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Physical Review Letters

Does a conservation law limit position measurements?

Masanao Ozawa

Published Oct 7, 1991• 31 citations
Publisher

Abstract

The demonstrations of Wigner and others, that observables which do not commute with additive conserved quantities cannot be measured precisely, are reexamined. A proposed new formulation of the claim is shown to be valid for observables with a continuous spectrum whenever the conserved quantities are bounded. However, a countermodel is constructed, and it suggests that the position can be measured as precisely as the momentum even though the measuring interaction conserves the total linear momentum.

DOI
10.1103/physrevlett.67.1956

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The demonstrations of Wigner and others, that observables which do not commute with additive conserved quantities cannot be measured precisely, are reexamined. A proposed new formulation of the claim is shown to be valid for observables with a continuous spectrum whenever the conserved quantities are bounded. However, a countermodel is constructed, and it suggests that the position can be measured as precisely as the momentum even though the measuring interaction conserves the total linear momentum.
According to a new extension to an old theory, a particle's position cannot be measured precisely even if its momentum is not measured simultaneously.By Charles Day H eisenberg's uncertainty principle limits the precision with which two observables that do not commute with each other can be simultaneously measured.The Wigner-Araki-Yanase (WAY) theorem goes further.If observables A and B do not commute, and if observable A is conserved, observable B cannot be measured with arbitrary precision even if A is not measured at all.In its original 1960 formulation, the WAY theorem applied only to observables, such as spin, whose possible values are discrete and bounded.Now Yui Kuramochi of Kyushu University and Hiroyasu Tajima of the University of Electro-Communications-both in Japan-have proven that the WAY theorem also encompasses observables, such as position, that are continuous and unbounded [1].Besides resolving the decades-long problem of how to deal with such observables, the extension will likely find practical applications in quantum optics.The difficulty of extending the WAY theorem arose from how an
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.
In quantum mechanics, conservation laws are usually understood only statistically and only in the context of unitary evolution: during such evolution, if an observable commutes with the Hamiltonian, its expectation value and Born-rule probability distribution remain unchanged. Individual measurement outcomes, however, disrupt even this weak form of statistical conservation. Recently, though, it has been argued that a stronger, non-statistical form of conservation, valid at the level of individual measurement outcomes, can be recovered. The proposal relies on the claim that any superposition of eigenstates of a conserved quantity must ultimately be prepared relative to another physical system, which serves as the relevant reference frame and remains correlated with the system being prepared. On this enlarged description, the change observed in the measured system is claimed to be exactly compensated by an opposite change in the preparer, thereby restoring conservation at the level of individual measurement outcomes. We analyze this proposal and argue that it faces significant conceptual difficulties. We show that the proposed mechanism is not adequately justified and, moreover, does not establish conservation in individual measurement outcomes as claimed.