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When Wyner and Ziv Met Bayes in Quantum-Classical Realm

Mohammad Aamir Sohail, Touheed Anwar Atif, S. Sandeep Pradhan

TL;DR

This work reframes lossy source coding with side information in quantum and classical settings through rate-channel theory, replacing local distortion criteria with a single-letter posterior channel and a block-error constraint. For QC-QSI, it derives a single-letter inner bound on the achievable rate in terms of $I(X;R|B)_\sigma$, with $\sigma^{XRB}$ built from a reconstruction distribution $P_X$ in $\mathcal{A}(\rho^{AB},\mathcal{W}_{X\rightarrow RB})$, and proves achievability using Winter's measurement compression and sequential decoding. For C-CSI, it obtains an analogous inner bound $R \ge I(X;U) - I(U;Z)$ via randomized encoding and likelihood encoders within $\mathcal{A}(P_{XZ},W_{X|YZ})$. A connecting theorem shows that a rate-channel protocol achieving the global error criterion also attains the standard rate-distortion function for a specific distortion $d(x,y) = -c\log_2 W_{X|Y}(x|y) + b(x)$, establishing a bridge between rate-channel theory and rate-distortion theory in both quantum-classical and classical settings.

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.

When Wyner and Ziv Met Bayes in Quantum-Classical Realm

TL;DR

This work reframes lossy source coding with side information in quantum and classical settings through rate-channel theory, replacing local distortion criteria with a single-letter posterior channel and a block-error constraint. For QC-QSI, it derives a single-letter inner bound on the achievable rate in terms of , with built from a reconstruction distribution in , and proves achievability using Winter's measurement compression and sequential decoding. For C-CSI, it obtains an analogous inner bound via randomized encoding and likelihood encoders within . A connecting theorem shows that a rate-channel protocol achieving the global error criterion also attains the standard rate-distortion function for a specific distortion , establishing a bridge between rate-channel theory and rate-distortion theory in both quantum-classical and classical settings.

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.

Paper Structure

This paper contains 17 sections, 19 theorems, 61 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Lossy Quantum-Classical Source Coding with Quantum Side Information
  4. Lossy Classical Source Coding with Classical Side Information
  5. Connection Between Rate-Channel Theory and Rate-Distortion Theory
  6. Proof of Theorem \ref{['thm:QC-QSI']}
  7. Proof of Theorem \ref{['thm:C-CSI']}
  8. Proof of Theorem \ref{['thm:connection']}
  9. Useful Lemmas
  10. Proof of Proposition \ref{['prop:binning']}
  11. Proof of Proposition \ref{['prop:covering']}
  12. Proof of Proposition \ref{['prop:encoding_error']}
  13. Proof of Proposition \ref{['prop:code_dependent_RV']}
  14. Proof of Proposition \ref{['prop:qc_NC']}
  15. Proof of Proposition \ref{['prop:qc_NP']}
  16. ...and 2 more sections

Key Result

Theorem 5

For a given $(\rho_{AB},\mathsf{X},\mathcal{W}_{\mathsf{X}\rightarrow R B})$ QC-QSI source coding setup, a rate $R$ is achievable if $\mathcal{A}(\rho^{AB},\mathcal{W}_{X\rightarrow R B})$ is non-empty and where the quantum mutual information is computed with respect to the following quantum-classical state, $\mathcal{A}$ is the set of reconstruction distributions defined as $\phi^{R AB} \text

Figures (2)

  • Figure 1: Illustration of Lossy QC-QSI Compression Protocol.
  • Figure 2: Illustration of Lossy C-CSI Compression Protocol.

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Theorem 9
  • Definition 10: Achievability
  • ...and 17 more