Tight uniform continuity bound for a family of entropies

2017/07/13 by Eric P. Hanson, Hanson, Eric P., Nilanjana Datta +1 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Computer and information sciences #FOS: Physical sciences #Information Theory (cs.IT) #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #Quantum chaos and dynamical systems #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.1707.04249

openalex publication_date 2017/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a tight uniform continuity bound for a family of entropies which includes the von Neumann entropy, the Tsallis entropy and the α-Rényi entropy, S_α, for α∈ (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 (ε-ball) of a given state, and thus was the minimal state in majorization order in the ε-ball. This minimal state satisfies a particular semigroup property, which we exploit to prove our bound.

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