# Fusion rules from entanglement

## TL;DR

This paper rigorously derives the algebraic structure of anyons, including fusion rules and quantum dimensions, from entanglement area laws in two-dimensional gapped systems, without assuming specific Hamiltonian details.

## Contribution

It provides a novel, rigorous derivation of anyon fusion rules and quantum dimensions solely from entanglement properties, establishing a deep link between entanglement and topological order.

## Key findings

- Fusion rules satisfy algebraic axioms of anyon theory
- Sub-leading entanglement entropy term equals log of total quantum dimension
- Existence of superselection sectors derived from entanglement constraints

## Abstract

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.