Average Entropy of a Subsystem

TL;DR

A simple proof of this conjecture is given that if a quantum system of Hilbert space dimension {ital nm} is in a random pure state then the average entropy of a subsystem of dimension{ital m} where {ital m}{le}{ital n} is {ital S}{sub {italm},{ital n}}=({summation}{ital k}={ital n}.

Abstract

It was recently conjectured by D. Page that if a quantum system of Hilbert space dimension $nm$ is in a random pure state then the average entropy of a subsystem of dimension $m$ where $m \leq n$ is $ S_{mn} = \sum^{mn}_{k=n+1}(1/k) - (m-1)/2n$. In this letter this conjecture is proved.