Maximal accuracy and minimal disturbance in the Arthurs-Kelly simultaneous measurement process

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 Cartright.