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 multiplicity 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, this term is shown to be equal to ln D, 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. They allow us to prove what we refer to as the isomorphism theorem, which enables us to define superselection sectors and fusion multiplicities without a Hamiltonian. These objects and the axioms of the anyon theory are shown to emerge from the structure and the internal self-consistency relations of an object known as the information convex.2. I(A : C) σ = 0 for the partition in Fig. 23.

Galois conjugation and multiboundary entanglement entropy

“…The fact that the invariants of the Galois action include both fusion and, on certain torus link complements, MEE is suggestive of a deeper relation between entanglement, fusion, and modular data. In fact, recent work [54,55] suggests that the entanglement entropy of [27,28] can be used to reconstruct the fusion rules and modular data of a TQFT. It would clearly be interesting to understand how MEE fits more precisely into this story.…”

“…We hope to return to some of these questions soon. The columns are labelled by the integrable representations, 0, 1, 2, 3, 54 and the rows are labelled by the Galois conjugations corresponding to 1, 2, 3, 4 ∈ Z × 5 . Let us now consider satellite links.…”

Toy model of boundary states with spurious topological entanglement entropy

“…Note that dual complementarity is not a necessary condition for Eq. (5). For example, one can consider E = E A 2 ⊗ id A 3 such that E is not the completely depolarizing channel, but A E = CI.…”

“…TEE is defined as the universal constant term γ [1]. The term γ is shown to be the logarithm of the total quantum dimension of the abstract anyon model under various conditions [1,[3][4][5].…”

Modular commutator in gapped quantum many-body systems

“…In such cases, one expects the two phases to admit the same chiral central charge. Using a part of the entanglement bootstrap program [30] dealing with gapped domain walls [31], we show that such a pair of phases has the same value of the modular commutator.…”