  We derive some of the axioms of the algebraic theory of anyon [A. Kitaev,
Ann. Phys., 321, 2 (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 $\ln \mathcal{D}$, where
$\mathcal{D}$ 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 follows 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.
