Convergence Properties of Good Quantum Codes for Classical Communication

TL;DR

This paper investigates the convergence properties of quantum codes used for classical communication over noisy quantum channels, extending classical results to the quantum setting and establishing the uniqueness of the optimal output distribution.

Contribution

It proves the uniqueness of the optimal output distribution for quantum codes and extends classical convergence results to the quantum domain using advanced mathematical techniques.

Findings

Uniqueness of the optimal output distribution in quantum codes.

Extension of vanishing error probability results to quantum codes.

Application of second-order converses using hypercontractivity in quantum settings.

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.