remains nonredundant regardless of how much total information is recorded. t ∼ γ −1 After the initial measurement has occurred, the dominant feature of dynamics is dissipation, which results from indirect (S -mediated) interaction between different parts of the environ− ment. On a timescale γ0 1 , dissipation eliminates nearly all of the nonredundant information. Redundancy peaks in this regime. t γ −1 Dissipation continues to decrease the amount of correlation between S and E , by destroying information that is now redundant. Eventually, almost no correlation remains, and redundancy drops to nil. We presented a theoretical model in Section 4.3 which explained some of these features (in particular, the nonredundant information left by measurement). We also used this model in conjunction with numerical results to examine where (in frequency space) the information is stored, 113 in Section 4.5. We concluded that resonant modes of the environment are (a) the richest source of information about S , and (b) crucial to quantum Darwinism, because they mediate dissipation. Finally, we demonstrated that redundancy and decoherence proceed on different timescales, which are determined by different parameters, in Section 4.4. These conclusions are relevant to experimental physics, and generally consistent with basic physical facts. QBM describes the interaction of an oscillating system with a quantum field. This simple model applies to many of the systems that physicists deal with on a regular basis. An example is an atom interacting with a radiation field. An atom with a natural frequency of 1 Ghz will emit (and absorb radiation) much more strongly around 1 Ghz than at any other frequency. From this perspective, our conclusion that resonant modes of E are the most important is unsurprising, and increases our confidence in the results. Many of our results confirm well-known facts. Some of our results, on the other hand, are surprising and counterintuitive. The relationship between decoherence and dissipation is interesting both for quantum technology, and for understanding the emergence of classicality. Our results show that redundancy thrives in the intermediate zone, between complete decoherence and complete dissipation. Our previous work, on spin-bath environments, did not show any such connection. The role of dissipation appears to be particularly important for continuous-variable systems. By studying baths both of oscillators and of spins, we have established firm evidence that common decoherence processes lead to redundant information storage. Many interesting questions remain to be investigated, however. A good question is why redundancy emerges differently for spin- 1 systems and for continuous (Gaussian state) systems. The appearance of non-redundant 2 information in QBM is one of the more obvious differences. We do not fully understand how INR is eliminated. Its rapid disappearance, which leaves behind only redundant information, appears to be a clear example of Quantum Darwinism. Other questions that will be examined in future work include • The behavior of non-Gaussian states (such as Schr¨dinger cat states). o • The connection between redundancy and pointer bases in QBM. • Whether non-Ohmic environments have different properties (e.g., high- or low-frequency modes in the environment might be more important). Another interesting approach to the same problems involves detectors. Detectors can be modeled as additional oscillators, similar to the central system, which are embedded in the same field. Each detector interacts with the field, so that the field modes provide a communication channel to the system. By treating the detectors as measurement apparatus, we can inquire how much information each detector can obtain about the system. Such ancillary systems allow a more experimentally motivated tensor-product decomposition of the environment. Finally, as mentioned in previous discussions of redundancy, we hope to develop a practical contextual measure of information. Our goal is to identify what an environment Ei knows – not just how much it knows – about the system of interest. Quantum mutual information is only one of several possible measures for simultaneous correlation. Ollivier et. al. proposed another measure of mutual information in [106], whose relationship to our I has been considered in the context of discord [107]. Another promising avenue of investigation involves measures of correlation at different times – i.e., “what does E at time t know about S at time t ?” Chapter 5 discusses this concept in more detail, and we hope to apply it to the QBM environment in the future. 114 115 Part III Objectivity and Decoherence: the Bigger Picture 117 Chapter 5 Pointer bases, operators, and algebras: the operator-sieve algorithm Researchers studying decoherence – the loss of quantum coherences in a system, induced by contact with an environment – realized early on that some states remain largely unaffected by decoherence. These pointer states often form a pointer basis – a preferred basis of states which do not themselves decohere, but which when superposed to form arbitrary states are affected by decoherence. To discover this pointer basis, the predictability sieve was proposed; it would sift through the set of states, selecting those which evolved most predictably (and thus suffered least from decoherence). As originally conceived, the predictability sieve is marvelously simple. However, it is difficult to implement generally, and leaves some important questions unanswered (for instance, in what sense do the pointer states form a basis?) In this chapter, we present the operator sieve, an algorithm to implement the predicability sieve. When a unique pointer basis exists, the operator sieve will identify it. However, our results go beyond the pointer-basis formalism. Our algorithm also identifies noiseless subspaces and noiseless subsystems 1 , and is well-suited to finding approximately noiseless structures. Most generally, our analysis indicates the concept of pointer observables, and their associated pointer algebra, underlie all of these phenomena. In Section 5.1, we outline the original predictability sieve, and point out some of its major shortcomings. In Section 5.2, we introduce superprojectors and density tensors, and present the operator-sieve algorithm. Section 5.3 analyzes some simple cases in detail, while in Section 5.4 we examine noiseless subspaces/subsystems and discuss pointer algebras. In Section 5.5, we apply the operator sieve to the spin bath models of decoherence examined in Chapter 3, delving deeper into the relationship between redundancy and decoherence. We conclude, in Section 5.6, by considering the implications and future applications of this work. 5.1 The Predictability Sieve The concept of a pointer basis was originally introduced by Zurek in 1981 [157], to denote the set of “classical” states in which the dial pointer on a measurement apparatus can be found, after it measures a quantum system. Superselection of the pointer basis requires the decohering intervention of an environment, and is thus referred to as Environment-Induced Superselection or 1 Also known, respectively, as decoherence-free subspaces and subsystems. 118 einselection [158]. Over time, the term “pointer basis” has come to describe the stable states of any open quantum system exhibiting einselection, not merely measurement apparatus. When observed, an open system with a pointer basis will always be found in one of its pointer states. Superpositions of pointer states rapidly decohere into incoherent mixtures. Such a mixed state is inevitably unpredictable, in that the outcome of a precise measurement made on it cannot be predicted with certainty. A pure state, in contrast, always admits some precise measurement whose result is deterministic. The insight that the evolution of pointer states is more predictable than that of non-pointer states led, in 1993, to the predictability sieve [162] for discovering pointer bases. To find a pointer basis using the predictability sieve, we simply make a list of every pure state in the system’s Hilbert space. After a time t, each pure state has evolved into a [probably mixed] state: |ψk (0) −→ ρk (t). We then sort the list {|ψk } by the amount of entropy, Hk (t) = −Tr (ρk ln ρk ) (5.2) (5.1) that each state has developed. At the top of the sorted list, we find the most predictable states. These form the pointer basis. This formulation is appealingly simple, but it has a number of drawbacks. The goal of this chapter is to remedy them. Several were noted when the sieve was originally proposed, so we will point them out in the original words of reference [162]: 1. There is no sharp boundary between pointer states and non-pointer states: “One may, nevertheless, ask where should one put a cut in the above list to define a border between the preferred classical set of states and the non-classical remainder. The answer to this question is somewhat arbitrary: Where exactly the quantum-classical border is erected is a subjective matter, to be decided by circumstances.” 2. We cannot predict the structure of the pointer states, or even whether they form a basis: “It should be pointed out that the qualifying preferred states will generally form (i) an overcomplete set and (ii) they may be confined to a subspace of [the Hilbert space] H.” 3. It’s not obvious how to account for variation of the pointer states over time: “One key ingredient of. . . the predictability sieve outlined above remains arbitrary: We have not specified the interval of time over which the evolution is supposed to take place. . . we expect that the list should not be too sensitive to this parameter.” 4. As originally stated, the predictability sieve is impractical. It requires the examination of infinitely many states. Even an approximate version, which demands only the covering of H to some fixed density, requires considering a set of states that grows exponentially with the dimension of H. These problems all stem from the over-generality of the original predictability sieve. As formulated, virtually nothing is assumed about the structure of the interaction with the environment. We must thus consider: (1) the ensemble of all pure states, in all its infinite detail; and (2) mappings of this ensemble to virtually all possible ensembles of mixed states. The goal of the predictability sieve is to find the global maxima of a function. In full generality, this is a hard problem. To simplify it, we need to identify structural properties of the function to be maximized. For instance, a function is much easier to maximize if it is known to be quadratic. The basic insight that we pursue in this chapter is that both decoherence processes and the space of possible states have a great deal of structure. By exploiting this structure, we simplify the original problem greatly. We can also deal constructively with cases where a pointer basis 119 does not even exist. The operator sieve thus achieves more operational generality than the original predictability sieve, by exploiting the non-generality of the decoherence processes it is designed to study. 5.2 The Operator-Sieve Algorithm In this section, we first motivate and then present the operator sieve algorithm. The algorithm itself is a simple mathematical construction. Most of the discussion after this section will focus on properties of the solution, and on the implications of our results. We begin by introducing some properties of quantum state ensembles in the abstract – e.g., without reference to decoherence. 5.2.1 Ensembles, basis identification, and the density superoperator We begin by considering a simple question: given an orthogonal basis B = {|i } for a d-dimensional Hilbert space H, how can we uniquely identify it? That is, what simpleminded (i.e., algorithmic) test will distinguish B from any other B = B (up to permutation of the elements)? This question is clearly relevant to the predictability sieve, since a perfect decoherence process2 produces a unique basis of pointer states which we would like to identify. Both B and B are [supernormalized] ensembles of states. An ensemble E is commonly described using its density operator, ρ: ρE = [ψi ,pi ]∈E pi |ψi ψi |. (5.3) We might try comparing ρB and ρB . However, since B and B are both complete orthonormal bases, ρB = ρB = 1l. The density operator for every basis is identical. Taking this idea a bit further, however, we consider instead the density superoperator, P . The density superoperator for an ensemble E is3 PE = [ψi ,pi ]∈E pi ||ψi ψi | |ψi ψi ||, (5.4) where ||ψ ψ| |ψ ψ|| is the projector, in the Hilbert-Schmidt space of operators, onto the operator |ψ ψ|. A more complete examination of P ’s properties will emerge in the course of the discussion. For now, we’re only interested in whether PB uniquely identifies the basis B . To show that it does, we consider Tr (PB PB ). 2 The phrase “perfect decoherence process” is used here to refer to a particular class of process. Specifically, we mean a process which (a) leaves a complete, orthonormal basis of pointer states unaffected, and (b) reduces every superposition of pointer states to a mixture. This is, of course, an idealization. Nonetheless, it is a common simplified view of decoherence. 3 The density superoperator is a new concept, the theory for which is currently being developed. A forthcoming paper by this author will present a complete treatment. In this dissertation, P is only used as a mathematical tool to characterize ensembles, and we present only enough background to explicate this use. For those seeking a richer physical interpretation: one interpretation of P is as a density matrix for an ensemble of cloned pairs of states; another is as the process where (1) a POVM measurement, whose outcomes are proportional to the states in the ensemble E , is performed on the input, and (2) the measured state is prepared as the output. 120 1. If B = B , then PB = PB , and Tr (PB PB ) = = ij 2 Tr PB Tr (||i i|j ij i| |i i|| ||j j | |j j ||) = = ij j |i 4 i|j j |i | i|j | = d. 2. On the other hand, if B = B then (letting B = {|i }), Tr (PB PB ) = ij Tr ||i | ij i|j i| |i i|| 4 |j j | |j j | = |. 2 The completeness of B and B implies that ij | i|j | = d. Since each individual overlap 2 | i|j | is in the range [0 . . . 1], and their sum is equal to d, the sum of their squares is less than 2 or equal to d, with equality only if every | i|j | is either 0 or 1. Since the equality condition contradicts the assumption that B and B are different, Tr (PB PB ) < d. We conclude that Tr (PB PB ) = d if and only if B = B , and therefore that the density superoperator P uniquely distinguishes orthogonal bases. This simple example indicates that P is well-suited to the analysis of pointer bases. If the ensemble of decohered states has a basis-like structure, then its density superoperator should reveal that structure. As we shall see, the connection is even more powerful; P for the ensemble of decohered states characterizes the decoherence process completely. 5.2.2 The Operator-sieve Algorithm To determine the pointer basis of a given decoherence process, we apply the following procedure: 1. Construct the ensemble of all possible post-decoherence states, Ed , using (a) the unitarily invariant ensemble of all pure states and (b) the decoherence process mapping |ψi ψi | −→ ρi . 2. Construct the density superoperator PEd for the post-decoherence ensemble. ˆ 3. Perform an eigendecomposition of PEd into its eigenvalues λi and eigenoperators Ai . 4. Use the properties of the eigendecomposition to identify the pointer basis, or, 5. alternatively, if no well-defined pointer basis exists, obtain a more general characterization of the pointer states. Steps 1 and 2 are, in the end, easy to perform. Step 3 is a straightforward (albeit computationally difficult) application of linear algebra. Steps 4 and 5 are the conceptual meat of the algorithm, and provide the theoretical insights we will explore in later sections. 121 The actual construction of Ed is only necessary in order to obtain the superoperator PEd , so it can be done symbolically. A decoherence process is a quantum operation, represented by a ˆ ˆ superoperator S : |ψ ψ| −→ ρ. The post-decoherence density superoperator can thus be written as: PEd = 1 dψ ˆ ˆ S [ | ψ ψ |] ˆ ˆ S [ |ψ ψ |] dψ, (5.5) ˆ ˆ where the integration measure dψ is the unitarily invariant Haar measure. Because S is linear, we can pull it out of the integral: PEd = 1ˆ ˆ S dψ ||ψ ˆ ψ | |ψ ψ || dψ S ˆ† (5.6) ˆ ˆ ˆ ˆ = S Puniform S † As has been shown elsewhere [116, 120], the density superoperator for the uniform ensemble is Puniform = ˆ ˆ ˆˆ 1l + |1l 1l| ,. d(d + 1) (5.7) ˆ ˆ ˆ It is an equally weighted sum of the identity superoperator 1l (whose action on an operator B is ˆˆ ˆ ˆ 1l B = B ), and the projector onto the identity operator. The latter is defined by the action |1l 1l| ˆ ˆˆ B = TrB Id). Combining Eqs. 5.6 and 5.7, we obtain PEd = ˆˆ ˆˆ S S † + d2 |ρS d(d + 1) ρS | , (5.8) ˆ ˆ where ρS = S ˆ ˆ . That is, ρS is the state into which S maps the totally-mixed state. For the ˆ ˆ ˆ ˆ ˆ moment, we assume that S is unital – that is, S [1l] = 1l – which immediately implies that (a) 1l is an eigenoperator of PEd , and (b) the only effect of the second term in Eq. 5.8 is to increase the corresponding eigenvalue. 1l d 5.3 Simple Examples ˆˆ ˆˆ We are left with PEd ∝ S S † , and the task of identifying a pointer basis from its eigendecomposition. This simplest example of steps 4-5 above can be summarized as follows:4 5.3.1 Perfect Pointer Bases ˆ ˆ Theorem 1. Suppose that S represents a “perfect decoherence” process, such that there exists a ˆ ˆ complete orthonormal basis of pointer states B = {|0 . . . |d } which are unaffected (i.e., S [|n n|] = 4 Theorem 1 is not really an original result, although we are not aware of this particular form having been used before. It is essentially a special case result of a theorem (Thm. 1 in [89]) for decoherence-free subspaces. In the simplest language possible, this theorem states that any Hilbert subspace on which all states evolve unitarily must be decoherence-free. Pointer basis states are simply 1-dimensional decoherence-free subspaces. The condition that ˆˆ ˆˆ |ψ ψ| is a λ = 1 eigenoperator of S S † is equivalent to saying that |ψ evolves unitarily. The main import of Theorem 1, however, is its context. We are particularly interested in identifying pointer bases – entities which are relatively uninteresting for quantum computing, and from the perspective of passive error correction. 122 (a) (b) (c) (d) Figure 5.1: These are some simple ensembles of states which can be obtained by acting on the uniform ensemble (diagram (a)) with a unital superoperator (CP-map). Diagram (b) shows incomplete, noiseless decoherence in the σz basis; the |↑ and |↓ states are perfectly preserved, the states around ˆ the equator (|± , |±i , etc.) are strongly decohered, and the rest of the states on the Bloch sphere “fill in” the ellipsoid. Diagram (c) shows the effect of decoherence into the σx basis – the situation ˆ is exactly the same as in (b) except that the preferred basis is different. Finally, diagram (d) shows nearly “perfect” decoherence in the σz basis; the entire Bloch sphere is essentially reduced to ˆ states co-diagonal with σz . The overarching principle here is that unital CP-maps always produce ˆ ellipsoidal ensembles centered at the origin. These ellipsoids’ properties are completely determined by their principal axes, which we obtain from the operator sieve. ˆˆ ˆˆ |n n|), but superpositions of the pointer states are completely decohered into mixtures. Then S S † has only two eigenvalues: λ = 1 and λ = 0. The λ = 1 eigenspace defines the set of pointer observables; in this case it is d-dimensional and contains every operator which is diagonal in the pointer basis. Each of the other d(d − 1) eigenoperators has eigenvalue 0. ˆ ˆ Proof. The proof is simple. Since projectors |n n| onto pointer basis states are unaffected by S , ˆ ˆ† ˆˆ each such projector is an eigenoperator of S S with eigenvalue λ = 1. Any diagonal operator can be written as a linear combination of these projectors, so the diagonal observables form a λ = 1 eigenspace. We also know that any density operator can be written as a sum of diagonal and offˆ ˆ diagonal matrices: ρ = ρd + ρod . S annihilates the off-diagonal part. Since density matrices span ˜ ˆˆ ˆˆ the space of operators, every off-diagonal operator must be annihilated by S S † , and thus has an eigenvalue λ = 0. In this way, the operator-sieve unambiguously identifies perfect pointer bases when they exist. We need only identify the λ = 1 space of pointer operators. Each projector |n n|, onto a pointer state, can be written as a linear combination of pointer operators. Alternatively, adopting the Heisenberg picture, we can ignore the pointer states entirely, focusing instead on the pointer observables themselves. Each pointer operator in the λ = 1 eigenspace represents a measurement which does not lose predictability through the decoherence process. That ˆ ˆ ˆ ˆ is, if the pointer observable X has a well-defined value before S occurs, then it evolves into some X 123 which has the same well-defined (that is, predictable) value after the decoherence process. ˆ ˆ This does not mean that X is unaffected by the operation! Rather, X can only evolve ˆ unitarily into X . Similarly, when we consider pointer states, |n may evolve into some |n ; unitary evolution is irrelevant to the predictability sieve, which sees only the loss of predictability incurred by non-unitary processes. The operator-sieve disregards such unitary evolution because it considers not ˆ ˆ ˆˆ ˆ ˆ ˆ† ˆ ˆ ˆˆ ˆ ˆˆ S but S S † . Any unitary evolution SU yields SU SU = 1l; thus, all unitary evolutions are equivalent in the context of the operator-sieve. 5.3.2 Overcomplete Pointer Bases One of the problems with the original formulation of the predictability sieve is that pointer ˆ ˆ bases are not always unique. The example above, of a completely unitary SU (see also Fig. 5.1a), ˆ ˆ† ˆˆ ˆˆ ˆˆ is an extreme case of this. Since S S = 1l, every operator is a λ = 1 eigenoperator of S S † . This particular case can be dealt with easily – we simply note that no decoherence is happening, therefore the process is really outside our scope. A more relevant case, however, is illustrated by the example of a spin-1 particle monitored ˆ by an environment which couples only to the magnitude of Jz . The environment can unambiguously ˆ determine if Jz = 0, but cannot distinguish between the Jz = ±1 states. Since Jz is a good pointer observable, the basis {|+1 , |0 , |−1 is a pointer basis. However, because the environment cannot 1 distinguish between |+1 and |−1 , the states √2 (|+1 ± |−1 ) are equally good pointer states – as, in fact, are any superpositions of the Jz = ±1 states. We face an unavoidable ambiguity; there are infinitely many pointer bases spanning this pointer subspace, each of which is equally good. This situation was alluded to early on, in [158]. This system is an example of a noiseless or decoherence-free subspace, which we discuss in the next section. For now, we will only examine how such non-unique pointer bases are reflected in the operator-sieve algorithm. Suppose an n-dimensional subspace Hn ⊂ H is entirely unaffected ˆ ˆ by the decoherence process represented by S . Every operator whose support is restricted to Hn is ˆ ˆ† ˆˆ then a λ = 1 eigenoperator of S S . The resulting subspace of operators is n2 -dimensional, and the full set of pointer observables is the union of all such subspaces. In the example above, the a 3-dimensional Hilbert space is partitioned into a 2-dimensional pointer subspace spanned by {|+1 , |−1 } and the 1-dimensional pointer subspace {|0 }. We can choose any 4 operators we wish to span the space of operators on the |±1 subspace; if we choose the Pauli spin matrices, then the five pointer observables are: 010 0 −i 0 100 000 100 O = 0 1 0 , 1 0 0 , i 0 0 , 0 −1 0 , 0 0 0 (5.9) 000 000 000 000 001 As before, any pure state which can be written as a linear combination of these operators is a pointer state. The advantage of the pointer-observable approach is that apparently different concepts – unique pointer bases, degenerate pointer subspaces, and combinations of the two – can be unified within the framework of a set of predictable observables. As we shall see in the next section, this framework is even more powerful when its algebraic properties are considered. 5.3.3 Incomplete Decoherence ˆ ˆ The previous examples assumed that the decohering superoperator S reduces any density matrix ρ to some ρd which is diagonal in a pointer basis, by annihilating all off-diagonal elements. ˆ ˆ This is equivalent to assuming that (as stated in the theorem on “perfect” decoherence), S has only λ = 1 and λ = 0 eigenspaces. In the real world, however, decoherence occurs gradually. The 124 (a) (b) (c) (d) Figure 5.2: Post-decoherence ensembles produced by dissipative processes (i.e., processes with no perfectly predictable pointer basis). Processes (a)-(c) are unital, as in Fig. 5.1, while (d) represents a non-unital process. Diagram (a) shows the effect of a totally depolarizing channel, which maps every input state to the maximally mixed state. Diagram (b) represents a partially depolarizing channel. Neither of these processes admits a preferred basis; every basis is equally unpredictable. Diagrams (c)-(d) represent processes which do have non-trivial pointer bases. In (c), we see the effect of a dissipative decoherence process. The |↑ and |↓ states are slightly unpredictable, but equatorial states (equal superpositions of |↑ and |↓ ) are much more affected. Finally, (d) shows a ˆ similar case to (c), except that 1l is not an eigenoperator. The principal axes of the process (and thus its pointer basis) remain the same as those in (c); even though the |↓ state is strongly affected, the σz observable remains the most predictable. ˆ off-diagonal elements of the density matrix decay smoothly toward zero. After a short time, superpositions of pointer states are only partially decohered. Additionally, the pointer states themselves eventually become unpredictable – albeit on a longer timescale than superpositions. These two effects – the incomplete diagonalization of initial states, and the eventual dissipation of on-diagonal terms – are reflected in the operator-sieve by intermediate eigenvalues λ ∈ (0 . . . 1). In general, although this sort of process can be handled by the predictability sieve, it is not consistent with a perfect pointer basis structure. However, we still have two techniques for extracting a set of pointer observables (and thus a pointer basis). If there exists a λ = 1 eigenspace of perfectly predictable observables, in addition to a number of λ < 1 eigenoperators (see Fig. 5.1b-c), then we can still take the λ = 1 eigenspace as ˆ ˆ the pointer observables. The operators with λ < 1, while not completely annihilated by S , are still clearly less predictable than the pointer observables. ˆˆ ˆ ˆˆ A more difficult situation occurs when no operators except 1l are fully preserved by S S † . In this case, it may be that there simply exists no pointer basis at all. An example is a totally depolarizing channel (see Fig. 5.2a), which maps every state to the maximally mixed state. Such a process has a trivial λ = 1 eigenspace, consisting solely of the identity operator, and a large null space. In more interesting cases, however, there may exist an imperfect pointer basis – that is, a basis of states which are not entirely predictable, but are nonetheless much better preserved than are superposition states (see Fig. 5.2c). Such a pointer basis can be recovered from the operator-sieve 125 (a) (b) (c) (d) Figure 5.3: This figure illustrates some of the stranger ensembles which a CP-map can (or case (c) cannot ) achieve. Diagram (a) shows the effects of a non-unital “refrigerator” process which has no ˆˆ ˆˆ pointer basis. Because all the eigenvalues of S S † are equal, no operator is more predictable than another – even though one of the eigenstates of σz is perfectly preserved. Diagram (b) represents ˆ the same class of decoherence process shown in Fig 5.1 (b)-(c), but with principal axes which don’t coincide with any of the standard σi operators. Finally, in (c)-(d), we examine processes ˆ which produce versions of the “Bloch pancake” – the uniform ensemble is reduced to a flat X − Y plane. The requirement that the perfectly preserved operators form an algebra indicates that (c) is ˆˆ ˆˆ impossible – it is, in fact, not CP. Diagram (d) is possible, however, since S S † has no unit eigenvalues 1 (λX = λY = 2 , λZ = 0). by selecting, as quasi-pointer observables, the eigenoperators with the largest eigenvalues. 5.4 Things Algebraic: Pointer Algebras and Noiseless Stuff So far, we have used the operator-sieve strictly to reproduce (efficiently) the results of the original predictability sieve – that is, to identify pointer bases when they exist. As a theoretical tool, however, our approach is more useful than that. In particular, the possible results of quantum open system dynamics are constrained by algebraic properties of the pointer observables. Our results indicate not only that the space of pointer observables is a more generally useful entity than a particular pointer basis, but that the algebraic properties of that space are even more fundamental. The operator-sieve is motivated by the observation that decoherence processes are always completely positive, linear maps – i.e., superoperators. This means that the set of pure states cannot evolve into an arbitrarily complex manifold of mixed states when the system decoheres. Instead, the ensemble of post-decoherence states forms a generalized ellipsoid in the Hilbert-Schmidt space of Hermitian operators. Figures 5.1-5.3 illustrate this behavior in the Bloch-sphere representation of qubit states. Such an ellipsoidal ensembles is completely described by its variance tensors – the density superoperator (P ) that we constructed in the previous section. The density superoperator, in turn, is most efficiently represented by its eigendecomposition. This is equivalent to describing the ellipsoidal 126 ensemble of post-decoherence states by its principal axes. The operator-sieve algorithm is obtained by noting that the purest states (after decoherence) must lie along the longest principal axis or axes of P . This observation reduces the complexity of the predictability sieve, from a search among infinitely many states, to a search among O(d3 ) states5 . The preceding analysis seems to indicate that the pointer observables could, in principle, be almost any subspace of observables. For a qubit (for instance), we might expect that there exists a process which annihilates σz , but preserves σy and σx . The resulting ellipsoidal manifold of states ˆ ˆ ˆ would form a “pancake” within the Bloch sphere (see Fig. 5.3c). Such a process violates the intuition that decoherence is equivalent to a measurement by an environment, since a measurement of one observable (e.g., σX ) destroys the values of all the conjugate observables (e.g., σy and σz ). Moreover, ˆ ˆ it can be shown that such a process violates complete positivity [33]. There exist entangled states (on an augmented Hilbert space) that the pancake process maps into nonpositive density matrices. Clearly, {σy , σx } is not a valid set of pointer observables. ˆˆ A simple and powerful theorem due to Kribs [82] clarifies the situation. It states: The fixed points of a unital, completely positive, trace-preserving map form an associative ˆˆ ˆ ˆˆ ˆˆ ˆ ˆˆ C ∗-algebra. S S † is a CP-map, and if S is unital then S S † is unital and trace-preserving. The fixed ˆ ˆ† ˆˆ points of S S are just the eigenoperators with λ = 1. Kribs’ theorem thus proves that the pointer observables for a unital process must form an associative algebra.6 This result provides a tremendous simplification to the task of classifying possible pointerobservable sets, or pointer algebras. For qubits, there are only three possible algebras: 1. {1l}, the 1-element trivial algebra. 2. {1l, σz }, the 2-element commutative algebra Z2 . 3. {1l, σz , σx , σy }, the full 4-element Pauli algebra. Each of these pointer algebras represents a familiar structure. The trivial algebra (1) represents the effect of a completely depolarizing channel. The commutative Z2 algebra (2) is the algebraic structure of a perfect pointer basis, and the quantum measurement process which produces it. The Pauli algebra (3), represents a perfectly preserved7 qubit. No other structures can exist. In particular, the pancake process is forbidden, because the set {1l, σx , σy } is not closed under multiplication. A particularly appealing consequence of this analysis is that each pointer algebra can be concisely described as an information preserving structure. The three possible qubit algebras preserve, respectively: 1. No information (0 bits). 2. A single classical bit (1 c-bit). 3. A single quantum bit (1 q-bit). From this information theoretic perspective, we can classify the three qubit algebras by their information-carrying ability: 1. I (the identity algebra). 2. C2 (one classical bit). 3. Q2 (one quantum bit). are d2 − 1 principal axes of the ellipsoid for arbitrary d, each of which consists of up to d extremal points to generalize this theorem to nonunital processes are in progress. 7 Or, in the context of quantum communication, transmitted. 6 Attempts 5 There 127 This is merely nomenclature, but it can be very useful. Many higher dimensional informationpreserving structures are formed out of these 2-dimensional building blocks. Higher dimensions support more possible algebras, but many of them can be decomposed into smaller structures. A qutrit, for instance, supports 5 information-preserving structures. In terms of the Gell-Mann matrices for SU (3) (see Appendix D.1 or [123]), they are: 1. {1l}, the 1-element trivial algebra. 2. {1l, λ8 }, the 2-element commutative algebra Z2 . 3. {1l, λ3 , λ8 }, the 3-element commutative algebra Z3 . 4. {1l, λ1 , λ2 , λ3 , λ8 }, a 5-element noncommutative algebra Q2 ⊕ I . 