From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions
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2023-07
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Journal Article
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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.
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published
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69 (7)
Pages / Article No.
4128 - 4144
Publisher
IEEE
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Subject
Uniform continuity bounds; entropies; infinite-dimensional systems; Fano-type inequality; Hölder continuity; Gibbs hypothesis
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