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

The Hellinger distance is a significant measure in quantum information theory, valued for its Riemannian and monotonic properties. It is simpler to compute than the Bures distance, a similar metric with applications in quantum information. Random quantum states play a critical role in secure communication, entanglement studies, quantum benchmarking, state tomography, and analyzing noisy quantum systems. While exact computations exist for distances like Bures and Hilbert-Schmidt distance measures, none exist for Hellinger distance for random density matrices. This work derives the mean and variance of the Hellinger distance between random density matrices, using Weingarten functions for unitary integrals. We also provide exact mean and square affinity results, proposing an approximate probability density function based on the gamma distribution for the Hellinger distance. Our analytical findings, confirmed through Monte Carlo simulations, show strong concordance with theoretical predictions.