Average Rényi entropy of a subsystem in random pure state

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 mn, 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 S_~α(m ,n ) analytically. When 1 ≪n , S_~α(m ,n ) reduces to lnm -α/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 S_~α(m ,n ) , we plot the lnm -dependence of the Rényi information derived from 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 →∞ . The physical implication of the result is briefly discussed.