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Convergence Properties of Good Quantum Codes for Classical Communication

Alptug Aytekin, Mohamed Nomeir, Lei Hu, Sennur Ulukus

TL;DR

The paper studies how the output statistics of quantum codes used for classical communication converge to the unique optimal output distribution of a quantum channel, extending classical results to the quantum setting. It proves that for good codes the per symbol quantum relative entropy between the actual and optimal output states vanishes, $\lim_{n\to\infty}\frac{1}{n}D(\omega_n\|\bar{\omega}_n)=0$, and develops a second order converse bound for block codes on strong converse channels, $\log(M_n) \le I(M;B^n)_\rho + 2\sqrt{n(|\mathcal{H}_B|-1)\log(1/(1-\epsilon))} + \log(1/(1-\epsilon))$, with a corresponding per letter convergence result for additive channels. The appendix provides a rigorous second order proof using weighted Rényi divergences and hypercontractivity of quantum depolarizing semigroups. Overall, the work extends classical empirical-output distribution results to quantum channels, clarifying the statistical structure of quantum codes and informing design for reliable classical communication over quantum media.

Abstract

An important part of the information theory folklore had been about the output statistics of codes that achieve the capacity and how the empirical distributions compare to the output distributions induced by the optimal input in the channel capacity problem. Results for a variety of such empirical output distributions of good codes have been known in the literature, such as the comparison of the output distribution of the code to the optimal output distribution in vanishing and non-vanishing error probability cases. Motivated by these, we aim to achieve similar results for the quantum codes that are used for classical communication, that is the setting in which the classical messages are communicated through quantum codewords that pass through a noisy quantum channel. We first show the uniqueness of the optimal output distribution, to be able to talk more concretely about the optimal output distribution. Then, we extend the vanishing error probability results to the quantum case, by using techniques that are close in spirit to the classical case. We also extend non-vanishing error probability results to the quantum case on block codes, by using the second-order converses for such codes based on hypercontractivity results for the quantum generalized depolarizing semi-groups.

Convergence Properties of Good Quantum Codes for Classical Communication

TL;DR

The paper studies how the output statistics of quantum codes used for classical communication converge to the unique optimal output distribution of a quantum channel, extending classical results to the quantum setting. It proves that for good codes the per symbol quantum relative entropy between the actual and optimal output states vanishes, , and develops a second order converse bound for block codes on strong converse channels, , with a corresponding per letter convergence result for additive channels. The appendix provides a rigorous second order proof using weighted Rényi divergences and hypercontractivity of quantum depolarizing semigroups. Overall, the work extends classical empirical-output distribution results to quantum channels, clarifying the statistical structure of quantum codes and informing design for reliable classical communication over quantum media.

Abstract

An important part of the information theory folklore had been about the output statistics of codes that achieve the capacity and how the empirical distributions compare to the output distributions induced by the optimal input in the channel capacity problem. Results for a variety of such empirical output distributions of good codes have been known in the literature, such as the comparison of the output distribution of the code to the optimal output distribution in vanishing and non-vanishing error probability cases. Motivated by these, we aim to achieve similar results for the quantum codes that are used for classical communication, that is the setting in which the classical messages are communicated through quantum codewords that pass through a noisy quantum channel. We first show the uniqueness of the optimal output distribution, to be able to talk more concretely about the optimal output distribution. Then, we extend the vanishing error probability results to the quantum case, by using techniques that are close in spirit to the classical case. We also extend non-vanishing error probability results to the quantum case on block codes, by using the second-order converses for such codes based on hypercontractivity results for the quantum generalized depolarizing semi-groups.
Paper Structure (8 sections, 7 theorems, 43 equations)

This paper contains 8 sections, 7 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Preliminaries
  4. Results
  5. Asymptotic Properties of Good Codes
  6. Properties of Good Codes for Strong Converse Channels
  7. Appendix
  8. Proof of Theorem \ref{['secondorder']}

Key Result

Lemma 1

For density matrices, $D(\rho\lVert\sigma)\geq0$, and moreover $D(\rho\lVert\sigma)=0$ iff $\rho=\sigma$.

Theorems & Definitions (17)

  • Remark 1
  • Definition 1: Good codes
  • Definition 2: Good codes for strong converse channels
  • Definition 3: Quantum relative entropy
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Remark 2
  • Theorem 1
  • Remark 3
  • ...and 7 more