5. {1l, λ1 , λ2 , λ3 , λ4 , λ5 , λ6 , λ7 , λ8 }, the full 9-element SU (3) algebra. The list includes I and C2 from above, as well as two rather obvious structures that are not found in qubits: C3 and Q3 . The remaining structure is somewhat curious. It arises from trying to implement Q2 in three dimensions, and is exactly the algebra given in Eq. 5.9. We denote this 5-dimensional algebra as Q2 ⊕ I , since it is similar, but not quite isomorphic, to the Q2 Pauli algebra. The Q2 ⊕ I algebra implies something rather counterintuitive. Any process which preserves a qubit embedded in a qutrit can also be used to preserve a classical trit, since Q2 ⊕ I contains C3 as well as Q2 . This is not an especially useful result in its own right, but it illustrates the fact that the marriage of algebraic analysis with decoherence theory can lead to powerful and unexpected results. Some of the information preserving structures seen above have been studied extensively already, as noiseless (or decoherence-free ) subsystems and subspaces [90, 47, 64, 74]. The derivation and analysis of these structures has focused on the algebra of error operators. The commutant of the error algebra defines the noiseless structure [64]. Our analysis in terms of pointer algebras provides a different approach to noiseless structures, based on the superoperator structure of quantum operations. This approach has two advantages. First, it unifies pointer bases and noiseless subspaces/subsystems into a single theory of information-preserving structures. Second, it allows us to consider imperfect structures, where the pointer operators have large but non-unit eigenvalues (e.g., λ = 1 − ). This has promise for analysis of experiments (e.g., as in [47]). It should be possible to analyze large and complicated systems using superoperator techniques, without knowing what underlying algebraic structure exists. 5.5 Application to a spin bath model Redundant storage in the environment E , of the the information (about a system S ) that is lost to decoherence, selects out a preferred basis for S . This was shown in Chapter 2 and in [106]. Environment-induced superselection of the pointer states has been recognized for some time as a standard feature of decoherence. The predictability sieve was introduced as a theoretical tool to identify pointer basis states, by selecting the states which are most predictable (i.e., develop the least entropy) over the course of a decoherence process. With the introduction of redundancy, there are two ways to identify the preferred basis. The relationship between these two sets of preferred states is not obvious. The existence of redundancy selects a basis because quantum information cannot be cloned. When two distinct chunks of the environment have equivalent information about S , that information must be classical, and therefore reveals a particular basis. The source of einselection is the manner of information storage in the environment – without knowing the detailed state of the environment, we cannot determine the preferred basis. On the other hand, redundancy is a property of a particular state |ψuniverse . We can determine the preferred basis from |ψuniverse without knowing anything about the process that produced it. 128 The predictability sieve, on the other hand, selects the preferred states by their predictability. The operator-sieve algorithm identifies a pointer basis from the process that maps initial pure states to final density matrices. A detailed knowledge of the environment and its state is unnecessary, and the pointer basis is a feature of the process, not of a particular state. Recent work [105] has explored the theoretical relationship of redundancy- and predictabilitybased einselection. In most circumstances the two are closely related. Essentially, when both redundancy and predictability criteria yield preferred bases, they must yield the same one. However, redundancy does not necessarily imply predictability, nor does the existence of a predictable pointer basis imply redundancy. In this section, we apply the operator-sieve algorithm to the dynamical decoherence/redundancy models of Chapter 3, to examine the redundancy-predictability connection. This serves both as an example of how the operator sieve can be used, and as a preliminary investigation into how the two methods of einselection may or may not be related. 5.5.1 Two mechanisms of einselection We view redundancy and predictability as criteria for identifying a set of preferred states. Each has an information-theoretic motivation: predictability seeks states of S which evolve predictably, while redundancy selects states of S which can be easily distinguished by measurements on the environment. An examination of the two criteria, however, reveals that they operate in very different ways, based on unrelated paradigms. Predictability The predictability sieve selects initial states |ψ0 of S , by comparing the amount of entropy8 that they develop. This is normally interpreted as a measure of how successfully we can predict ρt from |ψ0 , it is equally a measure of how well ρt retrodicts |ψ0 . In other words, the predictability sieve identifies states for which ρt has good information about ψt . At this time, the theory of quantum correlation does not extend to multiple times; we know of no measure for computing the mutual information between two states (at different times) of the same system. The standard interpretation is that if |ψ0 evolves into a pure state |ψt , then |ψt has perfect information about |ψ0 . Conversely, when |ψ0 evolves into a highly mixed ρt , accurate retrodiction of |ψ0 is impossible. It should be noted that this view is satisfactory only for unital (identity-preserving) processes. A refrigeration process, which maps all |ψ0 to the ground state |0 , is very predictable – but retrodiction is impossible, so |ψt yields no information about |ψ0 . In short, the predictability approach to pointer states is based entirely on the dynamics of the system. The environment acts only as a source of nonunitarity. It selects states based on the correlation between initial and final conditions of S . Redundancy Redundancy analysis selects states |ψi by comparing the number of distinct fragments (E{m} ) of the environment which can distinguish |ψi from |ψj . Unlike the predictability sieve, the redundancy algorithm makes no reference to the process which generated a state. However, it depends on: 1. Detailed knowledge of the environment’s state (more precisely, the joint state of system and environment). 2. The division of the environment E into subenvironments Ei . 8 Different measures of entropy may be used. Von Neumann’s quantum version of the Shannon entropy, H = Trρ log ρ, is often used because of its ubiquity in information theory. Other treatments, including the operator sieve, ˜ are based on linear entropy, H = 1 − Trρ2 , which more accurately characterizes the certainty of a independent measurements. 129 3. The observers’ knowledge of the initial state of E . At the core of this method is the nature of the system’s correlation with its environment. Instead of selecting a set of initial |ψ0 , the redundancy-based approach identifies final states |ψt which are redundantly correlated with the environment. There is no attempt to identify the |ψ0 which evolved into |ψt , nor do we care about how it happened. In fact, it is entirely possible that the set of states which are redundantly recorded in the environment have no connection whatsoever to initial states. A unitary evolution exists, for instance, which implements the map: |0 |1 S ⊗ |000 ⊗ |000 E −→ −→ S E 1 √ (|0 ⊗ |000 + |1 ⊗ |111 ) 2 1 √ (|0 ⊗ |010 − |1 ⊗ |101 ) . 2 (5.10) At the end of the process, the states |0 and |1 are clearly distinguishable by measurements on any of the three environments. However, they have no correlation at all with the initial states |0 and |1 , or any other initial states! In order to gain any information about the initial state of the system, a global measurement on the entire universe is required. The core of the difference between this behavior and the situation for pointer bases (above) is where information about S is stored. The predictability sieve looks for information in the state of S at a later time, while redundancy looks for information in various parts of the environment at the same time. In the latter case, requiring multiple copies of the information enforces its classicality . Connections and disconnections Ollivier et al [105] have proved that redundancy and predictability are related, in the following theorem: Theorem 2. If the basis B = {|i } is a pointer basis, and information about a basis B is recorded redundantly at the end of the decoherence process, then B and B are the same (or commute), up to a translation in time. Proof. When a pointer basis exists, it means that a set of final states {|it } are well-correlated with the corresponding initial states {|i0 }. The pointer states are unaffected by decoherence, which means that if the system is initialized in some pointer state |i0 it does not become entangled with the environment at all. The environment cannot have recorded information about an observable of which |it is not an eigenstate; therefore any preferred basis recorded in the environment must commute with the B = {|i }. This is a powerful and reassuring result. Without it, we might be in the uncomfortable situation of having multiple preferred bases... which defeats the purpose of having a preferred basis in the first place. However, it leaves open the question of whether the existence of one preferred basis implies the other. We answer this question in the negative – neither redundancy nor predictability implies the other – by considering a few examples. The example given in Eq. 5.10 proves that redundancy does not imply the existence of a pointer basis. Another, more intuitive example, follows. Consider a qubit which is first measured by E1 in the X-basis, and is then measured by E2 . . . EN in the Z-basis. Because both X and Z have been measured, the effect on S is that of a totally depolarizing channel, which has no pointer basis. Each of E2 . . . EN , however, has valid information about the basis {|0 , |1 }, so the Z-basis is redundantly recorded9 . Again, the crucial point is that redundancy is a property of correlations at the present time, and may be completely independent of the existence of correlations with an earlier time. 9 At the end of the day, E has no valid information about S ; its information was destroyed by the subsequent 1 Z-measurements. 130 It is even easier to show that the existence of a pointer basis does not imply redundancy. As pointed out in Chapter 3, redundancy is contingent on Huniverse respecting the tensor product structure of the environment. When a quantum-measurement process (which will always yield a good pointer basis) records its measurement in entangled modes of the environment, there will be no redundancy and no redundantly recorded basis. 5.5.2 How to analyze spin bath models The dynamical models of spin-bath decoherence in Chapter 3 provide an excellent test bed for the preferred-basis analysis presented here. In Chapter 3, we used quantitative redundancy to analyze four models of spin-bath decoherence: interaction-only, generalized quantum-measurement, dynamical-system, and multiple-measurement. We identify two categories of models: quantummeasurement models (the first two), and environment-entangling models (the last two). Quantum-measurement models are constructed so that the basis of the measurement operator M (e.g., Jz ) is always preserved. Unless there is no decoherence, a perfect pointer basis always exists. The conjugate observables (e.g., Jx and Jy ) are all destroyed equally. Quantum measurement models also generate substantial redundancy. Both Theorem 2 and a cursory analysis of the model show that the redundant basis is the eigenbasis of M. The environment-entangling models are both more realistic, and less easy to understand. The primary effect of including either (a) a system Hamiltonian (Hsys ), or (b) multiple measurements, is decay and disappearance of redundancy. The rate of decay increases with the extraneous interaction’s magnitude. The total amount of decoherence (measured by the system’s entropy, HS ) is also slightly reduced. The entropy reduction has different time-dependence from the redundancy decay, indicating that the two phenomena may be unrelated. HS is reduced more at short times, while redundancy decays over long timescales. We apply the operator sieve to these processes, with the aim of determining whether the disappearance of R over time is reflected in a corresponding decay of the pointer-basis structure. The operator sieve outputs: 1. The pointer basis; 2. The amount of depolarization, or purity decay, inflicted on the pointer basis states; 3. The amount of residual coherence, as measured by the remaining purity of the non-pointer observables. ˆˆ ˆˆ Each of these quantities is determined by the eigendecomposition of S S † . ˆ ˆ† ˆˆ ˆ S S has four eigenoperators (Ei ) with corresponding eigenvalues (λi ). One of the eigenˆ operators is 1l, with λ1l = 1. Ignoring the identity leaves three orthogonal Ei , which are unitarily equivalent to {Jx , Jy , Jz }. The operator with the largest eigenvalue is the pointer observable. Its eigenvalue λmax represents the amount of depolarization; λ = 1 indicates a perfectly preserved √ ˆˆ ˆˆ pointer basis, while λ = 0 means total depolarization. Because λmax is an eigenvalue of S S † (not ˆˆ ˆˆ ˆ ˆ S ), S maps its corresponding E to an observable E whose norm is ˆ E = ˆ λmax E . (5.11) √ We use D = 1 − λmax as a measure of depolarization. The other two λ represent residual coherences; a perfect quantum measurement process would have λmax = 1, and λ2 = λ3 = 0. If we define λ2 ≥ λ3 , then λ2 is an upper bound on the amount of residual coherence for a particular state, while λ3 is a lower bound. Since a randomly selected state will have some component of each, the larger λ2 is a good order-of-magnitude estimate √ for the typical residual coherence. Following the reasoning above, we use C = λ2 as a measure of residual coherence. 131 Depolarization in the Pointer Basis: H = E0Jy(S) + Hint 0.4 Purity Reduction 0.3 0.2 0.1 0 0 10 20 30 E0 = 0 E0 = 0.03 E0 = 0.06 E0 = 0.1 E0 = 0.15 E0 = 0.25 (a) 0.4 Purity Reduction 0.3 0.2 0.1 0 0 10 20 30 40 50 60 Time (arbitrary units) 70 80 90 100 Depolarization in the Pointer Basis: Jz⊗Jz - gd(Jx⊗Jx + Jy⊗Jy) gd = 0 gd = 0.01 gd = 0.02 gd = 0.05 gd = 0.1 (b) 0.4 Purity Reduction 0.3 0.2 0.1 0 0 10 20 30 40 50 60 Time (arbitrary units) 70 80 90 100 Depolarization in the Pointer Basis: H = Jz⊗Jz + gyJy⊗Jy gy = 0 gy = 0.01 gy = 0.02 gy = 0.05 gy = 0.1 (c) 40 50 60 Time (arbitrary units) 70 80 90 100 Figure 5.4: The depolarization of the pointer basis is plotted against time, for three different decoherence models. A quantum measurement Hamiltonian H = Jz ⊗ n kn Jz is “polluted” by an additional term. Plot (a) adds a system Hamiltonian, Hsys = E0 Jy . Plot (b) adds a dipole interaction term, Hint = gd Jy interaction term, Hint = (S ) ˆ ˆ (S ) ( n) ⊗ n kn Jy + Jx ⊗ ( n) (S ) n kn Jx ( n) . Plot (c) adds an asymmetric ⊗ Each plot represents a single simulation run. Depo√ ˆˆ ˆˆ larization is given by D = 1 − λmax , where λmax is the largest eigenvalue of S S † (not counting D (2−D ) λ1l ). The decoherence process reduces the purity (Trρ2 ) of a pointer state by ∼ 1 D. See 4 2 the text for analysis. (S ) gy Jy ( n) n kn Jy . 5.5.3 Results In Figure 5.4, we compare depolarization of the pointer basis (D) for: (a) evolving system models; (b) the “Z − Y ” multiple-measurement model; and (c) the dipole interaction multiplemeasurement model. Each model has a variable interaction energy. The interaction energy is denoted E0 , gy , or gd in the various models; we will refer it in general as g . The thick black line at 132 D = 0 represents the interaction-only model (g = 0), which preserves the pointer basis perfectly. For each model, depolarization rises to a maximum Dmax , then oscillates around Dmax with angular frequency ω = g . Dmax scales with the interaction energy. For the evolving system 1 model we measure Dmax 3g . 2 E0 , while for both of the multiple-measurement models Dmax An interesting effect (for which we have no good explanation) is that the dipole interaction and the Z − Y interaction have exactly the same scaling factor with g , despite the fact that the energy of √ interaction is gy for the Z − Y model and 2gd for the dipole interaction. The energy scaling is, however, apparent in the oscillation frequency of D. The timescale for maximum depolarization is the inverse interaction energy, τdepolarization = 1 . g (5.12) Redundancy levels for the same models (see Chapter 3) have only begun to decline by τdepolarization . Asymptotic behavior is reached on a substantially longer timescale. This partly confirms the conjecture made in Chapter 3: the decay of redundancy cannot be explained by analysis of the pointer-basis structure. To fully confirm the conjecture, we examine residual coherence in Fig. 5.5. Whereas depolarization means that information about the pointer observable is imperfectly preserved, residual coherences represent incomplete destruction of information about the conjugate operators. In terms of the information-preserving structures discussed previously, depolarization turns a Z2 algebra (e.g., a pointer basis) into an I (trivial) algebra. Residual coherences represent a small amount of a Q2 (qubit) structure overlaid on the Z2 structure. Residual coherence is a clear indication that the quantum measurement process has been stymied, since an effective measurement of Jz (for instance) must annihilate Jx and Jy . The data in Fig. 5.5 are more difficult to interpret than those in 5.4. Residual coherences are very small, and therefore hard to consistently characterize. Fortunately, rapid oscillations (due to individual environments measuring and unmeasuring the system) are minimized in several windows. Good examples are t = 20 . . . 26 and t = 60 . . . 70. A careful examination of the data reveals that the pattern seen in those windows is consistent across the entire time span. While the residual coherences fluctuate rapidly, they fluctuate above a baseline which, in the evolving-system and dipole-interaction models, is proportional to the interaction strength (g ). Like depolarization, the residual coherences 1 appear on a timescale τ ∼ g . Residual coherences have a major impact on redundancy. The importance of complete decoherence as a prerequisite for redundancy was pointed out in Chapter 3 (and discussed in more detail in Appendix B.5). Residual coherences, particularly when they are consistently above a threshold level as seen in Fig. 5.5a-b, impose an upper bound on redundancy. If the environment cannot measure the pointer observable well, then it can’t record it redundantly. Nonetheless, the observed residual coherences cannot explain the decay in redundancy, for the following three reasons. • The timescales are wrong. Residual coherences appear on a timescale given by g −1 , whereas redundancy persists for longer times. • Residual coherences behave quite differently in the two multiple-measurement models (Fig. 5.5b-c). This is probably because the dipole interaction has greater symmetry. However, the √ two models produce virtually identical redundancy decay profiles (when the factor of 2 in interaction energy is accounted for). This indicates that residual coherence and redundancy are not closely linked. • Most convincingly, we note that even the weakest interactions result in complete destruction of redundancy at sufficiently long times. Residual coherences, in contrast, rise to a level determined by g and stay there. Weak interactions induce little residual coherence. 133 Residual Coherence: H = E0Jy(S) + Hint 0.2 Purity E0 = 0 E0 = 0.03 E0 = 0.06 E0 = 0.1 E0 = 0.15 E0 = 0.25 0.1 0 0 10 20 30 (a) 40 50 60 Time (arbitrary units) 70 80 90 100 Residual Coherence: H = Jz⊗Jz - gd(Jx⊗Jx + Jy⊗Jy) gd = 0 gd = 0.01 gd = 0.02 gd = 0.05 Purity 0.1 0 0 10 20 30 (b) 40 50 60 Time (arbitrary units) 70 80 90 100 Residual Coherence: H = Jz⊗Jz + gyJy⊗Jy gy = 0 gy = 0.01 gy = 0.02 gy = 0.05 Purity 0.1 0 0 10 20 30 (c) 40 50 60 Time (arbitrary units) 70 80 90 100 Figure 5.5: The maximum residual coherence is plotted against time, for three different decoherence models. A quantum measurement Hamiltonian H = ZS ⊗ n kn Zn is “polluted” by one of the following terms. Plot (a) adds a system Hamiltonian Hsys = E0 YS . Plot (b) adds a dipole interaction term Hint = Id (YS ⊗ n kn Yn + XS ⊗ n kn Xn ). Plot (c) adds an asymmetric interaction term, Hint = Iy YS ⊗ n kn Yn . Residual coherence represents the maximum value of an off-diagonal √ element of ρS (when written in the pointer basis) after decoherence. It is calculated as C = λ2 , ˆˆ ˆˆ where λ2 is the 2nd largest eigenvalue of S S † . See the text for analysis. 5.5.4 Discussion Redundant information storage and perfect pointer bases are disturbed by the same dynamical effects. However, the pointer structure reaches an equilibrium state. Its “distance” from a perfect pointer basis grows with g , but is stable over time. Redundancy, in contrast, erodes steadily and eventually vanishes. Only the timescale depends on g . This supports the theory we proposed to explain redundancy decay in Chapter 3. Information about S , which was initially stored in local modes of the environment, diffuses slowly (at 134 2nd or 3rd order in H) into entangled modes. After a sufficiently long time, the conditional states of the environment are completely nonlocal – they appear to be randomly selected from the uniform ensemble. Storing information in uniformly selected states yields no redundancy at all (see Chapter 2). The redistribution of information is, however, irrelevant to the predictability of S . Decoherence of S depends only on the environment storing the information somewhere. Redundancy and predictability provide, in many cases, alternative routes to identifying the unique preferred basis. When they exist, they both reflect the same underlying informationpreserving structure – actually, the classical portion of it, since quantum information-preserving structures such as noiseless subspaces/subsystems do not fit into decoherence theory well. However, as we have seen, the two approaches to preferred bases are fundamentally different paradigms. Redundancy is local in time, but requires a global analysis of the system and its environments; predictability is local to the system, but requires information from initial and final times. This indicates an intriguing duality – identifying classicality requires nonlocal analysis either in time or in “space”. The common thread is information – information about the system at time t = tfinal which is accessible either from the environment at tfinal or from the system at t = 0. The thread which we believe will tie together both of the approaches viewed here is a unified theory of quantum correlation: how does system A(t) at time t provide information about system B (t ) at time t ? At this time, we have limited theories of (1) the information that A(t) has about B (t), and (2) the information that A(t) has about A(t ). A satisfactory theory should answer the question “How much information?” and also the question “Information about what?” Of particular interest is the ability to separate quantum correlations, or entanglement, from classical correlations. Entangled systems can provide information about multiple, incompatible properties of each other, whereas classically correlated systems are correlated only in a particular basis. 5.6 Conclusion: Implications and Applications We have presented the first fruits of an ongoing project to understand pointer bases better, and to construct a unified theory of information preserving structures. A great deal of work remains to be done – for instance, we do not yet have a systematic classification of all the possible pointer algebras in small Hilbert spaces. The results so far, however, are very exciting. The simple observations that (1) decoherence processes are superoperators, and (2) the effect of a superoperator on the ensemble of states is completely described by the density superoperator, have already proved valuable as a way of thinking about decoherence processes. The original purpose of this work was to produce an algorithmic version of the predictability sieve, and while some of the more speculative results and connections with decoherence-free structures are exciting, it should be noted that we succeeded in the original goal. The algorithm has been used to analyze tomographic data from NMR experiments (work in progress, not described here), and to analyze the pointer structure of spin-bath models and identify connections to redundancy. An additional benefit of the operator sieve is that we obtain not only the pointer states, but also quantitative information about (a) how complete the decoherence process is, and (b) how much depolarization has been inflicted on the pointer basis. In short, pointer algebras represent not only an intriguing theoretical avenue to explore, but also a very useful tool. 135 Chapter 6 Amplification in an upside-down oscillator 6.1 Introduction and Motivation Properties of a system become objective as information about them becomes available to many independent observers. One way to achieve this is through redundancy: the relevant information is recorded in many independent fragments of the environment. The observers are restricted to measuring a fixed set of fragments, which ensures their independence. This is not the only route to objectivity (and classicality), however. In experimental physics, quantum phenomena become objective through measurements which amplify the information about them. The canonical model of information amplification 1 is a photodetector. A photon – a single quantum of the electromagnetic field – triggers a cascading reaction which ultimately results in a classical pulse of current. The classical current can then be amplified, copied, etc., like any classical information. Of course, the amplification of information in the photon must come at a cost, for a single quantum can never be cloned [153]. By amplifying the photon-number observable, the photodetector annihilates2 information about conjugate observables such as the phase of the field. In this picture, amplification of information looks much like the generation of redundancy. In fact, the photodetector can be viewed as generating redundancy. Information, initially represented by a single quantum, ends up recorded in the positions of several trillion electrons. The difference between the photodetector and the redundancy-producing processes discussed in Chapters 2-4 is subtle but important: observers cannot measure the electrons independently. The pulse of current produced by a photodetector is a single collective system, not a collection of independent fragments, which nonetheless represents objective behavior. This example illustrates the rather confusing nature of amplification. The amplification of information to classical levels is akin to the emergence of redundancy, but not identical. Information in a single degree of freedom can be amplified to classical levels, but not divided up among multiple independent observers in any natural way. The photodetector is an ambiguous example; 1 Throughout this chapter, we use “amplification” to refer specifically to the amplification of information. This is related but not identical to energy amplification, or electromagnetic field amplification. Two distinctions are particularly important. First, the idea of information amplification is independent of the particular observable that is amplified. Thus, for instance, it is irrelevant whether a harmonic oscillator’s position or energy is amplified – information can be stored in either observable. Second, certain processes which amplify energy do not amplify information. For instance, a lasing tube acts on electromagnetic phase space by displacing the ground state to become a coherent state. This is an affine transformation on phase space, which does not make it easier to measure any observable. The linear phase space transformations that amplify information are squeezing transformations. 2 Technically, the information still exists. However, it is encoded in a joint mode of all the electrons making up the current pulse. For all practical purposes, it is totally inaccessible. 136 while it intuitively seems to represent amplification, it could very well be interpreted as generating redundancy instead if the current pulse were divided by a fanout gate. Our goal in this chapter is to examine a particular model of amplification in detail. We show that amplifying behavior can be produced by a single system. This result conclusively distinguishes amplification from redundancy. Our ultimate goal is to identify a cohesive way of analyzing amplification, much as partial information and quantitative redundancy have been used to analyze redundancy. 6.1.1 Sensitivity and chaos The distinguishing feature of information amplifiers is their sensitivity. An amplifying environment must be sensitive to tiny influences, which are then magnified into macroscopically visible phenomena. Most of the simple systems known to display such behavior are chaotic. Chaotic classical systems display a phenomenon known as sensitive dependence on initial conditions. Two copies of such a system, prepared in nearly identical states (two distinct points in phase space, separated by a very small distance), will evolve over time into widely separated states. In an idealized case, the distance between the two points in phase space grows as eλt , where λ is the largest Lyapunov exponent of the system. This cannot happen in quantum mechanics. Quantum mechanics is linear; two “nearly identical” states (i.e., states with a large initial overlap) remain nearly identical. Their overlap remains constant under unitary evolution. However, Peres [115] predicted an analogous phenomenon for quantum systems: two nearly identical systems, prepared in identical states but obeying slightly ˆ ˆ ˆ different Hamiltonians (i.e., H and H + δ H ), will evolve into two different states whose inner product ˆ is the quantization of a chaotic Hamiltonian. This idea has decays exponentially in time when H recently been studied and extended by Jalabert and Pastawski [70]. Our interest in this behavior stems from a simple model of decoherence, generated by an overall Hamiltonian (HS + HE + HSE ) where the interaction Hamiltonian (HSE ) is a product of a ˆ ˆ system operator (OS ) and an environment operator (OE ). When the system is in an eigenstate |sn of ˆ ˆ OS (with eigenvalue sn ), the environment evolves according to an effective Hamiltonian HE + sn OE . ˆ More generally, the system will be in a superposition of the eigenstates of OS , in which case the environment experiences a Hamiltonian that is conditional on the state of the system. Thus, if the initial state of the supersystem is |ΨSE (0) = ( n αn |sn (0) ) |ΨE (0) (where |ΨE (0) is simply the initial state of the environment), then after a time t the state will become |ΨSE (t) = n αn |sn (t) |εn (t) , where |εn (t) is the state into which the environment evolves if the system is in state |sn (equation (6.1)). When | εn (t)|εm (t) | = 1, no entanglement occurs, the system remains in a pure state, and there is no decoherence. When, however, | εn (t)|εm (t) | ∼ 0, superpositions of |sn and |sm are transformed into completely decohered mixtures. This model is tremendously simplified – it ignores many important phenomena, including the effect of the system Hamiltonian. However, it illustrates a key point: the rate at which an environment induces decoherence in a system depends on the decay rate of the overlap | εn (t)|εm (t) | , where ˆ |εj (t) = exp i(HE + sn OE )t |ΨE (0) (6.1) The sensitivity to perturbation of a chaotic environment’s Hamiltonian may cause rapid decoherence, even for very weak couplings. This is precisely what we would expect from an amplifying environment. This conclusion is encouraged by related analytic [166, 73] and numerical [124, 83, 93] studies. Some of its aspects are also beginning to be investigated experimentally [70, 111, 141, 88]. The nature of the chaotic evolution seems to be important for this conclusion [52, 119], and the physics of the related phenomena is still being debated [67, 68]. It is therefore useful to have an exactly solvable model that captures some of the features of quantum chaotic evolutions. A well-known feature of completely chaotic classical systems is that at every point in phase space there exist stable and unstable manifolds – directions along which a cell respectively 137 shrinks and grows exponentially. These manifolds fold in phase space, enabling a given region to be always stretching in some direction, yet remain within a bounded volume. We examine a system that exhibits such an exponential sensitivity to initial conditions, yet is analytically solvable: the inverted harmonic oscillator. We note that our model does not exhibit folding; we shall comment on the consequences of this shortcoming in due course. We shall also not discuss models with mixed phase spaces – our model is clearly too simple-minded for that. 6.2 Analysis of the Unstable Oscillator 2 2 2 The Hamiltonian for an inverted oscillator is H = 2p − M λ x , where we have replaced M 2 the parameter ω 2 for a simple harmonic oscillator (SHO) with −λ2 . Such an “oscillator” does not oscillate at all; it has an unstable fixed point at {x = 0, p = 0}, but accelerates exponentially away from the fixed point when perturbed. The price we pay for a solvable system that displays such sensitivity is that its phase space is unbounded; the kinetic energy of the system grows approximately as E ∝ e2λt . Although such a system is clearly nonphysical, it is an excellent short-time approximation for some real unstable systems. We believe that it can reproduce some of the behavior of a chaotic environment. Our model exhibits the same exponential sensitivity as a chaotic system. Real chaotic systems, of course, display folding in phase space. Local values of chaotic exponents can be quite different from their time-averages. The directions of stable and unstable manifolds vary from point to point. Our model misses these features. On the other hand, we are encouraged by the fact that this “integrable model of chaos” has been used previously [170], in a different context. Its validity has been confirmed by numerical simulations [83, 124, 94, 59, 93, 97, 98, 17]. We consider an inverted harmonic environment (IHE) consisting of one such oscillator, and couple it to a system consisting of a single SHO with mass m and frequency ω . This supersystem is linear, so it can be analyzed with the same master equation techniques used to treat quantum Brownian motion (QBM) and other linear problems. In this section, we discuss the novel method used to obtain the coefficients of the master equation. 