Skewness of von Neumann entanglement entropy

TL;DR

This paper derives an exact formula for the skewness of von Neumann entanglement entropy in quantum bipartite systems, enhancing understanding of its distribution's asymmetry and improving approximation methods.

Contribution

It provides the first exact expression for the third cumulant of von Neumann entropy, revealing distribution asymmetry and refining entropy distribution approximations.

Findings

Exact third cumulant formula for von Neumann entropy

Improved approximation of entropy distribution

New summation identities involving polygamma functions

Abstract

We study quantum bipartite systems in a random pure state, where von Neumann entropy is considered as a measure of the entanglement. Expressions of the first and second exact cumulants of von Neumann entropy, relevant respectively to the average and fluctuation behavior, are known in the literature. The focus of this paper is on its skewness that specifies the degree of asymmetry of the distribution. Computing the skewness requires additionally the third cumulant, an exact formula of which is the main result of this work. In proving the main result, we obtain as a byproduct various summation identities involving polygamma and related functions. The derived third cumulant also leads to an improved approximation to the distribution of von Neumann entropy.