Skein Theory and Topological Quantum Registers: Braiding Matrices and Topological Entanglement Entropy of Non-Abelian Quantum Hall States

Abstract

We study topological properties of quasi-particle states in the non-Abelian quantum Hall states. We apply a skein-theoretic method to the Read--Rezayi state whose effective theory is the SU(2)_K Chern--Simons theory. As a generalization of the Pfaffian (K=2) and the Fibonacci (K=3) anyon states, we compute the braiding matrices of quasi-particle states with arbitrary spins. Furthermore we propose a method to compute the entanglement entropy skein-theoretically. We find that the entanglement entropy has a nontrivial contribution called the topological entanglement entropy which depends on the quantum dimension of non-Abelian quasi-particle intertwining two subsystems.

42 pages, many eps files

References in corpus (32)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Anyons in an exactly solved model and beyond
• Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect
• Entanglement Entropy and Quantum Field Theory
• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Non-abelian statistics of half-quantum vortices in p-wave superconductors
• Geometric and Renormalized Entropy in Conformal Field Theory
• String-net condensation: A physical mechanism for topological phases
• Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level
• $2n$ Quasihole States Realize $2^{n-1}$-Dimensional Spinor Braiding Statistics in Paired Quantum Hall States
• Quantum Phases of Vortices in Rotating Bose-Einstein Condensates
• Interacting anyons in topological quantum liquids: The golden chain
• Electron correlation in the second Landau level; a competition between many, nearly degenerate quantum phases
• Entanglement entropy in fermionic Laughlin states
• Universal Quantum Computation with the nu=5/2 Fractional Quantum Hall State
• A Chern-Simons Effective Field Theory for the Pfaffian Quantum Hall State
• A Class of $P,T$-Invariant Topological Phases of Interacting Electrons
• Braid Topologies for Quantum Computation
• Quantum groups and nonabelian braiding in quantum Hall systems
• Topological Entanglement Entropy in the Quantum Dimer Model on the Triangular Lattice
• Interferometry of non-Abelian Anyons
• The two-eigenvalue problem and density of Jones representation of braid groups
• Topological Quantum Compiling
• Topological Quantum Computing with Only One Mobile Quasiparticle
• Large Fourier transforms never exactly realized by braiding conformal blocks
• Topologically protected quantum gates for computation with non-Abelian anyons in the Pfaffian quantum Hall state
• Wavefunctions for topological quantum registers
• Paired fractional quantum Hall states and the nu=5/2 puzzle
• Topological phases and topological entropy of two-dimensional systems with finite correlation length
• Towards a universal set of topologically protected gates for quantum computation with Pfaffian qubits

Cited by in corpus (13)

• Quantum Hall Physics - hierarchies and CFT techniques
• Optical Simulation of Yang-Baxter Equation
• Anyonic Entanglement and Topological Entanglement Entropy
• Direct Experimental Simulation of the Yang-Baxter Equation
• Matrix product states for anyonic systems and efficient simulation of dynamics
• Measures of entanglement in non-Abelian anyonic systems
• Information-theoretical formulation of anyonic entanglement
• Reaction-diffusion dynamics in a Fibonacci chain: Interplay between classical and quantum behavior
• ${\ell }_1$-norm and entanglement in screening out braiding from Yang-Baxter equation associated with ${\mathbb{Z}}_3$ parafermion
• Birman-Wenzl-Murakami Algebra and the Topological Basis
• Superactivation of Bell nonlocality in pure anyonic states
• Quantum Resource Theories of Anyonic Entanglement
• Birman-Wenzl-Murakami Algebra, Topological parameter and Berry phase