Superdense coding in the resource theory of asymmetry

We analyze the task of encoding classical information into a quantum system under the restriction by symmetry. Motivated by an analogy between the resource theories of asymmetry and entanglement, we ask whether an analog of superdense coding is possible in the former. I.e., we investigate whether the classical information capacity of an asymmetric state can be strictly larger than that of any symmetric state whereas the latter is a strictly positive constant. We prove that this is possible if and only if the unitary representation of the symmetry is non-Abelian and reducible. The result provides an information-theoretical classification of symmetries of quantum systems. We also discuss the possibility of superdense coding in other resource theories.