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Decomposition of quantum Markov chains and its applications. (English) Zbl 1391.68079
Summary: Markov chains have been widely employed as a fundamental model in the studies of probabilistic and stochastic communicating and concurrent systems. It is well-understood that decomposition techniques play a key role in reachability analysis and model-checking of Markov chains. (Discrete-time) quantum Markov chains have been introduced as a model of quantum communicating systems and also a semantic model of quantum programs. The BSCC (Bottom Strongly Connected Component) and stationary coherence decompositions of quantum Markov chains were introduced in [V. Umanità, Probab. Theory Relat. Fields 134, No. 4, 603–623 (2006; Zbl 1130.46039); S. Ying et al., Lect. Notes Comput. Sci. 8052, 334–348 (2013; Zbl 1390.68444); B. Baumgartner and H. Narnhofer, Rev. Math. Phys. 24, No. 2, 1250001, 30 p. (2012; Zbl 1258.81056)]. This paper presents a new decomposition technique, namely periodic decomposition, for quantum Markov chains. We further establish a limit theorem for them. As an application, an algorithm to find a maximum dimensional noiseless subsystem of a quantum communicating system is given using decomposition techniques of quantum Markov chains.

MSC:
68Q60 Specification and verification (program logics, model checking, etc.)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
68Q12 Quantum algorithms and complexity in the theory of computing
68Q85 Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)
68Q87 Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)
81P68 Quantum computation
81S25 Quantum stochastic calculus
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