Verlinde formula from entanglement

Abstract

We derive the Verlinde formula from a recently advocated set of axioms about entanglement entropy [B. Shi, K. Kato, I. H. Kim, arXiv:1906.09376 (2019)]. For any state that obeys these axioms, we can define a quantity that can be identified as the topological $S$-matrix of an abstract anyon theory. We show that the $S$-matrix is unitary and that it recovers the fusion multiplicities of the underlying anyon theory through the Verlinde formula. Importantly, we rigorously prove the modularity of the theory, which further implies that the mutual braiding statistics of anyons are nontrivial. The key to the proof is a generalized quantum state merging technique, which generates a topology beyond that of any subsystem of the original physical system.