Unveiling topological order through multipartite entanglement

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 ($N\ge 3$) 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 $n_h$, ${\sum }_{{\mu }_i=1}^{n_h}|I_{{\mu }_i}^{N_{{\mu }_i}}|=n_h\chi S_{topo}$. 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.

17 pages, 7 figures, 65 references

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