Wigner-Araki-Yanase Theorem for Continuous and Unbounded Conserved Observables

Abstract

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 LS of the system commutes with the implemented unitary US if LS is semibounded, while US†LSUS can shift up to possibly nonzero constant factor if the spectrum of LS is upper and lower unbounded. We give simple examples of the latter case, where LS is a momentum operator.Received 14 September 2022Revised 21 July 2023Accepted 29 August 2023DOI:https://doi.org/10.1103/PhysRevLett.131.210201Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum measurementsGeneral Physics

Topics & Concepts