On expansion of quantum quadratic stochastic processes into fibrewise Markov processes defined on von Neumann algebras

We give an expansion of a quantum quadratic stochastic process (q.q.s.p.) into a so-called fibrewise Markov process and prove that, conversely, such an expansion uniquely determines the quantum quadratic stochastic process. As an application, we give a criterion (in terms of this expansion) for the q.q.s.p. to satisfy the ergodic principle. Using this result, we prove that a q.q.s.p. satisfies the ergodic principle if and only if the associated Markov process satisfies this principle. The expansion obtained is used to introduce a new notion of conjugacy of two q.q.s.p.'s and to study the relation between this notion and the ergodic principle.