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Progress of Theoretical Physics

The Wigner-Araki-Yanase Theorem and Its Extension in Quantam Measurement with Generalized Coherent States

Shoju Kudaka, K. Kakazu

Published Jan 1, 1992• 6 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.87.61

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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.
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 (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.
The Wigner-Araki-Yanase (WAY) theorem shows that additive conservation laws limit the accuracy of measurements. Recently, various quantitative expressions have been found for quantum limits on measurements induced by additive conservation laws, and have been applied to the study of fundamental limits on quantum-information processing. Here, we investigate generalizations of the WAY theorem to multiplicative conservation laws. The WAY theorem is extended to show that an observable not commuting with the modulus of, or equivalently the square of, a multiplicatively conserved quantity cannot be precisely measured. We also obtain a lower bound for the mean-square noise of a measurement in the presence of a multiplicatively conserved quantity. To overcome this noise it is necessary to make large the coefficient of variation (the so-called relative fluctuation), instead of the variance as is the case for additive conservation laws, of the conserved quantity in the apparatus.
The Wigner-Araki-Yanase (WAY) theorem states that additive conservation laws imply the commutativity of exactly implementable projective measurements and the conserved observables of the system. Known proofs of this theorem are only restricted to bounded or discrete-spectrum conserved observables of the system and are not applicable to unbounded and continuous observables like a momentum operator. In this Letter, we present the WAY theorem for possibly unbounded and continuous conserved observables under the Yanase condition, which requires that the probe positive operator-valued measure should commute with the conserved observable of the probe system. As a result of this WAY theorem, we show that exact implementations of the projective measurement of the position under momentum conservation and of the quadrature amplitude using linear optical instruments and photon counters are impossible. We also consider implementations of unitary channels under conservation laws and find that the conserved observable L_{S} of the system commutes with the implemented unitary U_{S} if L_{S} is semibounded, while U_{S}^{†}L_{S}U_{S} can shift up to possibly nonzero constant factor if the spectrum of L_{S} is upper and lower unbounded. We give simple examples of the latter case, where L_{S} is a momentum operator.
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.