  Inspired by the theory of quantum information, I use two non-Hermitian random
matrix models - a weighted sum of circular unitary ensembles and a product of
rectangular Ginibre unitary ensembles - as building blocks of three new
products of random matrices which are generalizations of the Bures model. I
apply the tools of both Hermitian and non-Hermitian free probability to
calculate the mean densities of their eigenvalues and singular values in the
thermodynamic limit, along with their divergences at zero; the results are
supported by Monte Carlo simulations. I pose and test conjectures concerning
the relationship between the two densities (exploiting the notion of the
N-transform), the shape of the mean domain of the eigenvalues (an extension of
the single ring theorem), and the universal behavior of the mean spectral
density close to the domain's borderline (using the complementary error
function).
