Exact variance of von Neumann entanglement entropy over the Bures-Hall measure

TL;DR

This paper derives an explicit formula for the variance of von Neumann entanglement entropy over the Bures-Hall ensemble, enabling a Gaussian approximation of the entropy distribution in finite and large quantum systems.

Contribution

It provides the first explicit variance formula for von Neumann entropy in the Bures-Hall ensemble, enhancing understanding of quantum entanglement fluctuations.

Findings

Derived an explicit variance formula for von Neumann entropy

Established a Gaussian approximation for the entropy distribution

Conjectured the Gaussian as the limiting distribution for large systems

Abstract

The Bures-Hall distance metric between quantum states is a unique measure that satisfies various useful properties for quantum information processing. In this work, we study the statistical behavior of quantum entanglement over the Bures-Hall ensemble as measured by von Neumann entropy. The average von Neumann entropy over such an ensemble has been recently obtained, whereas the main result of this work is an explicit expression of the corresponding variance that specifies the fluctuation around its average. The starting point of the calculations is the connection between correlation functions of the Bures-Hall ensemble and these of the Cauchy-Laguerre ensemble. The derived variance formula, together with the known mean formula, leads to a simple but accurate Gaussian approximation to the distribution of von Neumann entropy of finite-size systems. This Gaussian approximation is also conjectured to be the limiting distribution for large dimensional systems.