In this article, we first obtain a characterization of relationships between pairwise linearly independent (PLI) rank-1 POVMs and Hamel bases for \({\mathbb {C}}^{d}\) . By utilizing the obtained relationships, a necessary and sufficient condition is provided for the set of incoherent states based on PLI rank-1 POVM to be empty. Secondly, we deduce a detailed characterization of incoherent pure states based on a PLI rank-1 POVM. Finally, by properties of rank-1 POVMs and equioverlapping measurements, we give some relationships between weak values of an observable and its trace and expectations.
