Fusion rules from entanglement

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.

arXiv comment57 pages, 28 figures, published version

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• Verlinde formula from entanglement
• Mixed-State Entanglement Measures in Topological Order
• Entropy scaling law and the quantum marginal problem
• Galois Conjugation and Multiboundary Entanglement Entropy
• Stability of topological purity under random local unitaries
• Boundary topological entanglement entropy in two and three dimensions
• Fusion Structure from Exchange Symmetry in (2+1)-Dimensions
• Remote detectability from entanglement bootstrap I: Kirby's torus trick
• Knots and entanglement