Entanglement properties of topological color codes

TL;DR

This paper analyzes the entanglement properties of topological color codes on 2D lattices, revealing their multipartite entanglement structure, topological order, and boundary-dependent entanglement entropy, including differences from the toric code.

Contribution

It provides a detailed calculation of entanglement entropy and topological order in color codes, highlighting their multipartite entanglement and boundary effects, and compares them with toric codes.

Findings

Qubit is maximally entangled with the system; pairs are not.

Entanglement entropy depends only on boundary degrees of freedom.

Topological subleading term in entropy is twice that of the toric code.

Abstract

The entanglement properties of a class of topological stabilizer states, the so called \emph{topological color codes} defined on a two-dimensional lattice or \emph{2-colex}, are calculated. The topological entropy is used to measure the entanglement of different bipartitions of the 2-colex. The dependency of the ground state degeneracy on the genus of the surface shows that the color code can support a topological order, and the contribution of the color in its structure makes it interesting to compare with the Kitaev's toric code. While a qubit is maximally entangled with rest of the system, two qubits are no longer entangled showing that the color code is genuinely multipartite entangled. For a convex region, it is found that entanglement entropy depends only on the degrees of freedom living on the boundary of two subsystems. The boundary scaling of entropy is supplemented with a topological subleading term which for a color code defined on a compact surface is twice than the toric code. From the entanglement entropy we construct a set of bipartitions in which the diverging term arising from the boundary term is washed out, and the remaining non-vanishing term will have a topological nature. Besides the color code on the compact surface, we also analyze the entanglement properties of a version of color code with border, i.e \emph{triangular color code}.