Joint measurements and Bell inequalities

Abstract

Joint quantum measurements of non-commuting observables are possible, if one accepts an increase in the measured variances. A necessary condition for a joint measurement to be possible is that a joint probability distribution exists for the measurement. This fact suggests that there may be a link with Bell inequalities, as these will be satisfied if and only if a joint probability distribution for all involved observables exists. We investigate the connections between Bell inequalities and conditions for joint quantum measurements to be possible. Mermin's inequality for the three-particle Greenberger-Horne-Zeilinger state turns out to be equivalent to the condition for a joint measurement on two out of the three quantum systems to exist. Gisin's Bell inequality for three co-planar measurement directions, meanwhile, is shown to be less strict than the condition for the corresponding joint measurement.