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Physical Review

Optimal Measuring Apparatus

Mutsuo M. Yanase

Published Jul 15, 1961• 182 citations
Publisher

Abstract

An upper limit for the accuracy of the measurement of a simple quantity which does not commute with a conserved quantity is obtained in terms of the "size" of the apparatus. The "size" of the apparatus is defined as the mean square value ${\ensuremath{\hbar}}^{2}{M}^{2}$ of the conserved quantity for the apparatus which is, in the example chosen, the $z$ component of the angular momentum. The measured quantity is the projection of a spin in a perpendicular direction. It is found that the probability of an unsuccessful measurement is at least $\frac{1}{8{M}^{2}}$.

DOI
10.1103/physrev.123.666

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An upper limit for the accuracy of the measurement of a simple quantity which does not commute with a conserved quantity is obtained in terms of the "size" of the apparatus. The "size" of the apparatus is defined as the mean square value \ensuremathℏ2M2{\ensuremath{\hbar}}^{2}{M}^{2}\ensuremathℏ2M2 of the conserved quantity for the apparatus which is, in the example chosen, the zzz component of the angular momentum. The measured quantity is the projection of a spin in a perpendicular direction. It is found that the probability of an unsuccessful measurement is at least 18M2\frac{1}{8{M}^{2}}8M21​.
A solvable model of an apparatus for measuring a spin component of a spin-1/2 particle is considered. The model enjoys full rotational invariance and, in the limit case in which a part of the apparatus becomes infinitely massive, works according to the ideal measurement scheme. The model is applied to the study of the deviations from the ideal measurement scheme imposed by the conservation of angular momentum. The detailed analysis of the model allows one to calculate the size of the various nonideal terms in the measurement scheme and to understand the physical mechanism which gives rise to such terms.
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.
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.
When a measurement is made on a system that is not in an eigenstate of the measured observable, it is often assumed that some conservation law has been violated. Discussions of the effect of measurements on conserved quantities often overlook the possibility of entanglement between the measured system and the preparation apparatus. The preparation of a system in any particular state necessarily involves interaction between the apparatus and the system. Since entanglement is a generic result of interaction, as shown by Gemmer and Mahler[1], and by Durt[2,3] one would expect some nonzero entanglement between apparatus and measured system, even though the amount of such entanglement is extremely small. Because the apparatus has an enormous number of degrees of freedom relative to the measured system, even a very tiny difference between the apparatus states that are correlated with the orthogonal states of the measured system can be sufficient to account for the perceived deviation from strict conservation of the quantity in question. Hence measurements need not violate conservation laws.