Negativity and topological order in the toric code

TL;DR

This paper investigates how the entanglement measure negativity reflects topological order in the toric code, revealing area-law behavior and a topological contribution linked to topological entropy, and distinguishes quantum from classical correlations.

Contribution

It provides an exact analytical expression for negativity in the toric code and demonstrates its sensitivity to quantum topological order, unlike classical models.

Findings

Negativity exhibits an area-law contribution in the toric code.

A topological contribution to negativity is present for non-trivial partitions.

Negativity vanishes for the classical 8-vertex model, highlighting quantum correlations.

Abstract

In this manuscript we study the behaviour of the entanglement measure dubbed negativity in the context of the toric code model. Using a method introduced recently by Calabrese, Cardy and Tonni [Phys. Rev. Lett. 109, 130502 (2012)], we obtain an exact expression which illustrates how the non-local correlations present in a topologically ordered state reflect in the behaviour of the negativity of the system. We find that the negativity has a leading area-law contribution, if the subsystems are in direct contact with one another (as expected in a zero-range correlated model). We also find a topological contribution directly related to the topological entropy, provided that the partitions are topologically non-trivial in both directions on a torus. We also show that the negativity captures only quantum contributions to the entanglement. Indeed, we show that the negativity vanishes identically for the classical topologically ordered 8-vertex model, which on the contrary exhibits a finite von Neumann entropy, inclusive of topological correction.