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

Since its conception, 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.