All Moments of the Uniform Ensemble of Quantum Density Matrices
Robert R. Tucci
Abstract
Given a uniform ensemble of quantum density matrices ρ, it is useful to calculate the mean value over this ensemble of a product of entries of ρ. We show how to calculate such moments in this paper. The answer involves well known results from Group Representation Theory and Random Matrix Theory. This quantum problem has a well known classical counterpart: given a uniform ensemble of probability distributions P=(P1, P2, ..., PN) where the Pj are non-negative reals that sum to one, calculate the mean value over this probability simplex of products of P components. The answer to the classical problem follows from an integral formula due to Dirichlet.
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