Quantum Rate Distortion, Reverse Shannon Theorems, and Source-Channel Separation

• Published in: IEEE transactions on information theory 2013-01, Vol.59 (1), p.615-630
• Main Authors: Datta, N., Min-Hsiu Hsieh, Wilde, M. M.
• Format: Article
• Language: English
• Subjects: Applied sciences; Channel coding; Channels; Classical and quantum physics: mechanics and fields; Coding, codes; Coherence; Computer memory; Data compression; Detection, estimation, filtering, equalization, prediction; Distortion; Exact sciences and technology; Information theory; Information, signal and communications theory; Physics; Quantum data compression; Quantum entanglement; Quantum information; quantum rate distortion; quantum Shannon theory; Rate-distortion; Receivers; reverse Shannon theorem; Separation; Signal and communications theory; Signal, noise; source-channel separation; Telecommunications and information theory; Theorems; Transmission
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2012.2215575

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 this paper, we prove that the quantum rate-distortion function is given in terms of the regularized entanglement of purification. We also determine a single-letter expression for the entanglement-assisted quantum rate-distortion function, and we prove that it serves as a lower bound on the unassisted quantum rate-distortion function. This implies that the unassisted quantum rate-distortion function is nonnegative and generally not equal to the coherent information between the source and distorted output (in spite of Barnum's conjecture that the coherent information would be relevant here). 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 condition for transmitting a memoryless source over a memoryless quantum channel up to a given distortion.