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Canadian Journal of Physics

ON A THEORY OF MEASUREMENT OF INCOMPATIBLE OBSERVABLES IN QUANTUM MECHANICS

Eduard Prugovečki

Published Jun 1, 1967• 44 citations
Publisher

Abstract

The empirical meaning of the simultaneity of measurement of several observables is carefully analyzed. General criteria sufficient to guarantee the internal consistency of a framework including simultaneous measurements of incompatible observables are put forward. The case is illustrated with the example of simultaneous measurements of position and momenta of particles in quantum mechanics. It is pointed out that, in general, the outcome of measurements of incompatible observables cannot be described statistically by means of the probability measures of mathematical statistics, which have primarily been introduced only with compatible observables in mind. The concept of complex probability measure is put forward. A generalization of the current interpretation of quantum mechanics is proposed that would also include the case of incompatible observables, by taking advantage of the concept of complex probability measure.

DOI
10.1139/p67-170

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The empirical meaning of the simultaneity of measurement of several observables is carefully analyzed. General criteria sufficient to guarantee the internal consistency of a framework including simultaneous measurements of incompatible observables are put forward. The case is illustrated with the example of simultaneous measurements of position and momenta of particles in quantum mechanics. It is pointed out that, in general, the outcome of measurements of incompatible observables cannot be described statistically by means of the probability measures of mathematical statistics, which have primarily been introduced only with compatible observables in mind. The concept of complex probability measure is put forward. A generalization of the current interpretation of quantum mechanics is proposed that would also include the case of incompatible observables, by taking advantage of the concept of complex probability measure.
Quantum mechanics predicts the joint probability distributions of the outcomes of simultaneous measurements of commuting observables, but the current formulation lacks the operational definition of simultaneous measurements. In order to provide foundations of joint statistics of local general measurements on entangled systems in a general theoretical framework, the question is answered as to under what condition the outputs of two measuring apparatuses satisfy the joint probability formula for simultaneous measurements of their observables. For this purpose, all the possible state changes caused by measurements of an observable are characterized, and the notion of disturbance in measurement is formalized in terms of operations derived by the measuring interaction.
We study the problem of a simultaneous measurement of incompatible observables in the framework of Prugoveckiis empirical theory of measurement (i.e., the theory where the irreducible errors in individual measurements ought to be taken into account). We propose a solution for the main problem of this theory: negative probability distributions that arise for some states in the probabilistic description of a simultaneous measurement. These probabilities have a meaningful frequency interpretation. However, limits of relative frequencies exist only in a generalized sense and this is only a chaotic oscillation from the ordinary point of view. We present a measurement (i.e., a preparation and determinative measurements) where negative and usual Kolmogorov probabilities might arise. We show that there are two roots of negative probabilities: disturbance effects and the finite exactness of a measurement. Resume : Nous Otudions le problme reliO aux mesures simultanOes diobservables
(1) Background: Over a dozen constructions have been proposed to untangle the foundations of quantum mechanics (QM). So many irreconcilable schemes raise the suspect that methodological requisites should be discussed before beginning a theoretical construction in QM. In particular, we identify the ensuing issues of methods: formal definitions of particles and waves; incompatible probability theories and the single case problem. (2) Methods of the present inquiry: We tackle the second and third issue with a novel and comprehensive probability theory. Next it applies the theoretical results to quantum physics. (3) Results of the present inquiry: The new probability theory proves how some effects - the impossibility of measuring, the collapse of outcomes, non-locality and counterfactual definiteness - are not exclusive to quantum mechanics but can be observed also in classical context. (4) Conclusions: A probability-based interpretation of quantum mechanics is derived; innovative answers are put forward about quantum dualism, the measurement problem, collapse and the relations between classical and quantum physics.
It is shown that the full unknown state of a spin-1/2 system, S, which, within Born's statistical interpretation, is meant as the state of an ensamble of identically prepared systems, can be determined with a simultaneous measurement with the help of an "assistant" system A whose initial state is known. The idea is to let S and A interact with each other in a known way during a proper interaction time, to measure simultaneously two observables, one of S and one of A and their correlation. One thus determines the three unknown components of the polarization vector of S by means of repeated experiments using a unique setting. In this way one can measure simultaneously all the non-commutative observables of S, which might seem prohibited in quantum mechanics.
It is shown that the full unknown state of a spin-1/2 system, S, which, within Born's statistical interpretation, is meant as the state of an ensamble of identically prepared systems, can be determined with a simultaneous measurement with the help of an "assistant" system A whose initial state is known. The idea is to let S and A interact with each other in a known way during a proper interaction time, to measure simultaneously two observables, one of S and one of A and their correlation. One thus determines the three unknown components of the polarization vector of S by means of repeated experiments using a unique setting. In this way one can measure simultaneously all the non-commutative observables of S, which might seem prohibited in quantum mechanics.
In this paper some consequences of the strengthened interpretative rules of quantum mechanics, which were proposed in an earlier paper, are obtained. It is also seen that, in general, the usual interpretative rules are too weak to obtain these results. For example, it is proved from the strengthened rules that if f:R → R is a Borel function which is also τ-definable, then for each observable A the procedure ``measure A and compute f(A outcome)'' is an f(A) measurement procedure. It is also shown that there exist Borel f and observables A such that the above procedure is not an f(A) measurement procedure. Two methods of measuring the sum A + B of two observables are then considered: The measurement of A and B on different systems followed by addition of the results and (if A and B commute) the simultaneous measurement of A and B on the same system followed by addition. It is proved from the strengthened rules that the first method is not a valid measurement procedure and the second is valid. Besides these, other processes such as procedures for preparing mixtures of different states, and the empirical generation of probability measures from outcome sequences are considered.