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 automata, 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 with a density matrix and a quantum channel such that
is a completely positive trace-preserving map, and a C*-algebra of bounded operators. The pair must obey the quantum Markov condition, that
for all .
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