In an earlier paper by one of us [K.-E. Hellwig (1981)], elements of discrete quantum stochastic processes which arise when the classical probability space is replaced by quantum theory have been considered. In the present paper a general formulation is given and its properties are compared with those of classical stochastic processes. Especially, it is asked whether such processes can be Markovian. An example is given and similarities to methods in quantum statistical thermodynamics are pointed out.

By a counter-example, we show that the set of the three spin components Sx, Sy, Sz of spin-1 systems is not informationally complete.

Based on three different types of affine mappings between the corresponding convex sets of states, three different kinds of relations between quantum mechanics and classical physics are reviewed.

It is shown that, to a certain extent, the statistical framework of Hilbert-space quantum mechanics can be reformulated in classical terms.