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qitd331 - Stochastic Quantum Dynamics I Born Rule Robert B...

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Stochastic Quantum Dynamics I. Born Rule Robert B. Gri ffi ths Version of 25 January 2010 Contents 1 Introduction 1 2 Born Rule 1 2.1 Statement of the Born Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.2 Incompatible sample spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Born rule using pre-probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Generalizations of the Born rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5 Limitations of the Born rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Two qubits 8 3.1 Pure states at t 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Properties of one qubit at t 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Correlations and conditional probabilities . . . . . . . . . . . . . . . . . . . . . . . . 10 References: CQT = Consistent Quantum Theory by Gri ffi ths (Cambridge, 2002), Chs. 8, 9, 10, 11. 1 Introduction Quantum dynamics, i.e., the time development of quantum systems, is fundamentally stochas- tic or probabilistic. It is unfortunate that this basic fact is not clearly stated in textbooks, including QCQI, which present to the student a strange mishmash of unitary dynamics followed by probabilities introduced by means of measurements . While the measurement talk is fundamentally sound when it is properly interpreted it can be quite confusing. Measurements: topic for separate set of notes What we will do is to introduce probabilities by means of some simple examples which show what one must be careful about when dealing with quantum systems, and the fundamental principles which allow the calculation of certain key probabilities related to quantum dynamics. 2 Born Rule 2.1 Statement of the Born Rule The Born rule can be stated in the following way. Let t 0 and t 1 be two times, and T ( t 1 , t 0 ) the unitary time development operator for the quantum system of interest. 1
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This system must be isolated during the time period under discussion, or at least it must not decohere, as otherwise the unitary T ( t 1 , t 0 ) cannot be defined. Assume a normalized state | ψ 0 at t 0 and an orthonormal basis {| φ k 1 } , k = 0 , 1 , . . . at t 1 . Then Pr( φ k 1 ) = Pr( φ k 1 | ψ 0 ) = | φ k 1 | T ( t 1 , t 0 ) | ψ 0 | 2 = ψ 0 | T ( t 0 , t 1 ) | φ k 1 φ k 1 | T ( t 1 , t 0 ) | ψ 0 (1) is the probability of | φ k 1 at t 1 given the initial state | ψ 0 at t 0 . In the argument of Pr() and sometimes in other places we omit the ket symbol; Pr( φ k 1 ) means the probability that the system is in the state | φ k 1 , or in the ray (one-dimensional subspace) generated by | φ k 1 , or has the property | φ k 1 corresponding to the projector | φ k 1 φ k 1 | = [ φ k 1 ]. The condition ψ 0 to the right of the vertical bar | is often omitted when it is evident from the context. (For more on conditional probabilities, see separate notes.) The final equality in (1) is a consequence of the general rule χ | A | ω * = ω | A | χ , together with the fact that T ( t 1 , t 0 ) = T ( t 0 , t 1 ). Mnemonic. In (1) subscripts indicate the time. In the matrix elements the time t j as an argument of T always lies next to the ket/bra corresponding to this time. It should be emphasized that the Born rule, (1), is on exactly the same level as the time- dependent Schr¨ odinger equation in terms of fundamental quantum principles. That is, it is a postulate or an axiom that does not emerge from anything else.
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