6.2.1 Obtaining the Coefficients of the Master Equation We begin with the standard master equation for a linear system coupled linearly to a linear environment (as derived in [26, 140, 66]), mωeff 1 ∂ x2 , ρ + 21 p2 , ρ + γeff [ˆ, {p, ρ}] − F (t) [ˆ, ρ] ˆˆ xˆ 2 mˆˆ 2 x ˆˆ ρ= i ˆ ∂t −f1 (t) [ˆ, [ˆ, ρ]] + f2 (t) [ˆ, [ˆ, ρ]] x xˆ x pˆ 2 . (6.2) This form is exact when all the terms in H = HS + HE + HSE are linear and quadratic in positions and momenta, provided that the coupling terms in HSE involve only position operators. Here and throughout, we use x and p to denote the position and momentum of the system, and yi and qi to denote the position and momentum of the ith environmental degree of freedom. To analyze a particular system-environment combination, we need to obtain the specific values of the time2 dependent coefficients ωeff , γeff , F, f1 , and f2 . Although the IHE has a single degree of freedom, we consider an arbitrary linear environment as long as possible in order to make the generality of our method explicit. At the end of the general analysis, we will specialize to the IHE. Since the Heisenberg equations of motion for the system and environment operators match exactly the classical equations of motion for the equivalent variables, we can obtain the coefficients of the master equation from its classical analogue, the equation of motion for the reduced system. This straightforward approach has been examined previously in [6], but primarily in the context of QBM-like systems with infinitely many degrees of freedom. 138 The supersystem is linear and Hamiltonian, so we can write its trajectories as z (t) = T (t)z (0), (6.3) where z is a 2N -dimensional vector of the form [x, p, y1 , q1 , y2 , . . .] and T (t) is a square 2N × 2N matrix. An equation of motion gives the derivatives of z (t) in terms of z (t) itself. We obtain this ˙ ˙ by first differentiating equation (6.3) to obtain z (t) = T z (0), then substituting the z (0) obtained by −1 inverting equation (6.3): z (0) = T z (t). This yields ˙ ˙ z (t) = T T −1 z (t) (6.4) This is an equation of motion for the supersystem; it gives the time derivatives of all the supersystem coordinates and momenta in terms of their values at time t. However, this is not the equation of motion that we need. When we trace over the environment to obtain the master equation, we assume that the environment’s state at time t is inaccessible; we know only its initial state. We need a different equation, one that respects this constraint. To obtain an equation of motion that provides x(t) and p(t) in terms of x(t), p(t), and the ˙ ˙ initial state of the environment, we define a new matrix Tp related to T : (Tp )ij = Tij for i ∈ {1, 2} δij for i > 2 (6.5) As we evolve the quantum state of the system, we presume that at all times we have access to knowledge of (1) the reduced density matrix of the system, and (2) the initial state of the environment. The corresponding classical state of knowledge is a vector zp (t) = [x(t), p(t), y1 (0), q1 (0), . . .]. This vector can be obtained using Tp : zp (t) = Tp (t)zp (0). By the same process that led to equation (6.4), we conclude: ˙ zp (t) = T˙p Tp−1 zp (t). (6.6) This yields x(t) and p(t), but in order to obtain the coefficients in the master equation, we need the ˙ ˙ derivatives of higher order powers of x and p. To do so, we make the simplifying assumption that all states are Gaussian. This is particularly convenient since Gaussian states form a closed set under linear evolution. Such states are completely described by linear and quadratic expectation values of x and p, so to characterize their evolution we need time derivatives of x2 , p2 , and xp as well as x and p. This is straightforward; we ˙ ˙ T define the symmetric variance tensor Vp = zp zp , which contains all the quadratic combinations of x and p, and transforms as Vp (t) = Tp Vp (0)TpT . The time derivative of Vp is thus given by: ˙ Vp = T˙p Tp−1 Vp + Vp T˙p Tp−1 T . (6.7) Now, we need to relate these quantities to the coefficients of the master equation. For ˆ∂ ˆ ˙ˆ any time-independent quantum operator A, ∂t A = Tr ρA . The derivatives of the relevant expectation values are obtained from the master equation: ∂x ∂t ∂p ∂t ∂ x2 ∂t ∂ p2 ∂t ∂ {x, p}/2 ∂t = p ˆ m (6.8) (6.9) (6.10) (6.11) 2 2 = −mωeff x − γeff p + F (t) ˆ ˆ = 2 {x, p}/2 ˆˆ m = −2mω 2 {x, p}/2 − 2γeff p2 + 2F (t) p + 2 2 f1 (t) ˆˆ ˆ ˆ 2 = −mωeff x2 + ˆ 12 p − eff {x, p}/2 + F (t) x + ˆ ˆˆ ˆ m f2 (t) (6.12) 139 We can apply the preceding classical analysis to the Heisenberg operators, then equate the results ˙ with those from equations (6.8-6.12). Equations (6.8-6.12) imply that the matrix Tp Tp−1 that gives the derivatives of the Heisenberg operators must take the form 0 1/m 0 0 ... 2 −mωeff −γeff Fy1 Fq1 . . . −1 0 0 0 0 ... , ˙ (6.13) Tp Tp = 0 0 0 0 ... . . . . .. . . . . . . . . . ˆ where we compute the net force F (t) as the expectation value of a force operator: F (t) = i Fyi (t)ˆi (0) + Fqi (t)ˆi (0). y q Thus, F (t) = i Fyi yi + Fqi qi . Using this matrix in equation (6.7) yields the time derivatives ˆ ˆ of the system variances x2 , p2 , and {x, p}/2 : ˆ ˆ ˆˆ ∂ x2 ˆ ∂t ∂ p2 ˆ ∂t ∂ {x, p}/2 ˆˆ ∂t 2 ˙ = (Vp )11 = {x, p}/2 ˆˆ m 2 ˙ ˆ = (Vp )22 = −2mωeff {x, p}/2 − 2γeff p2 + {F (t), p} ˆˆ ˆ ˆ (6.14) (6.15) = ˙ ˙ (Vp )12 + (Vp )21 2 12 1 ˆ p − γeff {x, p}/2 + ˆ ˆˆ {F (t), x} . ˆ m 2 (6.16) 2 = −mωeff x2 + ˆ Comparing with the results from equations (6.8-6.12), we can solve for f1 and f2 as f1 (t) f2 (t) = = 1 22 1 22 ˆˆ ˆ {F , p(t)} − 2 F ˆˆ ˆ {F , x(t)} − 2 F p(t) ˆ x(t) ˆ (6.17) (6.18) ˆ Clearly, these coefficients are nonzero only if the force operator F is correlated with (respectively) p or ˆ ˆ is always expressed in terms of the t = 0 operators of the environment (ˆi (0), qi (0)), these x. Since F ˆ y ˆ correlations occur because the evolution mixes system and environment operators: p(t) = i T2i zi (0) ˆ ˆ and x(t) = ˆ ˆ ˆ ˆ i T1i zi (0). By expanding x(t) and p(t) in this way and indulging in some tedious algebra, we obtain f1 (t) and f2 (t). They are most conveniently expressed as the contraction of two tensors, one of which is the initial variance tensor of the environment, VE = 2 ∆yi (0) ∆yi qi (0) ∆yi qi (0) 2 ∆qi (0) . (6.19) ˆˆ ˆˆ For lack of better notation, we use ∆yq ≡ 1 (ˆq + q y ) − y q throughout. Technically, this is 2 yˆ a second cumulant, ∆yq = y q , but we have adopted this notation because cumulant notation is ˆˆ not widely familiar. In terms of VE , the diffusion coefficients are N 2 f1 = i=1 N −1 i Tr mi Fyi T2,2i+1 Fyi T2,2i+2 mi Fyi T1,2i+1 Fyi T1,2i+2 mi Fqi T2,2i+1 Fqi T2,2i+2 mi Fqi T1,2i+1 Fqi T1,2i+2 VE VE (6.20) 2 f2 = i=1 −1 i Tr (6.21) (6.22) 140 We now have all the coefficients of the master equation in terms of elements of the two ˙ matrices Tp and Tp Tp−1 (note that while equations (6.20-6.21) are expressed in terms of T instead of Tp , they involve only the first two rows, which are identical between T and Tp ). While complete specification of Tp requires complete specification of the systems and their couplings, we can simplify the problem further. The underlying physics mandates that the matrix Tp be of a particular form. We first define the 2 × 2 matrix Mi = √ mi ˙ ms φi (t) √1 mi ms φi (t) ms ˙ mi φi (t) (6.23) ¨ mi ms φi (t) where mi is the mass of the ith degree of freedom in the supersystem, ms is the mass of the system, and φi (t) is a function determined by the form of HSE . We can then write Tp in 2 × 2 block form: M0 0 0 . . . M1 1 0 . . . 0 1 M2 0 0 . . . 0 0 ··· (6.24) Tp = 0 0 ··· .. . In order to calculate Tp−1 , we define Dij as the determinant of the 2 × 2 sub-matrix (Tp )[1,2],[i,j ] . Then Tp−1 can be written simply as ˙ φ0 1 − ms φ0 D23 −D13 D12 0 . . . D24 −D14 0 D12 . . . ··· Tp−1 ¨ 1 −ms φ0 = 0 D12 0 . . . ˙ φ0 0 0 . . . ··· ··· ··· .. . (6.25) ˙ By explicit computation, we verify that Tp Tp−1 takes the form of equation (6.13), with the coefficients given by: 2 ωeff = γeff Fyi Fqi 2 ... ¨2 ˙ φ0 − φ0 φ0 2 ˙ ¨ φ0 − φ0 φ0 ... ˙¨ φ0 φ0 − φ0 φ0 = ˙2 ¨ φ0 − φ0 φ0 ... √ 2˙ ¨ = ms mi φi − γeff φi + ωeff φi = N (6.26) (6.27) (6.28) (6.29) ms ¨ 2 ˙ φi − γeff φi + ωeff φi mi ms Tr mi ms Tr mi ¨ mi Fyi φi ˙ Fyi φi ˙ mi Fyi φi Fyi φi ¨ mi Fqi φi ˙ Fqi φi ˙ mi Fqi φi Fqi φi VE VE f1 = i=1 N (6.30) 2 f2 = i=1 (6.31) 141 6.2.2 Master Equation for the Inverted Harmonic Oscillator We begin with the supersystem Hamiltonian: H= √ p2 ms Ω2 2 q2 me Λ2 2 + x+ − y + α ms me xy 2ms 2 2me 2 (6.32) The time translation matrix T is obtained by diagonalizing the equations of motion, which yields two normal modes. This transformation is characterized by a new harmonic frequency ω , a new inverse frequency λ, and a mixing angle θ: ω2 λ2 tan θ = = = 1 2 1 2 Ω2 − Λ 2 + Λ 2 − Ω2 + Ω2 + Λ 2 − (Ω2 + Λ2 ) + 4α4 (Ω2 + Λ2 ) + 4α4 (Ω2 + Λ2 ) + 4α4 2 2 2 (6.33) (6.34) (6.35) 1 2α2 Using these quantities, the Tp -matrix can be expressed in the form of equation (6.24), where the φi (t) are given by: φ0 (t) φ1 (t) = = cos2 θ sin(ωt) sinh(λt) + sin2 θ ω λ sin 2θ sin(ωt) sinh(λt) − 2 ω λ (6.36) (6.37) Thus, for this case we immediately obtain the parameters of the master equation by direct substitution into equations (6.26-6.31). All terms share a common denominator, which we denote by D: D 2 ωeff = = = ω 2 − λ2 cos2 θ sin2 θ sin(ωt) sinh(λt) +ωλ 2 cos(ωt) cosh(λt) cos2 θ sin2 θ + cos4 θ + sin4 θ ωλ D ω 2 cos4 θ − λ2 sin4 θ 2 + sin4 2θ ω 2 − λ2 cos(ωt) cosh(λt) − 2ωλ sin(ωt) sinh(λt) (6.38) (6.39) (6.40) (6.41) (6.42) γeff Fy Fq ω 2 + λ2 sin2 2θ [λ sin(ωt) cosh(λt) − ω cos(ωt) sinh(λt)] 4D √ − ms me ωλ ω 2 + λ2 sin 2θ = cos2 θ cosh(λt) + sin2 θ cos(ωt) 2D ms ω 2 + λ2 sin 2θ =− ω cos2 θ sinh(λt) + λ sin2 θ sin(ωt) me 2D 142 The diffusion coefficients are naturally described in terms of the elements of the tensors in equations (6.30-6.31). Factoring out their common prefactor β = 4ms sin2 2θ ω 2 + λ2 , we obtain: 2D yy f1 yq f1 qy f1 qq f1 yy f2 yq f2 qy f2 qq f2 = me ωλβ cos2 θ cosh(λt) + sin2 θ cos(ωt) (λ sinh(λt) + ω sin(ωt)) = ωλβ cos θ cosh(λt) + sin θ cos(ωt) (cosh(λt) − cos(ωt)) = β ω cos θ sinh(λt) + λ sin θ sin(ωt) (λ sinh(λt) + ω sin(ωt)) β = ω cos2 θ sinh(λt) + λ sin2 θ sin(ωt) (cosh(λt) − cos(ωt)) me = me ωλβ cos2 θ cosh(λt) + sin2 θ cos(ωt) (cosh(λt) − cos(ωt)) = β cos θ cosh(λt) + sin θ cos(ωt) (ω sinh(λt) − λ sin(ωt)) = β ω cos2 θ sinh(λt) + λ sin θ sin(ωt) (cosh(λt) − cos(ωt)) β ω cos2 θ sinh(λt) + λ sin2 θ sin(ωt) (ω sinh(λt) − λ sin(ωt)) = me ωλ 2 2 2 2 2 2 2 (6.43) (6.44) (6.45) (6.46) (6.47) (6.48) (6.49) (6.50) 6.3 Results and Analysis Having obtained a master equation describing the evolution of a system coupled to an IHE (inverted harmonic environment), we now proceed to examine the consequences of that evolution. We have two tools for this analysis: on one hand, the master equation and its coefficients determine the instantaneous effects of the environment; on the other hand, we can explicitly evolve an initial state of the system to see how properties such as entropy and energy evolve. The master equation itself divides naturally into two parts corresponding to the two lines of equation (6.2); the terms in the first line produce renormalized unitary evolution (including external damping and forcing terms, which break unitarity), while the last two terms are diffusive and responsible for decoherence. We thus divide our analysis into three sections, addressing in turn the quasi-unitary portion of the master equation, the diffusive terms in the master equation, and the behavior of evolved observables. Our primary results are in equations (6.64-6.65), where we demonstrate the linear growth of entropy in the system at the rate set by the Lyapunov exponent and obtain an approximate decoherence timescale that turns out to be logarithmically dependent on the coupling. This implies that isolation from amplifying environments is, in some sense, exponentially difficult. In particular, it is harder to isolate a system from an amplifying environment than from the many harmonic oscillators comprising a QBM environment.3 . The reader who wishes to skip straight to the implications of amplification for decoherence may wish only to skim sections 6.3.1 and 6.3.2, which analyze the master equation in detail. 6.3.1 Unitary evolution The first five terms in the right-hand side of equation (6.2), 1 i 2 mωeff 2 1 γeff x ,ρ + ˆˆ p2 , ρ + ˆˆ [ˆ, {p, ρ}] − F (t) [ˆ, ρ] x ˆˆ xˆ 2 2m 2 (6.51) are exactly those for the evolution of an isolated harmonic oscillator subject to an external force F (t) and a damping force γeff (t). For convenience, although the term γeff [ˆ, {p, ρ}] breaks unitarity, x ˆˆ we refer to these terms in the master equation as the “unitary evolution” terms. Because the coefficients of all the terms except p2 , ρ are time-dependent, however, the ˆˆ 2 evolution induced by these terms is not necessarily intuitive. The time-dependence of ωeff and γeff 3 For QBM, the decoherence time is approximately quadratic in the coupling strength [140, 66] 143 2 Figure 6.1: Dependence of the master equation coefficients ωeff and γeff on coupling angle θ, system frequency ω , and the effective Lyapunov exponent λ of the environment. The base configuration is 2 ω = λ = ms = me = 1, θ = 10−3 : plot (a) shows the dependence of ωeff (t) on θ, plot (b) shows the 2 dependence of γeff (t) on θ, plot (c) shows the dependence of ωeff (t) on ω , and plot (d) shows the 2 dependence of ωeff (t) on λ. The units of time in all plots are identical but arbitrary. Q VT Q VT P UT P UT '(&$"  % #!  Q SR P ¢R '(&$"  % #!  Q SR P ¢R 4 2 A 9E D 0 I86 5HGF&@1) 42 A 9 0 786 5CB@1) 7642 0 8&531) q r0 t D @0 t 2 30 t Q Q Gh £s £` ba P YW X P YW X T TW T TW P P X X §¥ ¦ ¨ ©  § ¦ ¨ ©c  fe¥ d  QVT QVT PUT PUT '%# (&$!"   Q SR P¢R '%# (&$!"   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Since the various numerators in equations (6.39-6.50) do not change sign in sync with the denominator, all the coefficients not only vary from positive to negative, but diverge whenever D = 0. While this phenomenon is unexpected and counter-intuitive, we note that it does not indicate unphysical behavior. Detailed analysis (omitted here) shows that near a divergence in the coefficients, the system state ρ is forced to assume a form such that the effects of the superoperators in the master equation cancel each other out; thus, when the coefficients diverge, their effects sum to a perfectly finite value, and the evolution of the state is completely physical. We conclude from this that the divergences are not symptomatic of any physical phenomenon, but are rather a consequence of the system-environment paradigm that we have imposed on the supersystem. As the system and environment interact with each other, information about the initial x0 and p0 of the system is transferred to the environment (and vice-versa). At certain times, all information about a particular linear combination of x0 and p0 has been transferred to the environment, in return for which all information about some linear combination of y0 and q0 resides in the system. This is true only for an instant, but at that instant ρ(t) does not uniquely determine ˆ ρ(0), no differential equation for ρ can exist, and the master equation necessarily breaks down. The ˆ ˆ mathematical reflection of this is that the upper left 2 × 2 sub-matrix of Tp (which specifies the relationship between (x, p) and (x0 , p0 )) is instantaneously non-invertible, because D = 0. For t ∼ 0, D = 1. Assuming that the coupling is weak (θ 1), we can expand D for λ−1 t −2λ−1 log θ as D 1 + θ2 eλt ω 2 − λ2 sin(ωt) + 2ωλ cos(ωt) . ωλ (6.52) 1 until shortly before Since the fraction at the end of equation (6.52) is O(1), we conclude that D the first divergence occurs; further, the timescale of that divergence is tc −2 log θ + log λ ωλ . ω 2 + λ2 (6.53) While the first divergence may occur later than tc , it cannot occur earlier. Thus, we have a natural time scale in the system that divides time into two regions: an initial period, when t < tc ; and long times, when t > tc . In the initial period, the unitary coefficients of the master equation are well-approximated by their bare values, whereas in the long-time regime the coefficients’ values periodically diverge and are only indirectly related to their bare values. When t tc , the coefficients’ values are difficult to characterize. Finally, we note that the critical timescale is reflected in the diffusive coefficients as well (see Fig. 6.2), but not necessarily in physical quantities obtained from ρ(t) ˆ itself, as we expect from the argument that the divergences are not reflected in physical quantities. Nonetheless, tc is a useful time scale to keep in mind because in the initial period we can be assured that the renormalized unitary evolution is very similar to the bare unitary evolution; thus, the effects of interaction with the environment during this regime will be all but unnoticeable on classical scales. Beyond tc , the terms in the master equation that govern the evolution of largescale structures in phase space may depart dramatically from their uncoupled values; we cannot be certain. If we take as the criterion for our model’s relevance to a real chaotic environment that its effects be small on classical scales, then relevance is guaranteed for the initial period, but not for 145 long times. Thus, although we will examine the long time behavior of the model at times, we regard the initial (t < tc ) period as a candidate model of decoherence due to a chaotic environment. Another perspective on the same conclusion comes from the fact that any physical chaotic ˆ environment will have a bounded phase space. The operator OE that couples to a system operator ˆ ˆˆ OS will thus have a bounded spectrum, and the expectation value of the coupling HSE = OS OE will also be bounded by some value HSE max . As long as our model respects this constraint, we can consider it a plausible model for that environment; however, because the y operator is unbounded ˆ and y 2 increases exponentially in time, our model will eventually expand into a larger phase space, ˆ and the coupling will dominate the overall Hamiltonian. The critical time tc indicates roughly when HSE begins to dominate the overall Hamiltonian. For completeness, we consider briefly the effects of an environment which is not unstable – a simple harmonic oscillator or free particle. The simple harmonic environment does not have 2 divergences in the master equation coefficients. Instead, ωeff and γeff oscillate stably around ω 2 2 and 0 (respectively); ωeff is plotted in Fig. 6.3. The free particle is somewhat more interesting; a wave packet spreads slowly (∆x2 ∝ t) in the absence of a potential, so in this sense there is a weak irreversibility. We discuss this in more detail when we consider entropy, but as Fig. 6.3 shows, the divergences characteristic of the IHE do plague the free-particle environment. However, tc is no longer logarithmic in the coupling, but rather obeys a power-law. The free-particle environment (like the SHO environment) can be more effectively isolated from the system by reducing the coupling strength. 6.3.2 Diffusive terms The last two terms in equation (6.2), −f1 (t) [ˆ, [ˆ, ρ]] + f2 (t) [ˆ, [ˆ, ρ]] , x xˆ x pˆ (6.54) are non-unitary and diffusive, and produce decoherence. If ρ is transformed to a Wigner function ˆ W (x, p), then equation (6.2) becomes a Fokker-Planck type equation. The f1 or “normal diffusion” ∂2 ˙ term is seen to produce diffusion in momentum space according to W ∝ −f1 ∂ 2 p W , while the f2 or ˙ “anomalous diffusion” term tends to skew the state according to W ∝ −f2 ∂ ∂ W (see [140], where ∂x ∂p “anomalous” diffusion was first identified). This is also apparent in equations (6.8-6.12), where f1 x, ˆ ˆ appears only in ∂ t while f2 appears only in ∂ {∂tp} . In QBM, the coefficients f1 and f2 of these terms equilibrate to constant values or monotonically decreasing functions after an initial period; in our IHE model these coefficients, like those of the unitary terms, vary widely over time. Unlike the other coefficients, f1 and f2 are dependent on the state of the environment; however, they can each be written as the contraction of a 2 × 2 coefficient tensor with the variance tensor of the environment at t = 0 (see equations (6.43-6.50)): yy fn 1 qy 2 fn 1 yq 2 fn qq fn ∂ p2 ˆ fn (t) = Tr ∆y 2 ∆yq ∆yq ∆q 2 . (6.55) The absolute values of the four subcoefficients for each of f1 and f2 are plotted in Fig. 6.2, for the same base parameters as Fig. 6.1 and for three different coupling strengths. Each subcoefficient has a prefactor involving the masses of the system and environment, which is ignored in these plots; thus, if the system and environment masses are substantially different, one subcoefficient may be promoted over another. The most salient feature of the plots in Fig. 6.2 is that they appear identical – only at times shorter than λ−1 is any difference visible at all between the various coefficients. Examination of equations (6.43-6.50) confirms that for t λ−1 , all the subcoefficients are proportional to θ2 D−1 e2λt . This also explains the sharp distinction between short- and long-time behavior in Fig. 6.2. In the Figure 6.2: Absolute values of the diffusive master equation coefficients (f1 and f2 ) are plotted versus time, on a logarithmic scale (with arbitrary units), for three different values of the coupling angle (θ). Since f1 and f2 are dependent on the state of the environment, all four tensor components of each fn are plotted (note, however, that all eight components are nearly identical in this case). Plots (a)-(d) show the tensor components of f1 , while plots (e)-(h) show components of f2 . The base configuration is ω = λ = ms = me = 1. 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Divergences do appear in the coefficients for the free-particle environment, but their onset time is polynomial in θ−1 , and they do not appear in these plots except for the largest value of θ (π/5). @ EC @ EC 9 DC 9 DC &$" '%#!    @ BA 9 BA &$" '%#!    @ BA 9 BA 8 1 H( ) 476 1) 2H( 4 53 1) 2H( 8 1 H( ) 476 1) 2H( 4 53 1) 2H( @ @ £RQ P £ed c 9 HF G 9 A AF C CF G ¨¦ §¥ ©  9  ¥¦ U WXVV 9 a` Y 0F I I  @ EC @ EC 8 1 0( ) 476 20( 1) 4 53 20( 1) 8 1 0( ) 476 20( 1) 4 53 20( 1) 9 DC 9 DC &$" '%#!    @ BA 9 BA &$" '%#!    @ BA 9 BA @ @ £¡ ¤¢ £S TQ 9 HF G 9 C bA G ¥ §¨¦ ©  9  ¥¦ I 0F G 9 UV I WXV a` Y I HF  I 148 short-time regime, D 1, and each subcoefficient is well-approximated by f ∝ e2λt ; in the long-time regime, however, D θ2 eλt cos(ωt + φ), and f ∝ eλt sec(ωt + φ). Thus, on the log-plots in Fig. 6.2, we see two distinct lines with slopes 2λ and λ, the second of which is punctuated by the periodic divergences seen in all the master equation coefficients. In order to make this explicit, we examine the short-time (λ−1 < t < tc ) behavior of f1 , which is responsible for entropy production (see section 6.3.3), when menv msys . In this limit, f1 qq 1, θ 1 and D 1, we obtain: is dominated by f1 ; using the approximations sinh(λt) 2 f1 msys θ2 (ω 2 + λ2 ) 2λt e menv (6.56) The key point in equation (6.56) is that the coefficient of the term that produces diffusion in momentum (the primary factor in decoherence) increases exponentially with time and λ, but is only quadratic in the coupling strength. The change in the exponent of the f coefficients at t ∼ tc is physically relevant as well as mathematically sensible. In the short-time regime, the oscillatory dynamics of the system and the hyperbolic stretching of the environment proceed largely independently of one another; just as the environment induces only minor perturbations in the system, the system does not disturb the environment greatly. Thus, the stretching of the environment along its unstable manifold is reflected in the system as diffusion in one of the two phase-space dimensions. After tc , however, the interaction Hamiltonian begins to dominate the dynamics of the system. The overall dynamics become strongly coupled, and the unstable manifold of the environment rotates. Diffusion in the system is averaged over stable and unstable directions, and so the diffusion coefficients increase only as eλt . Nonetheless, because diffusion now occurs along all directions in phase space, the entropy of ρ continues to grow at the same rate (see Section 6.3.3). ˆ 6.3.3 Entropy of the Reduced Density Matrix Having analyzed the IHE master equation in detail, we turn to the behavior of ρ(t) and ˆ its properties. Because decoherence manifests as entropy production in ρ, we first calculate the Von ˆ Neumann entropy H (t) = Tr(ˆ ln ρ) of the reduced density matrix, and examine the dependence of ρˆ H (t) on various parameters. Entropy production can reflect not only the destruction of quantum coherences but also the destruction of large-scale “classical” structures in phase space. In order to verify the ability of the IHE to produce decoherence without destroying classical structures, we take the expectation value of energy, E = HS , as a convenient classical quantity. For appropriate initial conditions, H rises rapidly, while E remains relatively undisturbed. It is worth noting that because all the states we consider have x = p = 0, the constancy of E ˆ ˆ is probably an overly strong condition. Large (on classical scales) amounts of energy can be added to the system by a simple Galilean transformation at t = 0, which changes nothing of the analysis except for adding a constant offset to E (t). Entropy The canonical state to examine is a “Schr¨dinger Cat” state, typically a superposition of o widely separated coherent states. We have examined the effects of the IHE on such a state, but because the state is not itself Gaussian the analysis is quite messy in our ansatz. We discuss, instead, the behavior of a Gaussian state, squeezed to make it extended in x. In the presence of a diffusive decoherence process, the most significant features of a “cat” state are the interference fringes in W (x, p) that lie between the two Gaussian bumps. They are extended in x but equivalently narrow in p. A Gaussian state that is highly squeezed in momentum has similar small-scale structure, reflected in the strong off-diagonal correlations of ρ. ˆ 149 In general, H = Tr(ˆ log ρ) is difficult to compute. For Gaussian states, however, H can ρ ˆ be computed easily in terms of the state’s scaled area in phase space. The Heisenberg uncertainty relation ∆x∆p ≥ 2 provides a fundamental unit of phase space area. We thus define a0 = 2 , and define the phase space area occupied by a Gaussian state as a= ∆x2 ∆p2 − (∆xp) , 2 (6.57) and the scaled area as A = a/a0 .4 Every Gaussian pure state has an area a = a0 ; mixed states have a > a0 . It can be shown [169] that the entropy of a mixed state is given exactly by H= 1 [(A + 1) ln(A + 1) − (A − 1) ln(A − 1)] − ln(2). 2 (6.58) 0.31, A convenient approximation, which is exact for H = 0 and is always accurate to within 1−ln 2 is ˜ H = = ln(A) 1 ln(A2 ). 2 (6.59) The second version is useful because A2 is easily calculable as the determinant of the state’s variance tensor. This quantity is related to (but not identical to!) the linear entropy (ς = 1 − Trρ2 ) because ˜ Trρ2 = A−1 . Linear entropy is thus a first order approximation (in A−1 ) to H . Analysis of the H (t) Plots In Figs. 6.4-6.6, H (t) is plotted for a wide range of initial conditions and parameters. The initial conditions consist of the squeezing parameter r = ∆x and squeezing angle θ for both the ∆p system and the environment (large r and θ = 0 indicates a state extended in position and squeezed in momentum, while θ = π implies the reverse). The parameters of the master equation include the 2 bare ω and λ, the coupling θ, and mass-ratio = me /ms . The base parameters used in Figs. 6.4-6.6 are π , ms = 1, me = 1, rs = 4, re = 2, θs = 0, θe = 0. (6.60) ω = 1, λ = 1, θ = 64 Each plot varies one of these parameters, except for Fig. 6.5d. Fig. 6.5d plots the entropy for a non-inverted harmonic environment, and varies the coupling angle. All but one of the plots demonstrate the same basic behavior: entropy increases linearly as H = λt + H0 , with periodic modulations. The last plot in Fig. 6.5 shows S (t) for a stable environment; the entropy oscillates in time but does not increase irreversibly. Fig. 6.6 demonstrates entropy’s dependence on λ. For λ = 0 we obtain not constant entropy, but logarithmically increasing entropy. The λ = 0 environment is a free particle, whose wave packet spreads as ∆y 2 ∝ t. This mild irreversibility produces entropy growth that goes to zero only as t → ∞. For other values of λ, ∆y 2 grows as e2λt , and entropy grows linearly. Varying the squeezing parameter or squeezing angle of the environment (Fig. 6.4) changes only the initial jump in entropy, H0 . H0 is minimal for an unsqueezed environment (re = 1), and − increases approximately as log(re + re 1 ). The dependence on θe is minimal for = 1; for larger or smaller mass ratios, the dependence on the squeezing angle becomes more noticeable. It should be noted that the phrase “initial jump in entropy” refers precisely to the intercept of H (t) λt + H0 in the long time limit; H0 can be negative, which means only that the linear growth in entropy is postponed for a time −H0 /λ. Also in Fig. 6.4, it is apparent that the effect of varying rs and θs is merely to modify the periodic modulation of H (t). Since the system state rotates in phase space (according to the 4 See Section 6.2.1, just after equation 6.19, for an explanation of the (∆xp) notation. 150 Figure 6.4: Dependence of system entropy on the initial states of the system and the environment. π The base parameters are ω = λ = ms = me = 1, θ = 64 , rs = 4, re = 2, θs = 0, and θe = 0. Plot (a) shows the dependence of H (t) on re , the squeezing parameter of the environment; plot (b) shows the dependence of H (t) on rs ; plot (c) shows the dependence of H (t) on θe , the squeezing angle of the environment; and plot (d) shows the dependence of H (t) on θs . All entropies are dimensionless Von Neumann entropies (H = Trρ ln ρ). B EC B EC A DC A DC % # !   &$" % # !   &$" B B 9@TRQ' 5 0P URQ' 7 0P 5 T0 P ' 24 0 P SRQ' a 4 de0 P X a0 c bhP X 4 abhP 0X `0 UhP X £f gQ A A £H I) BEC BEC GF A GF A A DC  A DC  B B A A ¦§¥ ¦§¥ ¨© ¨© EC B EC B DC A DC A &$" %#!   &$" %#!   B B 9@61)' 5 0( 781)' 0( 5 60 ( ' 4 0( 231)' d a c e0 ( X 4 bY( a0 c abY( X 0 UY( X `0 X £¤¢ ¡ £W) V A A B EC B EC GF A A DC  A DC  B B A A §¥ ¦ ¦ §¥ ¨© ¨© GF A 151 Figure 6.5: Dependence of system entropy on the parameters of the master equation. The base π parameters for plots (a)-(c) are ω = λ = ms = me = 1, θ = 64 , rs = 4, re = 2, θs = 0, and θe = 0; π for plot (d) the parameters are λ = 4i, ω = ms = me = 1, θ = 64 , rs = 4, re = 2, θs = 0, and θe = 0. Plot (a) shows the dependence of H (t) on θ; plot (b) shows the dependence of H (t) on ms /me ; plot (c) shows the dependence of H (t) on ω ; and plot (d) shows the dependence of H (t) for a stable environment on θ. We emphasize that in plot (d), entropy oscillates but does not increase over time (compare with Fig. 6.6). The couplings are much larger for plot (d), in order to make H perceptible on the same scale as is used for the unstable environment. B EC B EC DC A DC A % # !   &$" % # !   &$" B B @ p5 0( 1)' 97 @ 0 )' ( @42 01)' ( 8 0( i)' WXU( P V T U( P R S( P £ hfg A A 2 Q (P £G IH B DC B DC $F A $F A A ¢C  A ¢C  B B A A ¦ §¥ ¦ §¥ ¨© ¨© B EC B EC A DC A DC &$" %#!  &$" %#!  567942 @ 0 ( )' 3997 8 0 ( )' 5 63 01)' ( 42 3 01)' ( B B W e( a d( a c 7 2(a b U( a £¤¢ ¡ £Y `H A A B DC B DC A¢C  A $F A¢C  B B A A ¦ §¥ ¦ §¥ ¨© ¨© A $F 152 Figure 6.6: Dependence of system entropy on the effective Lyapunov exponent. The base parameters π are ω = ms = me = 1, θ = 64 , rs = 4, re = 2, θs = 0, and θe = 0. The curve for λ = 0 shows logarithmic increase in entropy. 90 6 64 2 @590 % 80 !©¨     64 ¢57 6 64 ¢52 0 1$" # ) ' ($" # 2 3$" # % &$" # % 052 C 2 B0 A 7 % § ¦¥ ¤£ ¡ ¢ 6 D 153 dynamics) over time anyway, changing θs merely changes the phase of the oscillation in H (t), while rs determines the shape and amplitude of the oscillations. These become more dramatic as rs increases or decreases from 1 (which eliminates the modulation). Figures 6.5-6.6 show the effects of varying ω , λ, , and θ. Both θ and affect H0 without changing the character of H (t) in any other way. This is particularly interesting for the coupling θ; regardless of how small the coupling is, the entropy still grows as eλt . In particular, the plot for π θ = 1024 corresponds to a crossover time of tc > 15, yet entropy begins to grow linearly around t ∼ 2.5. Changing λ, as mentioned earlier, changes the rate of entropy production. Finally, the result of varying ω is a change in the periodic modulation; the frequency of this modulation is of course ω . The curve for ω = 0 is of special interest. Although the periodic modulation has vanished, H (t) rises not as λt but as λt + log(λt). The ω = 0 system is a free particle, so ∆x2 grows linearly in time and contributes a logarithmic term to the entropy. Conclusions regarding Entropy and Energy The linear growth of entropy seen in Figs. 6.4-6.6 indicates that the IHE model can continue to decohere a system long after other models, such as QBM, would have ceased to produce substantial entropy. However, it does not demonstrate that the IHE can produce decoherence faster than QBM. To examine this issue, we first derive a formula for the rate of entropy growth. The exact entropy (equation (6.58)) is difficult to work with; therefore we approximate ˜ H (t) with H (t) = 1 ln A2 from equation (6.59). Using equations (6.8-6.12)), we obtain 2 ∂2 (A ) = −γeff (t)A2 + 2 f1 (t)∆x2 − f2 (t)(∆xp) . ∂t (6.61) Not surprisingly, a positive dissipation coefficient γeff causes the state to shrink in phase space; however, as we saw in the analysis of the unitary terms, γeff can be both positive and negative, and thus expands the state as often as it shrinks it. We conclude that the γeff term contributes only to the periodic modulation of H (t). The effect of the f2 term is highly dependent on ∆xp, which can be positive, negative, or zero. However, because ∆xp oscillates around 0 as the system evolves, this term will contribute primarily to the periodic modulation as well. This leaves: ∂2 (A ) ∂t 2f1 (t)∆x2 (6.62) Since ∆x2 is strictly positive and f1 (t) in the short-time regime t < tc is positive (see equation (6.56)), this predicts monotonic growth in A2 , and thus H (t). In addition, we can use the previous analysis of f1 (t) to approximate f1 (t) κ2 θ2 e2λt (6.63) (where κ depends on ω , λ, , and the initial state of the environment, but not on time) and integrate equation (6.62), obtaining ˜ H (t)) = log (κ∆x2 ) + log θ + λt. (6.64) Equation 6.64 summarizes our most significant result. The amount of entropy produced is linear in t and in λ, but only logarithmically dependent on the properties of the initial state or on the coupling strength. Inverting equation (6.64) yields an approximate decoherence time, τd : 1 td Hd − log (κ∆x2 ) − log θ , (6.65) λ which is the time required to produce a certain amount (Hd ) of entropy (e.g., 1 bit for a Schr¨dinger o Cat state) from an initial pure state. td is only logarithmically dependent on the coupling (θ) and the initial properties of the environment (contained in κ). As the coupling becomes very weak, we expect only moderate increases in the decoherence time. Contrast this with the QBM model, where 154 the τd has a power-law dependence on the coupling strength. Isolating the system from a QBM environment (while still very difficult [160]) is not quite as hopeless as in the case studied here. Finally, we consider the system’s energy. If there is no coupling to the environment, then E is a constant of the motion; conversely, disruption of E indicates that effects of the environment are noticeable at classical scales. Since the system’s energy is given by E (t) = 1 2 ms ω 2 x2 + ˆ 1 p2 ˆ ms , (6.66) we can immediately calculate its time derivative as f1 (t) − γ p2 ˆ ∂ E (t) = ∂t ms (6.67) The system’s energy will grow, because of the f1 (t) term. 2 However, ∂A also contains f1 , but multiplied by ∆x2 . Thus, for states that are highly ∂t delocalized (e.g., Schr¨dinger cat states), H will grow more rapidly than E . In addition, the system’s o initial energy (E0 ) plays no role in ∂E . If the system’s initial energy is large, the added energy ∂t E (t) − E0 will (for some reasonable interval of time) be comparatively negligible. We conclude that initial states with large ∆x2 and large E0 (i.e., superposition states over classical length scales) admit a relatively long interval of time, over which energy remains constant while entropy grows rapidly. An example is given in Fig. 6.7. 