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.

arXiv comment17 pages, 7 figures, 65 references

References in corpus (43)

• Fault-tolerant quantum computation by anyons
• Anyons in an exactly solved model and beyond
• Area laws for the entanglement entropy - a review
• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• String-net condensation: A physical mechanism for topological phases
• Entanglement spectrum of a topological phase in one dimension
• Majorana Zero Modes and Topological Quantum Computation
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• An RVB phase in the triangular lattice quantum dimer model
• Quantum entanglement in condensed matter systems
• Z_2 Gauge Theory of Electron Fractionalization in Strongly Correlated Systems
• Models for gapped boundaries and domain walls
• General Relationship Between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States
• Entanglement spectrum in one-dimensional systems
• Topological Order and Conformal Quantum Critical Points
• Quasi-particle Statistics and Braiding from Ground State Entanglement
• Bipartite entanglement and entropic boundary law in lattice spin systems
• Entanglement entropy and entanglement spectrum of the Kitaev model
• Entanglement and topological entropy of the toric code at finite temperature
• Disentangling Entanglement Spectra of Fractional Quantum Hall States on Torus Geometries
• Entanglement Entropy of Gapped Phases and Topological Order in Three dimensions
• Topological quasiparticles and the holographic bulk-edge relation in 2+1D string-net models
• Adiabatic Preparation of Topological Order
• Foliated Field Theory and String-Membrane-Net Condensation Picture of Fracton Order
• Entanglement Spectrum and Entanglement Thermodynamics of Quantum Hall Bilayers at nu=1
• Irreducible multi-particle correlations in states without maximal rank
• Geometric phase of an atom in a magnetic storage ring
• Information-theoretical analysis of topological entanglement entropy and multipartite correlations
• Entanglement spectrum: Identification of the transition from vortex-liquid to vortex-lattice state in a weakly interacting rotating Bose-Einstein condensate
• Correlated spin liquids in the quantum kagome antiferromagnet at finite field: a renormalisation group analysis
• Gauge field entanglement of Kitaev's honeycomb model
• Fibonacci topological order from quantum nets
• Holographic unitary renormalization group for correlated electrons -- II: insights on fermionic criticality
• Magnetisation plateaux of the quantum pyrochlore Heisenberg antiferromagnet
• Entanglement of formation and monogamy of multi-party quantum entanglement
• Locality of Edge States and Entanglement Spectrum from Strong Subadditivity
• Fermionic criticality is shaped by Fermi surface topology: a case study of the Tomonaga-Luttinger liquid
• Holographic entanglement renormalisation of topological order in a quantum liquid
• Irreducible many-body correlations in topologically ordered systems
• Non-perturbative approach to quantum liquid ground states on geometrically frustrated Heisenberg antiferromagnets

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