Entropy fluctuation formulas of fermionic Gaussian states

TL;DR

This paper derives explicit formulas for the variance of von Neumann entropy in fermionic Gaussian states, resolving a recent conjecture and introducing new mathematical tools for simplifying complex summations.

Contribution

It provides the first exact formulas for entropy variance in fermionic Gaussian states, including a resolution of a recent conjecture, using a novel simplification framework.

Findings

Exact variance formulas for fermionic Gaussian states without particle number constraints.

Resolution of a recent conjecture on entropy variance.

Development of new summation and hypergeometric transformation techniques.

Abstract

We study the statistical behaviour of quantum entanglement in bipartite systems over fermionic Gaussian states as measured by von Neumann entropy. The formulas of average von Neumann entropy with and without particle number constrains have been recently obtained, whereas the main results of this work are the exact yet explicit formulas of variances for both cases. For the latter case of no particle number constrain, the results resolve a recent conjecture on the corresponding variance. Different than existing methods in computing variances over other generic state models, the key ingredient in proving the results of this work relies on a new simplification framework. The framework consists of a set of new tools in simplifying finite summations of what we refer to as dummy summation and re-summation techniques. As a byproduct, the proposed framework leads to various new transformation formulas of hypergeometric functions.