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

TL;DR

This paper investigates how including free evolution in the Arthurs-Kelly model affects the accuracy of simultaneous quantum measurements of position and momentum, revealing that free dynamics reduce measurement precision.

Contribution

It extends the Arthurs-Kelly model by incorporating free Hamiltonian dynamics, analyzing their impact on measurement accuracy and retrodictive predictions.

Findings

Free evolution causes spreading of measurement probability densities.

Inclusion of free Hamiltonian increases the product of error variances.

Measurement accuracy decreases due to free dynamics.

Abstract

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.