Simple and Tighter Derivation of Achievability for Classical Communication over Quantum Channels

TL;DR

This paper introduces a simplified, tighter derivation of one-shot achievability bounds for classical communication over quantum channels, improving existing results and applicable to various quantum information tasks.

Contribution

It presents a novel, elegant proof technique using the pretty-good measurement, removing the need for the Hayashi-Nagaoka inequality, and achieves the tightest known bounds for c-q channel coding.

Findings

Provides a closed-form one-shot bound similar to Holevo-Helstrom Theorem.

Yields unified asymptotic results in large, small, and moderate deviation regimes.

Achieves the tightest known one-shot capacity bounds, improving third-order rates.

Abstract

Achievability in information theory refers to demonstrating a coding strategy that accomplishes a prescribed performance benchmark for the underlying task. In quantum information theory, the crafted Hayashi-Nagaoka operator inequality is an essential technique in proving a wealth of one-shot achievability bounds since it effectively resembles a union bound in various problems. In this work, we show that the pretty-good measurement naturally plays a role as the union bound as well. A judicious application of it considerably simplifies the derivation of one-shot achievability for classical-quantum (c-q) channel coding via an elegant three-line proof. The proposed analysis enjoys the following favorable features. (i) The established one-shot bound admits a closed-form expression as in the celebrated Holevo-Helstrom Theorem. Namely, the error probability of sending $M$ messages through a c-q channel is upper bounded by the minimum error of distinguishing the joint channel input-output state against $(M-1)$ decoupled products states. (ii) Our bound directly yields asymptotic results in the large deviation, small deviation, and moderate deviation regimes in a unified manner. (iii) The coefficients incurred in applying the Hayashi-Nagaoka operator inequality are no longer needed. Hence, the derived one-shot bound sharpens existing results relying on the Hayashi-Nagaoka operator inequality. In particular, we obtain the tightest achievable $\epsilon$-one-shot capacity for c-q channel coding heretofore, improving the third-order coding rate in the asymptotic scenario. (iv) Our result holds for infinite-dimensional Hilbert space. (v) The proposed method applies to deriving one-shot achievability for classical data compression with quantum side information, entanglement-assisted classical communication over quantum channels, and various quantum network information-processing protocols.