Modulus of continuity of the quantum $f$-entropy with respect to the trace distance

TL;DR

This paper extends Audenaert's exact modulus of continuity result from von Neumann entropy to a broad class of quantum entropy functions defined by convex functions, using Schur majorization.

Contribution

It generalizes the known continuity bounds of quantum entropy to a wider class of entropy functions beyond von Neumann entropy.

Findings

Extended Audenaert's result to arbitrary convex functions f

Provided explicit bounds on the modulus of continuity for these entropy functions

Utilized Schur majorization in the proof

Abstract

A well-known result due to Fannes is a certain upper bound on the modulus of continuity of the von Neumann entropy with respect to the trace distance between density matrices; this distance is the maximum probability of distinguishing between the corresponding quantum states. Much more recently, Audenaert obtained an exact expression of this modulus of continuity. In the present note, Audenaert's result is extended to a broad class of entropy functions indexed by arbitrary continuous convex functions $f$ in place of the Shannon--von Neumann function $x\mapsto x\log_2x$. The proof is based on the Schur majorization.