Geodesic
Quantum Markov states and quantum hidden Markov states
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 13-23

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The present article is a continuation of our previous paper “Remarks on quantum Markov states” (Funct. Anal. Appl. 49, No. 3 (2015), 205–209). We prove some propositions from that paper and define quantum Markov states and quantum hidden Markov states. Some connections are established with other definitions of these notions. We consider such states for lattices and graphs. We also consider an example with the Cayley tree. 
@article{ZNSL_2018_468_a1,
     author = {Z. I. Bezhaeva and V. I. Oseledets},
     title = {Quantum {Markov} states and quantum hidden {Markov} states},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {13--23},
     publisher = {mathdoc},
     volume = {468},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a1/}
}
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Z. I. Bezhaeva; V. I. Oseledets. Quantum Markov states and quantum hidden Markov states. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 13-23. http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a1/