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

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 sourcechannel 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.Index Terms-quantum rate distortion, reverse Shannon theorem, quantum Shannon theory, quantum data compression, source-channel separation

Quantum predictive filtering

“…In conventional rate-distortion coding, the goal is to encode the information in X with minimal distortion [23,[25][26][27][28]. But in practice, it may be difficult to define an appropriate distortion measure.…”

“…Classically, this problem does not arise, and one can always make a copy of X, but quantum mechanically, no such physical process exists in general [29]. We therefore consider a protocol similar to that used in quantum rate distortion coding [25][26][27][28]: we take the input to the problem to be a purification of the state ρ X , by introducing a second quantum system that acts as a reference, R. Information is then encoded in the memory by mapping XR to M R. Subsequently, M R is mapped to M Y via the relevance channel.…”

“…Here we depart from the approach used in quantum rate distortion coding, as we do not use the usual qubit encoding rate to quantify cost, and the usual entanglement fidelity to quantify quality [25][26][27][28]. Instead, we measure quality by predictive power, i.e., the relevant information quantifying the correlations between M and Y .…”

“…The protocol we consider is similar to that used in quantum rate distortion coding [25][26][27][28]. Starting from the purification, ψ XR , information is encoded in the memory by a map,…”

On Zero-Error Communication via Quantum Channels in the Presence of Noiseless Feedback

“…Special thanks to Will Matthews for pointing out an error in our first proof of Shannon's Conjecture. We are also grateful to an anonymous reviewer for pointing out to us that the second part of the proof of Lemma 12 was presented in [21] to show the superadditivity of mutual information.…”