Average Rényi Entropy of a Subsystem in Random Pure State

Abstract

In this paper we examine the average Rényi entropy $S_α$ 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 Rényi entropy analytically for $m = α= 2$. We compare this analytical result with the approximate average Rényi entropy, which is shown to be very close. For general case we compute the average of the approximate Rényi entropy $\widetilde{S}_α (m,n)$ analytically. When $1 \ll n$, $\widetilde{S}_α (m,n)$ reduces to $\ln m - \fracα{2 n} (m - m^{-1})$, which is in agreement with the asymptotic expression of the average von Neumann entropy. Based on the analytic result of $\widetilde{S}_α (m,n)$ we plot the $\ln m$-dependence of the quantum information derived from $\widetilde{S}_α (m,n)$. It is remarkable to note that the nearly vanishing region of the information becomes shorten with increasing $α$, and eventually disappears in the limit of $α\rightarrow \infty$. The physical implication of the result is briefly discussed.