论文标题
Sarkar-Kumar的猜想平均纠缠熵的证明
Proof of Sarkar-Kumar's Conjectures on Average Entanglement Entropies over the Bures-Hall Ensemble
论文作者
论文摘要
Sarkar和Kumar最近猜想了[J.物理。答：数学。理论。 $ \ textbf {52} $，295203（2019）]，对于希尔伯特尺寸的两部分系统$ mn $，量子纯度的平均值和von neumann dimension $ m \ m \ leq n $的von neumann entropy the Bures-hall Mesase \ frac {2n（2n+m）-m^{2} +1} {2n（2mn-m^2+2）} \ end {equation*}和\ \ begin {equination*} ψ_{0} \ left（mn- \ frac {m^2} {2} +1 \右）我们在这项工作中证明了上述猜想的公式。证据的关键要素是Forrester和Kieburg关于Bures-Bures合奏与Bertola，Gekhtman和Szmigielski研究的Cauchy-Laguerre Biorthogonal合奏的发现。
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*} ψ_{0}\left(mn-\frac{m^2}{2}+1\right)-ψ_{0}\left(n+\frac{1}{2}\right), \end{equation*} respectively, where $ψ_{0}(\cdot)$ 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.
