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

$E:{ℬ}\otimes {ℬ}\to {ℬ}$

is a completely positive trace-preserving map, and ${ℬ}$ a C^*-algebra of bounded operators. The pair must obey the quantum Markov condition, that

$\mathrm{Tr}\rho (b_1\otimes b_2)=\mathrm{Tr}\rho E(b_1,b_2)$

for all $b_1,b_2\in {ℬ}$.

See also

• Quantum walk

References

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