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

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.