6.4 Conclusion The model we have studied is probably the simplest possible amplifying environment. The reward for drastic simplification is that we obtain an exact solution.5 The resulting conclusions are significant, and we expect them to be quite generally applicable. We have shown that an unstable environment with a single degree of freedom can produce decoherence more readily than the canonical QBM environment, which requires infinitely many degrees of freedom (and a much larger Hilbert space). The IHE also has an infinite-dimensional Hilbert space (its phase space volume is unbounded), but a bounded system could simulate the inverted oscillator arbitrarily well up to a certain time. Such a system would produce the same results over the time of interest, but would not be analytically soluble beyond that time. A brief comment is in order on the principle that enables our “small” (in terms of degrees of freedom and available Hilbert space dimension) environment to be so effective at amplifying information about S . A good measure of an environment’s decohering efficacy is the amount of entropy it can produce in the system, and the rate at which that entropy is produced. For pure initial states of the environment, entropy can only be produced through entanglement; the total entropy that can be produced is limited by size of the environment’s Hilbert space. Thus, in the long run, larger environments can decohere more effectively. However, the rate at entropy is produced is limited by the rate at which the environment can explore its phase space. A collection of harmonic oscillators is a stable system, and small perturbations (due to the state of the coupled system) do not force any one oscillator to explore a large volume of its phase space. The inverted oscillator, when perturbed, explores its phase space much more efficiently. We view this work as the first step in a larger project: understanding amplifying environments. A closely related goal is to determine the role that chaotic systems play in producing decoherence. The results presented here show conclusively that an unstable environment can (1) decohere a system in a manner markedly different from the standard (QBM) models of decoherence, 5 Although, in the end, we make simplifying assumptions in order to get “thumbnail results” such as equation (6.65). 155 Figure 6.7: Comparison of H (t) and E (t) for an extended state. Since E grows as e2λt , we have 1 plotted 2 ln(E/E0 ) so that both curves have the same slope asymptotically; thus all quantities plotted are dimensionless. The noteworthy regime is t < 5, where the entropy grows steadily but E remains virtually constant. The parameters for this calculation are: ω = 10−5 , λ = 1, ms = me = 1, π and θ = 512 . The initial state of the system is squeezed by a factor of 104 , with the long axis located π at an angle of 64 to the x-axis; the environment is initially squeezed by a factor of 16 in p. ˆ ˆ 2 87 6  § ¥ ¡ 5 ©¨¦£¤¢ 4  &'$%# 1)0§ ( #!  ¢"   3 22 3 6 5 287 4 156 and (2) display unexpected behavior – e.g., the periodic breakdown of the master equation formalism due to singularities in the coefficients. While we believe we have sketched accurately the regime in which our toy model represents faithfully some aspects of the behavior of actual chaotic/amplifying environments, future work will examine such environments directly using numerical simulations. Numerical studies carried out previously [124, 83, 93, 97, 94, 158] show a range of behaviors. We hope that the exactly solvable model we have described will aid in the analysis of the relevance of chaos to decoherence. Another relevant class of models, which we do not examine here, includes environments with many chaotic degrees of freedom (e.g., the atmosphere of the earth). In addition to these obvious future directions, we intend to investigate other effects of amplification. The key step in any decoherence process is that of tracing over the environment; if this is not done, the state of the universe is an entangled pure state, not a mixed state. This is commonly justified by the argument that the environment is vast – the whole universe, potentially – and this size guarantees that some of the information that has been transferred to the environment has been irreversibly lost. Despite this (reasonable) justification, explicit models of open quantum systems are usually bipartite. A small system is coupled to a larger (but still small compared with the rest of the Universe) environment. If, however, one imagines a set of “concentric” environments E (i) , each much larger than the last, as a model for the entire universe, then the environment to which the system is coupled (E (0) ) serves not as an independent environment, but rather as a communication channel between the system and the greater Universe (see [158, 160], also [167]). In this model, the system’s entropy is not limited by the size of the environment (as it is in the bipartite system - environment) model, but only by its own size. A small local environment can lead to redundant records of the preferred observables of the system in the rest of the universe. Our analysis of the IHE model indicates that chaotic local environments may act as amplifiers [51], carrying information away from the system more efficiently than integrable environments. Such connections between amplification and redundancy represent some of the most interesting outstanding questions. 157 Chapter 7 Applying these results to experiments We have focused, thus far, on a theoretical analysis of how information about a system gets recorded in its environment. In this chapter, we present some related experimental procedures, which serve both to illustrate the theory’s purpose and – in principle – as a means of verifying its predictions. This dissertation is not primarily focused on experiment, nor is the author an expert on experimental physics. Our emphasis here is on showing that redundancy and objectivity are experimentally relevant, not on designing specific experiments to quantify them. The examples that follow are intended partly as gedanken -experiments, much like those that Einstein, Podolsky, and Rosen proposed in 1935 [43]. More than half a century passed before the EPR experiments could be performed, but their idea clarified the debate over quantum mechanics even in the absence of results. The EPR “paradox” motivated J.S. Bell to propose a more general experimental diagnosis of entanglement, which turned out to be more easily tested. In the same way, we hope to motivate the design of practical experiments that elucidate how information about quantum systems is stored and propagated. We begin with the general problem of measuring (in experiments) the information-theoretic properties discussed previously. The central concept is correlation, which has an unambiguous experimental interpretation. Once this groundwork has been laid, we proceed to discuss experimental tests of the three core concepts of this dissertation: (1) predictability, and the algebraic structure of pointer observables; (2) redundancy of stored information; and (3) amplification of information into the environment. We choose to discuss them in this order (as opposed to the order in which they are presented in the dissertation) because the techniques discussed in Chapter 5 are particularly germane to experimental tests. 7.1 Experimental detection of information Quantum information theory grew out of quantum communication theory, just as classical information theory is generally dated to Shannon’s seminal paper, A Mathematical Theory of Communication [133, 134]. The key idea is that information represents a resource, which can be used to communicate. Measures of information (e.g., mutual information) tell us what can be achieved asymptotically – that is, as the number of signals or systems used to communicate becomes infinite. This is a very useful – and physically well-motivated – way to define information. However, it is not necessarily transparent or even useful to an experimental physicist who wishes to measure something. In this way, information is much like entropy, its close mathematical relative. 158 Entropy’s physical relevance is unquestioned, but its derived properties (e.g., temperature) are the experimentally measurable parameters. Information is strongly associated with correlation, in the same way that entropy is associated with uncertainty. As we pointed out in the introduction to this dissertation, mutual information is a measure of the correlation between two systems. A scientist wishing to measure the amount of mutual information between a system and part of its environment should, therefore, investigate the degree to which their states – and measurements made on their states – are correlated. Similarly, to determine the information-preserving structure that characterizes a system, the scientist should investigate the correlation between information put into that system (by preparing its state), and information gotten out of that system (by measuring it later). In classical systems, this is straightforward. Systems are prepared, acted upon, and then measured in the most precise way possible. These optimal measurements reveal the state of the system; measurements made on the system and its environment reveal their joint state. Analysis of the resulting data reveals correlation, and whatever hypothesis we made about the amount of correlation can be tested. This picture begins to fall apart for quantum systems, because no “most precise” measurement exists. The scientist must choose one of many possible measurements to make; if he chooses the wrong measurement, the correlations will not be apparent. In simple cases, the experimenter can deduce from first principles what measurements should be made to detect and utilize correlations. A more powerful technique would allow him to “bootstrap” to the correct measurement, by first searching for and then measuring correlations. In this chapter, our goal is to demonstrate that redundancy, predictability, and amplification can be experimentally tested – not to set forth a prescription for making optimal use of the resources they represent (i.e. bootstrapping). Such a prescription falls rather under quantum measurement theory, and is outside the scope of this work. The experiments we suggest in the following pages provide relatively straightforward and easily analyzed results, which do not require sophisticated analysis of the possible measurements. 7.2 Detecting and characterizing information-preserving structures The defining property of an information-preserving structure (IPS), embedded within the dynamics of an evolving system, is that certain measurements before the evolution are perfectly correlated with certain measurements after the evolution. Quantum states define and are defined by the outcomes of measurements, so this property can also be phrased in terms of states which are preserved. By mapping out the correlation between measurements, an experimenter can determine which states are preserved. Armed with this knowledge, an information theorist can determine what information-processing tasks can be accomplished using the system of interest. Quantum process tomography [117, 100, 29, 103, 96] identifies the map between input and output states. The experimenter prepares an initial state |ψ0 ψ0 | (either by performing a measurement, or by ensuring that if a measurement were performed, its outcome would be guaranteed), allows it to evolve, and performs a measurement. The data obtained from repeating this process many times yields frequencies – i.e., posteriori probabilities – for the various measurement outcomes. From these frequencies, the post-evolution state ρf can be inferred (this is quantum state tomography). This entire process is then repeated for a different initial |ψ0 ψ0 | – in fact, for a set of many |ψ0 ψ0 |. From the inferred evolution of many different states, the superoperator which completely describes the system’s evolution can in turn be inferred. Successful inference of ρf is only possible when the final measurement is informationally complete – that is, when the set of all the POVM operators Ei , which correspond to different possible outcomes, spans the system’s Hilbert-Schmidt space (the space of all Hermitian operators). To successfully infer the process superoperator as well, the set of initial |ψ0 ψ0 | must also be informationally 159 complete. 2 Process tomography is quite demanding. At least DS different states must be prepared; 2 they must be measured via an apparatus with at least DS different outcomes; and in order to obtain decent statistics, each input-output coincidence must be given the chance to occur many 4 DS times. Determining the process to within a fractional error requires approximately 2 trials. If the efficiency of detection or state preparation is less than 1, even more trials are required. Inferring the superoperator for a 2-qubit gate thus requires at least 256 separate measurements, each of which must be repeated several times. Nonetheless, many experimental groups are performing successful process tomography. The Kwiat and White groups (at UIUC and Queensland, respectively) have used the technique to analyze photonic systems [4, 103]. The Cory group, at MIT, has mapped out the effect of quantum gates in NMR systems [143, 25, 47] – a particularly interesting case, because the input states are “quasipure” mixed states, and the measurements yield only expectation values. The Blatt group, in Innsbruck, has performed the same analysis of gates (including quantum teleportation) in ion traps [121], where the input and output states are much closer to pure states. These three are far from the only groups to demonstrate process tomography, but they illustrate the wide range of relevant experiments. We predicted two general features of open-system dynamics in Chapter 5: 1. Information stored within a perfect IPS can be extracted later with a fidelity limited only by state preparation and measurement techniques. 2. Quantum processes which do not admit a perfect IPS should nonetheless display some approximate algebraic structure, which corresponds to one of the associative algebras discussed in Chapter 5. Each of these predictions can be tested. The first is a logical conclusion of quantum theory; like (e.g.) quantum teleportation, its demonstration in experiments is primarily important as a proof-ofprinciple. The second is less obvious; processes with no algebraic structure at all are possible. An example is the “pancake process” for a single spin (see Fig. 5.3), where |↑ and |↓ are completely depolarized, but the eigenstates of Jx and Jz are only 50% depolarized. We conjecture that this kind of process will not occur in a naturally decohering system. Prediction (1) above has already been tested in several systems (for instance, [85]). Its verification is, in fact, an absolute prerequisite for usable quantum computing hardware1 . When the IPS is a decoherence-free subspace or subsystem, this is done by using process tomography to show ˆ ˆˆ ˆ ˆ ˆˆ ˆ that the superoperator S that maps initial states ρ0 into final states ρt is unitary (i.e., S S † = 1l). For classical IPSs (i.e., pointer bases), a less thorough tomographic process will suffice; each pointer state is prepared, allowed to evolve, and verified to be in the predicted final state2 . The underlying idea is: preservation of information is confirmed by perfect correlation between initial (t = 0) and final (t = tf ) measurements. Prediction (2) is more novel. Process tomography experiments have been primarily concerned with (a) small systems (e.g., one or two qubits), (b) verifying the existence of nearly-perfect IPSs such as pointer bases and DFSs, and (c) demonstrating control. One of the problems for experimentalists is that tomography not only requires doing a lot of different measurements, but leads to an embarrassment of data. The superoperator for a 2-qubit process contains 128 independent complex numbers, a mass of data that is not easy to interpret. Our IPS formalism provides a convenient way of condensing this superoperator and interpreting its important properties. To demonstrate this approach, we have analyzed data obtained [24] from an NMR experiment in the Cory group [25]. A 2-qubit gate was implemented, using nuclear spins of two hydrogen 1 Real physical systems do not generally admit perfect IPSs. However, quite high fidelities can be achieved. The fidelity required to implement quantum error correction (which can then be used to achieve arbitrarily low error rates, at the cost of extra Hilbert space) is above 99%. 2 The existence of functioning classical computers is, in fact, evidence that pointer bases can be used to store information reliably. 160 atoms (on an ensemble of molecules in liquid solution) as the qubits. Process tomography was used to map out the superoperator. A Hamiltonian and a relaxation superoperator were obtained from a set of superoperators at different times. We used this data to analyze the quantum process observed midway through the gate. ˆˆ ˆˆ Table 7.1 shows the 16 eigenobservables of S S † , ranked by their eigenvalues λi . Since this ˆ is a real, noisy, experimental process, there is no perfectly preserved observable (except 1l), and no clear division between “nearly preserved” observables and “mostly destroyed” observables. By inserting an arbitrary dividing line between λ = 0.928 and λ = 0.94, however, we can separate out the 6 best-preserved operators in Table 7.1 (black) from the rest of the field (gray). These six operators form an approximate IPS, with an algebraic (Q2 ⊕ C2 ) structure (see Chapter 5). The Q2 portion represents the (approximate) decoherence-free subsystem spanned by |↑↓ and |↓↑ , which this experiment was engineered to produce. The C2 portion indicates that the states |↑↑ and |↓↓ are also good pointer states – but that superpositions that involve these states are not well-preserved. This system can thus be used to (approximately) preserve either one qubit or two c-bits. The perfect analogue to this approximate IPS applies the following mask to the density matrix: 1000 0 1 1 0 (7.1) 0 1 1 0 . 0001 The matrix elements corresponding to a 1 are preserved; those corresponding to a 0 are eliminated. We present this preliminary result to illustrate that (1) it is possible to test our conjecture, (2) experiments so far seem to confirm it, and (3) the pointer algebra framework really does provide a convenient and powerful way to extract meaning from process tomography. In this case, the analysis indicates that the NMR process of [25] simultaneously preserves a 4-state pointer basis and a 2-state decoherence-free subsystem. The system was engineered to produce the DFS, but there are alternative structures which contain a DFS. Other possibilities include Q2 ⊗ I1 (4 elements; one qubit), Q2 ⊗ C2 (8 elements; one qubit and one c-bit), and Q2 ⊗ Q2 = Q4 (16 elements; two qubits). Extreme caution should be taken in interpreting this data – it is presented to demonstrate the applicability of the theory, not as a conclusive analysis of this particular experiment. The original data has been manipulated in several ways prior to pointer-algebra analysis, and some of those manipulations may not be fully justified. For example, na¨ tomographic techniques generally ıve yield nonphysical results: nonpositive density matrices, or superoperators that are not completely positive. Techniques such as “maximum likelihood estimation” [71, 103] are widely used to constrain the results; these techniques may distort the result in ways that are significant to pointer-algebra analysis. More rigorous methods of analyzing tomographic data should enable a more confident application of this analysis. 7.3 Detecting and characterizing redundancy We discussed information preserving structures first, in the previous section, to emphasize one point: informational properties are reflected in the correlations of measurements. The amount of information that system A “has” about system B is a measure of how well certain measurements on A will be correlated with certain measurements on B . The statement “A has nearly complete information about B ” has a direct experimental equivalent: “If a measurement (Mb ) on B is correlated with a measurement (Me ) on almost all of the environment, then there exists a measurement (Ma ) on A alone, so that Ma is just as correlated with Mb as Me is.” ˆ ˆ The consequences of redundancy are even easier to state. If the information about S is 1 R-fold redundant in E , then only R of the environment need be measured in order to predict the ˆ ˆ outcome of a measurement on S as well as is possible. A certain amount of complexity has been 161 λ 1 .965 .965 0.14 0 0 0 1.4 0 0 0 1 0 0 0 ˆ M(1...8) 0 1 0 0 0 1.5 −0.08 0 0 −0.26 0.01 0 0 0 0 0 0 0 −i 0 0 0.03 1.0 0 0 −0.85 −0.01 0 0 0 0 1 0 0 0 −i 0 0 1 0 0 −0.08 −1.3 0 0 0.01 0.22 0 0 i 0 0 0 1. 0 −0.03 0 0 −0.01 −1.1 0 −.02(2+i) 0 0 −.02 .02(1−2i) 0 0 .02i 0 0 0 0 0 0 0 0 0 0 0 1.0 0 1 −.02 0 0 i −.02i 0 0 0 0 1 0 0 0 −0.34 0 0 0 −1.4 λ .926 .928 .928 .926 0 0.07+0.57i −0.01(1+i) 0 0 1+0.04i −0.03 0 0 1 −.01(3+i) 0 0 −.18+i .01(1−i) 0 ˆ M(9...16) 0.07−0.57i 0 0 −0.03i 1−0.04i 0 0 0.03 1 0 0 .01(1−i) −.19−i 0 0 .01(3i−2) 0 0.03i i 0 −0.03i 0 0 0.02−.04i 0.03 0 0 −0.04−0.02i −0.03 0 0 0.92+0.09i .01(i−3) 0 0 .38−.53i .01(1+i) 0 0 −.57−.77i −i 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 i 0 0 0 0 0.02+.04i 0 0 0 −0.04+0.02i 0 0 −0.01(1−i) 0 0 −i 0 0.03−0.01i 0.92−0.09i 0 0 .01(1+i) .38+.53i 0 0 −.01(2+3i) −.57+.76i 0 0 0.03i i 0 .951 .951 .940 .928 .928 0 0 0 0 0.99 0 0 0 0 0 .879 .879 .925 .925 0 0.03 1 0 0 0 0 1 0 0 0 −i 0 0 0 0 0 0 0 0 −.02(2−i) 0 0 0 .02(1+2i) 0 Table 7.1: Eigenoperators of a 2-qubit gate process in NMR. An arbitrary division at λ = 0.93 separates an approximate information-preserving-structure of 6 observables (colored black) from the other 10 observables (colored gray). These 6 observables form a pointer algebra (Q2 ⊕ C2 ) that consists of a 4-element Pauli algebra (representing a single qubit) plus two pointer states. The system was tailored to produce the 1-qubit DFS, but the emergence of a 6-element Q2 ⊕ C2 algebra rather than Q2 ⊗ I1 (4 elements) or Q2 ⊗ C2 (8 elements) is intriguing. 162 ˆ ˆ hidden in the phrase “as well as is possible” – most measurements on S cannot be predicted at all, since they correspond to observables which are not predictable! An experimentalist must know (1) what observables of the system are actually predictable, and (2) what measurements (out of those that he can perform) is correlated with them. It is possible to “bootstrap” this knowledge, by using tomography. The basic idea is that by making informationally complete measurements on both S and a fixed fragment of E , a scientist with no a priori knowledge of how they are correlated can deduce the amount and type of correlation. He can then design the best measurement to extract information. Showing how this is done is relatively straightforward, but not simple. In this section we will avoid this issue, by showing how to measure redundancy when the type of correlation is known. 7.3.1 How to measure redundancy generically In order to measure how redundantly information about a central system is stored in its surroundings, an experimenter needs to have a certain degree of control over some quantum system. In the next section, we discuss various systems in which this experiment could be performed; here we are concerned with the general prerequisites. The experimental system will consist of (1) a central system S which can be viewed as a single spin3 , and (2) ancillary systems of the same kind, which form the “environment” E . This is not a traditional environment, over which the experimenter has no control; instead, we control the environment to test specific predictions. The prerequisites are: • The ability to make precise and arbitrary measurements on the central system. Our theory provides a prescription for how to obtain information about S by measuring its surrounds; this information’s validity can only be confirmed by measuring the central system itself. • The ability to measure the “environment”. At the very least, this should include local measurements on each subenvironment. The assumption here is that classic non-demolition measurements can be made, which also prepare the “environment” in a nearly-pure state. This is not the case in NMR, where only expectation values can be measured, or in optics, where measurement of the electromagnetic field generally destroys its state. In these cases, the ability to prepare a pure or pseudopure state is an additional requirement. • The ability to generate controlled interactions between S and E . This implies two different things: first, that for some set of Hamiltonians {Hi }, the experimenter can guarantee that Hi is implemented; second, that all possible Hi can be generated in this way. The first condition guarantees unitarity, while the second determines how controllable the unitary evolution is. Unitarity is a strict requirement for our purposes. Nonunitary evolution is like having E initialized in a mixed state; it implies either ignorance of what is happening or that something external is interfering. Both will be disastrous. The ability to generate arbitrary unitary evolutions is less important. We require that particular unitaries be implemented, for which complete control is sufficient, but not necessary. Given this level of control over a set of systems, we can test the following hypothesis (implied by the results of Chapter 3): “Measurement interactions of the form MS ⊗ n REn record the system’s state redundantly and robustly. More complicated interactions, involving (a) multiple M ⊗ R terms, or (b) system evolution, do not.” A multitude of other predictions emerge from Chapter 3 and Chapter 4, which can be measured and tested by similar experiments. An experiment to test this hypothesis should consist of the following steps: 3 It need not actually be a spin, but it should have a finite Hilbert space on which operators analogous to angular momentum operators can be defined. Under the right conditions, this could even be a harmonic oscillator (see [55]). 163 1. Prepare the “environment” in a product state. The results of the experiment will depend (as discussed in the context of pliability in Chapter 3) on what this initial state is. Several experiments should be performed, with different initial environment states, and the final results should be averaged over these initial environment states. 2. Prepare the central system in a pure state. The precise identity of this state is unimportant, but it should not commute with the interaction between S and E that we generate in the next step. If it does, then the “measured” observable is already well-determined; no further information can be obtained. 3. Using unitary control, allow S to interact with E . This is the key step, and distinct experiments must be performed with different forms of interaction in order to test the hypothesis above. (a) In the first experiment, each subsystem En of the environment should be allowed to (S ) (E ) interact with S , for a random time, according to a Hamiltonian H = Jz ⊗ Jz n . It is immaterial whether the interactions are simultaneous (likely the most convenient scheme in NMR) or sequential (as is appropriate for trapped ions). (b) In the second experiment, each En should be allowed to interact with S according to a more complex Hamiltonian. Our results indicate that the form of this Hamiltonian is largely unimportant. An ideal example is the Heisenberg interaction, H = J(S ) · J(En ) . The simulations in Chapter 3 assumed simultaneous interaction with all the En ; again, we believe that our results should extend to sequential interactions, especially if each En interacts repeatedly with S . 4. Perform two measurements, and record the results: (a) Measure the central system. If the pointer basis is known (e.g., when the interaction Hamiltonian is of the form H = Jz ⊗ Jz , eigenstates of Jz are a good pointer basis), then the pointer basis should be measured. If not, then a wide range of different measurements should be attempted in different trials; we’ll identify which one was the pointer basis a posteriori. (b) Measure some fraction f of the “environment”. For instance, measure m out of Nenv subsystems. During the course of the experiment, the fraction f should be varied from 0 to 1 – that is, we measure m = 1, 2 . . . Nenv subenvironments successively. The goal is to extract the basis whose measurement result is most correlated with the measurement on S . This measurement may be obvious, or it may be computable using knowledge of what the interaction is, or it may be entirely unknown a priori. In the latter case, many different measurements should be made, and the “most useful” one can be extracted a posteriori (this is a basic form of tomography). 5. Repeat the prepare - interact - measure steps many times – enough to do statistical analysis on the measurement results. The number of times required depends upon whether optimal measurements or random measurements were made. For optimal measurements, n ∼ 100 measurements should be entirely sufficient to identify the level of correlation. If random measurements were made, each distinct measurement should be repeated n times, and then the pair of measurements (one on S , one on a fragment of E ) that is the most correlated should be chosen. 6. Analyze the degree of correlation – denote it C – between the outcomes of (a) the measurement on S and (b) the measurement on the fragment E{f } of the environment. The precise measure of correlation used to calculate C should be largely unimportant; classical mutual information is one candidate, as is collision probability. 164 The final result consists how correlation (between the maximally correlated measurements on S and a fragment E{f } of the environment) varies with the fraction f of E that was measured: 1. If information is stored redundantly, then C will reach a maximum value at some f increasing f thereafter will not increase C . 2. If not, then C will continue to increase until the entire environment has been captured. This simple result is robust to a wide range of interfering effects – for instance, to changing the form of Hint , or to redefining the measure of correlation represented by C . 1, and 7.3.2 Systems suited to exploring redundancy The experiment proposed in the previous section is a template. It provides a protocol which can be implemented in virtually any experimental quantum system that is (1) sufficiently large, (2) sufficiently well-controlled, and (3) sufficiently well-isolated. The requirements for performing this experiment (or any of the related experiments to, e.g., measure how redundantly something is recorded) are a subset of the DiVincenzo requirements for building a quantum computer [39] (note that this reference also provides an excellent, if slightly out-of-date, summary of what can be achieved with a wide variety of technologies). As such, this sort of experiment is an ideal testing ground for quantum information processors in development. NMR: The most mature quantum information processing technology at this time is NMR (nuclear magnetic resonance) of complex molecules in liquid solution [102, 136]. The advantages of NMR are that exquisite control over unitary evolution is possible, and that decoherence times are reasonably long. The major disadvantages are that pure states can never be prepared (“pseudopure” states, of the form ρ ∝ 1l+ |ψ ψ| are used instead), and that projective measurements are impossible. Additionally, scalability is strictly limited, for the polarization of a pseudopure state decreases exponentially as the molecule’s size increases. Redundancy-testing experiments nonetheless map very well to NMR, because 1. Pseudopure states can be used to store information just as well as pure states, provided that the level of polarization is taken into account. 2. Unitary control, which is crucial to our experiment, is possible. 3. Projective measurements are unnecessary; we use measurement only to obtain information, not to prepare states. 