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.

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