Noncommutative Markov chains associated to a preassigned evolution: an application to the quantum theory of measurement
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 Neumann for the quantum measurement process. The generalization consists in that this formalism ''intertwines'' Schrödinger's and von Neumann's quantum measurement evolution which, of course, can be achieved only in a discontinuous way. The discontinuity corresponds to the act of (virtual) measurement. The measurement is represented by a projection onto an algebra which the compatibility equations imply to be abelian. The case of continuous spectrum is not excluded, but in this case the projection cannot be faithful (i.e. nonzero on all positive observables). An explicit formula for the joint expectations at different times is derived and it is shown that this formula satisfies a kind of ''stochastic correspondence principle'' in the sense that, when the algebras are abelian, these joint expectations coincide with the ones associated to a classical dynamical system with random initial distribution, in the case of the quantum measurement process, and to the ones associated to a classical dynamical system with fully determined initial distribution (i.e. a Dirac delta function) in the case of usual quantum systems.
scientific article; zbMATH DE number 3599284