Rotational covariance restricts available quantum states

Quantum states of angular momentum and spin generally are not invariant under rotations of the reference frame. Therefore, they can be used as a resource of relative orientation, which is encoded in the asymmetry of the state under consideration. In this paper we introduce the analytical characterization of the rotational information by parameterizing the group characteristic function by polynomial functions. By doing so, we show that the set of states achievable through transformations lacking a reference frame (rotationally covariant ones) admits an analytical characterization and can be studied through the use of semidefinite optimization techniques. We demonstrate the developed methods via examples, and provide a physical scenario in which a reference-independent operation performs a metrologically useful operation: the preparation of a state of light improving interferometer sensitivity, which equivalently can be realized as a postprocessing step.