Unfortunately, as pointed out early on in the development of quantum statistical mechanics by John von Neumann and others, this kind of argument does not carry over to quantum mechanics. In quantum mechanics, the ensemble cannot support an ever-finer mixing process, because of the finite dimensionality of the relevant portion of Hilbert space. Instead of converging closer and closer to the equilibrium ensemble (time-averaged ensemble) as in the classical case, the density matrix of the quantum system will constantly show evolution, even showing recurrences. Developing a quantum version of the "H"-theorem without appeal to the "Stosszahlansatz" is thus significantly more complicated.