Nondemolition Principle of Quantum Measurement Theory

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 Quantum measurement theory, based on the ordinary von Neumann or a generalized reduction postulate, was never an essential part of quantum physics but rather of metaphysics. First, this was because the orthodox quantum theory had always dealt with a closed quantum system while the object of measurement is an open system due to the interaction with the measurement apparatus. Second, the superposition principle of quantum mechanics, having dealt with simple algebras of observables, is in contradiction with the von Neumann projection postulate while it may be not so in the algebraic quantum theory with the corresponding superselection rules. Third, due to the dynamical tradition in quantum theory going on from the deterministic mechanics, the process of the measurement was always considered by theoretical physicists as simply just an ordinary interaction between two objects while any experimentalist or statistician knows that this is a stochastic process, giving rise to the essential difference between a priori and a posteriori description of the states. The last and most essential reason for such an unsatisfactory status of the quantum measurement theory was the limitations of the projection postulate applicable only to the instantaneous measurement of the observables with the discrete spectra, while the real experiments always have a finite duration and the most important observation is the measurement of the position having the continuous spectrum. Date: Received August 31, 1992 .