Average R\'{e}nyi Entropy of a Subsystem in Random Pure State

TL;DR

This paper analytically investigates the average Rényi entropy of a subsystem in a random pure state, revealing how quantum information depends on subsystem size and Rényi parameter, with implications for understanding quantum entanglement.

Contribution

It provides an analytical computation of the average Rényi entropy for subsystems in random pure states, including asymptotic behavior and dependence on the Rényi parameter.

Findings

Analytical expression for average Rényi entropy when m=α=2.

Approximate average Rényi entropy closely matches the analytical result.

Quantum information derived from Rényi entropy diminishes with increasing α, disappearing at infinity.

Abstract

In this paper we examine the average R\'{e}nyi entropy $S_{\alpha}$ of a subsystem $A$ when the whole composite system $AB$ is a random pure state. We assume that the Hilbert space dimensions of $A$ and $AB$ are $m$ and $m n$ respectively. First, we compute the average R\'{e}nyi entropy analytically for $m = \alpha = 2$. We compare this analytical result with the approximate average R\'{e}nyi entropy, which is shown to be very close. For general case we compute the average of the approximate R\'{e}nyi entropy $\widetilde{S}_{\alpha} (m,n)$ analytically. When $1 \ll n$, $\widetilde{S}_{\alpha} (m,n)$ reduces to $\ln m - \frac{\alpha}{2 n} (m - m^{-1})$, which is in agreement with the asymptotic expression of the average von Neumann entropy. Based on the analytic result of $\widetilde{S}_{\alpha} (m,n)$ we plot the $\ln m$-dependence of the quantum information derived from $\widetilde{S}_{\alpha} (m,n)$. It is remarkable to note that the nearly vanishing region of the information becomes shorten with increasing $\alpha$, and eventually disappears in the limit of $\alpha \rightarrow \infty$. The physical implication of the result is briefly discussed.