When Wyner and Ziv Met Bayes in Quantum-Classical Realm

TL;DR

This paper explores lossy quantum-classical source coding with side information, introducing a new formulation using a posterior channel, and establishes bounds and connections to classical rate-distortion theory.

Contribution

It introduces a novel posterior channel formulation for quantum-classical source coding with side information and derives single-letter bounds on performance.

Findings

Derived an inner bound on asymptotic performance using mutual information.

Established a connection between rate-distortion and rate-channel theories.

Formulated a block error constraint for the quantum setting.

Abstract

In this work, we address the lossy quantum-classical source coding with the quantum side-information (QC-QSI) problem. The task is to compress the classical information about a quantum source, obtained after performing a measurement while incurring a bounded reconstruction error. Here, the decoder is allowed to use the side information to recover the classical data obtained from measurements on the source states. We introduce a new formulation based on a backward (posterior) channel, replacing the single-letter distortion observable with a single-letter posterior channel to capture reconstruction error. Unlike the rate-distortion framework, this formulation imposes a block error constraint. An analogous formulation is developed for lossy classical source coding with classical side information (C-CSI) problem. We derive an inner bound on the asymptotic performance limit in terms of single-letter quantum and classical mutual information quantities of the given posterior channel for QC-QSI and C-CSI cases, respectively. Furthermore, we establish a connection between rate-distortion and rate-channel theory, showing that a rate-channel compression protocol attains the optimal rate-distortion function for a specific distortion measure and level.