Optimal continuity bound for the von Neumann entropy under energy constraints

TL;DR

This paper establishes a mathematically optimal continuity bound for the von Neumann entropy under energy constraints, advancing quantum information theory by providing a comprehensive solution to a previously limited problem.

Contribution

It introduces a globally optimal semicontinuity bound for von Neumann entropy under general energy constraints, solving an open problem in quantum information theory.

Findings

Derived an optimal Fano-type inequality for infinite alphabet variables.

Established optimal semicontinuity bounds for Shannon entropy.

Provided a complete solution for the continuity of von Neumann entropy under energy constraints.

Abstract

Using techniques proposed in [Sason, IEEE Trans. Inf. Th. 59, 7118 (2013)] and [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)], and based on the results from the latter, we construct a globally optimal continuity bound for the von Neumann entropy. This bound applies to any state under energy constraints imposed by arbitrary Hamiltonians that satisfy the Gibbs hypothesis. This completely solves the problem of finding an optimal continuity bound for the von Neumann entropy in this setting, previously known only for pairs of states that are sufficiently close to each other. Our main technical result, a globally optimal semicontinuity bound for the von Neumann entropy under general energy constraints, leads to this continuity bound. To prove it, we also derive an optimal Fano-type inequality for random variables with a countably infinite alphabet and a general constraint, as well as optimal semicontinuity and continuity bounds for the Shannon entropy in the same setting. In doing so, we improve the results derived in [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)].