Average Rényi entropy of a subsystem in random pure state

Abstract

In this paper, we examine the average Rényi 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 mn, respectively. First, we compute the average Rényi entropy analytically for $m = \alpha = 2$. 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 ${\widetilde{S}}_{\alpha } (m,n)$ 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 Rényi 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.