  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 [1] and
also a semantic model of quantum programs [2]. The BSCC (Bottom Strongly
Connected Component) and stationary coherence decompositions of quantum Markov
chains were introduced in [3, 4, 5]. 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.
