  The density matrix formalism is a fundamental tool in studying various
problems in quantum information processing. In the space of density matrices,
the most well-known measures are the Hilbert-Schmidt and Bures-Hall ensembles.
In this work, the averages of quantum purity and von Neumann entropy for an
ensemble that interpolates between these two major ensembles are explicitly
calculated for finite-dimensional systems. The proposed interpolating ensemble
is a specialization of the $\theta$-deformed Cauchy-Laguerre two-matrix model
and new results for this latter ensemble are given in full generality,
including the recurrence relations satisfied by their associated bi-orthogonal
polynomials when $\theta$ assumes positive integer values.
