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The simultaneous measurement approach of Arthurs and Kelly has been asignificant tool for the better understanding of the measurement process inquantum mechanics. This model considers a strong interaction Hamiltonian bydiscarding the free evolution part. In this work, we study the effect of thefull dynamics -- taking into account the free Hamiltonian -- on the optimallimits of retrodictive and predictive accuracy of the simultaneous measurementprocess of position and momentum observables. To do that, we consider a minimumuncertainty Gaussian state as the system under inspection, which allows tocarry out an optimal simultaneous measurement. We show that the inclusion ofthe free Hamiltonian induces a spreading on the probability density of themeasurement setting, which increases the value of the product of the variancesof the so-called retrodictive and predictive error operators, this isequivalent to a reduction in the accuracy of the measurement.

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