Title: de Finetti Theorems for Braiding Parafermions

Abstract: The de Finetti theorem is a powerful tool relating symmetry under the permutation group and independence of random variables. It also plays a significant role in quantum information theory. We prove a new type of de Finetti theorem for the braid group acting on the parafermion algebra. We show that a braid-invariant state is extremal if and only if it is a product state. Furthermore, we provide an explicit characterization of braid-invariant states, such that the parafermion algebras generates a factor under the Gelfand-Naimark-Segal construction.