Complementarity for Generalized Observables

We examine basic properties of complementarity by using the most general description of quantum observables as positive-operator measures. We show that, in general, two observables can be complementary or not depending on the measure of fluctuations adopted and that complementarity is not a symmetric relation. This occurs because the states that determine the measured statistics do not necessarily coincide with the minimum uncertainty states for the same observable. We also show that there are observables without a complementary observable and that complementarity is not preserved by the Neumark extensions.