Quantum Markov chain

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

• Gudder, Stanley. "Quantum Markov chains." Journal of Mathematical Physics 49.7 (2008): 072105.