It is shown that Page's formula for the average entropy S'ffl,n of a subsystem of dimension m ::; n of a quantum system of Hilbert space dimension mn in a pure state [Phys. Rev. Lett. 71 A quantum system AB with Hilbert space dimension mn in a pure state (PAB = 11/!)(1/!1) has entropy SAB = 0. However, if AB is divided into two subsystems A and B, of dimension m and n, respectively (without loss of generality, we can take m ::; n), the entropy of the subsystems, SA = S B, is greater than zero unless A and Bare uncorrelated in the quantum sense (PAB = PA®PB) (1,2]. A convenient measure of the amount of entropy that arises from this coarse graining is the average (SA) = Srn,n of the entropy SA over all pure states of the total system AB, the average being defined with respect to the unitarily invariant Haar measure on the space of unit where Xi ~ 0, ~m(x) is the Vandermonde determinant of m variables,and, for positive integer z,where I' is Euler's constant. As conjectured by Page [2], Eq. (1) is equivalent toThe first proof of this conjecture was given by Foong and Kanno [3]. Here we show that a simpler proof can be achieved by noting that the second term in the right-hand side of (1) can be written as a one-dimensional integral in terms of the one-point correlation function of a Laguerre ensemble of complex Hermitian random matrices (see, e.g., Ref.[4]), whose explicit expression readily follows from a well-known result of random matrix theory [5,6]. Taking into account the symmetry between the m variables Xi, Eq. {1) can be written as {5)
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