Average Entropy of a Subsystem

TL;DR

There is less than one-half unit of information, on average, in the smaller subsystem of a total system in a random pure state.

Abstract

If a quantum system of Hilbert space dimension $mn$ is in a random pure state, the average entropy of a subsystem of dimension $m\leq n$ is conjectured to be $S_{m,n}=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n}$ and is shown to be $\simeq \ln m - \frac{m}{2n}$ for $1\ll m\leq n$. Thus there is less than one-half unit of information, on average, in the smaller subsystem of a total system in a random pure state.