Relating the Bures Measure to the Cauchy Two-Matrix Model

Abstract

The Bures metric is a natural choice in measuring the distance of density operators representing states in quantum mechanics. In the past few years a random matrix ensemble and the corresponding joint probability density function of its eigenvalues was identified. Moreover a relation with the Cauchy two-matrix model was discovered but never thoroughly investigated, leaving open in particular the following question: How are the kernels of the Pfaffian point process of the Bures random matrix ensemble related to the ones of the determinantal point process of the Cauchy two-matrix model and moreover, how can it be possible that a Pfaffian point process derives from a determinantal point process? We give a very explicit answer to this question. The aim of our work has a quite practical origin since the calculation of the level statistics of the Bures ensemble is highly mathematically involved while we know the statistics of the Cauchy two-matrix ensemble. Therefore we solve the whole level statistics of a density operator drawn from the Bures prior.MSC 2010: 33-XX, 60B20

Bures–Hall ensemble: spectral densities and average entropies

“…Interestingly, the above jpd also connects to the O(1) matrix model [53,54], as was discovered by Bertola et al while investigating the Cauchy two-matrix model [47]. Later on, Forrester and Kieburg demonstrated the explicit relationship between the unrestricted Bures-Hall ensembles and the Cauchy two-matrix model in [46]. This is very interesting since the former constitutes a Pfaffian point process, while the latter corresponds to a determinantal point process.…”

“…when n odd. As shown in the Appendix A, the Pfaffian in (21) can be evaluated to a yield a compact result for the normalization factor as [46] C = 2 n 2 +2αn π n/2 n j=1 Γ(j + α + 1/2) Γ(j + 1)Γ(j + 2α + 1)…”

“…for k = 1, ..., r. The g jk and H jk within the Pfaffian in (26) are as in (19), (22) and (23). In reference [46], the r-point correlation function for the unrestricted Bures-Hall ensemble has been derived by exploiting its relationship with the Cauchy twomatrix ensemble. It involves the Pfaffian of a 2r × 2r antisymmetric matrix with kernels involving integral over certain Meijer G-functions.…”

Exact evaluations of some Meijer G-functions and probability of all eigenvalues real for the product of two Gaussian matrices

“…Their study goes back to as early as 1950's where the focus was on exploring the behavior of dynamical systems, and the accompanying questions related to stochastic differential equations and Lyapunov exponents [1][2][3][4][5]. In the last few years there has been a revival in interest in their investigation because of their fascinating integrability properties [6][7][8][9][10][11][12][13][14][15][16][17][18][19] and identification of new problems where they can be applied, such as random graph states [20], combinatorics [21], quantum entanglement [14] and multilayered multiple channel telecommunication [9,22].…”

“…In the finite dimensionality case the determinants were found to involve certain Meijer G-functions. Meijer G-functions have also appeared as correlationkernels in product of Ginibre matrices or of truncated unitary matrices [8-10, 18, 19], Bures and Cauchy two-matrix models [16], and very recently in the results for product of a Wigner matrix and a Wishart matrix [23]. We give a brief introduction to Meijer G-functions in the next section.…”