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.

## Contents

* 1 Introduction
* 2 Formal statement
* 3 See also
* 4 References

## 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 [math]\displaystyle{ (E,\rho) }[/math] with [math]\displaystyle{ \rho }[/math] a density matrix and [math]\displaystyle{ E }[/math] a quantum channel such that

[math]\displaystyle{ E:\mathcal{B}\otimes\mathcal{B}\to\mathcal{B} }[/math]

is a completely positive trace-preserving map, and [math]\displaystyle{ \mathcal{B} }[/math] a C*-algebra of bounded operators. The pair must obey the quantum Markov condition, that

[math]\displaystyle{ \operatorname{Tr} \rho (b_1\otimes b_2) = \operatorname{Tr} \rho E(b_1, b_2) }[/math]

for all [math]\displaystyle{ b_1,b_2\in \mathcal{B} }[/math].

## See also

* Quantum walk

## References

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

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