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arXiv:2109.00581 · The European Physical Journal Plus

Effects of the free evolution in the Arthurs–Kelly model of simultaneous measurement and in the retrodictive predictions of the Heisenberg uncertainty relations

J. A. Mendoza-Fierro, L. M. Arévalo Aguilar, Víctor Manuel Velázquez Aguilar

Published Sep 1, 2021• 4 citations
arXivHTMLPDF
DOI
10.1140/epjp/s13360-021-01943-6

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The accuracy of the Arthurs-Kelly model of a simultaneous measurement of position and momentum is analysed using concepts developed by Braginsky and Khalili in the context of measurements of a single quantum observable. A distinction is made between the errors of retrodiction and prediction. It is shown that the distribution of measured values coincides with the initial-state Husimi function when the retrodictive accuracy is maximized (this being the result first obtained by Busch, using somewhat different methods). It is further shown that the distribution of measured values is related to the final-state anti-Husimi function (the P representation of quantum optics) when the predictive accuracy is maximized. The disturbance of the system by the measurement is also discussed. A class of minimally disturbing measurements is characterized. It is shown that the distribution of measured values then coincides with one of the smoothed Wigner functions described by Cartwright.
A numerical illustration of a universally valid Heisenberg uncertainty relation, which was proposed recently, is presented by using the experimental data on spin-measurements by J. Erhart et al. [Nat. Phys. 8, 185 (2012)]. This uncertainty relation is closely related to a modified form of the Arthurs–Kelly uncertainty relation, which is also tested by the spin-measurements. The universally valid Heisenberg uncertainty relation always holds, but both the modified Arthurs–Kelly uncertainty relation and the Heisenberg error–disturbance relation proposed by Ozawa, which was analyzed in the original experiment, fail in the present context of spin-measurements, and the cause of their failure is identified with the assumptions of unbiased measurement and disturbance. It is also shown that all the universally valid uncertainty relations are derived from Robertson's relation and thus the essence of the uncertainty relation is exhausted by Robertson's relation, as is widely accepted.