4. The inherent parallelism of NMR (all operations are effectively performed on trillions of molecules at the same time) reduces the need to repeat the experiment many times. Extending the template experiment to an NMR setting is trivial. Each individually addressable nuclear spin on the molecule being used is a subsystem; spin operators are directly controllable. Two-body interactions are completely controllable, and can be implemented simultaneously as opposed to sequentially. Ion traps: Ions in Paul traps are one of the most promising scalable technologies for quantum information processing [75, 87, 13, 28, 121]. The advantages and disadvantages of ion traps are in some sense orthogonal to those of NMR: pure states can be prepared and measured, but control is somewhat limited because the interaction between qubits (stored in the energy levels of each ion) is mediated by local interactions. By using collective modes as a bus for information, and by shuttling ions around in a trap with complicated geometry, universal control appears to be achievable. Their scalability is manifest (just add more ions). The one major drawback at this time is that measurement effectively ends the computation, because it heats ancillary modes of the system. Overall, however, ion traps are an excellent example of the “traditional” model of quantum 165 computation (in contrast to more exotic schemes which replace unitary evolution with measurements [101], or use postselection to make qubits interact [77]). Ion traps are an excellent setting for experiments exploring redundancy. The individual subsystems are discrete and well-defined, and the Pauli spin operators are straightforward to implement. In general, two-qubit gates are performed sequentially, but creative use of “bus” modes may make it possible for S to interact simultaneously with the entire environment. Pure initial states can be prepared, and arbitrary measurements can be made on the final state (by preceding local measurements with joint unitaries). The “environment as a witness” paradigm is particularly applicable because it provides a possible solution to the post-measurement heating problem. If the central system’s state is reliably correlated with several fragments of the environment – which are independent of S and can be physically removed from its vicinity – then it is possible, by measuring one such fragment, to find out the system’s state without actually disturbing its ancillary degrees of freedom. Like quantum teleportation (recently demonstrated in ion traps [121, 13]), this sort of explicitly indirect measurement may eventually be a valuable component in general quantum circuits. This is far from an exhaustive list of contexts in which experiments to explore quantum Darwinism could be performed. Other systems – e.g., optical lattices, or solid state qubits like the example suggested in Chapter 3– may provide more interesting applications of the theory, because they incorporate uncontrollable spin degrees of freedom. This is the situation that our analysis of information storage is really intended to deal with. In order to test the theory, however, the most advanced and controllable quantum systems are apropos. Just which technology is most advanced is an area of furious debate, but NMR and ion traps are certainly among the most promising. 7.4 Experimental implications of amplification The third major topic in this dissertation is the amplification of information by sensitive environments. This particular research program is less advanced than the others; Chapter 6 represents a first stab at characterizing what amplifying environments do. We have not yet investigated how such an environment stores information, merely how information leaks out of the system. In this section, we consider experimental testing of the rapid decoherence that sensitive environments should produce. The most straightforward prediction of Chapter 6 is that isolating a system from a chaotic or sensitive environment should be virtually impossible. Another way to put this is that superpositions of distinguishable states should be unstable in the presence of a sufficiently sensitive detector. However, in practice detectors must have limited sensitivity, because they too exist within a busy universe. A detector that is sensitive to arbitrarily small perturbations tends to get triggered, not by the system of interest, but by some other source of noise. This is why a lasing cavity does not make an effective single-photon detector. If the cavity is pumped to the point where a single incoming photon will trigger a cascade, then it will probably trigger itself through spontaneous emission. The challenge to experiments that test decoherence in sensitive environments is thus, ironically, decoherence itself. A truly sensitive environment “measures” everything around it – not just the central system – and hence decoheres. This can be avoided by making the S −E coupling stronger than any coupling to the greater universe. Then, however, the distinguishing feature of a sensitive environment (its ability to decohere S even when the coupling is small) is obscured. Practical quantum measurement devices – e.g., photodetectors – are actually metastable. They are designed to be insensitive to small fluctuations in the measured quantity. Thus, thermal vacuum fluctuations do not trigger a photodetector; instead, it only cascades in the presence of approximately n = 1 photon. The most promising investigations of truly sensitive environments are probably in chaotic many-spin systems, particularly in the context of the Loschmidt echo [111]. Numerical studies [86, 34] indicate that chaotic systems do produce rapid (exponential) decoherence. In principle, 166 these predictions should be readily testable – chaotic spin baths are easy to implement, because multi-body systems tend to be chaotic. Unfortunately, the condensed matter systems that can be straightforwardly coupled to these baths tend to decohere very rapidly in any case. In a noisy system, identifying a particular environment’s contribution to decoherence is difficult. Furthermore, distinguishing different rates of decoherence in a finite-dimensional system (i.e., a single spin/qubit) is also hard. The fact that some quantum systems do maintain coherences for startlingly long times is a strong indicator that infinitely sensitive environments – like the inverted oscillator we considered in Chapter 6– do not actually exist. Experimental evidence would be a welcome assistance in resolving the conflict between this rather obvious conclusion and the preponderance of theoretical arguments for the sensitivity of chaotic environments. What is needed is: 1. Experiments which test the theory that strongly coupled chaotic environments produce exponentially rapid decoherence. 2. Exploration of weaker coupling regimes. At some level of coupling, the theory should break down, and decoherence should become negligible. To date, however, no experiments which combine the requisites of (1) precise control and isolation, and (2) a genuinely chaotic “environment,” have emerged. Possible candidates for a chaotic environment include billiards [95], and driven optical lattices [137]. Quantum chaotic systems have become controllable within the last 10 years, but achieving precise coupling between a quantum chaotic system and another system without interference from external decoherence is still a challenge. 7.5 Conclusions In this chapter, we have presented a survey of experimental tests for aspects of information storage. Just as this dissertation only scratches the surface of interesting theoretical questions about information storage in decoherence, this chapter considers only a few of the possible experiments. For instance, we have considered how to measure correlations between the initial and final states of S , and between the final states of S and E . What measurements of E can tell us about prior states of S is another interesting question, and one very relevant to quantum information processing. Using unitary control to alter how information is stored – for instance, to engineer particular information preserving structures – is another fascinating area. These and other projects remain to be explored. For now, we have illustrated that the theory discussed earlier is experimentally accessible. Conversely, these experiments provide a useful benchmark for improving practical quantum measurements. Quantum control technology has been driven by the challenge to implement resources – for instance, entanglement, quantum communication, teleportation, high-fidelity gates. Metrology has been driven by some of the same challenges, in the quest to measure single quantum systems accurately. The indirect measurement paradigm presented here, which takes advantage of what the environment “knows” about the system of interest, provide a novel way of leveraging control to improve measurements. The challenge of efficiently extracting the environment’s information – or preventing it from obtaining information in the first place – should drive experimental developments in both areas. 167 Chapter 8 Conclusions and further work Good scientific work has a soul – a distant but shining goal, often implied instead of stated, that underlies and motivates the work. The work presented here is motivated by two distinct (yet complementary) visions. Our first vision is to be able to say, ”Microscopic systems obey quantum laws; macroscopic things are classical; and this is why.” The ultimate goal is to (1) list the symptoms of quantumness and classicality; (2) explain how and why the transition occurs; and (3) identify where and when it occurs. In this sense, we are motivated by pure science – “Why is the world like this?” – rather than practical applications. The second underlying goal is to identify the loopholes in the “macroscopic things are classical” rule. Quantum information technology can only exist if the quantum/classical boundary is pushed far into the realm of “big” things. We are scouting out the territory, trying to identify a sort of Northwest Passage1 that will allow scalability for quantum technology. Decoherence is the engine for both of these programs. It was originally intended to explain quantum measurement and classicality, but in recent years its mitigation been identified [39] as one of the five most important challenges to quantum computing. This work in this thesis includes both enhancements of decoherence theory (Chapters 5 and 6), and extensions beyond decoherence (Chapters 2-4). This chapter, therefore, is not really the “conclusion” of the project. It is more of a checkpoint, where we catch our breath, ponder what has been accomplished, and prepare for the next steps forward. 8.1 What has been accomplished Redundancy The bulk of this work has sought to answer the question, ”Do physical, decoherenceproducing environments record information redundantly?” We began by establishing the infrastructure necessary to pose the question: • We showed that the environment (E ) must have a locality (i.e., tensor-product) structure. The preferred locality structure is fixed by its interaction with the central system (S ). • We analyzed the role of quantum mutual information (I ), in measuring how much information a fragment of E has about S . Only a fraction 1 (1 − δ ) of the total IS :E can be demanded of 2 each fragment. 1 For those not familiar with early American history, the fabled Northwest Passage was a putative stretch of sea, north of North America, which explorers hoped would allow ships to pass between the Atlantic and Pacific Oceans without going all the way down to Tierra del Fuego. It was never found, for it does not exist. We hope that the analogy is not that good. 168 • We constructed two tools – partial information plots, and quantitative redundancy – to be used in analyzing how information is stored in E . PIPs show, in an obvious fashion, whether the environment (1) stores information redundantly, (2) encodes information in entanglement between fragments, or (3) distributes information uniformly and nonredundantly. • We used the shape of characteristic PIPs to argue that information, in the presence of redundancy, is divided into three parts. 1. Nearly half the total information stored in E is redundant (easy to obtain from small fragments), therefore objective, and arguably classical. 2. The other (almost) half of IS :E is almost impossible to obtain, very fragile with respect to losing part of E , and appears to be a purely quantum effect. 3. Between these two regimes lies a small amount of nonredundant information, which is neither easy nor absurdly difficult to obtain. It is obtained gradually, in proportion to the fraction of E that has been captured. These concepts are essential for understanding how information is recorded. Chapters 3 and 4 (and part of Chapter 2) consist of using these tools to attack a range of decoherence models, then seeing what pops out. Underlying these investigations is the “Environment as a Witness” paradigm. It can only be applied to models and experiments where the environment is accessible – e.g., not abstract models of decoherence based on dynamical semigroups or Lindblad equations. Such physical models fall mostly into two categories: spin baths, and oscillator baths (Brownian motion). We have analyzed both of these canonical decoherence models. Our investigation of spin bath models begins in Chapter 2. We use a very simple model, for a DS -dimensional system decohered by Nenv DE -dimensional environments, to understand basic properties of redundancy. Dynamics are neglected entirely; instead, the system’s pointer states are correlated by fiat with (1) random states of the whole environment, or (2) random product states of the subenvironments. The major results are: • When states are chosen randomly from the universe’s Hilbert space, information about S is not stored redundantly. Instead, it is encoded into entangled modes of E . Capturing less than 1 1 2 of the environment yields virtually nothing, but if more than 2 is captured, all information (both quantum and classical) is available. • When pointer states of S are correlated with random product states of E , information is stored redundantly. We derive a fairly accurate (to within ∼ 10%) formula for Rδ (the redundancy of (1 − δ ) of the available information), with the following properties: 1. Redundancy scales linearly with the number of subenvironments. This leads us to define ∂Rδ and use specific redundancy : rδ ≡ ∂Nenv . 2. The information deficit, δ , has only a small effect on redundancy. As long as δ 1, it appears logarithmically. The redundancy of “almost all” the information is not particularly sensitive to the definition of “almost”. 3. The information capacity of S and the individual Ei govern the amount of redundancy. A system’s information capacity is the logarithm of its Hilbert space dimension. A fragment of E can provide sufficient information about S only if the fragment’s capacity is greater than the system’s capacity. Redundancy thus scales roughly as Nenv times the ratio cE /cS , with a logarithmic correction from δ . • In spin bath models with redundancy, non-redundant information is insignificant. Small fragments provide virtually all the available information about S , dividing IS :E perfectly into redundant and [locally] inaccessible portions. 169 Dynamical redundancy The preceding results stem from a model that generates states randomly instead of dynamically. We proceed to consider several dynamical models in Chapter 2. Our focus is on how the dynamics affect redundancy, not on the structure of the model. Accordingly, the system is always a single spin, and the subenvironment are also single spins. We neglect, for instance, models where the system consists of multiple coupled spins (see [86]). Results from this study include: • Simple “interaction only” models, where S interacts via Hint = Jz ⊗ Jz with each Ei , store information redundantly. Redundancy develops more slowly than decoherence: τR is set by the average response time of the Ei , whereas τD is set by the fastest response time. • The theory developed for the “random-state model” (Chapter 2) works well for the interactiononly model. However, the dynamical model displays less redundancy. • We introduce pliability – an environment’s tendency to be imprinted by the system – to account for the change in redundancy. We show that the dynamical model is less pliable than the random-state model, and predict the reduction in Rδ to within 1%. • Adding internal dynamics for the Ei produces a class of “quantum measurement” models. We show that the subenvironments’ internal dynamics can either dynamically decouple them from the system (reducing redundancy drastically), or make them superpliable (increasing redundancy). Which effect depends on the relative orientation of Henv and Hint : random orientation leads to superpliability; if they are orthogonal, dynamical decoupling ensues. • Introducing a weak system Hamiltonian (Hsys ) causes redundancy to decay away on a timescale ||Hsys ||−1 . Redundancy decays and vanishes because Hsys induces an indirect (3rd order in H) entangling interaction between the Ei . Information that was initially stored in local modes of E is shifted irreversibly into entangled modes. • The same effect ensues from adding another interaction Hamiltonian (e.g., Hint → Jz ⊗ Jz + gy Jy ⊗ Jy ) to form a “multiple measurement” model. Multiple measurements also induce entangling interactions between subenvironments. Redundancy decays more rapidly, and we conjecture that this is because entanglement occurs at 2nd order in H. We also examine the development of redundancy in quantum Brownian motion. In this model, the system is a harmonic oscillator, and it is coupled to a bath of harmonic oscillators. The QBM model produces friction, noise, and decoherence for the central system. We find that the QBM model does produce redundancy, but its properties are different from spin bath models: • Numerical simulations show that redundancy in QBM develops much later than decoherence. Its timescale is set by dissipation, which is nearly irrelevant to the initial measurement (i.e., decoherence-producing) process. • Dissipation not only sets the timescale for redundancy, but appears to be directly responsible for its emergence. Using a theoretical model for the decoherence process, we show that QBM environments “measure” the system in a way that leaves exactly 2 ln 2 bits of non-redundant information. This INR makes the amount of redundancy ambiguous (Rδ depends strongly on δ ), but is destroyed rapidly by dissipation. • Eventually, dissipation destroys almost all the information recorded in E . Unambiguous redundancy thrives in the intermediate times when INR has been destroyed, but IS :E remains. • Systems which do not resonate with some frequency band of E are unaffected by dissipation. As a result, free particles and oscillators whose frequency is greater than any found in E do not develop massive redundancy. 170 • By examining the amount of information stored in different frequency bands of the environment, we show that resonant modes hold the most information. We conjecture that the resonant modes propagate redundant information throughout the environment at the expense of nonredundant information, inducing both dissipation and “Quantum Darwinism.” Only the fittest information, selected by the environment, survives dissipation. Redundancy is only one element in our project to understand objectivity and classicality. Amplification, another route to objectivity, appears to be closely related to redundancy. The most general goal of all is to find the best of way of describing and analyzing how information about S is stored – be it in the environment, or in fragments of the environment, or in the system itself at a later time. Chapters 5 and 6 deal with these issues. The operator sieve Decoherence has been understood since its inception in terms of predictability. Predictability identifies properties of the system that remain unaffected by decoherence – that is, information which remains local to S even when S interacts with E . Redundancy, on the other hand, identifies properties of the system that end up shared between S and E . In Chapter 5, we introduced the operator sieve, an improved version of the predictability sieve. We also illustrated how the operator sieve can be used to compare the redundancy- and predictability-based perspectives on decoherence. The operator sieve is both an improved algorithm for determining pointer bases, and a new way of looking at information. The results presented in Chapter 5 include: • We introduce the density superoperator to capture properties of state ensembles that the density matrix misses. We show that the density superoperator is well-suited to algorithmically distinguishing between orthonormal bases of states. • An arbitrary quantum operation (e.g., a decoherence process) can be represented using a density superoperator. Perfect pointer bases can be unambiguously identified using this procedure. • The analysis in terms of density superoperators is expanded, to identify im perfect pointer bases. Degenerate pointer subspaces are discussed, and identified with noiseless or decoherencefree subspaces and subsystems. • We show that perfect pointer bases and decoherence-free substructures must obey a simple algebraic constraint. The pointer states must form a pointer algebra. The algebraic structures in a given Hilbert space represent the possible information-preserving structures. We identify all the information-preserving structures in 2- and 3-dimensional Hilbert spaces, and relate them to familiar concepts such as qubits and c-bits. • We apply the operator sieve to the dynamical spin bath models considered in Chapter 3. The results confirm that predictability is largely unaffected by the decay of redundancy. Amplification When information becomes redundant, it is available from many independent sources. These sources are typically microscopic – a single spin, or a narrow frequency band of a quantum field. Amplification is a complementary process which turns this microscopic information into macroscopic information, recorded in effectively classical observables. We begin to study amplification in Chapter 6 by completely analyzing a simple amplifying environment, using the traditional tools of decoherence. A central oscillator system is coupled to an unstable harmonic environment – that is, an “inverted oscillator.” The environment is very sensitive to the central system’s position; like a pencil balanced on its point, a tiny perturbation causes it to accelerate in one direction. We obtain the following results: 171 • A master equation for the central system, describing its evolution while being monitored by the unstable environment. Two properties of the master equation are new and interesting 1. The master equation coefficients periodically diverge to infinity. This occurs in all parameter regimes. Divergences represent the instantaneous loss of all information about a particular observable of S . At that instant, the master equation paradigm breaks down. 2. The coefficients of diffusive terms (which cause decoherence) increase exponentially in time. Brownian motion environments, in contrast, have bounded and approximately constant diffusive terms. • Numerical results and approximation formulae for the system’s entropy as a function of time. The results confirm that entropy increases linearly in time, without bound. • We conclude that the unstable environment shows all the symptoms of amplification that we expect. In addition, we show that a large amount of decoherence (measured by entropy gain) can ensue before the interaction with E is visible on a classical scale (measured by the system’s energy). 8.2 The next steps We intend to continue this project, understanding decoherence and classicality through the environment-as-a-witness paradigm, along three distinct avenues. First, there are a few outstanding questions about redundancy. Second, a great deal of work remains in understanding amplification – we have only begun to explore it. Third, we hope to develop a more comprehensive theory of how information about one system is stored in another. Chapter 5 is a first step in this direction; further development will furnish new ways of looking at redundancy. Unanswered questions about redundancy Our results indicate that redundant information storage is common, and that in physical models it is also transient. We would like to understand the mechanism by which dissipation destroys redundancy. In spin bath models, entanglement among the subenvironments is a key feature of redundancy decay. In order to understand the quantitative mechanism by which this intra-environment entanglement destroys redundancy, we need to first obtain some understanding of multiparty entanglement. This is currently a topic of intense interest, and few results, in the quantum information theory community. A complete understanding of multiparty entanglement is (in the near future) probably too much to ask. We suspect, however, that the measures of entanglement relevant to this problem may be more accessible. A rough program for investigating this is: 1. Find a good general way of determining the environment’s conditional states. Redundancy is destroyed, not necessarily by the environment’s states being entangled, but by their conditional (upon the system’s state) states being entangled. 2. The key feature of intra-environment entanglement that leads to nonredundancy is the impossibility of distinguishing two orthogonal entangled states by means of local measurements. We should determine whether any existing measure of multiparty entanglement captures this property. Much work on multiparty entanglement to date has dealt with special few-party cases [41]; states generated by very particular dynamics, such as phase transitions [109, 108]; special states such as cluster states, chains, or rings [8, 62, 30, 104]; or measures which are clearly not suitable for our purposes [131, 130]. The literature is vast, however, and may contain useful results. 172 3. Determine how initial product states evolve out of the branching-state ensemble. Ideally, the measure of entanglement which describes local indistinguishability of distinct states (see previous point) would evolve elegantly under the Hamiltonians we consider. This may well not be the case, in which case another measure of entanglement might be necessary in order to capture the evolution of distributed correlations. 4. Finally, a reasonable first step toward all of these goals is to extend our numerical simulation to include a model where the subenvironments interact with each other. This is probably a true property of the physical model described in Chapter 3 (an electron spin coupled to a bath of nuclear spins), although most decoherence models ignore Ei − Ej interactions because they do not directly affect the system. It would also be interesting to simulate larger systems, with multiple independent prop1 erties. Reference [86] suggests a possible model, where the central system consists of two spin- 2 particles. More complex systems can support decoherence-free subsystems and subspaces, for instance. Another promising project is to insert actual “detectors” into the QBM model of decoherence. The standard paradigm surrounds a central system with a homogenous quantum field. By sprinkling additional oscillators – essentially, extra systems – into this field, it is possible to simulate detection apparatus. By resonating with the field, the detectors obtain information (second hand) about S . This is a convenient and physically well-motivated way to improve upon the way that fragments of the environment are divided up among virtual observers. Finally, the work presented here has focused on the effects of varying the environment, and its interaction with the system. We have kept the system as fixed as possible in order to focus on different environments. In particular, we have not varied the system’s initial state very much in any of these investigations (except in confirming that more highly squeezed states produce greater redundancy in QBM; see Fig. 4.7). The measurability of different states remains to be investigated, particularly in QBM. In particular, non-Gaussian states such as “Schr¨dinger cats” o should be investigated. Amplification Whereas redundancy has been exhaustively examined, amplification deserves much further investigation from a microscopic perspective. Chapter 6 is a sort of reconnaissance of the field. The next step is to identify the tools and machinery needed to analyze various environments in a consistent manner. One approach is to use a generalized definition of subsystems (as developed in [12, 11]) to divide a single classical degree of freedom. This would facilitate an analysis in terms of partial information plots, with strong parallels to redundancy. Another line of investigation, which can be pursued in parallel with that outlined above, is to analyze more realistic environments. Chaotic systems, in particular, may produce amplification. There has been substantial debate on this question, however. Other amplifying systems include photomultiplier tubes, and the sort of apparatus that Schr¨dinger proposed for elimination of cats. We o hope to identify the characteristic property of amplifying systems, and to analyze simple examples. Understanding information storage Throughout this work, we have used quantum mutual information as a measure of correlation between systems. It is not clear whether QMI is the best measure of “What information does system A have about system B ?” An alternative measure was proposed and used by Ollivier et. al., in [106]. In Chapter 5, we discussed the difference between simultaneous information (e.g., what can ρA (t) tell us about ρB (t)?) and non-simultaneous information (e.g., what can ρS (t) tell us about ρS (0)?). A number of different measures of information are known (see, for instance, [48]). 173 By identifying which is the best measure for a particular purpose, we can gain different perspectives on how information is recorded in the environment. Chapter 5 indicates one avenue of investigation. The information-preserving structures that we introduced there are a catalog of how S can retain information about its initial state. These structures were derived using superoperators – both dynamical operations, and density superoperators that characterize ensembles. The same approach can be taken to analyze (1) how information about ρS (0) propagates to ρE (t), and (2) how measurements on ρE (t) are reflected on ρS (t). This investigation is still in its infancy, but shows great promise as a comprehensive theory of correlation between systems. We hypothesize that each information-preserving structure has a corresponding correlation structure. For instance, a quantum operation on a qubit may preserve (1) no information at all, (2) the Jz basis only, or (3) the entire quantum bit. An interaction process between a qubit system S and a qubit environment E might, correspondingly, produce (S ) (E ) (1) no correlation at all between S and E ; (2) [classical] correlation between Jz and Jz , but (S ) no correlation between other operators; or (3) [quantum] correlation between each Ji and the (E ) corresponding Ji , through entanglement. Further research will, we hope, facilitate the project of understanding how information flows into the environment. 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Decoherence, chaos, and the second law. Physical Review Letters, 72(16):2508 – 11, 1994. 184 Appendix A Supporting Material on Redundancy A.1 Details on quantifying redundancy If I is some information about a system S , then I is stored in the environment E with redundancy R if (and only if) E can be partitioned into at least R blocks – that is, E = E1 ⊗E2 ⊗ . . . ER – such that each Ei provides the information I . This begs several questions: 1. What is information about S , and how do we determine whether a given block Ei provides it? 2. How much of the information I must a particular block provide? 