Key points are not available for this paper at this time.
A sufficient condition for the quantum entropy, -Tr (), to be nondecreasing under a tracepreserving positive linear map is that the "totally random" state, 1, be stationary. It is argued that this condition will be satisfied by the quantum system consisting of the region external to one or more black holes. This would explain why a "generalized second law of thermodynamics" appears to hold in the presence of black holes.
Rafael D. Sorkin (Mon,) studied this question.