  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.
