  Sarkar and Kumar recently conjectured [J. Phys. A: Math. Theor.
$\textbf{52}$, 295203 (2019)] that for a bipartite system of Hilbert dimension
$mn$, the mean values of quantum purity and von Neumann entropy of a subsystem
of dimension $m\leq n$ over the Bures-Hall measure are given by
\begin{equation*} \frac{2n(2n+m)-m^{2}+1}{2n(2mn-m^2+2)} \end{equation*} and
\begin{equation*}
\psi_{0}\left(mn-\frac{m^2}{2}+1\right)-\psi_{0}\left(n+\frac{1}{2}\right),
\end{equation*} respectively, where $\psi_{0}(\cdot)$ is the digamma function.
We prove the above conjectured formulas in this work. A key ingredient of the
proofs is Forrester and Kieburg's discovery on the connection between the
Bures-Hall ensemble and the Cauchy-Laguerre biorthogonal ensemble studied by
Bertola, Gekhtman, and Szmigielski.
