Noncommutative Markov chains associated to a preassigned evolution: an application to the quantum theory of measurement
type: Item
instance of: scholarly article
title: Noncommutative Markov chains associated to a preassigned evolution: an application to the quantum theory of measurement
zbMATH Open document ID: 0385.60087
DOI: 10.1016/0001-8708(78)90012-9
author: L. Accardi
published in: Advances in Mathematics
publication date: 1978-01-01T00:00:00Z
review text: In previous papers the author introduced the concepts of quantum noncommutative stochastic process and of quantum Markov process. The structure of the latter was analyzed in the cases of discrete and continuous one dimensional parameter. It was shown that non-relativistic quantum mechanical systems have a simple characterization in the framework of quantum Markov theory. In the classical case a Markovian evolution uniquely determines a Markov process up to the initial distribution. This is not the case for quantum processes: the evolution alone does not determine the process (i.e. the joint expectations) up to the initial distribution. This is due to the fact that the quantum theoretical analogue of the concept of conditional expectation does not preserve all the properties it has in the classical case (in particular it will not be, in general, a projection operator). It is then natural to investigate how to classify those quantum Markov processes which are ''associated'' to a given quantum Markovian evolution (i.e. a quantum dynamical semigroup in the sense of Lindblad). In this context the fact that a Markov process is ''associated'' to a given evolution does not mean, as in the classical case, that the evolution canonically associated to the process is the given one, but has a broader, technical meaning which is described in the paper. The problem formulated above is analyzed in the paper and its solution is reduced to the classification of the solutions of two ''compatibility'' equations. These equations are then solved in the case when the initial Markovian evolution is of Schrödinger's type. It is proved that in this case only two classes of quantum Markov processes can arise: one corresponds to the processes associated to the usual non-relativistic quantum systems; the other one corresponds to the quantum measurement process, in the sense that the evolution deduced from the compatibility equations is a generalization of the evolution postulated by J. von Neum…
reviewed by: L. Accardi
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 81P15
zbMATH DE Number: 3599284
MaRDI profile type: Publication
full work available at URL: 0001-8708(78)90012-9
OpenAlex ID: W1964336954
cites work: The noncommutative Markovian property
cites work: Nonrelativistic quantum mechanics as a noncommutative Markov process
cites work: Q5565478
cites work: Transformations of states in operator theory and dynamics
cites work: Characterizations of conditional expectation as a transformation on function spaces
cites work: \(C^*\)-algebras and \(W^*\)-algebras.
cites work: Q5640289
cites work: On Projection Maps of von Neumann Algebras.
cites work: On the projection of norm one in $W^ *$-algebras
cites work: Q5847989
description: scientific article; zbMATH DE number 3599284