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Concept# Quantum Markov chain

Summary

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.
Introduction
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.
Formal statement
More precisely, a quantum Markov chain is a pair (E,\rho) with \rho a density matrix and E a quantum channel such that
:E:\mathcal{B}\otimes\mathcal{B}\to\mathcal{B}
is a completely positive trace-preserving map, and \mathcal{B} a C*-algebra of bounded operators. The pair must obey the quantum Markov condition, that
:\operatorname{Tr} \rho (b_1\otimes b_2) = \operatorname{Tr} \rho E(b_1, b_2)
for all b_1,b

Official source

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