  We study the properties of entanglement in two-dimensional topologically
ordered phases of matter. Such phases support anyons, quasiparticles with
exotic exchange statistics. The emergent nonlocal state spaces of anyonic
systems admit a particular form of entanglement that does not exist in
conventional quantum mechanical systems. We study this entanglement by adapting
standard notions of entropy to anyonic systems. We use the algebraic theory of
anyon models (modular tensor categories) to illustrate the nonlocal
entanglement structure of anyonic systems. Using this formalism, we present a
general method of deriving the universal topological contributions to the
entanglement entropy for general system configurations of a topological phase,
including surfaces of arbitrary genus, punctures, and quasiparticle content. We
analyze a number of examples in detail. Our results recover and extend prior
results for anyonic entanglement and the topological entanglement entropy.
