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