Exact mean and variance of the squared Hellinger distance for random density matrices

TL;DR

This paper derives exact formulas for the mean and variance of the squared Hellinger distance between random quantum density matrices, providing insights into quantum state similarity measures.

Contribution

It presents the first exact analytical expressions for the mean and variance of the Hellinger distance for random density matrices, including related affinity measures.

Findings

Analytical formulas for mean and variance of Hellinger distance

Approximate PDF of Hellinger distance using gamma distribution

Monte Carlo simulations confirm analytical results

Abstract

The Hellinger distance between quantum states is a significant measure in quantum information theory, known for its Riemannian and monotonic properties. It is also easier to compute than the Bures distance, another measure that shares these properties. In this work, we derive the mean and variance of the Hellinger distance between pairs of density matrices, where one or both matrices are random. Along the way, we also obtain exact results for the mean affinity and mean square affinity. The first two cumulants of the Hellinger distance allow us to propose an approximation for the corresponding probability density function based on the gamma distribution. Our analytical results are corroborated through Monte Carlo simulations, showing excellent agreement.