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

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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

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More precisely, a quantum Markov chain is a pair   with   a density matrix and   a quantum channel such that

 

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

 

for all  .

See also

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References

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