A 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)] 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 $m\leq n$ is given by \begin{equation*} -ψ_{1}\left(mn+1\right)+\frac{m+n}{mn+1}ψ_{1}\left(n\right)-\frac{(m+1)(m+2n+1)}{4n^{2}(mn+1)}, \end{equation*} where $ψ_{1}(\cdot)$ is the trigamma function. We give a proof of this formula.