The uncertainty relation for joint measurement of position and momentum

We prove an uncertainty relation, which imposes a bound on any joint measurement of position and momentum. It is of the form (∆P )(∆Q) ≥ C~, where the ‘uncertainties’ quantify the difference between the marginals of the joint measurement and the corresponding ideal observable. Applied to an approximate position measurement followed by a momentum measurement, the uncertainties become the precision ∆Q of the position measurement, and the perturbation ∆P of the conjugate variable introduced by such a measurement. We also determine the best constant C, which is attained for a unique phase space covariant measurement.