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 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 ≥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}=χ 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 nearestneighbor 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_{topo}, χ and n_{h}, ∑μ_{i}=1_{nh}I_{μ}^{iNμi} =n_{h}χ S_{topo} . This constrains the concomitant measurement of several multipartite information on any complicated CSS. The N 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 manyparticle entanglement and correlations within a topologically ordered state of matter.
 Publication:

Physical Review A
 Pub Date:
 May 2022
 DOI:
 10.1103/PhysRevA.105.052428
 arXiv:
 arXiv:2112.02253
 Bibcode:
 2022PhRvA.105e2428P
 Keywords:

 Quantum Physics;
 Condensed Matter  Strongly Correlated Electrons
 EPrint:
 17 pages, 7 figures, 65 references