Exact mean root fidelity and mean-square Bures distance between random density matrices

Distance measures between quantum states play a crucial role in various aspects of quantum information theory and have applications in a variety of physical problems. Among various distance measures, Bures distance stands out as an excellent candidate for distinguishability measures between quantum states due to its several notable features. In this work, we derived exact analytical results for the mean root fidelity and the mean-square Bures distance using the random matrix theory (RMT) techniques for the pairs of a fixed density matrix and a random density matrix, and two random density matrices, taken from the Hilbert-Schmidt probability measure. The key idea of our calculation was to reformulate the problems in terms of known RMT ensembles with the aid of the Laplace transform approach. We also obtained the spectral density for the product of the above-mentioned pairs of density matrices. We compared our random matrix theory-based analytical results with Monte-Carlo-based numerical simulations and found excellent agreement. Additionally, we corroborated these analytical results by contrasting them with the mean square Bures distance between random density matrices generated via coupled kicked tops.