Simple proof of Page’s conjecture on the average entropy of a subsystem

Abstract

It is shown that Page's formula for the average entropy ${\mathit{S}}_{\mathit{m},}$n of a subsystem of dimension m\ensuremath{\le}n of a quantum system of Hilbert space dimension mn in a pure state [Phys. Rev. Lett. 71, 1291 (1993)] can be written in terms of the one-point correlation function of a Laguerre ensemble of random matrices. This leads to a proof of Page's conjecture, ${\mathit{S}}_{\mathit{m},}$n=${\mathit{tsum}}_{\mathit{k}=\mathit{n}+1}^{\mathit{m}\mathit{n}}$1/k-m-1/2n, which is simpler than that given by Foong and Kanno [Phys. Rev. Lett. 72, 1148 (1994)].