Random density matrices: analytical results for mean root fidelity and mean square Bures distance

Abstract

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.

Distinguishing Random and Black Hole Microstates

“…We note that an exact expression was recently found for the fidelity of two random density matrices, consistent with our large-N results[65]. Our result is complementary as the exact expression is very complicated and not tractable at large-N .…”

Gap probabilities for the Bures-Hall Ensemble and the Cauchy-Laguerre Two-Matrix Model

“…A number of statistics have been calculated for the Bures-Hall ensemble which are global in nature and only involve the one or two-point correlations such as fidelity [50], quantum purity E[Tr(ρ 2 )] [50], [34], [14], [28] and averages of the von Neumann entropy E[−Tr(ρ log ρ)] [35] and its higher moments [39], [52], [51], [30]. In [16] the smallest eigenvalue distribution of the fixed-trace Laguerre beta-ensemble, i.e.…”