Covariant measurements and uncertainty relations

Abstract The concept of covariant measurement is a generalization of the well-known notion of imprimitivity system. This concept allows mathematical description of such physical quantities as angle, phase etc. in the framework of quantum theory. In this paper the structure of covariant measurements is described in the cases of finite-dimensional or irreducible representation of the symmetry group. A noncommutative analogue of Hunt-Stein theorem in mathematical statistics is proved, giving a characterization of the most precise measurements. As examples, measurements of angle, phase and orientation are considered. Rigorous uncertainty relations are obtained in these cases.