Moments of quantum purity and biorthogonal polynomial recurrence
Shi‐Hao Li, Lu Wei · Journal of Physics A Mathematical and Theoretical · 2021
Abstract The Bures–Hall ensemble is a unique measure of density matrices that satisfies various distinguished properties in quantum information processing. In this work, we study the statistical behavior of entanglement over the Bures–Hall ensemble as measured by the simplest form of an entanglement measure—the quantum purity. The main results of this work are the exact second and third moment expressions of quantum purity valid for any subsystem dimensions, where the corresponding results in the literature are limited to the scenario of equal subsystem dimensions. In obtaining the results, we have derived recurrence relations of the underlying integrals over the Cauchy–Laguerre biorthogonal polynomials that may be of independent interest.