Continuity of the von Neumann Entropy

Dėnes Petz

Максим Евгеньевич Широков

Максим Евгеньевич Широков

The universal method of proving continuity of the von Neumann entropy on subsets of positive trace-class operators are considered. It makes possible to re-derive the known conditions of continuity of the entropy in the more general forms and to obtain the several new conditions. This method is based on the special approximation of the von Neumann entropy by the increasing sequence of concave continuous unitary invariant functions. Existence of this approximation is a corollary of the general property of the set of quantum states as a convex topological space called the strong stability property and considered in the first part of the paper.

Wilhelm Ochs

Becker, S., Nilanjana Datta, Michael G. Jabbour +1 more

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)].

Juan Pablo

Abstract The von Neumann entropy is one of the most relevant quantities in information theory, this makes it’s estimation a subject of research in the recent years. In the present paper we use the transport cost to make an improvement in the uniform continuity bound presented by Winter for the von Neumann entropy in infinite dimensional density matrices.