Skewness of von Neumann entropy over Bures-Hall random states

TL;DR

This paper derives a closed-form formula for the skewness of von Neumann entropy in bipartite quantum states over the Bures-Hall ensemble, enhancing the understanding of its distribution's asymmetry.

Contribution

It provides the first exact formula for the third cumulant of von Neumann entropy, improving distribution approximations for quantum entanglement measures.

Findings

Exact third cumulant formula derived

Enhanced approximation of von Neumann entropy distribution

New summation identities involving polygamma functions

Abstract

We study the degree of entanglement, as measured by von Neumann entropy, of bipartite systems over the Bures-Hall ensemble. Closed-form expressions of the first two cumulants of von Neumann entropy over the ensemble have been recently derived in the literature. In this paper, we focus on its skewness by calculating the third cumulant that describes the degree of asymmetry of the distribution. The main result is an exact closed-form formula of the third cumulant, which leads to a more accurate approximation to the distribution of von Neumann entropy. The key to obtaining the result lies on finding a dozen of new summation identities in simplifying a large number of finite summations involving polygamma functions.