Vivo-Pato-Oshanin关于冯诺伊曼熵涨落猜想的证明
A Proof of Vivo-Pato-Oshanin's Conjecture on the Fluctuation of von Neumann Entropy
摘要：
最近Vivo、Pato和Oshanin [Phys. Rev. E 93，052106（2016）]证明，对于纯态下希尔伯特维数$mn$的量子系统，维数$m\leq n$的子系统的冯诺伊曼熵的方差由\begin{equation*} -ψ_{1}\left(mn+1\right)+\frac{m+n}{mn+1}ψ_{1}\left(n\right)-\frac{(m+1)(m+2n+1)}{4n^{2}(mn+1)}, \end{equation*}给出，其中$ψ_{1}(\cdot)$是三伽马函数。我们给出了这个公式的证明。
英文摘要：
It was recently conjectured by Vivo, Pato, and Oshanin [Phys. Rev. E 93, 052106 (2016)] that for a quantum system of Hilbert dimension $mn$ in a pure state, the variance of the von Neumann entropy of a subsystem of dimension $m\leq n$ is given by \begin{equation*} -ψ_{1}\left(mn+1\right)+\frac{m+n}{mn+1}ψ_{1}\left(n\right)-\frac{(m+1)(m+2n+1)}{4n^{2}(mn+1)}, \end{equation*} where $ψ_{1}(\cdot)$ is the trigamma function. We give a proof of this formula.
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