Variance of relative entropy over random states

Overview & Synopsis

Average relative entropy over major generic random states is recently studied. In this work, the investigation is extended by deriving the variances of relative entropy over the Hilbert--Schmidt and Bures--Hall ensembles, which characterize its fluctuations around the average value. Exact results for the variances of relative entropy between independent states drawn from the same ensemble and from two different ensembles are obtained. These results further indicate that the average relative entropy is typical in the considered limiting regimes. In addition to the matrix-level calculations similar to the average relative entropy, eigenvalue correlation functions of the ensembles are involved in the variance computations through the use of zonal polynomials.