Quantum channel coding: Approximation algorithms and strong converse exponents

TL;DR

This paper investigates quantum channel coding relaxations, establishing equivalences between assistance types, and provides approximation algorithms with bounds for success probabilities, advancing understanding of quantum communication limits.

Contribution

It introduces a semi-definite programming relaxation called meta-converse, proves an approximation ratio for measurement channels, and characterizes the strong converse exponent for quantum channels.

Findings

Equivalence between non-signaling assistance and meta-converse in success probabilities

Approximation ratio of (1 - e^{-1}) for measurement channels

Dimension-dependent approximation ratio for fully quantum channels

Abstract

We study relaxations of entanglement-assisted quantum channel coding and establish that non-signaling assistance and a natural semi-definite programming relaxation\, -- \,termed meta-converse\, -- \,are equivalent in terms of success probabilities. We then present a rounding procedure that transforms any non-signaling-assisted strategy into an entanglement-assisted one and prove an approximation ratio of $(1 - e^{-1})$ in success probabilities for the special case of measurement channels. For fully quantum channels, we give a weaker (dimension dependent) approximation ratio, that is nevertheless still tight to characterize the strong converse exponent of entanglement-assisted channel coding [Li and Yao, IEEE Tran.~Inf.~Theory (2024)]. Our derivations leverage ideas from position-based coding, quantum decoupling theorems, the matrix Chernoff inequality, and input flattening techniques.