Quantum state nonclassicality in weak measurements in view of Bochner's criteria

We introduce a new formulation of nonclassicality in weak measurements based on probabilistic behavior of “quasi-moments” of a weakly measured observable. New definition determines existence of classical probabilistic interpretation and can be applied equally to quantum systems without classical counterpart in the usual sense. We show that the only consistent approach to define classicality in weak measurements should be based on the proper behavior of “quasi-moments” according to Bochner's theorem.