Continuity of quantum entropic quantities via almost convexity

TL;DR

This paper introduces the ALAFF method to establish continuity bounds for quantum entropic quantities, providing new and tight bounds for various relative entropies and mutual information measures.

Contribution

The paper presents the ALAFF method, a novel approach for proving continuity bounds of quantum entropic quantities, including new bounds for the Belavkin-Staszewski entropy.

Findings

Recovered known bounds for Umegaki relative entropy

Derived new bounds for Belavkin-Staszewski entropy

Proved almost concavity for certain quantum entropies

Abstract

Based on the proofs of the continuity of the conditional entropy by Alicki, Fannes, and Winter, we introduce in this work the almost locally affine (ALAFF) method. This method allows us to prove a great variety of continuity bounds for the derived entropic quantities. First, we apply the ALAFF method to the Umegaki relative entropy. This way, we recover known almost tight bounds, but also some new continuity bounds for the relative entropy. Subsequently, we apply our method to the Belavkin-Staszewski relative entropy (BS-entropy). This yields novel explicit bounds in particular for the BS-conditional entropy, the BS-mutual and BS-conditional mutual information. On the way, we prove almost concavity for the Umegaki relative entropy and the BS-entropy, which might be of independent interest. We conclude by showing some applications of these continuity bounds in various contexts within quantum information theory.