Proof of Page’s conjecture on the average entropy of a subsystem

Abstract

Page conjectured very recently [Phys. Rev. Lett. 71, 1291 (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\ensuremath{\le}n is given by ${\mathit{S}}_{\mathit{m},}$n=${\mathit{tsum}}_{\mathit{k}=\mathit{n}+1}^{\mathit{m}\mathit{n}}$1/k-(m-1)/2n. We outline a proof of this elegant formula.