A Quantum Multiparty Packing Lemma and the Relay Channel

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.Comment: 20 page

Quantum intersection and union

In information theory, we often use intersection and union of the typical sets to analyze various communication problems. However, in the quantum setting it is not very clear how to construct a measurement which behaves analogous to intersection and union of the typical sets. In this work, we construct a projection operator which behaves very similar to intersection and union of the typical sets. Our construction relies on the Jordan's lemma. Using this construction we study the problem of communication over authenticated classical-quantum channels and derive its capacity. As another application of our construction, we study the problem of quantum asymmetric composite hypothesis testing. Further, we also prove a converse for the quantum binary asymmetric hypothesis testing problem which is arguably very similar in spirit to the converse given in the Thomas and Cover book for the classical version of this problem