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Progress of Theoretical Physics

A Reexamination of the Wigner and Araki-Yanase Theorem

S. Matsumoto

Published Jul 1, 1993• 5 citations
Publisher

Abstract

The limitation on the measurement of an operator imposed by the presence of one conservation law was discussed by Wigner and Araki-Yanase. But we do not know whether their results still hold or not when we consider two conservation laws or more at a time. In this article, a model of a quantum measuring process is discussed from the point of view that all three components of angular momentum should be conserved in the process.

DOI
10.1143/ptp/90.1.35

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The limitation on the measurement of an operator imposed by the presence of one conservation law was discussed by Wigner and Araki-Yanase. But we do not know whether their results still hold or not when we consider two conservation laws or more at a time. In this article, a model of a quantum measuring process is discussed from the point of view that all three components of angular momentum should be conserved in the process.
The relationship between the measurement of an operator and the presence of a conservation law is studied using generalized coherent states. It is shown that an operator which does not commute with a conserved quantity cannot be measured exactly; this is an extension of the theorem by Wigner, Araki and Yanase, expressing the conserved quantity in a more general form. Classical operators play an important role in this approach. We present a simple model of the approximate measurement of a particle-spin which does not commute with the (additive) conserved angular momentum. Also, using two other models we show that the condition of distinguishability of pointer-operators is consistent with the situation where the statistical operator in a physical system tends to just its mixed-part.
Measurement of a spin component of a spin-1/2 particle is reconsidered under the constraint that all three components of total angular momentum are conserved in the measuring process.We obtain an inequality by which we can estimate the amount of disturbance of the measured system caused by the measurement.Our formula has more information than the inequalities previously obtained by Yanase and others in the sense that it shows us how to choose the initial state of the apparatus which will make the measurement closest to the ideal.It is emphasized that the conventional interpretation of the Wigner-Araki-Yanase theorem, in which much attention is focussed only on the size of the apparatus, fails to take into account the initial state of the appratus.
The Wigner-Araki-Yanase (WAY) theorem is a well-known theorem regarding limitations of quantum measurement in the presence of additive conservation laws. Under the assumptions of the von Neumann measurement model, for which the system conserved quantity LSL_{S}LS​ is bounded, given a conserved total additive system plus apparatus quantity LSAL_{SA}LSA​, the measurement operator ESE_{S}ES​ must commute with LSL_{S}LS​. Prior proofs have exploited the properties of unitary evolution constrained by momentum conserving operations that tend to obscure the physical nature of the WAY theorem and as well lead to bounds on performance. As it is generally agreed that momentum is always exactly conserved in measurement, we instead develop a general angular momentum conserving model of measurement. This model is shown to lead to a simple explanation of the major implications of the WAY theorem and provides exact results of the effects of measurement based on the apparatus model. This is shown by both tracing the apparatus from the density matrix and also via a system-only channel model based on Kraus operators.
The ability to measure every quantum observable is ensured by a fundamental result in quantum measurement theory. Nevertheless, additive conservation laws associated with physical symmetries, such as the angular momentum conservation, may lead to restrictions on the measurability of the observables. Such limitations are imposed by the theorem of Wigner, Araki, and Yanase (WAY). In this paper a formulation of the WAY theorem is presented rephrasing the measurability limitations in terms of quantum incompatibility. This broader mathematical basis enables us to both capture and generalize the WAY theorem by allowing us to drop the assumptions of additivity and even conservation of the involved quantities. Moreover, we extend the WAY theorem to the general level of positive operator-valued measures.