Probabilistic approach to Page's formula for the entropy of a quantum subsystem

摘要

Page conjectured [Phys. Rev. Lett. 71, 1291-1294 (1993)] that if a quantum system of Hilbert space dimension mn is in a random pure state, the average entropy of a subsystem of dimension m≤n should be given by the simple and elegant formula S{sub}(m,n) = (∑{sub}(k=n+1)){sup}mn (1/k-(m-1)/2n). This formula appeared to be true and was first proved by Foong and Kanno [Phys. Rev. Lett. 72, 1148-1151 (1994)] by using Fourier transform, and next by Sanchez-Ruiz [Phys. Rev. E 52, 5653-5655 (1995)] and by Sen [Phys. Rev. Lett. 77, 1-3 (1996)] by using random matrix theory connected with generalized Laguerre polynomials. Adopting this last viewpoint, we detail the probabilistic approach to this problem. Especially, viewing any Gaussian vector as a product of a uniformly distributed unitary vector by an Erlang distribution, we give a new insight for the different entropies introduced by Page.