Average Rényi entropy of a subsystem in random pure state

In this paper, we examine the average Rényi entropy \(S_{\alpha }\) $S_{\alpha }$ 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 \(m = \alpha = 2\) $m=\alpha =2$ . 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 \({\widetilde{S}}_{\alpha } (m,n)\) ${\tilde{S}}_{\alpha }(m,n)$ analytically. When \(1 \ll n\) $1\ll n$ , \({\widetilde{S}}_{\alpha } (m,n)\) ${\tilde{S}}_{\alpha }(m,n)$ reduces to \(\ln m - \frac{\alpha }{2 n} (m - m^{-1})\) $lnm-\frac{\alpha }{2n}(m-m^{-1})$ , which is in agreement with the asymptotic expression of the average von Neumann entropy. Based on the analytic result of \({\widetilde{S}}_{\alpha } (m,n)\) ${\tilde{S}}_{\alpha }(m,n)$ , we plot the \(\ln m\) $lnm$ -dependence of the Rényi information derived from \({\widetilde{S}}_{\alpha } (m,n)\) ${\tilde{S}}_{\alpha }(m,n)$ . It is remarkable to note that the nearly vanishing region of the information becomes shorten with increasing \(\alpha \) $\alpha$ and eventually disappears in the limit of \(\alpha \rightarrow \infty \) $\alpha \to \infty$ . The physical implication of the result is briefly discussed.