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.

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 $$b_1,b_2\in \mathcal{B}$$.