The uncertainty relation for joint measurement of position and momentum

Abstract

We prove an uncertainty relation, which imposes a bound on any joint measurement of position and momentum. It is of the form $(\Delta P)(\Delta Q)\ge C\hslash$, 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 $\Delta Q$ of the position measurement, and the perturbation $\Delta P$ 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.

arXiv comment17 pages, 4 figures; for Festschrift A.S. Holevo

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