Random density matrices: Analytical results for mean fidelity and variance of squared Bures distance

One of the key issues in problems related to quantum information theory is concerned with the distinguishability of quantum states. In this context, Bures distance serves as one of the foremost choices among various distance measures. It also relates to fidelity, which is another quantity of immense importance in quantum information theory. In this work we derive exact results for the average fidelity and variance of the squared Bures distance between a fixed density matrix and a random density matrix and also between two independent random density matrices. These results go beyond the recently obtained results for the mean root fidelity and mean of the squared Bures distance. The availability of both mean and variance also enables us to provide a gamma-distribution-based approximation for the probability density of the squared Bures distance. The analytical results are corroborated using Monte Carlo simulations. Furthermore, we compare our analytical results with the mean and variance of the squared Bures distance between reduced density matrices generated using coupled kicked tops and a correlated spin chain system in a random magnetic field. In both cases, we find good agreement.