Quantum interpolating ensemble: Average entropies and orthogonal polynomials

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 and physically relevant measures are the Hilbert-Schmidt ensemble and the Bures-Hall ensemble. In this work, we propose a generalized ensemble of density matrices, termed quantum interpolating ensemble, which is able to interpolate between these two seemingly unrelated ensembles. As a first step to understand the proposed ensemble, we derive the exact mean formulas of entanglement entropies over such an ensemble generalizing several recent results in the literature. We also derive some key properties of the corresponding orthogonal polynomials relevant to obtaining other statistical information of the entropies. Numerical results demonstrate the usefulness of the proposed ensemble in estimating the degree of entanglement of quantum states.