Some continuity properties of quantum R\'enyi divergences

TL;DR

This paper establishes a key continuity property of quantum Renyi divergences, proving a strong converse property in quantum channel discrimination and analyzing the continuity of various quantum Renyi divergences.

Contribution

It introduces a new continuity property of sandwiched Renyi divergences and applies it to prove the strong converse in quantum channel discrimination.

Findings

Proved the equality of two threshold values in quantum channel discrimination.

Established a new continuity property of sandwiched Renyi divergences.

Analyzed continuity properties of various quantum Renyi divergences.

Abstract

In the problem of binary quantum channel discrimination with product inputs, the supremum of all type II error exponents for which the optimal type I errors go to zero is equal to the Umegaki channel relative entropy, while the infimum of all type II error exponents for which the optimal type I errors go to one is equal to the infimum of the sandwiched channel R\'enyi $\alpha$-divergences over all $\alpha>1$. We prove the equality of these two threshold values (and therefore the strong converse property for this problem) using a minimax argument based on a newly established continuity property of the sandwiched R\'enyi divergences. Motivated by this, we give a detailed analysis of the continuity properties of various other quantum (channel) R\'enyi divergences, which may be of independent interest.