Entanglement negativity and topological order

We use the entanglement negativity, a measure of entanglement for mixed states, to probe the structure of entanglement in the ground state of a topologically ordered system. Through analytical calculations of the negativity in the ground state (s) of the toric code model, we explicitly show that the pure-state entanglement of a region A and its complement B is the sum of two types of contributions: boundary entanglement and long-range entanglement. Boundary entanglement is seen to be insensitive to tracing out the degrees of freedom in the interior of regions A and B, and therefore it entangles only degrees of freedom in A and B that are close to their common boundary. We recover the well-known result that boundary entanglement is proportional to the size of each boundary separating A and B and it includes an additive, universal correction. The second, long-range, contribution to pure-state entanglement appears only when A and B are noncontractible regions (e. g. , on a torus) and it is seen to be destroyed when tracing out a noncontractible region in the interior of A or B. In the toric code, only the long-range contribution to the entanglement depends on the specific ground state under consideration.