Characterization of informational completeness for covariant phase space observables

Each covariant phase space observable is determined by a unique positive trace one operator. We characterize the informational completeness of these observables in terms of the zero set of the Weyl transform of the generating operator, the relevant condition being that the zero set has dense complement. We then consider two other conditions, namely, that the zero set is of Lebesgue measure zero, or empty. We show that all three conditions are distinct, and show how they are related to approximation theorems in harmonic analysis.