Classification of the anyon sectors of Kitaev's quantum double model

• URL: http://arxiv.org/abs/2310.19661v3
• Date: Fri, 1 Dec 2023 17:32:13 GMT
• Title: Classification of the anyon sectors of Kitaev's quantum double model
• Authors: Alex Bols, Siddharth Vadnerkar
• Abstract summary: We give a complete classification of the anyon sectors of Kitaev's quantum double model on the infinite triangular lattice. As conjectured, the anyon sectors correspond precisely to the irreducible representations of the quantum double algebra of $G$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We give a complete classification of the anyon sectors of Kitaev's quantum double model on the infinite triangular lattice and for finite gauge group $G$, including the non-abelian case. As conjectured, the anyon sectors of the model correspond precisely to the irreducible representations of the quantum double algebra of $G$. Our proof consists of two main parts. In the first part, we construct for each irreducible representation of the quantum double algebra a pure state and show that the GNS representations of these pure states are pairwise disjoint anyon sectors. In the second part we show that any anyon sector is unitarily equivalent to one of the anyon sectors constructed in the first part. Purity of the states constructed in the first part is shown by characterising these states as the unique states that satisfy appropriate local constraints. These constraints are of two types, namely flux constraints and gauge constraints. The flux constraints single out certain string-net states, while the gauge constraints fix the way in which these string-nets condense. At the core of the proof is the fact that certain groups of local gauge transformations act freely and transitively on collections of local string-nets. The proof that the GNS representations of these states are anyon sectors relies on showing that they are unitarily equivalent to amplimorphism representations which are much easier to compare to the ground state representation. For the second part, we show that any anyon sector contains a pure state that satisfies all but a finite number of the constraints characterising the pure states of the first part. Using known techniques we can then construct a pure state in the anyon sector that satisfies all but one of these constraints. Finally, we show that any such state must be a vector state in one of the anyon sectors constructed in the first part.

Related papers

• Multipartite Embezzlement of Entanglement [44.99833362998488]
  Embezzlement of entanglement refers to the task of extracting entanglement from an entanglement resource via local operations and without communication. We show that finite-dimensional approximations of multipartite embezzling states form multipartite embezzling families. We discuss our results in the context of quantum field theory and quantum many-body physics.
  arXiv  Detail & Related papers  (2024-09-11T22:14:22Z)
• Critical Fermions are Universal Embezzlers [44.99833362998488]
  We show that universal embezzlers are ubiquitous in many-body physics. The same property holds in locally-interacting, dual spin chains via the Jordan-Wigner transformation.
  arXiv  Detail & Related papers  (2024-06-17T17:03:41Z)
• Classifying 2D topological phases: mapping ground states to string-nets [0.0]
  Two gapped ground states of lattice Hamiltonians are in the same quantum phase of matter, or topological phase, if they can be connected by a constant-depth quantum circuit. It is conjectured that the Levin-Wen string-net models exhaust all possible gapped phases with gappable boundary, and these phases are labeled by unitary modular tensor categories.
  arXiv  Detail & Related papers  (2024-05-27T17:36:17Z)
• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• Almost device-independent certification of GME states with minimal measurements [41.94295877935867]
  Device-independent certification of quantum states allows the characterization of quantum states present inside a device. A major problem in this regard is to certify quantum states using minimal resources. We consider the multipartite quantum steering scenario with an arbitrary number of parties but only one of which is trusted in the sense that the measurements performed by the trusted party are known.
  arXiv  Detail & Related papers  (2024-02-28T17:54:55Z)
• Entanglement of Quantum States which are Zero on the Symmetric Sector [0.0]
  We consider a quantum system of n qudits and the Clebsch-Gordan decomposition of the associated Hilbert space. We prove that any separable state must have a nonzero component on the symmetric sector. We identify a class of entanglement witnesses for these systems.
  arXiv  Detail & Related papers  (2023-11-28T22:48:24Z)
• $W$ state is not the unique ground state of any local Hamiltonian [0.0]
  characterization of ground states among all quantum states is an important problem in quantum many-body physics. We introduce a new class of simple states, including the $W$ state, that can only occur as a ground state alongside an exactly degenerate partner.
  arXiv  Detail & Related papers  (2023-10-16T18:00:01Z)
• The Schmidt rank for the commuting operator framework [58.720142291102135]
  The Schmidt rank is a measure for the entanglement dimension of a pure bipartite state. We generalize the Schmidt rank to the commuting operator framework. We analyze bipartite states and compute the Schmidt rank in several examples.
  arXiv  Detail & Related papers  (2023-07-21T14:37:33Z)
• Error bounds for Lie Group representations in quantum mechanics [44.99833362998488]
  We provide state-dependent error bounds for strongly continuous unitary representations of Lie groups. Our method works for any connected Lie group and the metric is independent of the chosen representation.
  arXiv  Detail & Related papers  (2022-11-15T23:55:53Z)
• Separability and entanglement in superpositions of quantum states [0.0]
  We study the superpositions of a pure entangled state and a pure product state, when the amplitudes corresponding to the states appearing in any superposition are nonzero. All such superpositions produce only entangled states if the initial entangled state has Schmidt rank three or higher. We find that conditional inseparability of superpositions help in identifying strategies for conclusive local discrimination of shared quantum ensembles.
  arXiv  Detail & Related papers  (2021-08-04T19:48:29Z)
• The split and approximate split property in 2D systems: stability and absence of superselection sectors [0.0]
  In 1D, gapped ground states have the split property with respect to cutting the system into left and right half-chains. In 2D, however, the split property fails to hold for interesting models such as Kitaev's toric code. We show that long-range entanglement, in a way that we will define precisely, is a necessary condition to have non-trivial superselection sectors.
  arXiv  Detail & Related papers  (2021-02-15T17:59:10Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.