3. In what ways can the Hilbert space of E be partitioned? The first of these questions is clearly fundamental; the other two have simple answers which turn out to be overly naive. We’ll address them in order. Information, about some unknown quantity x, is essentially a resource which permits the reduction of uncertainty about x. An environment E has information about system S if, by measuring E , we can reduce the system’s entropy Hs. A good measure, then, of the information I that E has about S , is the amount by which HS decreases when E is measured. Of course, this depends on (a) what measurement is made on E , and (b) what outcome is obtained. We would need to average over possible outcomes, and either optimize or randomize over feasible measurements. The mathematical difficulty of doing so makes this appealing measure of information useless for analytical calculations, and impractical for numerical ones. We are thus forced to come up with some other measure of information. Our alternative measure is the Quantum Mutual Information (QMI), a generalization of the well-known mutual information from classical information theory. Defined as IS :E = HS + HE − HSE , the QMI is a measure of the correlations between system and environment. When IS :E = 0, the two systems are uncorrelated – that is, ρSE = ρS ⊗ ρE . When S and E are classically correlated, so that ρSE = i pi |si si | ⊗ |ei ei |; si | s j = ei | ej = δij , (A.1) the mutual information IS :E is precisely the reduction in HS that can be achieved by measuring E , since HS = HE = HSE = − pi log pi (A.2) i 185 and a measurement of E in the Schmidt basis must reduce all the entropies to zero. Some confusion may ensue, however, when the QMI is used for entangled states. Unlike the classical mutual information, the QMI may be greater than the total entropy of either the system or environment, as in the case of a pure entangled state where HS = HE , but HSE = 0, so that IS :E = 2HS . This property is useful for a measure of correlation, since it diagnoses the fact that entangled states possess in some sense more correlation than classically correlated states; however, it challenges us to interpret the statement “Bit S has 2 bits of information about bit E ,” or more generally, to make sense of IS :E > HS . We discuss these features in more detail as they arise in practice. The second question, “How much information do we demand from each block Ei of the environment?” seems to have a natural answer: we should demand that each block supply all the information required to define the property of interest. In this case, the property of interest is the state of the system itself, so each block should supply an amount of information equal to HS – that is, enough to determine the state of the system without ambiguity. If the state ρSE of the “universe” is mixed, the entire environment E may not provide enough information to accomplish this, in which case the sensible choice is to demand that each Ei provide exactly as much information (IS :E ) as the whole environment E . In practice, however, we find that in virtually every interesting case, the smallest sub-environment which provides all the information that E provides is E itself! The problem is that information is never perfectly redundant – the vast majority may be, but tiny details are hidden in the dusty corners of the environment. This is easily fixed; instead of demanding IS :Ei = IS :E , we allow an information deficit δ , and demand IS :Ei = (1 − δ )IS :E . We then speak of Rδ , so that R0.1 is the redundancy of 90% of the total information. An appealing extension of this concept comes from demanding that each Ei supply information about a portion of the system. A complicated system has many independent properties, some of which might be recorded more redundantly than others. For instance, consider the 2-qubit system A ⊗ B . If information IA about state of A is recorded with low (say, RA = 2) redundancy, but information IB about the state of B is recorded with high (RB = 1000) redundancy, then the redundancy of the full 2 bits of information available about S is limited by RA . By setting a lower threshold, and demanding that each Ei supply only IB , the massive redundancy of IB could be established. However, this presents a new problem; since QMI is a purely quantitative measure, we have no way of determining whether an environment that provides 1 bit of information is providing information about A, B , or some combination. What is feasible is to note that whereas we can only choose 2 blocks that provide IS :E ∼ 2 bits, we can subdivide into 1000 blocks which each provide IS :E ∼ 1 bit of information. Since the information is not contextual – that is, we don’t know what it’s about – we have to make a worst-case assumption in computing R. We assume that 500 blocks provide IA and 500 provide IB ; thus, no single bit of information is more than 500-fold redundant, and R0.5 = 500. Generalizing this worst-case assumption leads to the formula Rδ ≥ (1 − δ )Nδ , where Nδ is the number of blocks having at least (1 − δ )IS :E information. Thus, we can vary the fraction x, to see where redundancy peaks. Small values of 1 − δ will typically yield larger Nδ , since smaller chunks of the environment yield sufficient information, but the worst-case assumption will reduce Rδ for large δ . Unfortunately, there remains one caveat to resolve, thanks to the quirks of the quantum mutual information. As we mentioned while introducing the QMI, for entangled states, IS :E may be up to twice the total entropy HS of the system. Nonetheless, we demanded that each environment provide no more than HS information – potentially as little as 1 IS :E . As we prove in Corollary 1, 2 of Theorem 3, in Appendix A.2, the environment can never be subdivided into chunks which all provide more than HS information about the system. An excellent example of this is a (N +1)-party GHZ state, 1 1 |ψ S E = √ |0 S ⊗ |000...0 E + √ |1 S ⊗ |111...1 E . (A.3) 2 2 It’s easy to verify that (1) HS = 1 bit; (2) for any chunk of the environment E ⊆ E , HE = 1 bit; and (3) HSE = 1 bit unless E is the whole environment E , in which case HSE = 2 bits. Thus, 186 whereas the full environment E has IS :E = 2 bits of QMI with the system, any sub-environment E has only 1 bit of information. The extra bit of information possessed by E is purely quantum information – it represents, essentially, the ability to measure an arbitrary observable on S , instead of merely the pointer observable σz . All the classical information about the system (1 bit) is clearly obtainable from any single environment, so its redundancy is R = N . Were we to demand more than HS (1 bit) of information from each subsystem, we would find no redundancy at all; this illustrates an important principle related to the no-cloning theorem [153]: Only classical information can be redundantly recorded. Unfortunately, this introduces a wrinkle into our redundancy calculation. Supposing that E can be divided into two chunks EA and EB , each of which has HS = 1 IS :E mutual information with 2 the system. Then we have no way (within this framework) of distinguishing between a GHZ-type state (|ψ ∝ |0S 0A 0B + |1S 1A 1B ) with R = 2, and a state where each of EA and EB has 1 bit of quantum information about distinct properties xA and xB of S – thus no information is redundantly stored. Of course, if the system is a qubit then this cannot happen (qubits don’t have two distinct properties), but in general we have to consider it. We note, however, that if three environments each have HS information, then they cannot all be independent – at least two of the environments must hold mutually redundant information, while the third could hold independent information. Again, we make a worst-case assumption, assuming that if N environments have full information, one of them is independent of the others. Our final, quantitative lower-bound formula for redundancy is: Rδ ≥ (1 − δ )Nδ − 1, where there exist Nδ chunks Ei such that IS :Ei ≥ (1 − δ )IS :E . (A.4) A.1.1 The importance of tensor product structures Concerning how the environment can be broken into blocks, it’s tempting to allow any decomposition of the environment’s Hilbert space, as E = E1 ⊗ E2 ⊗ . . . EN . This procedure, however, trivializes the problem; perfect recording of any information becomes equivalent to that information being maximally redundant. To see this, we consider the following state of an (N +1)-qubit universe: |ψ SE 1 = √ |0 2 S |0 E1 + |1 S |1 E1 ⊗ |000...0 E{2...N } (A.5) Clearly, the information about the system (which can be obtained only from E1 ) is not redundantly stored. If, however, we permit the division of the environment in any basis, then this is equivalent to the freedom to perform arbitrary unitaries on the environment before partitioning it. One such unitary transforms the state given above into an (N + 1)-party GHZ state – which clearly has N fold redundancy. In fact, every non-product state has maximal redundancy by this definition, using some tensor-product-structure partition of the environment – but there also exists a fine-grained partitioning which (as in the state above) yields no redundancy. We are led inexorably to the conclusion that objectivity is entirely dependent upon a preexisting tensor product structure of the environment. This reflects the fact, long known in the decoherence community, that decoherence is meaningless unless we first define what is the system and what is the environment. We now see that in order to proceed, the granular structure of the environment must be defined as well. Once this is done – that is, once we have established what Hilbert spaces correspond to E1 , E2 , etc. – the individual sub-environments can be clumped together to form larger or smaller chunks. Fortunately, the subdivision of the environment is usually easy and physically motivated – for example, the radical difference in difficulty between measuring a single photon using a photodetector, and performing a Bell measurement on a pair of photons, indicates that the electromagnetic field should be treated in terms of photons, and not of collective modes. 187 A.2 Properties of QMI: the Symmetry Theorem The symmetry theorem for QMI is important for understanding the shape of partial information plots. It says, in essence, that the amount of information that can be gained from the first few environments to be captured, is mirrored by the amount of information that can be gained from the last few environments. Thus, when capturing a small fraction of E yields much information, an equivalent amount of information cannot be gained without capturing the last outstanding bits of E. Theorem 3 (Mutual Information Symmetry Theorem). Let the universe be in a pure state |ψ S E , and let the environment E be partitioned into two chunks EA and EB . Then the total mutual information between the system and its environment is equal to the sum of the mutual informations between S and EA and between S and EB : that is, IS :E = IS :EA + IS :EB . Proof. We simply expand each mutual information as Ix:y = Hx + Hy − Hxy , and use the fact that if a bipartite system x ⊗ y has a pure state |ψ xy , then the entropies of the parts are equal; Hx = Hy . I S : EA + I S : EB = = = = = HS + HA − HSA + HS + HB − HSB H S + HA − H B + H S + HB − HA H S + HS HS + HAB − 0 I S :E Corollary 1. Under no circumstances can two sub-environments both have I > HS information about the system. If the universe is in a pure state, then the Symmetry Theorem states that any bipartite division of the environment will yield two chunks, at least one of which has I ≤ HS . Additionally, we note that a chunk has at least as much I about the system as any of its sub-chunks (that is, decreasing the size of a chunk cannot increase its I ). If we could find two chunks A and B with I > HS , then by subsuming the remainder of E into A we would have a bipartite division into A and B , each of which has I > HS – but this contradicts the Symmetry Theorem. The proof for a mixed state of the universe follows from the “Church of the Larger Hilbert Space” argument. We purify ρSE by enlarging the environment from E to E , and follow the same steps to show that E cannot have two subenvironments with I > HS . Since E is a subset of E , it too cannot have two such subenvironments. Corollary 2. For a pure state |ψ S E of the universe, the partial information plot (PIP) must be antisymmetric around the point (m = N , I = HS ). 2 This follows straightforwardly from the Symmetry Theorem. For each chunk E{m} of the environment that contains m individual environments, there exists a complementary chunk E{N −m} , containing the complement of E{m} , with N − m individual environments. The Symmetry Theorem implies that IS :E{m} + IS :E{N −m} = IS :E = 2HS . By averaging this equation over all possible chunks E{m} , we obtain an equation for the PIP: I (m) + I (N − m) = 2HS . This equation is equivalent to the stated Corollary. A.3 Miscellaneous approximations for both ensembles In this section, we note some interesting and potentially useful properties of “typical” states from the uniform and branching-state ensembles. These properties of the averaged partial information plots characterize not particular states, but rather the ensemble as a whole. 188 A.3.1 Useful properties of the uniform ensemble Figures 2.5,2.6 seems to indicate that (a) the rate of information gain around m = 1 N 2 is nearly invariant, and (b) away from m = 1 N , I (m) converges exponentially to either IS :E or 0. 2 By applying assorted approximations, we can confirm both observations. To compute the rate of information gain at the halfway point, we take the limit as N → ∞ of I (m)/ m: I 2DS = log(DE ) m log 2(ds − 1) (A.6) The factor log(DE ) is just the information capacity of a single sub-environment, so in terms of information capacity (c) captured, we get: I ccap = 2DS log 2(ds − 1) (A.7) This approximation is extremely good (to within 2%) even for very small N . Another set of approximations, primarily using Ψ(x) log(x − 1) for large x, yields an approximate result for I (m) in the region 0 m N/2: 2 DS − 1 2 I (m) DEm−N (A.8) 2 log 2DS Thus, we can characterize the PIPs for uniform ensembles as follows: I (ccap ) is essentially 2 DS cE zero for ccap 2 , rises with the information capacity as log 2(ds−1) ccap near that halfway point, then converges exponentially to I 2 log2 (DS ). A.3.2 Useful properties of branching states While the approximation that we computed for branching-state PIPs in Section 2.3.3 is, in E the end, rather complex, we can abstract a useful property from it. As m grows, the dij{m} (additive decoherence factors) also grow, and rapidly damp the off-diagonal terms. When the off-diagonal terms are damped to zero (that is, all the di j factors are effectively infinite), the density matrix is fully decohered, and I = HS . This is the origin of the “classical plateau”; ρE{m} approaches ρdiagonal exponentially as m increases. Since the entropy is a function of ρ, it also approaches HS , approximately as e−dm . This explains the weak dependence of Rδ on δ ; even reducing δ by an order of magnitude requires that m increase only slightly, because of the exponential convergence to the classical plateau. A.3.3 Perfect states The primary intuition that we obtain from the I (m) plots is that most states are “encoding” states, but an important sub-ensemble of states are “redundant” states. We are naturally led to ask whether “perfect” examples of each type of state exist – that is, a state that encodes information more redundantly than any other state, or a state that hides the encoded information better than any other state. The answer is somewhat surprising: whereas perfectly redundant states exist for any N and any DS , DE , perfect coding states apparently exist only for certain N (at least for DS = DE = 2). The perfectly redundant states are easy to understand; they are the generalized GHZ (and GHZ-like) states of the form: |ΨSE = α |0 S |0 Ei + β |1 S |1 Ei , (A.9) i i with the obvious generalizations to higher DS , DE . Of course, it’s necessary that DE ≥ DS . 189 A true GHZ state is invariant under interchange of any two subsystems; however, since mutual information is invariant under local unitaries, we only require that the states |0 Ei and |1 Ei be orthogonal. Clearly, such states exist for all N . Any sub-environment with 0 < m < N has exactly H (S ) information, but only by capturing the entire environment (m = N ) can we obtain the full I = 2H (S ). Thus, the information is stored with N -fold redundancy. A perfect coding state, on the other hand, would be one where I (m) = 0 for any m < N/2, and I (m) = ISE for m > N/2. An equivalent condition, for qubit universes, is the existence of two orthogonal states of N qubits, each of which is maximally entangled under all possible bipartite divisions. If such pairs of states exist, then the system states |0 and |1 can be correlated with them to produce the perfect coding state. It is known (as detailed in [130]) that such states only exist for N = 2, 3, 5, 6, and possibly for N = 7 (for N = 6, only a single state exists[27]). Thus, while for large N almost every state is an excellent coding state, perfect examples seem not to exist except for N = 2, 3, 5, (7?)! We are not aware of any results for non-qubit systems. A.4 QMI for the uniform ensemble: Page’s mean entropy formula Page’s formula [110, 132, 46, 125] for the mean entropy H (m, n) of an m-dimensional subsystem of an mn-dimensional system (where m ≤ n) is mn H (m, n) = k=n+1 1 m−1 − k 2n m−1 , 2n (A.10) (A.11) = Ψ(mn) − Ψ(n + 1) − where the latter expression is given in terms of the digamma Ψ function. For a ds -dimensional system in contact with N environments of size de , the average mutual information between the system and m sub-environments is N N IS :E{m} = H (ds , dN ) + H (dm , ds de −m ) − H (ds dm , de −m ). e e e (A.12) A.5 Entropy of a near-diagonal density matrix Suppose that the pure state π = |ψ ˆ ψ |, i| ψ whose components in the pointer basis are = si , (A.13) is subjected to decoherence. The off-diagonal elements are reduced according to πi,j −→ σi,j = γi,j πi,j , where γi,i = 1 for all i. The limiting point of the process, where γi,j = 0 for all i = j , is ρ: ρi,j = δij |si |2 . (A.15) (A.14) As the γi,j approach zero, σ converges to ρ. The partially decohered σ can be written as σ = ρ + ∆, where ∆ is strictly off-diagonal. ∆ is defined by ∆i,j = (1 − δij ) γi,j si s∗ . j (A.17) (A.16) 190 As σ approaches ρ, its entropy approaches the entropy of ρ. Our goal here is to write H (σ ) as a power series (in ∆) around H (ρ). The entropy of σ is ˜ H (σ ) = −Tr(σ ln σ ) = Tr H (σ ) where ˜ H (σ ) ≡ −σ ln σ. The difference between H (σ ) and H (ρ) is ˜ ˜ ˜ δH = Tr(δ H ) = Tr H (ρ + ∆) − H (ρ) . (A.20) (A.19) (A.18) ˜ We will seek a power series for δ H . Keeping in mind that its trace is the relevant quantity, we will discard traceless terms. A na¨ approach to expanding H (ρ + ∆) ıve It’s tempting to begin by expanding Eq. A.19 around σ = ρ. Using the MacLaurin series for −σ ln σ gives ˜ H = −∆ (1l + ln ρ) − ≈ − ∆2 ∆3 + 2 .... 2ρ 6ρ ∆n+2 (−1)n (n + 1)(n + 2) ρn+1 n=0 ∞ (A.21) (A.22) We discarded the first term because it is traceless.1 The problem with this expression is that matrix quotients are not well-defined. Expressions like ∆ are ambiguous. Either ∆ρ−1 or ρ−1 ∆ ρ are possible. In fact, both are nonsymmetric and therefore incorrect. Further experimentation with 1 1 symmetric orderings such as ρ− 2 ∆ρ− 2 also give incorrect results. The expansion in Eq. A.22 is an inappropriate generalization of a scalar expansion, and is ill-defined. We will take a different approach which (a) gives the correct result, and (b) turns out to define the correct representation of matrix quotients. The correct approach ˜ ˜ ˜ Instead of expanding H (σ ) around σ = ρ, we expand both H (σ ) and H (ρ) around the identity. ˜ δH ˜ ˜ = H (ρ + ∆) − H (ρ) ˜ ˜ = H (1l − (1l − ρ − ∆)) − H (1l − (1l − ρ)). The expansion around 1l is always well-defined, because 1l and its inverse commute with everything: ˜ H (1l − x) = x − ˜ Using this expansion in δ H yields ∞ xn+2 . (n + 1)(n + 2) n=0 ∞ (A.23) ˜ δ H = −∆ + n=0 (1l − ρ)n+2 − (1l − ρ − ∆)n+2 . (n + 1)(n + 2) (A.24) 1 That is, Tr (∆f (ρ)) = 0 for any f (ρ) that is co-diagonal with ρ (e.g., 1l + ln ρ). In the pointer basis, f (ρ) is diagonal and ∆ is strictly off-diagonal. The two matrices are therefore orthogonal, and their trace must be zero. 191 We once again discard ∆ because it is traceless, leaving only the sum. The two matrix powers within the sum can be rewritten using the identity n (1l + x)n = j =0 nn x, j (A.25) which yields ∞ n+2 ˜ δH = − n=0 j =0 (−1)j n+2 j (ρ + ∆)j − ρj . (A.26) In order to simplify this, we must introduce a new notation. Consider (x + y )p , where x and y may be either scalars or noncommuting matrices. For scalars, p (x + y )p = k=0 p k p−k ab . k (A.27) p p When x and y are noncommuting matrices, k xk y p−k is replaced by a sum over all of the k possible k p−k orderings of k x’s and p − k y ’s. We define the notation x ◦ y to describe this convention. e.g., x2 ◦ y 2 if x, y are scalars, x2 ◦ y 2 x2 y 2 + xyxy + xy 2 x + yx2 y + yxyx + y 2 x 6 = x2 y 2 . = (A.28) (A.29) Using this definition of a totally symmetric product, j (ρ + ∆)j = k=0 j ∆k ◦ ρj −k , k (A.30) ˜ and the entropy difference operator δ H is ∞ n+2 j −1 ˜ δH = − n=0 j =0 k=0 n+2 (−1)j (n + 1)(n + 2) j j ∆k+1 ◦ ρj −k−1 k+1 (A.31) Equation A.31 can be put into a more convenient form by rearranging the order of summation, ∞ ∞ n ˜ δH =− k=1 n=0 j =0 ∞ n+1 (−1)j +k+1 n+k+1 (n + k )(n + k + 1) j + k + 1 j+k+1 ∆k+1 ◦ ρj k+1 (A.32) − n=0 j =0 (−1)j n+2 (j + 1) ∆ ◦ ρj (n + 1)(n + 2) j+1 (A.33) The second line (i.e., the k = 0 term) can be discarded because Tr(∆ ◦ ρj ) = Tr(∆ρj ) = 0. We then perform the sum over j to obtain ∞ ˜ δH = − k=1 (−1)k n+k+1 ∆k+1 ◦ (1l − ρ)n . k+1 (n + k )(n + k + 1) n=0 ∞ (A.34) Expanding the binomial coefficients and simplifying leads to the following result: ∞ ˜ δH = k=1 (−1)k k+n−1 ∆k+1 ◦ (1l − ρ)n . k (k + 1) n n=0 ∞ (A.35) 192 We have come full circle. The sum over n in Eq. A.35 is just the MacLaurin expansion for ρ−k around ρ = 1l. Equation A.35 can thus be written symbolically as ∞ ˜ δH = k=1 (−1)k ∆k+1 ◦ ρ−k , k (k + 1) (A.36) if the symmetric product ∆k+1 ◦ ρ−k is interpreted as “take the symmetric product of ∆k+1 with the power series representing ρ−k .” Essentially, what we have derived is the “correct” interpretation of the matrix quotient ∆k+1 . This result is interesting in its own right, but for now we are interested only in the leading ρk order (i.e., ∆2 ) term. Truncating the series at k = 1, we obtain the following simple result: ∞ δH ≈ −hf n=0 Tr ∆2 ◦ (1l − ρ)n + O ∆3 . (A.37) This is the simplest possible general form for δH . In order to perform the traces, we need to take advantage of the form of the symmetric product. From the definition of the symmetric product, we can write out explicit expressions for ∆k ◦ M n , for particular small values of k . ∆ ◦ Mn = 1 M p ∆M n−p n + 1 p=0 2 M q ∆M p ∆M n−p−q (n + 1)(n + 2) p=0 q=0 n n−p n (A.38) ∆2 ◦ M n = (A.39) The second case (for ∆2 ) is the useful one. We need the trace of the symmetric product, which can be simplified using the cyclic property of trace, Tr ∆2 ◦ M n = 1 Tr ∆M p ∆M n−p . n + 1 p=0 n (A.40) Together with Eq. A.37, this formula yields an explicit expression for δH : δH ≈ − 1 1 Tr ∆(1l − ρ)p ∆(1l − ρ)n−p 2 n=0 n + 1 p=0 ∞ n (A.41) We now insert specific forms for ρ and ∆, from Eqs. A.15 and A.17: DS −1 Tr ∆M p ∆M n−p = i,j,k,l=0 DS −1 ∆ij (1l − ρ)p ∆kl (1 − ρ)n−p jk li si s∗ sk s∗ γij γkl δjk δil 1 − |sj |2 j l i,j,k,l=0 p (A.42) n−p = = i,j =i 1 − |si |2 |γij |2 . (A.43) (A.44) |si |2 1 − |si |2 n−p |sj |2 1 − |sj |2 p Since the goal is to average over an ensemble of states, we replace |γij |2 with an average, |γ |2 , Tr ∆M p ∆M n−p = |γ |2 i |si |2 1 − |si |2 k p j n |sj |2 1 − |sj |2 n−p − |sk |4 1 − |sk |2 = |γ |2 Tr [ρ(1l − ρ)p ] Tr ρ(1l − ρ)n−p − Tr ρ2 (1l − ρ)n (A.45) 193 Inserting this expression into Eq. A.41 yields δH ≈ − |γ |2 2 1 Tr [ρ(1l − ρ)p ] Tr ρ(1l − ρ)n−p − Tr ρ2 (1l − ρ)n n + 1 p=0 n=0 ∞ n . (A.46) Finally, we can simplify this expression slightly by (1) taking advantage of the identity ∞ n=0 (1l − ρ)n = ρ−1 , and (2) rearranging the summation variables. |γ |2 − 2 |γ |2 2 |γ |2 2 Tr [ρ(1l − ρ)p ] Tr [ρ(1l − ρ)n−p ] − Tr ρ2 (1l − ρ)n n+1 n=0 p=0 n=0 Tr [ρ(1l − ρ)p ] Tr [ρ(1l − ρ)n ] − Tr ρ2 (1l − ρ)n n+p+1 n=0 p=0 n=0 Tr [ρ(1l − ρ)p ] Tr [ρ(1l − ρ)n ] −1 n+p+1 n=0 p=0 ∞ ∞ ∞ ∞ ∞ ∞ n ∞ δH ≈ (A.47) =− (A.48) =− (A.49) Equation A.49 is the simplest form we have been able to achieve, except in very special cases, for H (ρ + ∆) − H (ρ). A.6 Probability distributions for additive decoherence factors If |ψn and |ψn are selected at random from the uniform ensemble of n-dimensional quantum states, then the probability that | ψn |ψn | = x (for x ∈ [0 . . . 1]) is p(x) = 2(n − 1)x(1 − x2 )n−2 (A.50) The additive decoherence factor d is given by d = − log(x), so that x = e−d and d ∈ [0 . . . ∞]. The probability distribution transforms as p(d)dd = p(x)dx dx p(d) = p(x) dd = e−d p(x) = 2(n − 1)e−2d 1 − e−2d n−2 (A.51) The decoherence factor for a collection of subenvironments is simply the sum of d(i) over the contributing subenvironments. Ideally, we could obtain exact distributions pm (d) for a sum of m such d-factors. For qubits (n = 2), p(d) is a 1st-order Poisson distribution, so pm (d) is just the mth order Poisson distribution (for details, see [20]). For n = 2, however, no such simple description exists (see Figure A.1a). However, the distribution functions p(d) are well-approximated by Gaussian distributions, so we can treat the summing problem as a random walk. We compute the mean value d and variance ∆d = d2 − d for the single-system distribution, and extrapolate to multiple √ = m∆ d . 2 systems via dm = md and ∆dm 194 2 mean decoherence factor p(d) (probability) 1.5 1 D=2 D=3 D=4 D=5 D=6 D=8 D = 10 D = 12 D = 16 3 2.5 2 1.5 1 0.5 0 0 5 dD ∆dD D/4 Log(D)/2 + 0.29 10 15 20 25 30 35 40 45 50 D (Hilbert space dimension) 0.5 0 0 0.5 1 1.5 2 d = -log(|〈ψψ′〉|) 2.5 3 (a) (b) ¯ Figure A.1: Useful information for computing and visualizing d and ∆d. Plot (a) shows the probability distributions for dD , the additive decoherence factor for an D-dimensional state. For D > 2, p(d) is well approximated by a Gaussian. For D = 2, however, d is Poisson-distributed, which makes the Gaussian approximation rather poor. Plot (b) shows the mean value d¯ and the variance ∆dD D ¯ for values of D up to 50. As D becomes large, d approaches log(D)/2 + 0.29, while ∆d asymptotes ¯ to about 0.64. For small values of D, d appears to scale linearly as 1 D. By D = 6, however, the 4 linear behavior has given way to logarithmic growth. The mean d is given by: d = an expansion in binomial coefficients: d = = 2(n − 1) 0 ∞ n−2 ∞ 0 ∞ dp(d)dd. The integral is somewhat nontrivial, involving n−2 de−2d 1 − e−2d de−2d 0 n−2 k=0 dd 2(n − 1) n−2 k ∞ −e−2kd dd d −e−2(k+1)d dd = 2(n − 1) k=0 (−1)k n−2 k 0 = n−1 2 n−2 n−2 k=0 (−1)k (n − 2)! (k + 1)2 k !(n − 2 − k )! = k=0 1 2(k + 1) (A.52) = 1 (Ψ(n) + γEM ) 2 where Ψ(n) is the digamma function, and γEM = 0.5772 . . . is the Euler-Mascheroni constant. A virtually identical calculation for d2 yields n−2 ∆d 2 = k=0 2 1 2(k + 1)2 (A.53) = π Ψ1 (n) − 24 4 in terms of the trigamma function Ψ1 (n). The first few values of d and ∆d are given in Table 2.1. As n becomes large, d → 1 (log n + γEM ), and ∆d → √π . Figure A.1b illustrates this behavior, as 2 24 195 well as the fact that for small n, d appears to increase linearly as 1 n, until the logarithmic behavior 4 becomes apparent around n = 6. 196 Appendix B Supporting Material on Spin Bath Dynamics B.1 Overview of simulation techniques Our most important tool for investigating the dynamics of redundancy is numerical simulation. The data presented here were generated by evolving the joint state of a spin- 1 system and 2 an Nenv -system bath of spin- 1 environments. From the resulting time-dependent states, we extract 2 the entropy of the system, and compute redundancy for several values of the information deficit δ 1 1 (specifically, δ ∈ 1 , 1 , 4 , 1 , 10 ). Computing R requires a Monte Carlo sampling of all the possible 23 5 permutations of the subenvironments, to identify the average fragment size required for sufficient information. We find that 16 randomly selected permutations give a reasonably good value of m, but when possible we employ as many as 2048 to reduce the noise level. For certain variants of the model, computing the mutual information between the system and a subenvironment is the most computationally expensive step, so in these cases we use a relatively small number (16 to 32) of permutations. For each simulation run, we select particular values of the interaction coefficients kn , and particular initial states for the environments. These parameters both affect the results strongly, so we average together the results of many runs. Thus, while each dynamical simulation typically displays strong time dependence in the results, the average tends to wipe out time dependent features which depend on the details of the interaction coefficients for a particular run; the remaining time dependence is particular to the class of models being simulated, and rather interesting. The initial state of the system obviously has a great deal of effect on the results as well. We consistently set the initial state of the system to |+ , which maximizes the amount of information to be gained by the environment. From a computational standpoint, the various models outlined in section 3.1.1 fall into two classes: those which can be simulated efficiently, and those which require an expensive bruteforce approach. The interaction-only and quantum-measurement models can be efficiently simulated, precisely because of the structure of branching states (detailed in Chapter 2). We know ahead of time what the pointer basis states |i will be, so we can simply track its coefficient si and the environment state |Ei correlated with it. The model does not permit the environments to become entangled with each other, so |Ei is a product state; evolving the state of the universe comes down to keeping track ( of the states |Ei n) . The task scales linearly with Nenv , and so for the quantum measurement models we simulate the decohering effect of up to 128 environments without substantial trouble. When the model contains system dynamics or multiple measurements, the evolving state is no longer singly-branching; we cannot use the efficient algorithm detailed above to keep track of evolution. An analytic solution exists for the case of system dynamics, based on work by Melikidze 197 et al[92], but it can only be used to compute the system density matrix efficiently. Our study of redundancy requires that we compute the state of the entire universe in order to determine what the environments know about the system, so the analytic model is unfortunately not useful for us. We use a Chebyshev polynomial method, first implemented by Dobrovitski et al[40, 34, 86], to evolve the state of the universe in time. The Chebyshev method is probably not optimal for short time steps, but it has the great advantage of being a very general algorithm. The simulations presented here are based on a black-box implementation of the Chebyshev polynomial algorithm, which does not depend at all on the structure of the Hamiltonian. As such, the simulation code is extremely flexible; almost any imaginable Hamiltonian can be inserted into the code. We do take advantage of the sparseness of the Hamiltonian, but this is not necessary. In principle, environments of up to 20 spins can practically be simulated. Given the huge number of Monte Carlo trials we need to perform to get smooth data, however, the time required to evolve a universe with 106 or more states is prohibitive. The majority of our simulations implement 12 spins in the environment – sufficient to see the effects of redundancy. Future research will be less exploratory in nature, and optimization should make focused examination of 16-20 spins in the environment possible. 198 Initial rise in R10% (Interaction-Only model) 30 25 20 R10% 15 10 5 1 0.2 0.4 0.6 0.8 Time (arbitrary units) Nenv = 6 Nenv = 12 Nenv = 24 1 1.2 R10% 10 100 Initial rise in R10% (Interaction-Only model) 0.2 0.4 0.6 0.8 Time (arbitrary units) 1 1.2 Nenv = 48 Nenv = 80 Nenv = 128 Nenv = 6 Nenv = 12 Nenv = 24 Nenv = 48 Nenv = 80 Nenv = 128 (a) (b) Figure B.1: Here, we examine the initial rise in redundancy for interaction-only environments. Plots (a) and (b) plot R(t) (on linear and logarithmic scales, respectively), comparing large (Nenv = 128) and small (Nenv = 6) environments. Asymptotic redundancy Ravg (obtained from the simulations) is plotted as thin straight lines. In each case, R(t) rises to Ravg by a time τR ≈ 0.6 . . . 0.9. The redundancy timescale is determined by the average interaction strength ( kn ), not Nenv . Initial rise in HS (Interaction-Only model) 1 0.8 HS (bits) 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 Time (arbitrary units) Nenv = 6 Nenv = 12 Nenv = 24 Nenv = 48 Nenv = 80 Nenv = 128 1 1.2 Figure B.2: Unlike redundancy (Fig. B.1), HS reaches its asymptotic value at a rate that decreases 2 as Nenv . Larger environments cause faster decoherence. A small amount of redundancy can saturate HS , so a large environment produces strong decoherence long before maximum redundancy. −1 B.2 Timescales 199 B.3 Pliability If each subenvironment records information about the system independently, then the redundancy of the recorded information depends only upon (1) the total number Nenv of subenvironments, and (2) the average number m of subenvironments required to secure “sufficient” information – that is, all but a small deficit δ of the total information available. We refer to a collection of m subenvironments, E{m} , as a fragment of size m, so m is the mean fragment size for a particular state. The amount of information provided by a subenvironment is equivalent to its ability to reduce the appropriate off-diagonal coherence in ρS . A single subenvironment E (n) , evolving into ( ( either |Ei n) or Ej n) conditional on the system being in (respectively) |i or |j , contributes a mul( n) ( ( tiplicative decoherence factor γij = Ej n) |Ei n) to the coherence term between |i and |j of the system. A more convenient representation of an environment’s decohering power is in terms of the ( n) ( n) additive decoherence factor, dij = − log γij . When several subenvironments (e.g., E1 . . . Em ) are combined into a fragment, their d-factors add together: m d({m}) ≡ d(1...m) = n=1 d ( n) (B.1) This additivity permits us to model the process of concatenating randomly chosen subenvironments into a fragment as a biased Gaussian random walk. After m steps, d({m}) is (approximately) a Gaus√ sian random number with mean value d({m}) = md and variance ∆d({m}) = m∆d. Here, d and ∆d are the mean and variance (respectively) of the distribution of single-subenvironment d-factors. This random-walk model was used in Chapter 2 to derive an expression for the mean fragment size, in terms of DS (the Hilbert-space dimension of the system), δ (the permitted information deficit), and HS (the entropy of the system): m= log(DS − 1) − log (2δHS ) ∆d2 1 + 2 + 2. 