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Quantum Markov chains. (English) Zbl 1152.81457
Summary: A new approach to quantum Markov chains is presented. We first define a transition operation matrix (TOM) as a matrix whose entries are completely positive maps whose column sums form a quantum operation. A quantum Markov chain is defined to be a pair \((G,E)\) where \(G\) is a directed graph and \(E=[E_{ij}]\) is a TOM whose entry \(E_{ij}\) labels the edge from vertex \(j\) to vertex \(i\). We think of the vertices of \(G\) as sites that a quantum system can occupy and \(E_{ij}\) is the transition operation from site \(j\) to site \(i\) in one time step. The discrete dynamics of the system is obtained by iterating the TOM \(E\). We next consider a special type of TOM called a transition effect matrix. In this case, there are two types of dynamics, a state dynamics and an operator dynamics. Although these two types are not identical, they are statistically equivalent. We next give examples that illustrate various properties of quantum Markov chains. We conclude by showing that our formalism generalizes the usual framework for quantum random walks.

MSC:
81S25 Quantum stochastic calculus
46L53 Noncommutative probability and statistics
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