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NONDEMOLITION PRINCIPLE OF QUANTUM MEASUREMENT THEORY

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

Abstract. We give an explicit axiomatic formulation of the quantum measurement theory which is free of the projection postulate. It is based on the generalized nondemolition principle applicable also to the unsharp, continuous– spectrum and continuous-in-time observations. The “collapsed state–vector” after the “objectification ” is simply treated as a random vector of the a posteriori state given by the quantum filtering, i.e., the conditioning of the a priori induced state on the corresponding reduced algebra. The nonlinear phenomenological equation of “continuous spontaneous localization ” has been derived from the Schrödinger equation as a case of the quantum filtering equation for the diffusive nondemolition measurement. The quantum theory of measurement and filtering suggests also another type of the stochastic equation for the dynamical theory of continuous reduction, corresponding to the counting nondemolition measurement, which is more relevant for the quantum experiments. 1. The status of quantum measurement theory

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•Jan 1, 1971•0 citations
Abstract. We give an explicit axiomatic formulation of the quantum measurement theory which is free of the projection postulate. It is based on the generalized nondemolition principle applicable also to the unsharp, continuous– spectrum and continuous-in-time observations. The “collapsed state–vector” after the “objectification ” is simply treated as a random vector of the a posteriori state given by the quantum filtering, i.e., the conditioning of the a priori induced state on the corresponding reduced algebra. The nonlinear phenomenological equation of “continuous spontaneous localization ” has been derived from the Schrödinger equation as a case of the quantum filtering equation for the diffusive nondemolition measurement. The quantum theory of measurement and filtering suggests also another type of the stochastic equation for the dynamical theory of continuous reduction, corresponding to the counting nondemolition measurement, which is more relevant for the quantum experiments. 1. The status of quantum measurement theory
A brief presentation is given of the basic concepts in quantum probability theory in analogy to the classical ones, as well as the necessary information about noncommutative stochastic integration and its explicit representation in the Fock space. Algebraic differential relations that define quantum diffusion motion with nondemolition observation are derived. In the Markov case we obtain the stochastic equation of quantum diffusion filtering which is analogous to the Zakai equation of classical nonlinear filtering. A quantum linear stochastic model with continuous observation is given for which this equation is reduced to a linear stochastic quantum filtering equation of Kalman-Busi type and to the operator Riccati equation. We also consider an example of observing the coordinate of a free quantum particle, for which the solution of a stationary quantum filtering problem is obtained.
A brief presentation of the basic concepts in quantum probability theory is given in comparison to the classical one. The notion of quantum white noise, its explicit representation in Fock space, and necessary results of noncommutative stochastic analysis and integration are outlined. Algebraic differential equations that unify the quantum non Markovian diffusion with continuous non demolition observation are derived. A stochastic equation of quantum diffusion filtering generalising the classical Markov filtering equation to the quantum flows over arbitrary *-algebra is obtained. A Gaussian quantum diffusion with one dimensional continuous observation is considered.The a posteriori quantum state difusion in this case is reduced to a linear quantum stochastic filter equation of Kalman-Bucy type and to the operator Riccati equation for quantum correlations. An example of continuous nondemolition observation of the coordinate of a free quantum particle is considered, describing a continuous collase to the stationary solution of the linear quantum filtering problem found in the paper.
Measurements continuous in time have been consistently introduced in quantum mechanics and applications worked out, mainly in quantum optics. In this context a quantum filtering theory has been developed giving the reduced state after a measurement when a certain trajectory of the measured observables is registered (the a posteriori states). In this paper a new derivation of filtering equations is presented for the cases of counting processes and of measurement processes of diffusive type. It is also shown that the equation for the a posteriori dynamics in the diffusive case can be obtained, by a suitable limit, from that in the counting case. Moreover, the paper is intended to clarify the meaning of the various concepts involved and to discuss the connections among them. As an illustration of the theory, simple models are worked out.
We develop a rigorous treatment of discontinuous stochastic unitary evolution for a system of quantum particles that interacts singularly with quantum "bubbles" at random instants of time. This model of a "cloud chamber" allows to watch and follow with a quantum particle along the trajectory in the cloud chamber by sequential unsharp localization of spontaneous scatterings of the bubbles. Thus, the continuous reduction and spontaneous localization theory is obtained as the result of quantum filtering theory, i.e., a theory describing the conditioning of the a priori quantum state by the measurement data. We show that in the case of indistinguishable particles the a posteriori dynamics is mixing, giving rise to an irreversible Boltzmann-type reduction equation. The latter coincides with the nonstochastic Schroedinger equation only in the mean field approximation, whereas the central limit yields Gaussian mixing fluctuations described by stochastic reduction equations of diffusive type.
It is shown that many dissipative phenomena of "old" quantum mechanics which appeared 100 years ago in the form of the statistics of quantum thermal noise and quantum spontaneous jumps, have never been explained by the "new" conservative quantum mechanics discovered 75 years ago by Heisenberg and Schroedinger. This led to numerous quantum paradoxes which are reconsidered in this paper. The development of quantum measurement theory, initiated by von Neumann, indicated a possibility for resolution of this interpretational crisis by divorcing the algebra of the dynamical generators from the algebra of the actual observables. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of past observables with the potential future. This rule, together with the self-compatibility of measurements insuring the consistency of histories, is called the nondemolition principle. The application of this causality condition in the form of the dynamical commutation relations leads to the derivation of the generalized von Neumann reductions, usharp, instantaneous, spontaneous, and even continuous in time. This gives a quantum probabilistic solution, in the form of the dynamical filtering equations, of the notorious measurement problem which was tackled unsuccessfully by many famous physicists starting with Schroedinger and Bohr. The simplest Markovian quantum stochastic model for the continuous in time measurements involves a boundary-value problem in second quantization for input "offer" waves in one extra dimension, and a reduction of the algebra of "actual" observables to an Abelian subalgebra for the output waves.
The central problem in the quantum theory of measurement, how to describe the process of state reduction in terms of the quantum mechanical formalism, is solved on the basis of the relativity of quantal states, which implies that once the apparatus is detected in a well-defined state, the object state must reduce to a corresponding one. This is a process termed by Schrodinger disentanglement. Here, it is essential to observe that Renninger's negative result does constitute an actual measurement process. From this point of view, Heisenberg's interpretation of his microscope experiment and the Einstein-Podolsky-Rosen arguments are reinvestigated. Satisfactory discussions are given to various experimental situations, such as the Stern-Gerlach-type experiment, successive measurements, macroscopic measurements, and Schrodinger's cat. Finally it is proposed to regard a state vector in quantum mechanics as an irreducible physical construct, in Margenau's sense, that is not further analyzable both mathematically and conceptually.