Quantum Markov chain

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. 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 ( E , ρ ) {displaystyle (E, ho )} with ρ {displaystyle ho } a density matrix and E {displaystyle E} a quantum channel such that is a completely positive trace-preserving map, and B {displaystyle {mathcal {B}}} a C*-algebra of bounded operators. The pair must obey the quantum Markov condition, that for all b 1 , b 2 ∈ B {displaystyle b_{1},b_{2}in {mathcal {B}}} .