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

The Hellinger distance between quantum states is a significant measure in quantum information theory, known for its Riemannian and monotonic properties. It is easier to compute than the Bures distance, another measure that shares these properties, particularly in information-theoretic applications. Furthermore, in these applications, random quantum states are crucial for securing communication, studying entanglement, benchmarking quantum systems, aiding in quantum state tomography, and understanding how quantum systems behave in noisy environments. Recent works have computed exact results for distance measures such as Bures and Hilbert-Schmidt for random density matrices; however, no such results exist for the Hellinger distance. 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. Our derivation utilizes Weingarten functions to perform the necessary unitary group integrals and is supported by existing results for eigenvalue moments of random density matrices. 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.