Proof of Sarkar-Kumar's Conjectures on Average Entanglement Entropies over the Bures-Hall Ensemble

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-m2+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.