Fusion rules from entanglement

We derive some of the axioms of the algebraic theory of anyon (Kitaev, 2006) from a conjectured form of entanglement area law for two-dimensional gapped systems. We derive the fusion rules of topological charges and show that the multiplicities of the fusion rules satisfy these axioms. Moreover, even though we make no assumption about the exact value of the constant sub-leading term of the entanglement entropy of a disk-like region, this term is shown to be equal to lnD, where D is the total quantum dimension of the underlying anyon theory. These derivations are rigorous and follow from the entanglement area law alone. More precisely, our framework starts from two local entropic constraints which are implied by the area law. From these constraints, we prove what we refer to as the “isomorphism theorem.” The existence of superselection sectors and fusion multiplicities follow from this theorem, even without assuming anything about the parent Hamiltonian. These objects and the axioms of the anyon theory are shown to emerge from the structure and the internal self-consistency relations of the information convex sets. •A derivation of the anyon theory from the entanglement area law is initiated.•Basic concepts of the algebraic theory of anyon are defined from a quantum state.•The topological charges/superselection sectors of the anyon theory are defined.•The fusion rules of anyons are defined and shown to satisfy the expected properties.•The well-known formula of topological entanglement entropy is reproduced.