Abstract: A quantum Markov chain(qMC) is a reformulation of the ideas of a classical Markov chain which has many applications as a statistical model in real-world processes. In this report, we firstly review the development of Markov chains and present the detail of stationary distributions and the limit theorem for Markov chains. In the quantum world, we show some results of long-term behaviours, including the quantum limit theorem, of quantum Markov chains. Analogous to the classical counterpart, we introduce irreducible quantum Markov chains and aperiodic quantum Markov chains respectively, and develop algorithms to check them. As a consequence, we obtain that an irreducible and aperiodic quantum Markov chain has a limiting state.

Date: 16 December 2015