tqi14 - Phys. Rev. A 76 (2007) 062320; arXiv:0707.3752...

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Phys. Rev. A 76 (2007) 062320; arXiv:0707.3752 Types of Quantum Information Robert B. Grifths * Department of Physics, Carnegie-Mellon University, Pittsburgh, PA 15213, USA Quantum, in contrast to classical, information theory, allows for diFerent incompatible types (or species) of information which cannot be combined with each other. Distinguishing these incompat- ible types is useful in understanding the role of the two classical bits in teleportation (or one bit in one-bit teleportation), for discussing decoherence in information-theoretic terms, and for giving a proper de±nition, in quantum terms, of “classical information.” Various examples (some updating earlier work) are given of theorems which relate diFerent incompatible kinds of information, and thus have no counterparts in classical information theory. PACS numbers: I. INTRODUCTION Despite an enormous number oF publications in the ±eld oF quantum inFormation (see [1, 2] For useFul intro- ductions), neither the Fundamental principles underlying the subject, nor its connection with classical inForma- tion theory as developed by Shannon and his successors [3], is altogether clear. On the one hand there has been some dispute [4, 5] about whether Shannon’s ideas can be applied at all in the quantum domain. On the other hand there have been suggestions that the connection with Shannon’s ideas occurs only For macroscopic sys- tems or asymptotically large N (number oF transmissions, or whatever) limits, as in what is sometimes called “quan- tum Shannon” theory [6, 7]. The author’s position is that, to the contrary, there are perFectly consistent ways oF applying the basic ideas oF classical inFormation the- ory to small numbers (even one) oF microscopic quantum systems provided attention is paid to the Hilbert space structure oF quantum theory, and probabilities are intro- duced in a consistent Fashion. And, Further, that this approach has advantages in that simple systems are sim- pler to think about than complicated systems, so it is useFul to develop some intuition as to how they behave. One goal is to understand both classical and quantum in- Formation theory in Fully quantum terms, since the world is (most physicists believe) Fundamentally quantum me- chanical. The basic strategy oF this paper is based on the idea that quantum inFormation comes in a variety oF incom- patible types or species . Each type or species reFers to a certain class oF (typically microscopic) mutually- compatible properties oF a quantum system. As long as the discussion oF inFormation is limited to a single type, all the usual Formalism and intuition provided by clas- sical inFormation theory apply directly to the quantum * Electronic address: rgrif@cmu.edu domain. On the other hand, incompatible types oF quan- tum inFormation cannot be combined, as this makes no sense in the context oF standard Hilbert space quantum mechanics. Since in classical inFormation theory there is only a single type or species oF inFormation, or, equiva- lently, all di²erent types are compatible with each other,
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tqi14 - Phys. Rev. A 76 (2007) 062320; arXiv:0707.3752...

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