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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Limitations on quantum measurements. I. Determination of the minimal amount of nonideality and identification of the optimal measuring apparatuses

G. C. Ghirardi, F. Miglietta, A. Rimini, T. Weber

Published Jul 15, 1981• 49 citations
Publisher

Abstract

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.

DOI
10.1103/physrevd.24.347

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•Aug 16, 2016•48 citations
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.
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.
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.
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 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​.
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.