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

• Main Authors: Mendoza-Fierro, J A, Aguilar, L M Arévalo, Aguilar, V M Velázquez
• Format: Journal Article
• Language: English
• Published: 01-09-2021
• Subjects: Physics - Quantum Physics
• DOI: 10.48550/arxiv.2109.00581

European Physical Journal Plus (2021) 136:954 The simultaneous measurement approach of Arthurs and Kelly has been a
significant tool for the better understanding of the measurement process in
quantum mechanics. This model considers a strong interaction Hamiltonian by
discarding the free evolution part. In this work, we study the effect of the
full dynamics -- taking into account the free Hamiltonian -- on the optimal
limits of retrodictive and predictive accuracy of the simultaneous measurement
process of position and momentum observables. To do that, we consider a minimum
uncertainty Gaussian state as the system under inspection, which allows to
carry out an optimal simultaneous measurement. We show that the inclusion of
the free Hamiltonian induces a spreading on the probability density of the
measurement setting, which increases the value of the product of the variances
of the so-called retrodictive and predictive error operators, this is
equivalent to a reduction in the accuracy of the measurement.