Decomposition of quantum Markov chains and its applications

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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 crucial 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. This thesis presents a new decomposition technique, namely periodic decomposition, for quantum Markov chains. This decomposition further helps us find sufficient and necessary conditions for limiting states of quantum Markov chains. To confirm the power of these decomposition techniques, we apply them to characterizing the one-shot zero-error capacity of quantum channels, finding the structure of quantum decoherence-free subsystems against quantum noises and super-activating quantum memory with entanglement via modeling the underlying quantum systems by quantum Markov chains.
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