  Bures distance holds a special place among various distance measures due to
its several distinguished features and finds applications in diverse problems
in quantum information theory. It is related to fidelity and, among other
things, it serves as a bona fide measure for quantifying the separability of
quantum states. In this work, we calculate exact analytical results for the
mean root fidelity and mean square Bures distance between a fixed density
matrix and a random density matrix, and also between two random density
matrices. In the course of derivation, we also obtain spectral density for
product of above pairs of density matrices. We corroborate our analytical
results using Monte Carlo simulations. Moreover, we compare these results with
the mean square Bures distance between reduced density matrices generated using
coupled kicked tops and find very good agreement.
