Unveiling topological order through multipartite entanglement

Abstract

It is well known that the topological entanglement entropy (${S}_{\mathrm{topo}}$) of a topologically ordered ground state in two 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 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 ($N\ensuremath{\ge}3$) is a topological invariant equal to the product of ${S}_{\mathrm{topo}}$ and the Euler characteristic of the CSS embedded on a planar manifold, $|{I}^{N}|=\ensuremath{\chi}{S}_{\mathrm{topo}}$. 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-neighbor subsystems. While the addition of a handle between further neighbor subsystems causes ${I}^{N}$ to vanish, the multipartite information measures of the two smaller annular CSS emergent from this deformation again yield the same topological invariant. For a general CSS with multiple holes (${n}_{h}>1$), we find that the sum of the distinct, multipartite information measured on the annular CSS around those holes is given by the product of ${S}_{\mathrm{topo}}, \ensuremath{\chi}$ and ${n}_{h}, {\ensuremath{\sum}}_{{\ensuremath{\mu}}_{i}=1}^{{n}_{h}}|{I}_{{\ensuremath{\mu}}_{i}}^{{N}_{{\ensuremath{\mu}}_{i}}}|={n}_{h}\ensuremath{\chi}{S}_{\mathrm{topo}}$. This constrains the concomitant measurement of several multipartite information on any complicated CSS. The $N\text{th}$ order irreducible 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. Thus, our results offer important insight into the nature of the many-particle entanglement and correlations within a topologically ordered state of matter.