Tight uniform continuity bound for a family of entropies

Abstract

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

arXiv comment16 pages, 4 figures. v2: added missing definition of Tsallis entropy, corrected minor typos

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• Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints
• Some general properties of unified entropies
• Smoothed generalized free energies for thermodynamics
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Cited by in corpus (3)

• From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions
• A tight uniform continuity bound for the Arimoto-Rényi conditional entropy and its extension to classical-quantum states
• Smooth entropy in axiomatic thermodynamics