Unveiling topological order through multipartite entanglement

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 ($N\ge 3$) is a topological invariant equal to the product of $S_{\mathrm{topo}}$ and the Euler characteristic of the CSS embedded on a planar manifold, $|I^N|=\chi S_{\mathrm{topo}}$. 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 $S_{\mathrm{topo}}, \chi$ and $n_h, {\sum }_{{\mu }_i=1}^{n_h}\left|I_{{\mu }_i}^{N_{{\mu }_i}}\right|=n_h\chi S_{\mathrm{topo}}$. 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.

Figure 1

(a) CSS with $N$ members placed in an annular structure ($\left\{{\mathcal{A}}_N^{\left(a\right)}\right\}\equiv \{A_1,...,A_N\}$), and where each individual subsystem is simply connected. (b) An open CSS ($\left\{{\mathcal{A}}^{\left(1b\right)}\right\}$) formed out of $N$ number of simply connected subsystems, such that their union does not form a closed annular structure. (c) CSS ($\left\{{\mathcal{A}}^{\left(1c\right)}\right\}$) composed of $N$ subsystems, where $N-1$ form a closed CSS ($\left\{{\mathcal{A}}_{N-1}^{\left(a\right)}\right\}$) and the last is an isolated island $A_N$. (d) CSS ($\left\{{\mathcal{A}}^{\left(1d\right)}\right\}$) formed from $N+1$ subsystems created by joining the (appendage) subsystem $A_{N-1}$ with the closed annular CSS $\left\{{\mathcal{A}}_{N-1}^{\left(a\right)}\right\}$.

Figure 2

(a) CSS $\left\{{\mathcal{A}}_N^{\left(2a\right)}\right\}$ represents the case where individual subsystems have holes or multiple self handles, but there is no intersubsystem handles. (b) CSS representing the case where there are handles among nearest-neighbor subsystems, but no connection among further-neighbor subsystems. (c) CSS with a connection between two subsystems ($A_m$ and $A_n$) that lie beyond nearest neighbor ($n-m>1$).

Figure 3

Summary of various results for the $N$-partite information $I^N$ corresponding to different CSS topologies placed on a planar manifold presented in Sec. 3. Please refer to the text for details.

Figure 4

CSS with five subsystems ($\left\{{\mathcal{A}}_5^{\left(2\right)}\right\}=\{A,B,C,D,E\}$) and embedded on planar manifold. The two holes in the CSS are denoted by $C_1$ and $C_2$.

Figure 5

CSS ($\left\{A_3^{\left(LW\right)}\right\}$) composed of three subsystems $A,B,C$, and where $B$ is comprised of two disjoint islands [33].

Figure 6

Figures displaying the graphs equivalent to various CSS topologies. (a) The simple annular CSS ($\left\{{\mathcal{A}}_N^{\left(a\right)}\right\}$) with $N$ subsystems. (b) Graph [$\mathrm{\Gamma }=\mathrm{{\rm Y}}\left(\left\{{\mathcal{A}}_N^{\left(a\right)}\right\}\right)$] corresponding to the CSS ($\left\{{\mathcal{A}}_N^{\left(a\right)}\right\}$, (a), where each subsystem $A_i$ is replaced by a node $i$ and the wall between two subsystems $A_i,A_{i+1}$ is replaced by an edge. $\mathrm{\Gamma }=\mathrm{{\rm Y}}\left(\left\{{\mathcal{A}}_N^{\left(a\right)}\right\}\right)$ has a total of $N$ nodes, as well as $N$ edges. (c) An open CSS ($\left\{{\mathcal{A}}_N^{\left(o\right)}\right\}$) with $N$ subsystems. (d) Graph [$\mathrm{\Gamma }=\mathrm{{\rm Y}}\left(\left\{{\mathcal{A}}_N^{\left(o\right)}\right\}\right)$] corresponding to the CSS $\left\{{\mathcal{A}}_N^{\left(o\right)}\right\}$, containing $N$ number of nodes and $N-1$ number of edges.

Figure 7

In general, a CSS has $n_h$ number of holes, $N$ subsystems, and $d_{nn}$ partitions. (a) An example of a CSS ($\left\{{\mathcal{A}}_N^{\left(5\right)}\right\}$) with $n_h=5$. (b) Graph corresponding to $\left\{{\mathcal{A}}_N^{\left(5\right)}\right\}$, where each vertex represents a subsystem and each edge represents a partition.