A quantum quadratic stochastic process (of type either \(A\) or \(B\)) on a von Neumann algebra \(M\) with a fixed normal state \(\omega\) is a family of unital normal completely positive operators \(\{P^{s,t}:M \to M \otimes M, s,t \geq 0, t-s \geq 1\}\) which take values in the symmetric part of the tensor product and satisfy certain modifications (of type either \(A\) or \(B\)) of the Kolmogorov-Chapman equations, which for all \(s,\tau,t \geq 0\), \(\tau-s \geq 1\), \(t-\tau \geq 1\), determine the operator \(P^{s,t}\) in terms of the operators \(P^{s,\tau}\), \(P^{\tau,t}\) and states \((\omega\otimes \omega)\circ P^{0,s}\), \((\omega\otimes \omega)\circ P^{0,\tau}\). The author proves that in both cases one can associate to such a process in a canonical way its two so-called marginal processes, which are quantum processes in the usual sense (albeit with the increments defined only for the intervals of the length greater or equal to 1), one on \(M\) and one on \(M\otimes M\). In fact in all but one of the four arising cases the marginal processes are Markov. It is shown that one can reconstruct the original quantum quadratic process out of its marginal processes and that it shares some natural ergodic properties with its marginal processes.
For the entire collection see [Zbl 1197.81034].