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Quantum Measurements in thePresence of Symmetry

Leon Loveridge

Published Jul 1, 2012• 4 citations
OpenAlex

Abstract

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.

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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 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.
In quantum theory, physically measurable quantities of a microscopic system are represented by self-adjoint operators. However, not all of the self-adjoint operators correspond to measurable quantities. The superselection rule is a criterion to distinguish measurable quantities. Any measurable quantity must obey the superselection rules. By contraposition, any quantity which does not obey the superselection rules cannot be measured. Although some of superselection rules were proved, the raizon d'être of the superselection rules has been still obscure. In this paper we deduce the superselection rules from an assumption on symmetry property of measurement process. We introduce the notion of covariant indicator, which is a macroscopic observable whose value indicates the value of a microscopic object observable. We prove that if an object system has a quantity that is conserved during the measurement process, other quantities that do not commute with the conserved quantity are non-measurable by the covariant indicator. Our derivation of superselection rules is compared with the uncertainty relation under the restriction by a conservation law. An implication of the color superselection rule for the color confinement is discussed. It is also argued that spontaneous symmetry breaking enables a measurement that the superselection rule prohibits.
We propose that observables in quantum theory are properly understood as representatives of symmetry-invariant quantities relating one system to another, the latter to be called a reference system. We provide a rigorous mathematical language to introduce and study quantum reference systems, showing that the orthodox “absolute” quantities are good representatives of observable relative quantities if the reference state is suitably localised. We use this relational formalism to critique the literature on the relationship between reference frames and superselection rules, settling a long-standing debate on the subject.
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.
We propose that observables in quantum theory are properly understood as representatives of symmetry-invariant quantities relating one system to another, the latter to be called a reference system. We provide a rigorous mathematical language to introduce and study quantum reference systems, showing that the orthodox "absolute" quantities are good representatives of observable relative quantities if the reference state is suitably localised. We use this relational formalism to critique the literature on the relationship between reference frames and superselection rules, settling a long-standing debate on the subject.