Kurtosis of von Neumann entanglement entropy

TL;DR

This paper derives an exact formula for the fourth cumulant of von Neumann entanglement entropy in bipartite quantum systems, enhancing understanding of its distribution and tail behavior.

Contribution

It presents the first exact expression for the fourth cumulant of von Neumann entropy, improving finite-size distribution approximations and supporting the Gaussian limit conjecture.

Findings

Exact formula for the fourth cumulant derived

Enhanced finite-size distribution approximation

Supporting evidence for Gaussian limit of entropy distribution

Abstract

In this work, we study the statistical behavior of entanglement in quantum bipartite systems under the Hilbert-Schmidt ensemble as assessed by the standard measure - the von Neumann entropy. Expressions of the first three exact cumulants of von Neumann entropy are known in the literature. The main contribution of the present work is the exact formula of the corresponding fourth cumulant that controls the tail behavior of the distribution. As a key ingredient in deriving the result, we make use of newly observed unsimplifiable summation bases that lead to a complete cancellation. In addition to providing further evidence of the conjectured Gaussian limit of the von Neumann entropy, the obtained formula also provides an improved finite-size approximation to the distribution.