Proof of Sarkar-Kumar's Conjectures on Average Entanglement Entropies over the Bures-Hall Ensemble
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Sarkar and Kumar recently conjectured [J. Phys. A: Math. Theor. 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≤ n over the Bures-Hall measure are given by 2n(2n+m)-m^2+1/2n(2mn-m^2+2) and ψ_0(mn-m^2/2+1)-ψ_0(n+1/2), respectively, where ψ_0(·) 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.
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