Geodesic
Ergodic properties of discrete quadratic stochastic processes defined on von Neumann algebras
Izvestiya. Mathematics , Tome 64 (2000) no. 5, pp. 873-890

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The paper contains necessary and sufficient conditions under which the ergodic principle and the regularity condition hold for discrete quantum quadratic stochastic processes defined on von Neumann algebras. A connection between these processes and Markov processes is established. 
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     author = {N. N. Ganikhodzhaev and F. M. Mukhamedov},
     title = {Ergodic properties of discrete quadratic stochastic processes defined on von {Neumann} algebras},
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N. N. Ganikhodzhaev; F. M. Mukhamedov. Ergodic properties of discrete quadratic stochastic processes defined on von Neumann algebras. Izvestiya. Mathematics , Tome 64 (2000) no. 5, pp. 873-890. http://geodesic.mathdoc.fr/item/IM2_2000_64_5_a0/