From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions

Simon Becker, Nilanjana Datta, Michael G. Jabbour*

*Corresponding author for this work

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Abstract

We prove a variety of improved uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality. We then employ this classical result to derive a tight energy-constrained continuity bound for the von Neumann entropy. To deal with more general entropies in infinite dimensions, e.g. α-Rényi and α-Tsallis entropies, we develop a novel approximation scheme based on operator Hölder continuity estimates. Finally, we settle an open problem raised by Shirokov regarding the characterisation of states with finite entropy.

Original languageEnglish
JournalIEEE Transactions on Information Theory
Volume69
Issue number7
Pages (from-to)4128-4144
ISSN0018-9448
DOIs
Publication statusPublished - 2023

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