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 ${\tilde{S}}_{\alpha }(m,n)$ analytically. When $1\ll n$, ${\tilde{S}}_{\alpha }(m,n)$ reduces to $\ln m-\frac{\alpha }{2n}(m-m^{-1})$, which is in agreement with the asymptotic expression of the average von Neumann entropy. Based on the analytic result of ${\tilde{S}}_{\alpha }(m,n)$ we plot the $\ln m$-dependence of the quantum information derived from ${\tilde{S}}_{\alpha }(m,n)$. 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 $\alpha \to \infty$. The physical implication of the result is briefly discussed.

14 pages, 3 figures, will appear in QIP