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. 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 $(E,\; ho)$ with $ho$ a density matrix and $E$ a quantum channel such that

:$E:mathcal\{B\}otimesmathcal\{B\}\; omathcal\{B\}$

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 :$operatorname\{Tr\}\; ho\; (b\_1otimes\; b\_2)\; =\; operatorname\{Tr\}\; ho\; E(b\_1,\; b\_2)$

for all $b\_1,b\_2in\; mathcal\{B\}$.