Quantum Markov chain explained

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 thatE: l{B} ⊗ l{B}\to l{B}

is a completely positive trace-preserving map, and

l{B} a C^*-algebra of bounded operators. The pair must obey the quantum Markov condition, that \operatorname{Tr} \rho (b _{1 ⊗} b _{2)} = \operatorname{Tr} \rho E(b _{1,} b _{2)}

for all

b _{1,b} _{2\in} l{B} .

See also

• Quantum walk

References

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