2d 2d (B.2) The physics of the decoherence model is contained entirely in the distribution of d-factors for the subenvironments – which, in the Gaussian approximation, is described by d and ∆d. The model is valid even for particular states, where d can take on Nenv well-defined values. Its real power, however, is the ability to predict typical redundancies when the actual conditional states of the environment are unknown, by averaging over an entire ensemble of them. The ensemble averages d and ∆d for a dynamically-produced ensemble of conditional states characterize the pliability of that environment. The dependence of m on d and ∆d is, unfortunately, not simple enough to reduce to a single parameter – in particular, the deficit δ appears in a nontrivial way. However, only the first term in Eq. B.2 depends on anything other than the pliability of the environment (d and ∆d). When it dominates, the mean fragment size is inversely proportional to d, so d essentially characterizes the pliability of the environment. The other terms become significant only when ∆d d, or log(DS − 1) − log (2δHS ) d. Although the additive form of the decoherence factor is most useful for computing mean fragment size (and therefore the amount of redundancy), its multiplicative form γ is also useful. In particular, |γ |2 determines the expected amount of decoherence from a single environment, so m |γ |2 determines the expected decoherence from m environments. Expressing this in terms of d, this means that for computing average residual coherences – that is, the expected magnitude of off-diagonal terms after ρS has been decohered by the environment – the significant quantity is 1 d = 2 log(e−2d ). Qualitatively, the difference between d and d is that d tends to be dominated 200 by ineffective environments, with small d-factors, whereas d is dominated by effective environments. The relationship between average residual coherences and redundancy is nontrivial, and we will consider it later. B.3.1 Calculating pliability for the interaction-only model The simplest model of decoherence is the interaction-only model of Eq. 3.6, which generates quantum measurement dynamically: DS H= i=1 |i i| iS ⊗ Hi , (E ) If S is in the state |i , the environment evolves according to HE . We can assume that the the i conditional Hamiltonians are local, DS −1 Nenv H= i=0 |i i|S ⊗ n=1 H(n) i, (B.3) and that the initial state is a product state of the system S and each subenvironment En : DS |ψ0 = i=1 si |i S ⊗ |E (1) (t=0) ⊗ |E (2) (t=0) ⊗ . . . |E (Nenv ) (t=0) (B.4) After a time t, this initial state evolves into: DS |ψt = i=1 (si |i S (1) (2) ( ⊗ |Ei (t) ⊗ |Ei (t) ⊗ . . . |Ei Nenv ) (t) ) (B.5) where the conditional states of each subenvironment are ( |Ei n) (t) = exp −iHi t |E (n) (t=0) , ( n) (B.6) and thus the decoherence factors can be computed as γij ( n) = = Ej (n) (t)|Ei (n) (t) E (n) (t=0)| exp iHj t exp −iHi t |E (n) (t=0) . ( n) ( n) (B.7) The Hamiltonian in Eq. B.3 is very general, placing no restrictions on how different states of the system affect the environments’ dynamics. Most physical interaction Hamiltonians, including the one we simulate here (Eq. 3.7), are of the form H = MS ⊗ RE + 1lS ⊗ VE = i (B.8) R ( n) + 1lS ⊗ n µi |i i| i ⊗ n V ( n) (B.9) The first term, M ⊗ R, determines all the conditional evolution of the environments. M, the measurement operator, picks out the pointer basis of the system and the strength µi with which each pointer state |i influences the environments. R(n) , the recording operator, determines how that information is written into the nth individual subenvironment, in conjunction with the subenvironment’s internal Hamiltonian V(n) . The decoherence factors are: γij = ( n) i(V(n) +µj R(n) )t −i(V(n) +µi R(n) )t (n) E (n) (t=0)| e e |E (t=0) . (B.10) 201 When the recording operator R(n) commutes with that subenvironment’s self-Hamiltonian V ,V factors out of Eq. B.10. We refer to this as an “interaction-only” model, despite the presence of environment dynamics, because the environment dynamics doesn’t affect decoherence in any detectable way. By contrast, the addition of a V term which does not commute with R can cause substantial changes. For an interaction-only model, the decoherence factors for E (n) depend only on R(n) and the environment’s initial state: ( n) ( n) γij ( n) = = i(µj −µi )R(n) t (n) E (n) (t=0)| e |E (t=0) DE (B.11) (B.12) |ψk |2 ei(µj −µi )ωk t k=1 where ωk is the k th eigenvalue of R, |rk is the corresponding eigenstate, and ψk = rk |E (n) (t=0) . The environment’s power to produce redundancy is best characterized by d and ∆d, whereas total decohering power is related to |γ |2 . Since the latter is much easier to calculate, we will consider it first. The initial state of the environment is clearly an important factor in determining pliability; a state which is unbiased with respect to R will maximize pliability, while an eigenstate of R will have zero pliability to the system. We assume that the initial states of the En are distributed randomly and uniformly throughout the environment’s Hilbert space. We can average over the initial state using a formula from [116, 120]: ˆ ψ | A |ψ ˆ ψ | B |ψ ˆˆ ˆˆ Tr AB + TrATrB = D(D + 1) (B.13) ˆˆ ˆ ˆ We substitute A = U ≡ exp i (µj − µi ) R(n) t and B = U † to obtain |γij |2 (t) = DE + |TrU | DE (DE + 1) 2 + DE + 1 2 (B.14) cos [(µj − µi ) (ωk − ωl ) t] DE = k=l DE (DE + 1) (B.15) (B.16) At t =, the environment has had no chance to evolve into distinct conditional states; there is no decoherence and |γ |2 = 1. With increasing t, however, the time-dependent term declines rapidly due to destructive interference of the various frequencies involved. It drops to zero on a timescale of tdecoherence ∼ |µj − µi | Tr(R2 ) − (TrR)2 , (B.17) which is obtained simply by recognizing that the frequency differences ωk − ωl are distributed symmetrically about their mean, and thus that the cosine terms in Eq. B.14 will cancel out when the RMS value of their arguments reaches π . If we average |γ |2 over t, the time-dependent terms 2 vanish entirely, and we are left with 2 |γ |2 = . (B.18) DE + 1 Equation B.18 predicts the total decohering effect on the central system. It also provides a baseline for redundancy analysis. Even though redundancy depends on the additive decoherence factor rather than γ , we expect that qualitative conclusions based on |γ |2 will still hold for redundancy in most cases. In particular, we notice that the time-averaged value of |γ |2 in Eq. B.18 is consistently larger (indicating less decoherence) than our previous result for randomly chosen conditional states. 202 When the conditional states are chosen randomly (as opposed to being dynamically generated), we − found that |γ |2 = DE 1 . This leads us to expect that the dynamical pliability of a given environment will generally be less than the value predicted by averaging over random conditional states. Since d is a transcendental function of γ , the sort of analytic averaging we did above is simply impossible for d and ∆d. For certain simple examples, analytic integration is possible. This is, however, more in the nature of a parlor trick, since few cases are integrable. We turn to numerical integration to examine the pliability of interaction-only models in more detail. The dependence of Eq. B.12 on the eigenvalues ωk of R is intrinsic, and so we have to pick a particular form for R. The dependence is actually on the distribution of frequency differences, ωk − ωl , since the decoherence factors can be written in terms of |γ |2 : γij ( n) 2 DE = k,l=1 |ψk |2 |ψl |2 cos ((µj − µi ) (ωk − ωl ) t) (B.19) Spin Hamiltonians of the form H = n · J are used in the numerical simulations we present here. ˆ Like harmonic oscillators and bosonic fields, spin Hamiltonians have linearly-spaced spectra, where the energies (or frequencies) are given by ωk = kω + ω0 . For now, we specialize to linear-spectrum operators for R and V, although later we’ll consider how these results may extend to arbitrary Hamiltonians. For a linear spectrum, the distribution of frequency differences is simple to calculate. The frequency differences are integer multiples of ω from −(DE − 1)ω . . . (DE − 1)ω , with nω appearing 2(DE − n) times. This spectral density forms a discretized triangular step (see Figure B.3), which becomes continuous in the limit DE → ∞. Since |γ |2 from Eq. B.19 is just its Fourier transform, we can do the transform in the continuous limit and obtain an excellent approximation to Eq. B.14: |γ |2 (t) DE 1 + DE + 1 DE + 1 sin DE 2 ωt DE 2 ωt 2 . (B.20) The actual spectral density is discrete, not continuous, so |γ |2 experiences a strong recurrence at π t = 2π , as shown in Fig. B.4. For times less than ω , Eq. B.20 describes the average decohering ω power of the environment well. The main lesson to be gleaned from Fig. B.4, however, is that for moderately large environments driven by recording Hamiltonians with linear spectra, the expected − decohering power agrees closely with the random-state average |γ |2 = DE 1 most of the time, but short and dramatic recurrences at regular intervals drive the time-averaged decohering power to |γ |2 | = 2/(DE + 1). The behavior of d and ∆d, which we plot in Fig. B.5, adheres fairly closely to what we expect from the time-dependence of |γ |2 (Fig. B.4). The additive decoherence factor, like the multiplicative one, tracks the random-state average closely for most of a full period, with the expected recurrence at t = 2π/ω causing a sharp and dramatic drop. In other words, for most of a full period, the environment provides a consistent amount of decohering power – but for short periods of time in each cycle, it loses essentially all its decohering power due to recurrences. This is a natural consequence of the environment’s linear spectrum. If we had chosen a different spectrum for R, with incommensurate energies, then the recurrence would be broken up and mixed throughout the period. However, when we consider environments made of many such subenvironments, the subenvironments will in general be out of phase with one another, and the specific time-dependence of the decohering power is thus less important than its time average. Figure B.6 shows the time-averaged values of d and ∆d as a function of the subenvironment’s size DE , with the calculated random-state average for reference. Dynamical averages of d are consistently slightly below the random-state theory, but converge towards the latter as DE becomes large. For the special case DE = 2, we can actually compute the dynamical average analytically (through one of the “parlor-trick” calculations alluded 203 18 16 14 12 Multiplicity 10 8 6 4 2 0 -8 -6 -4 -2 0 2 4 Frequency Difference (∆ω) 6 8 Figure B.3: This figure illustrates the distribution of frequency differences |ωi − ωj | for an single 7 subenvironment with a linear spectrum ωk = omega0 + kω . The specific system shown is a spin- 2 ˆ system with H = Z , so it has 8 frequencies and 64 frequency differences. The triangular-step distribution is ubiquitous among linear-spectrum systems, however. For large DE , the actual distribution approaches the continuous distribution outlined by the dashed line. Modeling a finite system such 7 as this spin- 2 system by the continuous one gives a good short-time approximation to its decohering power. to previously) as d = 1 − log 2 0.306, which agrees with the numerical results. By contrast, the random-state average is d = 0.5; thus, for low-dimensional environments, the difference in decohering power between dynamical averaging and random-state averaging can be as much as 40%. These results provide a solid baseline for estimating the decohering power of an environment whose recording operator R commutes with its self-Hamiltonian V. Of course, we have not provided an exhaustive survey of every possible form of R – a task beyond the scope of this work. However, applying these techniques to a specific environment not considered here should be fairly simple. Our main purpose in this section is to lay a foundation for examining redundancy in large environments, and the environments we consider here are sufficient for that purpose. Our analysis cannot be complete, however, without considering the far more general case where R does not commute with V. B.3.2 The effect of an environment’s dynamics on its pliability When the environment’s self-Hamiltonian V(n) plays a nontrivial role in the process of recording information about the system, some surprising effects can occur. In this section, we 204 Numerical |γ|2(t): interaction-only 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 (a) |γ|2 (decoherence factor) Dε = 2 Dε = 3 Dε = 4 Dε = 6 Dε = 10 Dε = 20 Dε = 128 1 2 3 4 Time (arbitrary units) Theoretical |γ|2(t) 5 6 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 Time (arbitrary units) 5 6 (b) Figure B.4: These plots show the time-dependence of |γ |2 for spin-j environments (where j ranges from 1 to 127 ) in the interaction-only model. The frequency spacing of the interaction Hamiltonian 2 2 is ∆ω = 1, so that a full recurrence occurs at t = 2π . Plot (a) shows numerical results, while plot (b) shows the simple theoretical model obtained from Eq. B.20. The simple model reproduces the actual behavior of |γ |2 (t) quite well, until the recurrence time. Plot (a) illustrates that although the environment’s decoherence-producing power is usually right at the random-state average of 1 |γ |2 = DE , the dramatic recurrences at t = 2πn are responsible for raising the time-average of |γ |2 2 to DE +1 . consider a “fairly general” model of such environments, where V(n) , R(n) = 0. A fully general model of these dynamics is, again, beyond our scope – that would be equivalent to a complete and general theory of the Loschmidt echo, which is a thriving subfield of quantum dynamics research. The majority of work on the Loschmidt echo (and its synonym, fidelity decay) is concerned with the spectral properties of V, and the effect that they have on |γ |2 . We avoid this complication by simply extending the work of the previous section, retaining the focus on linear Hamiltonians (e.g., spin-j particles and harmonic oscillators). |γ|2 (decoherence factor) Dε = 2 Dε = 3 Dε = 4 Dε = 6 Dε = 10 Dε = 20 Dε = 128 205 d(t): Interaction-only 3.5 d (decoherence factor) 3 2.5 2 1.5 1 0.5 0 0 1 2 3 4 Time (arbitrary units) ∆d(t): Interaction-only 1.2 1 0.8 ∆d 0.6 0.4 0.2 0 0 1 2 3 4 Time (arbitrary units) 5 6 Dε = 2 Dε = 3 Dε = 4 Dε = 6 Dε = 10 Dε = 20 Dε = 128 random states 5 6 Dε = 2 Dε = 3 Dε = 4 Dε = 6 Dε = 10 Dε = 20 Dε = 128 random states (a) (b) Figure B.5: In these plots, we depict the time dependent behavior of d and ∆d, which characterize the additive decoherence factor, and therefore redundancy. Plot (a) illustrates d(t), while plot (b) shows ∆d. For sufficiently large environments, both coefficients hover around their random-state averages for most of a full period, before dropping to zero at the recurrence time. The spike around t = π is of some interest; it illustrates the point made earlier that d is more influenced by those cases where the environment is highly effective at decohering the system. When t = π , the possibility of obtaining γ = 0 is nonzero, and this influences d much more than it influences |γ |2 . The decoherence factors are derived from Eq. B.10: γij = ( n) i(V(n) +µj R(n) )t −i(V(n) +µi R(n) )t (n) |E (t=0) E (n) (t=0)| e e . When R(n) fails to commute with V(n) , we can separate R into two parts, one of which does commute with V and the other of which is maximally noncommuting. Written in the eigenbasis of V, these are (respectively) diagonal and off-diagonal operators, so we let R = Rdiag + Roffdiag . ˆ For a concrete realization of this, we consider a spin-j environment (DE = 2j + 1), where V = Z ˆ ˆ and R = µ cos θZ + sin θX . The relative strength of the recording operator is thus µ, and θ determines how much of R commutes with the environment’s self-Hamiltonian. 1 Pliability of spin- 2 environments Figure B.7 shows the (numerically) time-averaged value of |γ |2 , for a qubit environment which experiences the conditional Hamiltonians ˆ ˆ ˆ H± = Z ± µ cos θZ + sin θX . (B.21) 206 Theoretical d: interaction-only models 3 d (decoherence factor) 2.5 2 1.5 1 0.5 0 0 20 interaction-only random-state 40 60 80 100 Dε (environment size) 120 140 ∆d 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0 Theoretical ∆d: interaction-only models interaction-only random-state 20 40 60 80 100 Dε (environment size) 120 140 (a) (b) Figure B.6: When a system’s full environment E consists of many quasi-independent subenvironments Ei , the individual subenvironments will typically evolve at different rates. Their lack of coherence means that their redundancy will be determined not by the subenvironments’ time-dependence, but by the time-averaged values of d and ∆d we illustrate here. Plot (a) compares the dynamical average of d, for the interaction-only model, to the random-state average we derived previously, while plot (b) does the same for ∆d. The behavior of d is much more important for computing redundancy. The dynamical average d is consistently slightly below the random-state average, because the evolution operators U = e−iHt are biased toward 1l instead of being uniformly distributed. The difference is most noticeable for DE = 2, where an analytic calculation yields d = 1 − ln(2), almost 40% below the random-state value. Plot (b) indicates that the fluctuations ∆d are slightly higher in the dynamical case than in the random-state analysis. Fluctuations in the initial state |Ei (t=0) combine with the fluctuations in time, visible in Fig. B.5, to produce the slightly higher dynamical value of ∆d. The strength of the recording Hamiltonian is varied from µ = 0 to µ = 16 (relative to the strength ˆ of V, since V = 1 · Z ), with several different values of θ. For θ = 0, of course, R commutes with V, and we’re back to the results of the previous section – in particular, |γ |2 is independent of µ, which only affects the timescale. As θ increases, a sharp peak at µ = 1 appears, becoming wider as θ increases further. Then, at θ = π/2, the entire character of the data changes. In particular, at µ = 0, the environment’s average decohering power vanishes entirely. Understanding and explaining these results requires separate consideration of three cases: θ = 0, θ = π/2, and 0 < θ < π/2. When θ = 0, as stated above, the interaction-only theory holds because R and V commute. H+ and H− have different eigenvalues, but identical eigenvectors. The key quantity Tr eitH+ e−itH− is thus dependent only on their spectra; it oscillates between +DE and −DE as a sum of sinusoidal terms as discussed in the previous section. On the other hand, when θ = π/2, R and V are not only noncommuting but completely orthogonal – R is entirely off-diagonal. As a result, the eigenspectra of H+ and H− are identical, and they differ only by their eigenvectors. When µ is small, the difference between H+ and H− is proportionally small. The conditional states e−itH+ |ψ0 and e−itH− |ψ0 evolve around slightly different axes on the Bloch sphere at the same rate, and never diverge very far from one another. The state of the environment is thus “pinned” in place by its self-Hamiltonian, and |γ |2 never drops very far below 1. Once µ ≥ 1, the recording operator begins to dominate H± . The self-Hamiltonian becomes a mere perturbation, and we return (nearly) to the interaction-only regime. The intermediate regime where µ ∼ 1 is difficult to describe, but as we see in Fig. B.7, the decohering power is greater than the interaction-only value. Because the conditional states move on nontrivial orbits around the Bloch sphere, they explore more of the total Hilbert space than is permitted by the simple dynamics 207 Time-averaged |γ|2 vs. µ,θ: Qubits 1 |γ|2 (decoherence factor) 0.9 0.8 0.7 0.6 0.5 0 1 2 3 4 µ 5 6 7 8 θ= 0 θ = 0.02π θ = 0.1π θ = 0.2π θ = 0.4π θ = 0.5π Figure B.7: When we extend the interaction-only model by adding a non-commuting Hamiltonian Vi for Ei , its decoherence-producing behavior becomes much more complicated. Qubits, or spin1 2 particles, provide a simple example which can be qualitatively extended to higher DE . In this figure we examine the dependence of |γ |2 (averaged over time and initial states) on the strength (µ) and orientation (θ), relative to Vi , of the recording operator Ri . At θ = 0, R and V are perfectly aligned (they commute), and we recover the interaction-only model, where |γ |2 doesn’t depend on the strength µ of the interaction. When θ = π , however, R and V are orthogonal, 2 and decoherence is dramatically suppressed by “pinning” of the environment’s conditional state, for µ << 1. At intermediate θ, the pinning effect is transient (and therefore vanishes from the timeaveraged result). Instead, we see an amplified capacity for decoherence which peaks at µ = 1, due to the interaction between the diagonal and off-diagonal components of R. As µ becomes very large, it overwhelms the self-energy of the environment, and we return asymptotically to the interaction-only regime. Note: The numerical data are represented only by points; the lines on the plot are the analytical calculation of Eqs. B.22-B.23. 208 of the interaction-only case. When θ is neither 0 nor π/2, H+ and H− differ in both their eigenvalues and their eigenvectors. Because the eigenvalues are different, we are guaranteed that the “pinning” effect seen above will not occur; the conditional states will diverge on a timescale given roughly by the inverse of ||H+ || − ||H− || regardless of their eigenvectors. The discrepancy between the eigenvectors also contributes to the divergence, but on a timescale given by the inverse of ||H+ || + ||H− ||, and with a reduced amplitude due to the pinning effect mentioned above. The rule of thumb for understanding the rather complicated dynamical effects surveyed above is that Rdiag acts to perturb the eigenvalues of V, while Roffdiag perturbs the eigenvectors. When µ is small, perturbations in the eigenvalues lead to slow (t ∼ µ−1 ) but large-amplitude oscillations in γ , while eigenbasis perturbations due to Roffdiag are small-amplitude (∼ µ2 /||V||) but rapid. If we average the results over arbitrarily long times, the large oscillations in γ due to Rdiag are the dominant effect if a diagonal component of R exists. In practice, we can assume that R will always have some diagonal component – true pinning of the environment state occurs only when θ is precisely π/2. However, if the diagonal Rdiag is sufficiently small, the timescale for the large-amplitude oscillations may be so long that the off-diagonal Roffdiag is the only producer of decoherence on the timescale of interest. In the intermediate regime around µ ∼ 1, neither the self-Hamiltonian nor the recording Hamiltonian can be regarded as a perturbation, and the dynamics is difficult to describe in any simple fashion. We can, however, use Eq. B.14, with U = eitH+ e−itH− , to compute |γ |2 . This calculation, while not difficult per se, is rather arduous – even employing the computer algebra system Maple, one of the intermediate steps is an expression with over 900 terms. In the interest 1 of preserving both space and the readers’ interest, we provide only the results here. For spin- 2 environments (DE = 2), we obtain µ4 − 2µ2 cos θ + 1 if θ = (µ2 + 2µ cos θ + 1) (µ2 − 2µ cos θ + 1) µ4 + 2 π =2 2 if θ = 2 (µ2 + 1) 11 2 = + |TrU | 36 = 2 |TrU | |TrU | 2 π 2 (B.22) (B.23) (B.24) 2 |γ |2 These formulae are reproduced as the solid lines in Fig. B.7, which reproduce the numerical results perfectly. This serves primarily to verify that the numerical results are accurate, and to give us greater confidence in the numerical results for d and ∆d. Not surprisingly, given the complexity of the dynamical processes involved, the additive measures of pliability, d and ∆d, are not amenable to analytic calculations. Figure B.8 shows the numerical results. The results are in good agreement with our expectations from the preceding analysis of the multiplicative decoherence factor, with d consistently slightly greater than d = − 1 log |γ |2 . Particularly interesting is the fact that the dynamical average of d exceeds the random2 state average near µ = 1. In other words, correctly chosen dynamics can actually produce more redundancy than is obtained by correlating the pointer states of S with randomly chosen states of the environment. We note that this is something of a subtle point, because this “optimally decohering” dynamical evolution does not produce a greater average level of decoherence in the system. When µ = 1, the average decoherence per environment (|γ |2 ) is exactly the same as in the random-state average. However, the distribution over many environments is slightly different; essentially, a given environment is slightly more likely (in the dynamical case) to have either a lot of information or very little information about the system, which increases the expected redundancy. 209 Time-averaged d vs. µ,θ: Qubits 0.6 d (decoherence factor) 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 µ 5 θ= 0 θ = 0.02π θ = 0.1π θ = 0.2π θ = 0.4π θ = 0.5π 6 7 8 Average ∆d 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 Time-averaged ∆d vs. µ,θ: Qubits θ= 0 θ = 0.02π θ = 0.1π θ = 0.2π θ = 0.4π θ = 0.5π 1 2 3 4 µ 5 6 7 8 (a) (b) Figure B.8: In this figure, we illustrate the behavior of the additive decoherence factor d, which is more relevant to redundancy than its multiplicative cousin γ . We have no general analytical calculation for d, however, so the connecting lines in plots (a)-(b) represent only a visual aid. We show the same parameter regime as in Fig. B.7, and d behaves essentially as we would expect. As θ is increased from zero to π , the flat interaction-only profile grows a sharp peak at µ = 1, 2 which broadens as θ grows. What is not evident in the time-averaged behavior is a growing (with θ) “pinning” effect at short times. At θ = π , the pinning timescale becomes infinite, and the effect 2 is suddenly visible even in the time-averaged d. While ∆d displays similar behavior, it has no real effect on redundancy. Of particular note, however, is the fact that near µ = 0, d actually exceeds the random-state value. This indicates that it is possible, in principle, to enhance redundancy even above the random-state value by choosing the subenvironments’ Hamiltonians appropriately. By contrast, the net decoherence inflicted upon S is determined by |γ |2 , which only barely achieves its random-state value at µ = 1 (see Fig. B.7). Pliability of spin-j environments The preceding analysis of spin- 1 environments is complete – by varying µ and θ, we can 2 explore the entire parameter space of possible interactions. As the size DE of the individual environments grows, however, the set of possible interactions grows rapidly. However, if we restrict the possible interactions to spin-type Hamiltonians, then by staying within the SU (2) manifold we can 1 expect that the larger subenvironments – spin-j particles, where j > 2 – will display fairly similar behavior. The algebraic properties of the angular momentum operators Jx , Jy , Jz are the same, and the spin-coherent state manifold that they generate is large enough to span the entire Hilbert space. Figure B.9 confirms this, although there are small but interesting differences in the precise behavior of d and ∆d for each value of DE . In particular, environments with µ = 1 remain the most pliable. In Fig. B.10, we compare the maximum pliability (at µ = 1) for various DE to the interaction-only pliability, and the previously calculated random-state pliability. The maximum pliability is actually slightly greater than the random-state calculation, although this discrepancy vanishes for larger DE . We also note that as DE grows, the peak around µ = 1 becomes broader. This tends to support the conjecture that large environments have generic behavior – they achieve the random-state value of pliability with little dependence on µ and θ. This disregards the effect of pinning, when R and V are exactly orthogonal. Pinning is very sensitive to θ, however – if R has even a tiny diagonal component, then it destroys the pinning effect. One residual effect that we do expect to see, however, is a delayed onset of redundancy for small θ, because of the long timescale on which Rdiag acts. While the general profile of d and |γ |2 versus µ remains the same as DE increases, the numerical results indicate some odd features around the µ = 1 peak itself, which might indicate 210 Time-averaged d vs. µ,θ: Denv = 3 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 µ 5 0.8 0.7 Average ∆d 0.6 0.5 0.4 0.3 0.2 0.1 0 8 0 d (decoherence factor) Time-averaged ∆d vs. µ,θ: Denv = 3 θ= 0 θ = 0.02π θ = 0.1π θ = 0.2π θ = 0.4π θ = 0.5π 6 7 θ= 0 θ = 0.02π θ = 0.1π θ = 0.2π θ = 0.4π θ = 0.5π 1 2 3 4 µ 5 6 7 8 (a) Time-averaged d vs. µ,θ: Denv = 4 0.8 d (decoherence factor) 1 Average ∆d 0.8 0.6 0.4 0.2 0 0 1 2 3 4 µ 5 θ= 0 θ = 0.02π θ = 0.1π θ = 0.2π θ = 0.4π θ = 0.5π 6 7 8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 (b) Time-averaged ∆d vs. µ,θ: Denv = 4 θ= 0 θ = 0.02π θ = 0.1π θ = 0.2π θ = 0.4π θ = 0.5π 3 4 µ 5 6 7 8 (c) Time-averaged d vs. µ,θ: Denv = 5 d (decoherence factor) 1.2 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 µ 5 θ= 0 θ = 0.02π θ = 0.1π θ = 0.2π θ = 0.4π θ = 0.5π 6 7 8 Average ∆d 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 (d) Time-averaged ∆d vs. µ,θ: Denv = 5 θ= 0 θ = 0.02π θ = 0.1π θ = 0.2π θ = 0.4π θ = 0.5π 3 4 µ 5 6 7 8 (e) (f ) Figure B.9: Plots (a)-(f ) present precisely the same data as Fig. B.8, but for environments with 3-, 4-, and 5-dimensional Hilbert spaces. The same general behavior is evident; d rises above the interaction-only value with a maximum at µ = 1, declining back to the interaction-only value sharply for small θ and slowly for large θ. For θ = π , pinning is still observed for µ < 1. The fluctuations 2 (∆d) display rather complicated behavior, but the scale of the complications is negligible in the overall scheme of redundancy. Finally, we note two interesting (but not terribly relevant) trends as DE grows. While the net difference in d between µ = 0 and µ = 1 seems to remain roughly constant, at approximately 0.23, the width of the peak around µ = 1 grows with DE . Large environments, in other words, seem to be more sensitive to small off-diagonal components of R – but not to their precise magnitude. This indicates that for sufficiently large DE , we may expect to see “generic” redundancy behavior, independent of almost everything about R and V. 211 Baseline values for d vs. Denv d (additive decoherence factor) 3 2.5 2 1.5 1 0.5 0 0 20 0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 20 40 60 80 100 Dε (subenvironment size) 120 dynamical davg at µ=1 interaction-only davg random-state davg Difference between dµ=1 and drandom state 40 60 80 Dε (subenvironment size) dµ=1 - drandom state 100 120 Figure B.10: The precise values of d, for various times and interaction strengths, are distilled in a simple summary in this figure. We plot the minimum (µ = 0) and maximum (µ = 1) dynamical average values of d for environment sizes from DE = 2 to DE = 128. We also provide the theoretical random-state average computed previously for comparison. For large environments, the dynamical range of d is quite small compared to typical values; for qubit (spin- 1 ) environments, however, 2 the discrepancy between interaction-only (µ = 0) and maximally pliable (µ = 1) environments is nearly 40%. The maximum dynamical value, however, tracks the random-state prediction extremely closely; the inset shows the difference, which is never more than 10%, and drops rapidly with increasing environment size. Our general conclusion is that for “large” environments, virtually every calculation gives effectively identical results – i.e., we may as well use the analytical randomstates average. However, when the environment consists of small systems such as spin- 1 particles, a 2 detailed understanding of the dynamics is crucial. This figure also ignores the pinning effect, which can be substantial if the recording operator R is orthogonal to the environment’s Hamiltonian V. instabilities or flaws in the numerical methods. To check this, we calculated the analytic average value of |γ |2 for DE = 3: |γ |2 |TrU | 2 θ= π 2 = = |TrU | 2 θ= π 2 = 1 1 2 + |TrU | , (B.25) 4 12 8 2 6 4 2 4 2 2 3µ − (8 cos θ + 4)µ + (20 cos θ − 24 cos θ + 22)µ − (4 cos θ + 2)µ + 3 , 2 2 (1 + 2µ cos θ + µ2 ) (1 − 2µ cos θ + µ2 ) (B.26) 3µ8 − 4µ6 + 42µ4 − 12µ2 + 9 . (B.27) 4 (µ2 + 1) 212 Time-averaged |γ|2 vs. µ,θ: Denv = 3 1 0.9 |γ|2 (decoherence factor) 0.8 0.7 0.6 0.5 0.4 0.3 0 1 2 3 4 µ 5 6 7 8 θ= 0 θ = 0.02π θ = 0.1π θ = 0.2π θ = 0.4π θ = 0.5π Figure B.11: In order to verify that our understanding of the non-commutative dynamics is valid for DE > 2, we plot the numerical values of |γ |2 obtained for a spin-1 system, and compare them to the analytical result in Eqs. B.26-B.25. The analytical result, displayed as a solid line, fits the data perfectly, indicating that the curious bumps and lumps in the data reflect actual phenomena, instead of numerical error. This has no great scientific relevance in itself, but serves a vital function as validation of the numerical data presented in Fig. B.9. For larger DE , this calculation becomes increasingly difficult. However, by comparing the numerical results (Fig. B.11) with Eq. B.22-B.24, we verify that the odd kinks and flat spots in the numerical results are (at least for DE = 3) genuine features of the dynamics. B.3.3 Redundancy in simple spin models The goal of our investigation into pliability, of course, is to predict the amount of redundancy that emerges from decoherence processes. An additional goal is to understand when the environment’s Hamiltonian may significantly affect the amount of redundancy. In this section, we convert the results for pliability into concrete predictions for the level of redundancy in spin-bath environments. Equation B.2 lets us calculate the number of subenvironments that need to be concatenated in order to obtain nearly-complete information: mδ = log(DS − 1) − log (2δHS ) ∆d2 1 + 2 + 2, 2d 2d 213 Theoretical r10% vs. Dε 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 5 10 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 5 r (specific redundancy) r (specific redundancy) Theoretical r1% vs. Dε random-state average interaction-only µ=1 15 Dε 20 25 30 random-state average interaction-only µ=1 10 15 Dε 20 25 30 (a) (b) Figure B.12: The ultimate goal of the entire investigation into pliability is, of course, a quantitative prediction for redundancy. In these plots, we combine all the previous results to predict the specific redundancy (r), or redundancy (R) per subenvironment, as a function of the subenvironment size. Plot (a) shows the prediction for a 10% information deficit (δ ), and plot (b) shows the prediction for a deficit of 1%. Unsurprisingly (in light of previous results for the mean additive decoherence factor d), the maximum dynamical redundancy is achieved for µ = 1, when the environment Hamiltonian and the recording operator have equal magnitudes (and are neither perfectly parallel nor perfectly orthogonal). At µ = 1, the dynamical redundancy is indistinguishable from the random-state average, while in the absence of nontrivial dynamics for the subenvironments, the dynamical redundancy 1 is lower – by up to a factor of 2 in the spin- 2 case. in terms of which the specific redundancy rδ is rδ = 1−δ . mδ (B.28) Using the data shown in Figs. B.9-B.10, we compute specific redundancies both for spin- 1 particles 2 (Fig. B.13) and larger subenvironments (Fig. B.14). Ignoring the fine details of the data, we can make several generalizations. • The random-state average computed previously in our discussion of the branching-state ensemble is a good, if rough, estimate of dynamical redundancy for almost all cases. As a rule, r falls between the interaction-only model’s prediction and the random-state average. • The most dramatic deviation from the just-mentioned rule of thumb occurs when the recording 1). In this case, operator R is perfectly orthogonal to V (θ = π ), and small in magnitude (µ 2 decoherence (and therefore redundancy) is nonexistent at µ = 0, but rises roughly linearly to the interaction-only level at µ ∼ 1. • Pliability and specific redundancy are maximal when |µi − µj |R and V are of equal magnitude. The difference between this rmax and the random-state average rrand is substantial for DE = 2, but becomes less significant as the environment gets larger (Fig. B.12). As explained in section 3.1.1, our focus here is primarily on time-independent and asymptotic quantities. However, the effect of transient pinning requires some consideration, both because it has substantial ramifications for quantum technology and because it appears in the simulation data of the next section. The recording operator R can be decomposed into Rdiag + Roffdiag , where the former commutes with V and the latter is orthogonal to it. When Rdiag = 0, the conditional Hamiltonians H+ and H− have identical eigenvalues, and the conditional states are only induced 214 Theoretical r10% vs. µ,θ: Qubits 0.35 r (specific redundancy) 0.3 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 4 µ 5 θ= 0 θ = 0.02π θ = 0.2π θ = 0.5π 6 7 8 r (specific redundancy) 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 Theoretical r1% vs. µ,θ: Qubits θ= 0 θ = 0.02π θ = 0.2π θ = 0.5π 1 2 3 4 µ 5 6 7 8 (a) (b) Figure B.13: This figure shows the full spectrum of possible dynamical r for qubit (spin- 1 ) envi2 ronments. The most interesting feature is the dramatic reduction in the environment’s decohering (and redundancy-producing) power when θ = π and the interaction strength µ is small. While all 2 other cases produce a level of redundancy consistent with Fig. B.12, this particular interaction (e.g., where R and V are orthogonal) can result in the near-complete elimination of both redundancy and decoherence. In practice we expect that R will never be completely orthogonal to V – but if they are nearly-orthogonal, then the pinning effect may persist for a long enough time to reduce short-time redundancy substantially. to diverge by the difference in eigenvectors, resulting in pinning. If ||Rdiag || ||Roffdiag ||, then the conditional states display both slow and fast oscillation. The fast oscillation, due to the eigenbasis difference, has a small amplitude and doesn’t contribute much to decoherence. Thus, decoherence is driven primarily by the slow oscillation due to Rdiag . In this situation, the effects of Roffdiag can be largely ignored, and the timescale for decoherence and redundancy is set by Rdiag . Rdiag is co-diagonal with V , so the decoherence timescale from Eq. B.17 is valid, with the addition of a factor sin θ. We predict a timescale, for the development of maximal pliability, of tdecoherence ∼ sin θ|µj − µi | Tr(R2 ) − (TrR)2 (B.29) 215 Theoretical r10% vs. µ,θ: Dε = 3 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 4 µ 5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 r (specific redundancy) r (specific redundancy) Theoretical r1% vs. µ,θ: Dε = 3 θ= 0 θ = 0.02π θ = 0.2π θ = 0.5π 6 7 8 θ= 0 θ = 0.02π θ = 0.2π θ = 0.5π 1 2 3 4 µ 5 6 7 8 (a) Theoretical r10% vs. µ,θ: Dε = 4 r (specific redundancy) r (specific redundancy) 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 µ 5 θ= 0 θ = 0.02π θ = 0.2π θ = 0.5π 6 7 8 0.5 0.4 0.3 0.2 0.1 0 0 1 2 (b) Theoretical r1% vs. µ,θ: Dε = 4 θ= 0 θ = 0.02π θ = 0.2π θ = 0.5π 3 4 µ 5 6 7 8 (c) Theoretical r10% vs. µ,θ: Dε = 5 0.6 r (specific redundancy) r (specific redundancy) 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 (d) Theoretical r1% vs. µ,θ: Dε = 5 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 µ 5 θ= 0 θ = 0.02π θ = 0.2π θ = 0.5π 6 7 8 θ= 0 θ = 0.02π θ = 0.2π θ = 0.5π 3 4 µ 5 6 7 8 (e) (f ) Figure B.14: For completeness, we present the same data as in Fig. B.13, but for larger (DE = 3, 4, 5) environments. This serves primarily to verify that the intuitions gleaned from studying 2-dimensional systems are still valid for larger systems; the data are qualitatively the same. Two trends are worth noting, however. As DE grows, the difference between rµ=1 and rµ=0 decreases; at the same time, the peak in r at µ = 1 becomes broader. This confirms our general intuition that for extremely large environments, specific redundancy becomes virtually independent of the details of R and V – total Hilbert space dimension becomes the only important parameter. In this case, however, the range of spectral complexity for R and V becomes very great To fully understand the behavior of large environments would involve a detailed study of the possible spectral effects, which is another field entirely. 