Disentangling (2+1)d Topological States of Matter with the Entanglement Negativity

TL;DR

This paper investigates how entanglement negativity constrains the structure of topological ground states in (2+1)d systems, revealing conditions for disentangling and differences between Abelian and non-Abelian states.

Contribution

It introduces a method using entanglement negativity to identify when topological states can be factorized into simpler components, highlighting distinctions between Laughlin and Moore-Read states.

Findings

Entanglement negativity determines necessary conditions for disentangling topological states.

The disentangling condition is sufficient for Laughlin states.

Moore-Read states cannot be disentangled even when conditions are met.

Abstract

We use the entanglement negativity, a bipartite measure of entanglement in mixed quantum states, to study how multipartite entanglement constrains the real-space structure of the ground state wavefunctions of $(2+1)$-dimensional topological phases. We focus on the (Abelian) Laughlin and (non-Abelian) Moore-Read states at filling fraction $\nu=1/m$. We show that a combination of entanglement negativities, calculated with respect to specific cylinder and torus geometries, determines a necessary condition for when a topological state can be disentangled, i.e., factorized into a tensor product of states defined on cylinder subregions. This condition, which requires the ground state to lie in a definite topological sector, is sufficient for the Laughlin state. On the other hand, we find that a general Moore-Read ground state cannot be disentangled even when the disentangling condition holds.