  We develop an approach based on edge theories to calculate the entanglement
entropy and related quantities in (2+1)-dimensional topologically ordered
phases. Our approach is complementary to, e.g., the existing methods using
replica trick and Witten's method of surgery, and applies to a generic spatial
manifold of genus $g$, which can be bipartitioned in an arbitrary way. The
effects of fusion and braiding of Wilson lines can be also straightforwardly
studied within our framework. By considering a generic superposition of states
with different Wilson line configurations, through an interference effect, we
can detect, by the entanglement entropy, the topological data of Chern-Simons
theories, e.g., the $R$-symbols, monodromy and topological spins of
quasiparticles. Furthermore, by using our method, we calculate other
entanglement measures such as the mutual information and the entanglement
negativity. In particular, it is found that the entanglement negativity of two
adjacent non-contractible regions on a torus provides a simple way to
distinguish Abelian and non-Abelian topological orders.