216 B.4 Detailed R(t) plots for quantum-measurement models In this appendix, we show plots of redundancy Rδ as it evolves over time, for quantummeasurement models of spin bath interactions. Three figures – Figs. B.15-B.17 – plot R10% (t) for 1 environments consisting of 12 (Fig. B.15), 48 (B.16) and 128 (B.15) spin- 2 particles. Each figure contains plots for three different variants of the quantum-measurement model, as well as a reference plot of the interaction-only model’s behavior: 1. Assorted dynamically-decoupled environments, where the individual subenvironments’ Hamiltonians are perfectly orthogonal to the recording operator. 2. Superpliable environments, where the subenvironment Hamiltonians are oriented at an angle of ∼ π to the recording operator. This allows the environment’s conditional states to explore 4 the entire available Hilbert space. 3. Transiently decoupled environments, where the subenvironment Hamiltonians are almost – but not quite – orthogonal to the recording operator. As a result, the environments appear to be dynamically decoupled until the non-orthogonal portion of R comes into play, at which point the redundancy climbs to superpliable levels. All these models are discussed in detail in the main body of the text. Here, our goal is simply to present the data for interested readers. Of particular interest is the fact that all three figures are virtually identical, except for the scaling of the R-axis. This confirms that redundancy scales linearly with the size of the environment, and therefore that specific redundancy is the appropriate quantity to describe information storage in spin bath environments. 217 Averaged R10%(t); Nenv = 12, V = Jx 2.5 2 R10% 1.5 1 0.5 0 v= 0 v = 0.1 v = 0.2 v = 0.4 v = 0.8 v = 1.6 v = 3.2 v = 12.8 0 5 10 15 20 25 30 35 40 (a) Averaged R10%(t); Nenv = 12, V = .6Jx + .8Jz 2.5 2 R10% 1.5 1 0.5 0 v= 0 v = 0.1 v = 0.2 v = 0.4 v = 0.8 v = 1.6 v = 3.2 v = 12.8 0 5 10 15 20 25 30 35 40 45 50 (b) Averaged R10%(t); Nenv = 12, V = .995Jx + .1Jz. Part (I) 2.5 2 R10% 1.5 1 0.5 0 v= v= v= v= v= 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.4 0.8 (c) Averaged R10%(t); Nenv = 12, V = .995Jx + .1Jz. Part (II) 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 30 Time (arbitrary units) 35 40 45 50 R10% v= 0 v = 1.6 v = 3.2 v = 6.4 v = 12.8 (d) Figure B.15: R10% (t) is shown for three different quantum measurement models with Nenv = 12, where each environment has a Hamiltonian V whose magnitude is v . In plot (a), V (v Jx ) is perfectly orthogonal to the recording operator R (Jz ); in plot (b) V is split into roughly equal parts which (respectively) are orthogonal to R, and commute with R); in plots (c)-(d), V is mostly orthogonal to R but contains a small co-diagonal term. Compared to the interaction-only results (solid black lines), we see that V induces oscillations in R(t), which decays away with increasing t as the environments get out of phase. The asymptotic redundancy Ravg is decreased by large v due to pinning in plot (a), but increased by v ∼ 1 in plot (b) due to enhancement of the environments’ pliability. We see in plots (c)-(d) that when R and V are nearly-orthogonal, the two effects merge to produce pinning at short times and enhanced pliability at longer times. 218 Averaged R10%(t); Nenv = 48, V = Jx 12 10 8 6 4 2 0 0 5 10 15 20 25 30 35 40 45 50 v= 0 v = 0.1 v = 0.2 v = 0.4 v = 0.8 v = 1.6 v = 3.2 (a) 14 12 10 8 6 4 2 0 R10% Averaged R10%(t); Nenv = 48, V = .6Jx + .8Jz v= 0 v = 0.1 v = 0.2 v = 0.4 v = 0.8 v = 1.6 v = 3.2 v = 6.4 0 5 10 15 20 25 30 35 40 45 50 (b) 14 12 10 8 6 4 2 0 R10% Averaged R10%(t); Nenv = 48, V = .995Jx + .1Jz. Part (I) R10% v= 0 v = 0.1 v = 0.2 v = 0.4 v = 0.8 0 5 10 15 20 25 30 35 40 45 50 (c) 14 12 10 8 6 4 2 0 Averaged R10%(t); Nenv = 48, V = .995Jx + .1Jz. Part (II) R10% v= 0 v = 1.6 v = 3.2 v = 6.4 0 5 10 15 20 25 30 Time (arbitrary units) 35 40 45 50 (d) Figure B.16: R10% (t) is shown for three different quantum measurement models with Nenv = 48, where each environment has a Hamiltonian V whose magnitude is v . In plot (a), V (v Jx ) is perfectly orthogonal to the recording operator R (Jz ); in plot (b) V is split into roughly equal parts which (respectively) are orthogonal to R, and commute with R); in plots (c)-(d), V is mostly orthogonal to R but contains a small co-diagonal term. Compared to the interaction-only results (solid black lines), we see that V induces oscillations in R(t), which decays away with increasing t as the environments get out of phase. The asymptotic redundancy Ravg is decreased by large v due to pinning in plot (a), but increased by v ∼ 1 in plot (b) due to enhancement of the environments’ pliability. We see in plots (c)-(d) that when R and V are nearly-orthogonal, the two effects merge to produce pinning at short times and enhanced pliability at longer times. 219 Averaged R10%(t); Nenv = 128, V = Jx 35 30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 40 45 50 v= 0 v = 0.1 v = 0.2 v = 0.4 v = 0.8 v = 1.6 v = 3.2 (a) 40 35 30 25 20 15 10 5 0 R10% Averaged R10%(t); Nenv = 128, V = .6Jx + .8Jz v= 0 v = 0.1 v = 0.2 v = 0.4 v = 0.8 v = 1.6 v = 3.2 v = 6.4 0 5 10 15 20 25 30 35 40 45 50 (b) 40 35 30 25 20 15 10 5 0 R10% Averaged R10%(t); Nenv = 128, V = .995Jx + .1Jz. Part (I) R10% v= 0 v = 0.1 v = 0.2 v = 0.4 v = 0.8 0 5 10 15 20 25 30 35 40 45 50 (c) 40 35 30 25 20 15 10 5 0 Averaged R10%(t); Nenv = 128, V = .995Jx + .1Jz. Part (II) R10% v= 0 v = 1.6 v = 3.2 v = 6.4 0 5 10 15 20 25 30 Time (arbitrary units) 35 40 45 50 (d) Figure B.17: R10% (t) is shown for three different quantum measurement models with Nenv = 128, where each environment has a Hamiltonian V whose magnitude is v . In plot (a), V (v Jx ) is perfectly orthogonal to the recording operator R (Jz ); in plot (b) V is split into roughly equal parts which (respectively) are orthogonal to R, and commute with R); in plots (c)-(d), V is mostly orthogonal to R but contains a small co-diagonal term. Compared to the interaction-only results (solid black lines), we see that V induces oscillations in R(t), which decays away with increasing t as the environments get out of phase. The asymptotic redundancy Ravg is decreased by large v due to pinning in plot (a), but increased by v ∼ 1 in plot (b) due to enhancement of the environments’ pliability. We see in plots (c)-(d) that when R and V are nearly-orthogonal, the two effects merge to produce pinning at short times and enhanced pliability at longer times. 220 Entropy (HS) in the Interaction-Only model (10 run avg.) 1 0.01 0.0001 1 - HS (bits) 1e-06 1e-08 1e-10 1e-12 1e-14 0 5 10 15 20 25 Time (arbitrary units) 30 35 40 Nenv = 6 Nenv = 12 Nenv = 24 Nenv = 48 Figure B.18: As Nenv increases, the increase in redundancy is reflected in decoherence which is more and more complete. The difference between HS and its theoretical maximum H = 1 bit scales as e−Nenv , so that by Nenv = 48, the numerical error of the simulation has almost completely washed out ∆H . B.5 Dependence of redundancy on complete decoherence In this appendix, we briefly discuss the connection between decoherence and redundancy. The existence of a qualitative connection is obvious – without decoherence, no information is stored, and redundancy is meaningless. More interesting is the question of a quantitative connection – how the amount of decoherence in a system is related to the amount of redundancy. We begin by showing that complete decoherence is a prerequisite for redundancy in interactiononly and quantum-measurement models of decoherence. After discussing the requirements for extending this proof to more general contexts, we consider the relationship between decoherence and the decline of redundancy in dynamical-system and multiple-measurement models. We show that the decline of redundancy cannot be explained by insufficiency of decoherence, which tends to support our conjecture that redundancy decays because information is moved into entangled modes of the environment. B.5.1 Branching-state models (interaction-only, quantum-measurement) Consider a decoherence process where a perfect pointer basis (1) exists, and (2) is known. Let the pointer states be |0 and |1 , without loss of generality (the following argument is trivially generalizable to number DS of pointer states). The system’s initial state can then be written in the 221 pointer basis, as |ψ0 (S ) = α |0 + β |1 . (B.30) Decoherence occurs when the environment becomes correlated with the system. The joint state is a branching state, ( (2) (1) ( (2) (1) |ψfinal = α |0(S ) |E0 |E0 . . . |E0N ) + β |1(S ) |E1 |E1 . . . |E1N ) , (B.31) where the division of the environment into Ei . . . EN is a division into fragments that provide sufficient (i.e., all but δ ) information. In other words, we have skipped over the process of concatenating subenvironments into fragments, jumping straight to the fragment decomposition. Each fragment provides all but δ of the total information, by assumption. This implies that by tracing over any Ei , the density matrix of S and the remaining environments can be reduced to nearly-diagonal form. This, in turn, is equivalent to the statement E0 |E1 (i) (i) ≤ O(δ ), (B.32) for all i. Here, O(δ ) means “something approximately as small as δ ”; the exact bound is complicated and only obfuscates the reasoning. The reduced state of the system alone is obtained by tracing over all the Ei : ρS = |α|2 α∗ β Γ∗ αβ ∗ Γ |β |2 (B.33) where the off-diagonal elements are suppressed by Γ (1) (1) = E0 |E1 N ≤ O(δ ) . E0 |E1 (2) (2) ... E0 (N ) |E1 (N ) (B.34) (B.35) Conclusion: if 1 − δ of the total information is N -fold redundant, then the off-diagonal coherences must be suppressed by a factor of approximately δ N . Complete decoherence annihilates the off-diagonal elements completely, so we have shown that redundancy can only occur when decoherence is exponentially close to completeness. In Fig. B.18, we plot the difference between HS (t) and its theoretical maximum value of H = 1 bit, for interaction-only models with various Nenv . The results agree with the simple derivation above: as redundancy increases, decoherence becomes exponentially more complete. To go further than this analysis – for instance, to make a sweeping statement about redundancy and completeness of decoherence – requires sorting out several rather subtle points. In particular, we must identify what the environment has information about. When secondary environments, which measure the primary environments instead of the system itself, are involved, then the subenvironments’ conditional states may become correlated with non-orthogonal states of the system. In such a case, it can be argued that the environment has redundant but incomplete information about a Hermitian observable – or that the environment has redundant and complete information about a non-Hermitian observable. To avoid embroilment in this controversy, we confine the present discussion where all environment measure the system directly, a pointer basis exists, and near-complete decoherence is definitely a prerequisite for redundancy. B.5.2 Dynamical-system and multiple-measurement models The preceding discussion indicates a powerful relationship between decoherence and redundancy. We are led to inquire whether the decay of redundancy in dynamical-system and multiplemeasurement models is reflected somehow in the completeness of decoherence. To investigate this, we plot HS (t), for the models considered in Sections 3.4 and 3.5, in Figs. B.19-B.21. 222 Entropy vs. Time (10-run avg): H = E E0 = 0 E0 = 0.03 E0 = 0.06 E0 = 0.1 E0 = 0.15 E0 = 0.25 1 - Hsys (bits) 0.01 0.001 0.0001 0 (a) 2 4 6 8 Time (arbitrary units) 10 26 28 30 32 34 Entropy vs. Time (single run): H = E 0.01 1 - Hsys (bits) 0.0001 1e-06 1e-08 0 (b) E0 = 0 E0 = 0.03 E0 = 0.06 E0 = 0.1 E0 = 0.15 E0 = 0.25 2 4 6 8 Time (arbitrary units) 10 26 28 30 32 34 Figure B.19: We examine the effect of system dynamics (Hsys ∝ E0 ) on the the amount of total decoherence, measured by the system’s entropy (HS ). Plot (a) shows HS averaged over 10 different simulations, while plot (b) shows the results of a single simulation. The averaged results indicate that Hsys causes a reduction in entropy at short times, but at at longer times the difference has become insignificant. This fails completely to explain the redundancy results in Fig. 3.10. By examining a single run, however, we can observe the true effect of system dynamics. The presence of Hsys prevents decoherence from surpassing a certain level of completeness – that is, the environment never has perfectly reliable information about S . This cutoff effect cannot explain the time-dependent decay of R, however, because the cutoff effect does not increase with time. The data speak for themselves. A slight reduction in HS , compared to the interaction-only model, is apparent – but it completely fails to explain the observed reduction in redundancy. First, the reduction in completeness is too small to account for the drastic decline in redundancy. Second, and more importantly, the two phenomena have opposite time dependence. Redundancy is lost slowly over time, whereas HS is reduced mostly at short times. After a few decoherence timescales, HS is very close to 1 bit – entirely sufficient to support redundancy. We conclude that the dynamical-system and multiple-measurement models do produce nearly-complete decoherence, although it is suppressed for a short while. The loss of redundancy occurs entirely independently of any effects in decoherence. This tends to confirm the hypothesis that redundancy decays because the information stored in the environment is transferred from localized modes of E into entangled modes. We note in passing that HS is not always a good indicator of decoherence’s completeness. Maximal HS does imply complete decoherence – but complete decoherence can occur even if HS is not maximal, when the system’s initial state is not maximally measurable. Entropy differences on the order of 2−10 bits are plotted in the figures, and these can never be detected experimentally. A much better approach is the use of process tomography to determine how successfully coherence are suppressed. This approach is taken in Chapter 5, where we confirm these results more rigorously. 223 Entropy vs. Time (10-run avg): H = gy = 0 gy = 0.01 gy = 0.02 gy = 0.04 gy = 0.05 gy = 0.1 1 - Hsys (bits) 0.01 0.001 0.0001 0 (a) 2 4 6 8 Time (arbitrary units) 10 26 28 30 32 34 Entropy vs. Time (single run): H = J 0.01 1 - Hsys (bits) 0.0001 1e-06 1e-08 0 (b) gy = 0 gy = 0.01 gy = 0.02 gy = 0.04 gy = 0.05 gy = 0.1 2 4 6 8 Time (arbitrary units) 10 26 28 30 32 34 Figure B.20: These plots show the amount of decoherence (measured by HS ) produced by the multiple-measurement model using a Y-Z interaction. The format of the plots is identical to Fig. B.19 and Fig. B.21; plot (a) shows 10-run averages while plot (b) shows data from a single run. The primary conclusion we draw from this data is the same as for the case of system dynamics; adding multiple measurements does reduce the amount of decoherence, but (1) not enough to explain the loss of redundancy, and (2) not with the same time dependence as the destruction of R. More interestingly, by contrasting this data with Fig. B.21, we see a much more disorderly dependence on gy than on gd , which probably reflects the higher level of symmetry in the dipole model. The corresponding redundancy data in Figs. 3.12-3.13 displays no such evidence of symmetry. 224 Entropy vs. Time (10-run avg): H = Jz gy = 0 gy = 0.01 gy = 0.02 gy = 0.05 gy = 0.1 1 - Hsys (bits) 0.01 0.001 0.0001 0 (a) 2 4 6 8 Time (arbitrary units) 10 26 28 30 32 34 Entropy vs. Time (single run): H = Jz 0.01 1 - Hsys (bits) 0.0001 1e-06 1e-08 0 (b) gy = 0 gy = 0.01 gy = 0.02 gy = 0.05 gy = 0.1 2 4 6 8 Time (arbitrary units) 10 26 28 30 32 34 Figure B.21: These plots duplicate Figure B.20, but show the results of simulating a dipole interaction H = Jz ⊗ Jz − gd (Jx ⊗ Jx + Jy ⊗ Jy ). Of primary interest is the greater disorder in Fig. 3.12, which is not reflected in redundancy; see the caption of Fig. 3.12 for details. 225 Appendix C Supporting Material on Quantum Brownian Motion C.1 Quantum Brownian motion: Details of Rδ ’s dependence on various parameters In this appendix, we present the dependence of Rδ , HS , and INR on the parameters of the environment. The effects of varying γ0 , Λ, and Nenv are presented in Fig. C.1. None of the results presented here are particularly surprising, in light of the discussion in the main body of the chapter; they primarily confirm that redundancy and total information behave in the way we have outlined previously for a relatively wide parameter range. We note three features: 1. The coupling constant (γ0 ) sets the timescale for redundancy. It also affects the maximum level of redundancy achieved, but only slightly. This is due to a change in the thermal (asymptotic) levels of IS :E and INR . 2. Changing the cutoff frequency (Λ) produces only small changes in IS :E , and affects redundancy to a moderate degree. When Λ is increased, there are more high-frequency bands that take part in redundancy. These high-frequency bands tend to store non-redundant information, so redundancy decreases with increased Λ. The effect is moderate, and is less important as the state is made more measurable (i.e., ∆x is increased). Thus, for “classical” superpositions of positions, very large Λ will not affect the emergence of massive redundancy. 3. Changing the number of environments (Nenv ), which also changes the frequency spacing (ω0 ) has a limited effect. As Nenv becomes large, it also becomes irrelevant – the behavior of information storage approaches and saturates at a continuum limit. Total information IS :E is almost unaffected, while redundancy quickly reaches continuum behavior. For sufficiently small Nenv , of course, redundancy is reduced because Rδ cannot possibly be greater than Nenv ! As long as Nenv Rδ , however, its precise value is largely irrelevant. 226 IS:ε(t): Varying Λ 8 7 6 IS:ε (nits) 5 4 3 2 1 0 0 10 20 30 Time 40 50 60 Λ= 4 Λ= 8 Λ = 16 R10% 25 20 15 10 5 0 0 10 Λ= 4 Λ= 8 Λ = 16 R10%(t): Varying Λ 20 30 Time 40 50 60 (a) INR(t): Varying Λ 2.5 2 INR (nits) 1.5 1 0.5 0 0 10 20 30 Time 40 50 60 Λ= 4 Λ= 8 Λ = 16 R10% 14 12 10 8 6 4 2 0 0 10 (b) R10%(t): Varying Nenv Nenv = 64 Nenv = 128 Nenv = 256 Nenv = 512 20 Time 30 40 50 (c) IIS:ε(t): Varying γ 8 7 6 IS:ε (nits) 5 4 3 2 1 0 0 10 20 30 40 50 Time 60 70 80 90 R10% (d) R10%(t): Varying γ 12 10 8 6 4 2 0 0 10 20 30 40 50 Time 60 70 80 90 γ = 0.05 γ = 0.1 γ = 0.2 γ0 = 0.05 γ0 = 0.1 γ0 = 0.2 (e) (f ) Figure C.1: The effects of varying: (a-b) the cutoff frequency (Λ), (c-d) the total number of bands (Nenv , e.g. frequency resolution), or (e-f ) the coupling constant (γ0 ). See text (Appendix C.1) for discussion. 227 Differential Information: ωsys = 0, T = 2 ∆IS:ε 2 1.5 1 0.5 0 0 0.05 8 4 12 0.1 fraction 0.15i of ε already captured ωε 16 ∆IS:ε 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 Differential Information: ωsys = 0, T = 8 ω ω 16 12 0.05 8 4 0.1 fraction 0.15i of ε already captured ωε (a) Differential Information: ωsys = 0, T = 20 ∆IS:ε 0.5 0.4 0.3 0.2 0.1 0 0 ∆IS:ε 0.6 0.5 0.4 0.3 0.2 0.1 0 0 (b) Differential Information: ωsys = 0, T = 40 ω ω 16 12 0.05 8 4 0.1 fraction 0.15i of ε already captured ωε 16 12 0.05 8 4 0.1 fraction 0.15i of ε already captured ωε (c) (d) Figure C.2: Differential information for a free particle (ωS = 0), coupled to the standard QBM environment (Eq. 4.23). Differential mutual information (DMI) represents the additional information yielded by a particular band (ω ) if that band is captured after a fraction f of the total environment has already been captured. DMI thus combines the frequency-resolution of local information with the contextual dependence of partial information (i.e., PIPs). See text (Appendix C.2) for discussion. C.2 Quantum Brownian motion: Differential information In this appendix, we present plots (Figs. C.3, C.4, and C.5) of the differential information for our base environment (see Eq. 4.23), coupled to systems oscillating at frequencies of ωS = 1, 4, 16. We discussed differential information in Sec. 4.5; ∆I (ω, f ) is the amount of additional information contributed by the band ω , after a fraction f of the bands (randomly chosen) has already been captured. Differential information allows us to identify which bands of the environment retain their usefulness even after quite a bit of information has already been captured. Differential information data is presented in four figures, consisting of four plots each. Each figure shows ∆I (ω, f ) at four different times over the evolution of the universe. By viewing the individual plots within a figure as a time series, we can form a picture of how information flows between subenvironments over time. Each plot’s profile along the ωE axis indicates the relative amount of information provided by different bands of the environment. For instance, information about the free particle (Fig. C.3) becomes concentrated in the low-frequency environments as time progresses. This is indicated by a reduced ∆I for ωE > 0, regardless of f . At the same time, variation along the f axis indicates how contextual a given subenviron- 228 Differential Information: ωsys = 1, T = 2 ∆IS:ε 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 ∆IS:ε 2.5 2 1.5 1 0.5 0 0 Differential Information: ωsys = 1, T = 8 ω ω 16 12 0.05 8 4 0.1 fraction 0.15i of ε already captured ωε 16 12 0.05 8 4 0.1 fraction 0.15i of ε already captured ωε (a) Differential Information: ωsys = 1, T = 20 ∆IS:ε 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 ∆IS:ε 0.3 0.25 0.2 0.15 0.1 0.05 0 0 (b) Differential Information: ωsys = 1, T = 40 ω ω 16 12 0.05 8 4 0.1 fraction 0.15i of ε already captured ωε 16 12 0.05 8 4 0.1 fraction 0.15i of ε already captured ωε (c) (d) Figure C.3: Differential information for a low-frequency (ωS = 1) oscillator, coupled to the standard QBM environment (Eq. 4.23). Differential mutual information (DMI) represents the additional information yielded by a particular band (ω ) if that band is captured after a fraction f of the total environment has already been captured. DMI thus combines the frequency-resolution of local information with the contextual dependence of partial information (i.e., PIPs). See text (Appendix C.2) for discussion. ment’s information is. A given band ω may have information which is unique – e.g., capturing that band always yields novel information regardless of what is already possessed – or information which is redundant. If the information in band ω is redundant, then it will become much less useful after a small fraction f 1 has been captured. For example, the plots for ωS = 1, 4 (Figs. C.3,C.4) show a rapid decline in ∆I for f > 0. This indicates redundancy. For ωS = 0, 16, however, certain bands provide unique – i.e., nonredundant – information. This is apparent because ∆I does not drop rapidly with f for those bands. We present these plots of differential mutual information because it provides a unique perspective into information storage. With DMI, we can identify which bands have redundant information, and which bands have non-redundant information. However, this should be considered work in progress. The data is thus presented with, as it were, no warranty or guarantee of fitness – which is why it is in the Appendix. 229 Differential Information: ωsys = 4, T = 2 ∆IS:ε 2.5 2 1.5 1 0.5 0 0 ∆IS:ε 3.5 3 2.5 2 1.5 1 0.5 0 0 Differential Information: ωsys = 4, T = 8 ω ω 16 12 0.05 8 4 0.1 fraction 0.15i of ε already captured ωε 16 12 0.05 8 4 0.1 fraction 0.15i of ε already captured ωε (a) Differential Information: ωsys = 4, T = 20 ∆IS:ε 2.5 2 1.5 1 0.5 0 0 ∆IS:ε 1 0.8 0.6 0.4 0.2 0 0 (b) Differential Information: ωsys = 4, T = 40 ω ω 16 12 0.05 8 4 0.1 fraction 0.15i of ε already captured ωε 16 12 0.05 8 4 0.1 fraction 0.15i of ε already captured ωε (c) (d) Figure C.4: Differential information for a high-frequency (ωS = 4) oscillator, coupled to the standard QBM environment (Eq. 4.23). Differential mutual information (DMI) represents the additional information yielded by a particular band (ω ) if that band is captured after a fraction f of the total environment has already been captured. DMI thus combines the frequency-resolution of local information with the contextual dependence of partial information (i.e., PIPs). See text (Appendix C.2) for discussion. 230 Differential Information: ωsys = 16, T = 2 ∆IS:ε 5 4 3 2 1 0 0 ∆IS:ε 5 4 3 2 1 0 0 Differential Information: ωsys = 16, T = 8 ω ω 16 12 16 12 0.05 0.1 fraction 0.15i of ε already captured ωε 0.05 0.1 fraction 0.15i of ε already captured ωε (a) Differential Information: ωsys = 16, T = 20 ∆IS:ε 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 ∆IS:ε 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 (b) Differential Information: ωsys = 16, T = 40 ω ω 16 12 16 12 0.05 0.1 fraction 0.15i of ε already captured ωε 0.05 0.1 fraction 0.15i of ε already captured ωε (c) (d) Figure C.5: Differential information for an ultra-high-frequency (ωS = 16) oscillator, coupled to the standard QBM environment (Eq. 4.23). Differential mutual information (DMI) represents the additional information yielded by a particular band (ω ) if that band is captured after a fraction f of the total environment has already been captured. DMI thus combines the frequency-resolution of local information with the contextual dependence of partial information (i.e., PIPs). See text (Appendix C.2) for discussion. 231 Cyclic Mutual Information: ωsys = 0, T = 2 IS:ε f Cyclic Mutual Information: ωsys = 0, T = 8 IS:ε f 16 12 ω0 8 4 0.8 0.6 0.4 fraction of εi already captured 1 16 12 ω0 8 4 0.8 0.6 0.4 fraction of εi already captured 1 (a) Cyclic Mutual Information: ωsys = 0, T = 20 IS:ε f (b) Cyclic Mutual Information: ωsys = 0, T = 60 IS:ε f 16 12 ω0 8 4 0.8 0.6 0.4 fraction of εi already captured 1 16 12 ω0 8 4 0.8 0.6 0.4 fraction of εi already captured 1 (c) (d) Figure C.6: Cyclic information for a free particle (ωS = 0), coupled to the standard QBM environment (Eq. 4.23). Cyclic mutual information (CMI) is another way of jointly representing partial information and local information (like differential information, Figs. C.2-C.5). The mutual information between S and a wide band of the environment consisting of [ωmin . . . (ωmin + f Λ) mod Λ] is plotted, as a function of ωmin and f . C.3 Quantum Brownian motion: Cyclic PIPs In Figs. C.6-C.9, we plot cyclic mutual information, another way (like differential information) of representing the local and contextual information at the same time. The mutual information between S and a wide band of the environment consisting of [ωmin . . . (ωmin + f Λ) mod Λ] is plotted, as a function of ωmin and f . A “classical plateau” is visible for ωS = 1 and ωS = 4, in Figs. C.7 and C.8, but not in Figs. C.6 and C.9. 232 Cyclic Mutual Information: ωsys = 1, T = 2 IS:ε f Cyclic Mutual Information: ωsys = 1, T = 8 IS:ε f 16 12 ω0 8 4 0.8 0.6 0.4 fraction of εi already captured 1 16 12 ω0 8 4 0.8 0.6 0.4 fraction of εi already captured 1 (a) Cyclic Mutual Information: ωsys = 1, T = 20 IS:ε f (b) Cyclic Mutual Information: ωsys = 4, T = 60 IS:ε f 16 12 ω0 8 4 0.8 0.6 0.4 fraction of εi already captured 1 16 12 ω0 8 4 0.8 0.6 0.4 fraction of εi already captured 1 (c) (d) Figure C.7: Cyclic information for a low-frequency oscillator (ωS = 1), coupled to the standard QBM environment (Eq. 4.23). Cyclic mutual information (CMI) is another way of jointly representing partial information and local information (like differential information, Figs. C.2-C.5). The mutual information between S and a wide band of the environment consisting of [ωmin . . . (ωmin + f Λ) mod Λ] is plotted, as a function of ωmin and f . 233 Cyclic Mutual Information: ωsys = 4, T = 2 IS:ε f Cyclic Mutual Information: ωsys = 4, T = 8 IS:ε f 16 12 ω0 8 4 0.8 0.6 0.4 fraction of εi already captured 1 16 12 ω0 8 4 0.8 0.6 0.4 fraction of εi already captured 1 (a) Cyclic Mutual Information: ωsys = 4, T = 20 IS:ε f (b) Cyclic Mutual Information: ωsys = 4, T = 60 IS:ε f 16 12 ω0 8 4 0.8 0.6 0.4 fraction of εi already captured 1 16 12 ω0 8 4 0.8 0.6 0.4 fraction of εi already captured 1 (c) (d) Figure C.8: Cyclic information for a high-frequency oscillator (ωS = 4), coupled to the standard QBM environment (Eq. 4.23). Cyclic mutual information (CMI) is another way of jointly representing partial information and local information (like differential information, Figs. C.2C.5). The mutual information between S and a wide band of the environment consisting of [ωmin . . . (ωmin + f Λ) mod Λ] is plotted, as a function of ωmin and f . 234 Cyclic Mutual Information: ωsys = 16, T = 2 IS:ε f Cyclic Mutual Information: ωsys = 16, T = 8 IS:ε f 16 12 ω0 8 4 0.8 0.6 0.4 fraction of εi already captured 1 16 12 ω0 8 4 0.8 0.6 0.4 fraction of εi already captured 1 (a) Cyclic Mutual Information: ωsys = 16, T = 20 IS:ε f (b) Cyclic Mutual Information: ωsys = 16, T = 60 IS:ε f 16 12 ω0 8 4 0.8 0.6 0.4 fraction of εi already captured 1 16 12 ω0 8 4 0.8 0.6 0.4 fraction of εi already captured 1 (c) (d) Figure C.9: Cyclic information for an ultra-high-frequency oscillator (ωS = 16), coupled to the standard QBM environment (Eq. 4.23). Cyclic mutual information (CMI) is another way of jointly representing partial information and local information (like differential information, Figs. C.2-C.5). The mutual information between S and a wide band of the environment consisting of [ωmin . . . (ωmin + f Λ) mod Λ] is plotted, as a function of ωmin and f . 235 Appendix D Supporting Material on Pointer algebras D.1 The Gell-Mann matrices The Gell-Mann matrices, λ1 . . . λ8 , together with the identity 1l, form a basis for the 3dimensional representation of SU (3). The parameterization presented here is obtained from Introduction to Quantum Field Theory, by L. H. Ryder [123]. 1 1l = 0 0 0 1 0 0 0 , 1 0 0 , 0 0 1 , 0 0 λ1 = 1 0 0 λ4 = 0 1 0 λ7 = 0 0 1 0 0 0 0 0 0 0 , 0 1 0 , 0 0 −i 0 λ2 = i 0 0 , 000 0 0 −i 0 , λ5 = 0 0 i0 0 1 1 λ8 = √ 3 0 0 0 0 1 0 0 −2 10 λ3 = 0 −1 00 0 λ6 = 0 0 0 0 1 0 0 0 −i , i 0 ...
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Decoherence and Beyond - Decoherence and Beyond by Robin...

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