Average Entropy of a Quantum Subsystem
Siddhartha Sen · Physical Review Letters · 1996
It was recently conjectured by D. Page that if a quantum system of Hilbert space dimension $\mathrm{nm}$ is in a random pure state then the average entropy of a subsystem of dimension $m$ where $m\ensuremath{\le}n$ is ${S}_{m,n}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\left(\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}n+1}^{\mathrm{mn}}1/k\right)\ensuremath{-}(m\ensuremath{-}1)/2n$. In this Letter a simple proof of this conjecture is given.