Average relative entropy of random states

Abstract

Relative entropy serves as a cornerstone concept in quantum information theory. In this work, we study relative entropy of random states from major generic state models of Hilbert-Schmidt and Bures-Hall ensembles. In particular, we derive exact yet explicit formulas of average relative entropy of two independent states of arbitrary dimensions from the same ensemble as well as from two different ensembles. One ingredient in obtaining the results is the observed factorization of ensemble averages after evaluating the required unitary integral. The derived exact formula in the case of Hilbert-Schmidt ensemble complements the work by Kudler-Flam (2021 Phys Rev Lett 126 171603), where the corresponding asymptotic formula for states of equal dimensions was obtained based on the replica method.