Continuity of the von Neumann entropy

TL;DR

This paper introduces a new method to prove the continuity of the von Neumann entropy on subsets of quantum states, generalizing known conditions and deriving new ones using an approximation approach.

Contribution

It presents a novel approximation technique for the von Neumann entropy based on finite rank decompositions and explores the strong stability property of quantum states.

Findings

Re-derivation of known continuity conditions for entropy

Introduction of new continuity conditions

Establishment of the approximation method for entropy

Abstract

A general method for proving continuity of the von Neumann entropy on subsets of positive trace-class operators is considered. This makes it possible to re-derive the known conditions for continuity of the entropy in more general forms and to obtain several new conditions. The method is based on a particular approximation of the von Neumann entropy by an increasing sequence of concave continuous unitary invariant functions defined using decompositions into finite rank operators. The existence of this approximation is a corollary of a general property of the set of quantum states as a convex topological space called the strong stability property. This is considered in the first part of the paper.