Spectral statistics for the difference of two Wishart matrices

Abstract

In this work, we consider the weighted difference of two independent complex Wishart matrices and derive the joint probability density function of the corresponding eigenvalues in a finite-dimension scenario using two distinct approaches. The first derivation involves the use of unitary group integral, while the second one relies on applying the derivative principle. The latter relates the joint probability density of eigenvalues of a matrix drawn from a unitarily invariant ensemble to the joint probability density of its diagonal elements. Exact closed form expressions for an arbitrary order correlation function are also obtained and spectral densities are contrasted with Monte Carlo simulation results. Analytical results for moments as well as probabilities quantifying positivity aspects of the spectrum are also derived. Additionally, we provide a large-dimension asymptotic result for the spectral density using the Stieltjes transform approach for algebraic random matrices. Finally, we point out the relationship of these results with the corresponding results for difference of two random density matrices and obtain some explicit and closed form expressions for the spectral density and absolute mean.

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

“…Pure state σ: In this case, working in the eigenbasis of σ, it is evident that the only nonzero eigenvalue of the matrix √ σW √ σ equals one of the diagonal elements of W . The diagonal elements of W are independent and identically distributed as gamma random variable with PDF (see, e.g., [52]),…”

“…An example is the celebrated Page formula which gives the average von Neumann entropy associated with bipartite system [35]. For distance measures also, there has been some noteworthy progress in the context of random density matrices [20,[45][46][47][48][49][50][51][52], however much remains to be explored.…”

Random density matrices: Analytical results for mean root fidelity and the mean-square Bures distance

“…A widely used probability measure is the Hilbert-Schmidt measure on the set of the finite-dimensional mixed-sates [27][28][29][30][31][34][35][36][37][38][39][40][41][42][43][44][45]. Statistical investigation of various distance measures involving these Hilbert-Schmidt random states has become an active area of research due to its fundamental as well as applied aspects [21,[46][47][48][49][50][51][52][53][54].…”