Tight uniform continuity bound for a family of entropies

TL;DR

This paper establishes a precise and tight uniform continuity bound for a broad family of entropies, including von Neumann, Tsallis, and Rényi entropies, with conditions for equality that are consistent across the family.

Contribution

It introduces a unified tight continuity bound for multiple entropies and characterizes equality conditions, extending previous work on minimal states in majorization order.

Findings

Derived a tight uniform continuity bound valid for several entropy measures.

Identified necessary and sufficient conditions for equality in the bound.

Connected the bound to a minimal state with a semigroup property in majorization.

Abstract

We prove a tight uniform continuity bound for a family of entropies which includes the von Neumann entropy, the Tsallis entropy and the $\alpha$-R\'enyi entropy, $S_\alpha$, for $\alpha\in (0,1)$. We establish necessary and sufficient conditions for equality in the continuity bound and prove that these conditions are the same for every member of the family. Our result builds on recent work in which we constructed a state which was majorized by every state in a neighbourhood ($\varepsilon$-ball) of a given state, and thus was the minimal state in majorization order in the $\varepsilon$-ball. This minimal state satisfies a particular semigroup property, which we exploit to prove our bound.