Boundary algebras of the Kitaev Quantum Double model

• URL: http://arxiv.org/abs/2309.13440v1
• Date: Sat, 23 Sep 2023 17:56:38 GMT
• Title: Boundary algebras of the Kitaev Quantum Double model
• Authors: Mario Tomba and Shuqi Wei and Brett Hungar and Daniel Wallick and Kyle Kawagoe and Chian Yeong Chuah and David Penneys
• Abstract summary: We prove the LTO axioms for Kitaev's Quantum Double model for a finite group $G$. We identify the boundary nets of algebras with fusion categorical nets associated to $(mathsfHilb(G),mathbbC[G],$ or $(mathsfRep(G),mathbbCG)$ depending on whether the boundary cut is rough or smooth.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The recent article [arXiv:2307.12552] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev's Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev's Quantum Double model for a finite group $G$. We identify the boundary nets of algebras with fusion categorical nets associated to $(\mathsf{Hilb}(G),\mathbb{C}[G])$ or $(\mathsf{Rep}(G),\mathbb{C}^G)$ depending on whether the boundary cut is rough or smooth respectively. This allows us to make connections to work of Ogata on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial $G$-symmetry protected topological phase ($G$-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3+1)D Quantum Double model associated to an abelian group.

Related papers

• Volichenko-type metasymmetry of braided Majorana qubits [0.0]
  This paper presents different mathematical structures connected with the parastatistics of braided Majorana qubits. Mixed-bracket Heisenberg-Lie algebras are introduced.
  arXiv  Detail & Related papers  (2024-06-02T21:50:29Z)
• 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)
• Local topological order and boundary algebras [0.0]
  We introduce axioms for locally topologically ordered quantum spin systems in terms of nets of local ground state projections. For a locally topologically ordered spin system on $mathbbZk$, we define a local net of boundary algebras on $mathbbZk-1$. We construct a canonical quantum channel so that states on the boundary quasi-local algebra parameterize bulk-boundary states without reference to a boundary Hamiltonian.
  arXiv  Detail & Related papers  (2023-07-24T06:38:48Z)
• Quantum Current and Holographic Categorical Symmetry [62.07387569558919]
  A quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance. The condition for quantum currents to be superconducting is also specified, which corresponds to condensation of anyons in one higher dimension.
  arXiv  Detail & Related papers  (2023-05-22T11:00:25Z)
• Algebraic Aspects of Boundaries in the Kitaev Quantum Double Model [77.34726150561087]
  We provide a systematic treatment of boundaries based on subgroups $Ksubseteq G$ with the Kitaev quantum double $D(G)$ model in the bulk. The boundary sites are representations of a $*$-subalgebra $Xisubseteq D(G)$ and we explicate its structure as a strong $*$-quasi-Hopf algebra. As an application of our treatment, we study patches with boundaries based on $K=G$ horizontally and $K=e$ vertically and show how these could be used in a quantum computer
  arXiv  Detail & Related papers  (2022-08-12T15:05:07Z)
• Quantum teleportation in the commuting operator framework [63.69764116066747]
  We present unbiased teleportation schemes for relative commutants $N'cap M$ of a large class of finite-index inclusions $Nsubseteq M$ of tracial von Neumann algebras. We show that any tight teleportation scheme for $N$ necessarily arises from an orthonormal unitary Pimsner-Popa basis of $M_n(mathbbC)$ over $N'$.
  arXiv  Detail & Related papers  (2022-08-02T00:20:46Z)
• Boundary and domain wall theories of 2d generalized quantum double model [1.3965477771846408]
  The quantum double lattice realization of 2d topological orders based on Hopf algebras is discussed in this work. To generalize the model to a 2d surface with boundaries and surface defects, we present a systematic construction of the boundary Hamiltonian and domain wall Hamiltonian.
  arXiv  Detail & Related papers  (2022-07-08T15:44:47Z)
• Matrix product operator algebras I: representations of weak Hopf algebras and projected entangled pair states [0.5872014229110214]
  Matrix Product Operators (MPOs) represent operators acting on 1D systems. MPOs can be used to represent algebras of non-trivial symmetries. We show that MPOs can be used to construct Kitaev's quantum double models for Hopf algebras.
  arXiv  Detail & Related papers  (2022-04-12T16:43:15Z)
• Annihilating Entanglement Between Cones [77.34726150561087]
  We show that Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation.
  arXiv  Detail & Related papers  (2021-10-22T15:02:39Z)
• Quantum scrambling of observable algebras [0.0]
  quantum scrambling is defined by how the associated physical degrees of freedom get mixed up with others by the dynamics. This is accomplished by introducing a measure, the geometric algebra anti-correlator (GAAC) of the self-orthogonalization of the commutant of $cal A$ induced by the dynamics. For generic energy spectrum we find explicit expressions for the infinite-time average of the GAAC which encode the relation between $cal A$ and the full system of Hamiltonian eigenstates.
  arXiv  Detail & Related papers  (2021-07-02T14:30:58Z)
• Quantum double aspects of surface code models [77.34726150561087]
  We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry. We show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$.
  arXiv  Detail & Related papers  (2021-06-25T17:03:38Z)

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.