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American Journal of Physics

Perfect disturbing measurements

Toru Ohira, Philip Pearle

Published Aug 1, 1988• 58 citations
Publisher

Abstract

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.

DOI
10.1119/1.15502

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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.
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 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 presence of an additive-conserved quantity imposes a limitation on the measurement process. According to the Wigner-Araki-Yanase theorem, perfect repeatability and distinguishability of the apparatus cannot be attained simultaneously. Instead of repeatability, in this paper, the distinguishability in both systems is examined. We derive a trade-off inequality between the distinguishability of the final states on the system and the one on the apparatus. An inequality shows that perfect distinguishability of both systems cannot be attained simultaneously.
Measurement error and disturbance, in the presence of conservation laws, are analysed in general operational terms. We provide novel quantitative bounds demonstrating necessary conditions under which accurate or non-disturbing measurements can be achieved, highlighting an interesting interplay between incompatibility, unsharpness, and coherence. From here we obtain a substantial generalisation of the Wigner-Araki-Yanase (WAY) theorem. Our findings are further refined through the analysis of the fixed-point set of the measurement channel, some extra structure of which is characterised here for the first time.
We study the Wigner-Araki-Yanase theorem by utilizing a slightly modified version of a formulation recently proposed by Ozawa [Phys. Rev. Lett. 67, 1950 (1991)]. After characterizing the quantities that are conserved in the evolution of the total system (object plus apparatus), we suitably decompose the Hamiltonian and show that the Wigner-Araki-Yanase theorem can be formulated in terms of the conserved quantities of the object system and of some observables of the apparatus. A few examples clarify our general arguments.