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 ${\displaystyle (E,\rho )}$ 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 ${\displaystyle b_{1},b_{2}\in {\mathcal {B}}}$.

See also

• Quantum walk

References

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