In mathematics, the quantum Markov chain is a reformulation of the ideas of a classical Markov chain, replacing the classical definitions of probability with quantum probability.
Very roughly, the theory of a quantum Markov chain resembles that of a measure-many automaton, with some important substitutions: the initial state is to be replaced by a density matrix, and the projection operators are to be replaced by positive operator valued measures.
More precisely, a quantum Markov chain is a pair
a quantum channel such that is a completely positive trace-preserving map, and
The pair must obey the quantum Markov condition, that for all