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 , ρ ) {\displaystyle (E,\rho )} with ρ {\displaystyle \rho } a density matrix and E {\displaystyle E} a quantum channel such that
E : B ⊗ B → B {\displaystyle E:{\mathcal {B}}\otimes {\mathcal {B}}\to {\mathcal {B}}}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
Tr ρ ( b 1 ⊗ b 2 ) = Tr ρ E ( b 1 , b 2 ) {\displaystyle \operatorname {Tr} \rho (b_{1}\otimes b_{2})=\operatorname {Tr} \rho E(b_{1},b_{2})}for all b 1 , b 2 ∈ B {\displaystyle b_{1},b_{2}\in {\mathcal {B}}} .
See also
- Gudder, Stanley. "Quantum Markov chains." Journal of Mathematical Physics 49.7 (2008): 072105.