Sarkar and Kumar recently conjectured (2019 J. Phys. A: Math. Theor. 52 295203) 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 ##IMG## [http://ej.iop.org/images/1751-8121/53/23/235203/aab8d07ieqn1.gif] {$\frac{2n\left(2n\,+\,m\right)\,-\,{m}^{2}\,+\,1}{2n\left(2mn\,-\,{m}^{2}\,+\,2\right)}$} and ##IMG## [http://ej.iop.org/images/1751-8121/53/23/235203/aab8d07ieqn2.gif] {${\psi }_{0}\left(mn-\frac{{m}^{2}}{2}+1\right)-{\psi }_{0}\left(n+\frac{1}{2}\right),$} 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 et ...

中文翻译：

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关于Bures-Hall集合中平均纠缠熵的Sarkar-Kumar猜想的证明

Sarkar和Kumar最近猜想（2019 J. Phys。A：Math。Theor。52 295203），对于希尔伯特尺寸为mn的二分系统，量子纯度和尺寸为的子系统的von Neumann熵的平均值Bures–Hall度量由## IMG ##给出[http://ej.iop.org/images/1751-8121/53/23/235203/aab8d07ieqn1.gif] {$ \ frac {2n \ left（2n \ ，+ \，m \ right）\，-\，{m} ^ {2} \，+ \，1} {2n \ left（2mn \，-\，{m} ^ {2} \，+ \， 2 \ right）} $}和## IMG ## [http://ej.iop.org/images/1751-8121/53/23/235203/aab8d07ieqn2.gif] {$ {\ psi} _ {0} \ left（mn- \ frac {{m} ^ {2}} {2} +1 \ right）-{\ psi} _ {0} \ left（n + \ frac {1} {2} \ right），$ }，其中ψ0（⋅）是digamma函数。我们在这项工作中证明了上述猜想的公式。