Proof of Vivo-Pato-Oshanin's conjecture on the fluctuation of von Neumann entropy

It was recently conjectured by Vivo, Pato, and Oshanin [Phys. Rev. E 93, 052106 (2016)2470-004510.1103/PhysRevE.93.052106] that for a quantum system of Hilbert dimension mn in a pure state, the variance of the von Neumann entropy of a subsystem of dimension m≤n is given by-ψ₁(mn+1)+m+n/mn+1ψ₁(n)-(m+1)(m+2n+1)/4n²(mn+1),where ψ₁(·) is the trigamma function. We give a proof of this formula.