Average Entropy of a Subsystem

TL;DR

This paper investigates the average entropy of a subsystem in a quantum system in a random pure state, providing a conjectured formula and asymptotic approximation that quantify the typical entanglement entropy.

Contribution

It proposes a conjecture for the average entropy of a subsystem in a random pure quantum state and derives its asymptotic form for large subsystem size.

Findings

Average entropy approximates to 1a0m - 1a0m/(2n) for large systems

Less than half a unit of information is stored in the smaller subsystem on average

The conjectured formula matches numerical evidence for the entropy distribution

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.