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 ${displaystyle (E,rho )}$ with ${displaystyle rho }$ a density matrix and ${displaystyle E}$ a quantum channel such that

${displaystyle E:{mathcal {B}}otimes {mathcal {B}}to {mathcal {B}}}$

is a completely positive trace-preserving map, and ${displaystyle {mathcal {B}}}$ a C^*-algebra of bounded operators. The pair must obey the quantum Markov condition, that

${displaystyle operatorname {Tr} rho (b_{1}otimes b_{2})=operatorname {Tr} rho E(b_{1},b_{2})}$

for all ${displaystyle b_{1},b_{2}in {mathcal {B}}}$.

• Quantum walk