Quantum rate distortion, reverse Shannon theorems, and source-channel separation

Details Quantum rate distortion, reverse Shannon theorems, and source-channel separation Quantum rate distortion, reverse Shannon theorems, and source-channel separation

2011-08-24

arxiv.org/abs/1108.4940v1

Physics

Quantum Physics

18 pages, 4 figures

Scientific paper

We derive quantum counterparts of two key theorems of classical information theory, namely, the rate distortion theorem and the source-channel separation theorem. The rate-distortion theorem gives the ultimate limits on lossy data compression, and the source-channel separation theorem implies that a two-stage protocol consisting of compression and channel coding is optimal for transmitting a memoryless source over a memoryless channel. In spite of their importance in the classical domain, there has been surprisingly little work in these areas for quantum information theory. In the present work, we prove that the quantum rate distortion function is given in terms of the regularized entanglement of purification. Although this formula is regularized, at the very least it demonstrates that Barnum's conjecture on the achievability of the coherent information for quantum rate distortion is generally false. We also determine single-letter expressions for entanglement-assisted quantum rate distortion. Moreover, we prove several quantum source-channel separation theorems. The strongest of these are in the entanglement-assisted setting, in which we establish a necessary and sufficient codition for transmitting a memoryless source over a memoryless quantum channel up to a given distortion.

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