Article Dans Une Revue IEEE Transactions on Information Theory Année : 2023

From classical to quantum: uniform continuity bounds on entropies in infinite dimensions

Résumé

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 infinitedimensional 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 [1], [2] regarding the characterisation of states with finite entropy.

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hal-05022350 , version 1 (06-04-2025)

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Simon Becker, Nilanjana Datta, Michael G Jabbour. From classical to quantum: uniform continuity bounds on entropies in infinite dimensions. IEEE Transactions on Information Theory, 2023, 69 (7), pp.4128-4144. ⟨10.1109/TIT.2023.3248228⟩. ⟨hal-05022350⟩
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