A Quantum Multiparty Packing Lemma and the Relay Channel

TL;DR

This paper introduces a quantum multiparty packing lemma that generalizes the classical HSW theorem, enabling new quantum network communication schemes and providing both finite and asymptotic results.

Contribution

It presents a novel quantum packing lemma based on Sen's joint typicality, applicable to network scenarios, and derives quantum analogs of classical relay channel coding schemes.

Findings

Quantum packing lemma generalizes HSW theorem

Derived quantum coding schemes for relay channels

Provides finite blocklength and asymptotic results

Abstract

Optimally encoding classical information in a quantum system is one of the oldest and most fundamental challenges of quantum information theory. Holevo's bound places a hard upper limit on such encodings, while the Holevo-Schumacher-Westmoreland (HSW) theorem addresses the question of how many classical messages can be "packed" into a given quantum system. In this article, we use Sen's recent quantum joint typicality results to prove a one-shot multiparty quantum packing lemma generalizing the HSW theorem. The lemma is designed to be easily applicable in many network communication scenarios. As an illustration, we use it to straightforwardly obtain quantum generalizations of well-known classical coding schemes for the relay channel: multihop, coherent multihop, decode-forward, and partial decode-forward. We provide both finite blocklength and asymptotic results, the latter matching existing classical formulas. Given the key role of the classical packing lemma in network information theory, our packing lemma should help open the field to direct quantum generalization.