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

TL;DR

This paper establishes quantum analogs of classical information theory theorems, including rate distortion and source-channel separation, providing fundamental limits and conditions for lossy quantum data compression and transmission.

Contribution

It derives quantum rate distortion functions and source-channel separation theorems, extending classical results to the quantum domain with new formulas and bounds.

Findings

Quantum rate distortion function expressed via regularized entanglement of purification

Single-letter formula for entanglement-assisted quantum rate distortion

Necessary and sufficient conditions for quantum source transmission with distortion

Abstract

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 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 non-negative 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 codition for transmitting a memoryless source over a memoryless quantum channel up to a given distortion.