Page conjectured very recently [Phys. Rev. Lett. 71, 1291(1993] that if a quantum system of Hilbert space dimension mn is in a random pure state, the average entropy of a subsystem of dimension m & n is given by S "= P&" &1 -™ . We outline a proof of this elegant formula.PACS numbers: 05.30.Ch, 02.90. +p, 03.65.w, 05.90.+m Very recently Page [1] considered the problem of getting entropy out of a system in a pure quantum state. He took a quantum system AB of Hilbert dimension mn with normalized density matrix, and divided it into two coupled

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The simplest random evolution is the motion of a particle on a straight line with constant velocity suffering random collisions, which reverse the velocity. The position x(t) of the particle at time t is the stochastic process that defines Kac's path-integral solution of the telegrapher equation. We view Kac's prescription as a path-dependent time reparametrization, which associates to the time t spent by the particle in going from xo to x along a path co, the time r(co) the particle would have taken to go from xo to x without reversal. We compute explicitly the probability distribution of the random variable r. It is then possible to compute by simple quadrature any path integral with an integrand function of rico).

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