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Progress of Theoretical Physics

A Reexamination of the Wigner and Araki-Yanase Theorem

Shuichi Matsumoto

Published Jul 1, 1993• 2 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.90.35

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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 Wigner–Araki–Yanase theorem is often interpreted as meaning that there must be an error in measurement under certain conditions (when the commutator C between an additive conserved quantity and the discrete-spectrumed measured quantity does not vanish), and that the error may only be reduced by increasing the size of the apparatus. By explicit example, it is shown that it is possible to have a perfectly accurate measurement if the system being measured is disturbed, regardless of the size of the apparatus. Also illustrated is how the error in an imperfectly accurate measurement may be reduced by decreasing the magnitude of C, without affecting the size of the apparatus.
The problem of determining the minimal deviation from the ideal scheme in a measurement, compatible with the existence of additive conserved quantities, is reconsidered. In the case of the measurement of a spin component of a spin-1/2 particle, we rederive in a simple and rigorous way the bound previously obtained by Yanase. The procedure allows the derivation of definite equations characterizing an optimal measuring apparatus. A detailed discussion of the possible malfunctioning of the apparatuses is also given.
The Wigner-Araki-Yanase (WAY) theorem states a remarkable limitation to quantum mechanical measurements in the presence of additive conserved quantities. Discovered by Wigner in 1952, this limitation is known to induce constraints on the control of individual quantum systems in the context of information processing. It is therefore important to understand the precise conditions and scope of the WAY theorem. Here we elucidate its crucial assumptions, briefly review some generalizations, and show how a particular extension can already be obtained by a simple modification of the original proofs. We also describe the evolution of the WAY theorem from a strict no-go verdict for certain, highly idealized, precise measurements into a quantitative constraint on the accuracy and approximate repeatability of imprecise measurements.