On Strong Converse Theorems for Quantum Hypothesis Testing and Channel Coding

TL;DR

This paper presents a simplified proof of a key strong converse bound in quantum hypothesis testing using variational expressions and H"older's inequality, contributing to the theoretical foundations of quantum information theory.

Contribution

It provides an alternative, concise proof of a known strong converse bound, highlighting the role of variational expressions and H"older's inequality in quantum information theory.

Findings

Simplified proof of the one-shot strong converse bound

Demonstration that the variational expression follows from H"older's inequality

Enhanced understanding of the mathematical tools in quantum hypothesis testing

Abstract

Strong converse theorems refer to the study of impossibility results in information theory. In particular, Mosonyi and Ogawa established a one-shot strong converse bound for quantum hypothesis testing [Comm. Math. Phys, 334(3), 2014], which servers as a primitive tool for establishing a variety of tight strong converse theorems in quantum information theory. In this short note, we demonstrate an alternative one-line proof for this bound via the variational expression of measured R\'enyi divergences [Lett. Math. Phys, 107(12), 2017]. Then, we show that the variational expression is a direct consequence of H\"older's inequality.