On the exact variance of Tsallis entropy in a random pure state

TL;DR

This paper derives an exact formula for the variance of Tsallis entropy in bipartite quantum systems, providing explicit results for special cases and offering a new proof for von Neumann entropy variance.

Contribution

It introduces a precise variance formula for Tsallis entropy in random pure states, including simplified forms for special cases and a novel proof for von Neumann entropy variance.

Findings

Exact variance formula involving hypergeometric functions

Simplified variance expressions for quadratic entropy and small dimensions

New proof of von Neumann entropy variance

Abstract

Tsallis entropy is a useful one-parameter generalization of the standard von Neumann entropy in information theory. We study the variance of Tsallis entropy of bipartite quantum systems in a random pure state. The main result is an exact variance formula of Tsallis entropy that involves finite sums of some terminating hypergeometric functions. In the special cases of quadratic entropy and small subsystem dimensions, the main result is further simplified to explicit variance expressions. As a byproduct, we find an independent proof of the recently proved variance formula of von Neumann entropy based on the derived moment relation to the Tsallis entropy.