Toward a Proof of Entropy Increase in the Presence of Quantum Black Holes

A sufficient condition for the quantum entropy, $-\mathrm{Tr}\left(\rho \ln \rho \right)$, to be nondecreasing under a tracepreserving positive linear map is that the "totally random" state, $\rho \propto 1$, 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.