New type of solutions of Yang-Baxter equations, quantum entanglement and related physical models

Abstract

Starting from the Kauffman-Lomonaco braiding matrix transforming the natural basis to Bell states, the spectral parameter describing the entanglement is introduced through Yang-Baxterization. It gives rise to a new type of solutions for Yang-Baxter equation, called the type-II that differs from the familiar solution called type-I of YBE associated with the usual chain models. The Majorana fermionic version of type-II yields the Kitaev Hamiltonian. The introduced ${\ell }_1$ -norm leads to the maximum of the entanglement by taking the extreme value and shows that it is related to the Wigner's D-function. Based on the Yang-Baxter equation the 3-body S-Matrix for type-II is explicitly given. Different from the type-I solution, the type-II solution of YBE should be considered in describing quantum information. The idea is further extended to ${\mathbb{Z}}_3$ parafermion model based on $SU(3)$ principal representation. The type-II is in difference from the familiar type-I in many respects. For example, the quantities corresponding to velocity in the chain models obey the Lorentzian additivity $\frac{u+v}{1+uv}$ rather than Galilean rule $(u+v)$. Most possibly, for the type-II solutions of YBE there may not exist RTT relation. Further more, for ${\mathbb{Z}}_3$ parafermion model we only need the rational Yang-Baxterization, which seems like trigonometric. Similar discussions are also made in terms of generalized Yang-Baxter equation with three spin spaces $\{1,\frac{1}{2},\frac{1}{2}\}$.

arXiv comment20 pages, 1 figure; Conference paper for Group32 in Prague, 2018. arXiv admin note: text overlap with arXiv:1704.00252 by other authors

References in corpus (19)

• Non-Abelian Anyons and Topological Quantum Computation
• Unpaired Majorana fermions in quantum wires
• Quantifying Coherence
• Spin-orbit interactions of light
• Non-abelian statistics of half-quantum vortices in p-wave superconductors
• Non-Abelian statistics and topological quantum information processing in 1D wire networks
• 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
• Parafermionic edge zero modes in Z_n-invariant spin chains
• Topological phases with parafermions: theory and blueprints
• Free parafermions
• Braiding transformation, entanglement swapping and Berry phase in entanglement space
• Berry phase and quantum criticality in Yang--Baxter systems
• Algebra of Majorana Doubling
• Localization of unitary braid group representations
• Factorized Three-body S-Matrix Restrained by Yang-Baxter Equation and Quantum Entanglements
• Exact Solution of a Yang-Baxter Spin-1/2 Chain Model and Quantum Entanglement
• Multipartite $d-$level GHZ bases associated with generalized braid matrices
• ${\ell }_1$-norm and entanglement in screening out braiding from Yang-Baxter equation associated with ${\mathbb{Z}}_3$ parafermion