Quantum Markov chain - Alchetron, The Free Social Encyclopedia

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 ) with ρ a density matrix and E a quantum channel such that

E : B ⊗ B → 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

Tr ⁡ ρ ( b 1 ⊗ b 2 ) = Tr ⁡ ρ E ( b 1 , b 2 )

for all b 1 , b 2 ∈ B .

References

Quantum Markov chain Wikipedia

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Contents

Introduction

Formal statement

References