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arXiv:1303.6536 · WORLD SCIENTIFIC eBooks

QUANTUM MEASUREMENTS CONSTRAINED BY SYMMETRIES

Paul Busch, Leon Loveridge

Published Aug 14, 2013• 3 citations
arXivHTMLPDF

Abstract

We revisit the theorem of Wigner, Araki and Yanase (WAY) describing limitations to repeatable quantum measurements that arise from the presence of conservation laws. We will review a strengthening of this theorem by exhibiting and discussing a condition that has hitherto not been identified as a relevant factor. We will also show that an extension of the theorem to continuous variables such as position and momentum can be obtained if the degree of repeatability is suitably quantified.

DOI
10.1142/9789814518550_0083

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We revisit the theorem of Wigner, Araki and Yanase (WAY) describing limitations to repeatable quantum measurements that arise from the presence of conservation laws. We will review a strengthening of this theorem by exhibiting and discussing a condition that has hitherto not been identified as a relevant factor. We will also show that an extension of the theorem to continuous variables such as position and momentum can be obtained if the degree of repeatability is suitably quantified.
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 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.
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.
This thesis concerns how symmetries impinge on quantum mechanical measurements, and preclude certain self adjoint operators from representing observable quantities. After developing the requisite mathematical machinery and aspects of quantum measurement theory necessary for our analysis, we proceed to critically review the literature surrounding the remarkable theorem of Wigner, Araki and Yanase (WAY) that prohibits accurate and repeatable measurements of any observable not commuting with an additive conserved quantity, as well as discussing the conditions under which approximate measurements with approximate degrees of repeatability can be achieved. We strengthen the original statement of the WAY theorem and generalise it to the case of position measurements obeying momentum conservation, leading to a solution of a long-standing problem of Stein and Shimony. A superselection rule appearing as the existence of an observable which commutes with all others gives rise to a stronger restriction than the WAY theorem, yielding self adjoint operators which do not represent observable quantities. We analyse various perspectives on superselection rules, aiming to clarify different viewpoints appearing in the literature since the inception of the topic in 1952. We exploit an explicit description of relative phase observables which have been lacking in other contributions, delineating conditions under which relative and (prohibited) absolute phases become statistically close. By providing simple models we are able to mimic a number of attempts to overcome superselection rules, in order to highlight the generic features of such attempts. We show that the statistical proximity of absolute and relative quantities arises only when there is a highly localised phase reference, and that the superselection rule compatible relative phase factors between certain superpositions takes on the appearence of a forbidden relative phase factor in this limit. However, we argue that these relative phase factors can be determined fully within the confines of a superselection rule.
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.