# A Proof of Vivo-Pato-Oshanin's Conjecture on the Fluctuation of von Neumann Entropy

## TL;DR

This paper provides a rigorous proof for a conjecture regarding the variance of von Neumann entropy in bipartite quantum systems, confirming a specific formula involving special functions.

## Contribution

The paper offers the first complete proof of Vivo, Pato, and Oshanin's conjecture on the fluctuation of von Neumann entropy for quantum subsystems.

## Key findings

- Confirmed the conjectured variance formula for von Neumann entropy
- Validated the specific mathematical expression involving trigamma functions
- Contributed to the theoretical understanding of quantum entropy fluctuations

## Abstract

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*} -\psi_{1}\left(mn+1\right)+\frac{m+n}{mn+1}\psi_{1}\left(n\right)-\frac{(m+1)(m+2n+1)}{4n^{2}(mn+1)}, \end{equation*} where $\psi_{1}(\cdot)$ is the trigamma function. We give a proof of this formula.