Entanglement negativity and topological order

TL;DR

This paper investigates the entanglement structure in topologically ordered systems using entanglement negativity, revealing boundary and long-range contributions, with explicit calculations in the toric code model.

Contribution

It provides an analytical study of entanglement negativity in topological phases, distinguishing boundary and long-range entanglement contributions, and clarifies their behavior in the toric code model.

Findings

Boundary entanglement is proportional to boundary size and includes a universal correction.

Long-range entanglement exists only for non-contractible regions and depends on the ground state.

Long-range entanglement is destroyed when non-contractible regions are traced out.

Abstract

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 entanglement of a region $A$ and its complement $B$ is the sum of two types of contributions. The first type of contributions consists of \textit{boundary entanglement}, which we see to be insensitive to tracing out the interior of $A$ and $B$. It therefore entangles only degrees of freedom in $A$ and $B$ that are close to their common boundary. As it is well-known, each boundary contribution is proportional to the size of the relevant boundary separating $A$ and $B$ and it includes an additive, universal correction. The second contribution appears only when $A$ and $B$ are non-contractible regions (e.g. on a torus) and it consists of long-range entanglement, which we see to be destroyed when tracing out a non-contractible region in the interior of $A$ or $B$. Only the long-range contribution to the entanglement may depend on the specific ground state under consideration.