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Physics

Uncertainty beyond the Uncertainty Principle

Charles Day

Published Nov 21, 2023• 0 citations

Abstract

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

DOI: 10.1103/physics.16.s150

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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

In two articles, the authors claim that the Heisenberg uncertainty principle limits the precision of simultaneous measurements of the position and velocity of a particle and refer to experimental evidence that supports their claim. It is true that ever since the inception of quantum mechanics, the uncertainty relation that corresponds to a pair of observables represented by non-commuting operators is interpreted by many scientists and engineers, including Heisenberg himself, as a limitation on the accuracy with which observables can be measured. However, such a limitation cannot be deduced from the postulates and theorems of quantum thermodynamics.

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 continuous and unbounded extension of the Wigner--Araki--Yanase (WAY) theorem by Kuramochi and Tajima establishes a conservation-law obstruction to exact position measurements. We investigate how this result is modified when the measured system obeys a quadratic generalized uncertainty principle (GUP), [X,P] = iℏ(1 +βP2). First, we recover the intrinsic minimal-localization scale ∆Xmin = ℏ√β. Second, assuming an additive conserved momentum, a Yanase-compatible pointer, and the usual hypotheses underlying Ozawa's noise--conservation inequality, we derive a conditional GUP-modified bound on the root-mean-square measurement noise, ϵX ≥ℏ[1 +β⟨P2S⟩]/[2√(∆PS)2 +(∆PA)2]. The result reduces to the canonical bound when β → 0 and makes explicit the dependence on both the system state and apparatus resource. We distinguish this quantitative, model-dependent statement from the intrinsic minimal-length relation and identify the operator-domain and composite-system assumptions required for a full theorem.

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.