Information in black hole radiation for initial mixed states

The question of whether black holes deplete the Universe of information is usually addressed in terms of initial gravitationally collapsing pure states. Pure states are nongeneric, having null prior probability of occurrence. The more realistic case of initial mixed states is, thus, examined here. The average information in an m-dimensional system in a random mixed state is found to approach asymptotically, as m→∞, an upper limit of 1/6 nats, that is, 1/6 ln 2 or 0.240449 bits. Also, the average information in an m-dimensional subsystem A of an mn-dimensional system AB proves to be smaller if AB is in a random mixed state than in a random pure state. This finding reinforces the recently drawn conclusion of Page that information in black hole radiation may come out initially so slowly that it would never show up in an analysis perturbative in ${\mathit{M}}_{\mathrm{Planck}}$/M.