  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.
