Unveiling topological order through multipartite entanglement

TL;DR

This paper investigates the multipartite entanglement structure of topologically ordered states, revealing that N-partite information measures are topological invariants related to the Euler characteristic and robust against deformations.

Contribution

It introduces a comprehensive analysis of N-partite information in topologically ordered states, establishing its invariance and relation to topological features like Euler characteristic and holes.

Findings

N-partite information is a topological invariant proportional to the Euler characteristic.

Multipartite information remains robust under deformations such as holes and handles.

Sum of multipartite informations around holes scales with the number of holes, topological entropy, and Euler characteristic.

Abstract

It is well known that the topological entanglement entropy ($S_{topo}$) of a topologically ordered ground state in 2 spatial dimensions can be captured efficiently by measuring the tripartite quantum information ($I^{3}$) of a specific annular arrangement of three subsystems. However, the nature of the general N-partite information ($I^{N}$) and quantum correlation of a topologically ordered ground state remains unknown. In this work, we study such $I^N$ measure and its nontrivial dependence on the arrangement of $N$ subsystems. For the collection of subsystems (CSS) forming a closed annular structure, the $I^{N}$ measure ($N\geq 3$) is a topological invariant equal to the product of $S_{topo}$ and the Euler characteristic of the CSS embedded on a planar manifold, $|I^{N}|=\chi S_{topo}$. Importantly, we establish that $I^{N}$ is robust against several deformations of the annular CSS, such as the addition of holes within individual subsystems and handles between nearest-neighbour subsystems. For a general CSS with multiple holes ($n_{h}>1$), we find that the sum of the distinct, multipartite informations measured on the annular CSS around those holes is given by the product of $S_{topo}$, $\chi$ and $n_{h}$, $\sum^{n_{h}}_{\mu_{i}=1}|I^{N_{\mu_{i}}}_{\mu_{i}}| = n_{h}\chi S_{topo}$. The $N^{th}$ order irreducible quantum correlations for an annular CSS of $N$ subsystems is also found to be bounded from above by $|I^{N}|$, which shows the presence of correlations among subsystems arranged in the form of closed loops of all sizes. Our results offer important insight into the nature of the many-particle entanglement and correlations within a topologically ordered state of matter.