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

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If you have a question about this talk, please contact Prof. Ramji Venkataramanan .

It is known that the Shannon entropy is discontinuous for discrete random variables with a countably infinite alphabet. Analogously, in the quantum case, the von Neumann entropy is discontinuous for quantum states on an infinite-dimensional, separable Hilbert space. However, continuity can be restored by imposing natural constraints on the random variables (resp. quantum states). We obtain the first tight mean-constrained continuity bound on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality. This classical result can be used to derive a tight energy-constrained continuity bound for the von Neumann entropy. This is joint work with Simon Becker and Michael Jabbour: IEEE Trans. Inf. Th., vol. 69, no. 7, p. 4128-4144 (2023).

This talk is part of the Information Theory Seminar series.

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