Quantum Information Processing

, Volume 12, Issue 7, pp 2417–2426 | Cite as

The topological basis expression of four-qubit XXZ spin chain with twist boundary condition

  • Guijiao Du
  • Kang Xue
  • Chengcheng Zhou
  • Chunfang Sun
  • Gangcheng Wang


We investigate the XXZ model’s characteristic with the twisted boundary condition and the topological basis expression. Owing to twist boundary condition, the ground state energy will changing back and forth between \(E_{13}\) and \(E_{15}\) by modulate the parameter \(\phi \). By using TLA generators, the XXZ model’s Hamiltonian can be constructed. All the eigenstates can be expressed by topological basis, and the whole of eigenstates’ entanglement are maximally entangle states (\(Q(|\phi _i\rangle )=1\)).


Topological basis Quantum entanglement Knot theory 



This work was supported in part by NSF of China (Grant No. 11175043).


  1. 1.
    Nayak, C., Simon, S.H., Stern, A., Freedman, M., Sarma, S.D.: Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008)ADSMATHCrossRefGoogle Scholar
  2. 2.
    Hu, S.W., Xue, K., Ge, M.L.: Optical simulation of the Yang-Baxter equation. Phys. Rev. A 78, 022319 (2008)MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Ardonne, E., Schoutens, K.: Wavefunctions for topological quantum registers. Ann. Phys. 322, 201–235 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    Feiguin, A., Trebst, S., Ludwig, A.W.W., Troyer, M., Kitaev, A., Wang, Z.H., Freedman, M.H.: Interacting anyons in topological quantum liquids: the golden chain. Phys. Rev. Lett. 98, 160409 (2007)ADSCrossRefGoogle Scholar
  5. 5.
    Nayak, C., Simon, S.H., Stern, A., Freedman, M., Sarma, S.D.: Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008)Google Scholar
  6. 6.
    Hikami, K.: Skein theory and topological quantum registers: braiding matrices and topological entanglement entropy of non-Abelian quantum Hall states. Ann. Phys. 323, 1729–1769 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Jones, V.F.R.: A polynomial invariant for links via von Neumann algebras. Bull. Am. Math. Soc. 129, 103–112 (1985)CrossRefGoogle Scholar
  8. 8.
    Wadati, M., Deguchi, T., Akutsu, Y.: Exactly solvable models and knot theory. Phys. Rep. 180, 247–332 (1989)MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Kauffman, L.H.: An invariant of regular isotopy. Trans. Am. Math. Soc. 318(2), 417–471 (1990)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Abramsky, S.: Temperley-Lieb Algebra: From Knot Theory to Logic and Computation via Quantum Mechanics. arX-iv:0910.2737 (2009)Google Scholar
  11. 11.
    Temperley, H.N.V., Lieb, E.H.: Relations between the ‘Percolation’ and ‘Colouring’ Problem and other Graph-Theoretical Problems Associated with Regular Planar Lattices: Some Exact Results for the ‘Percolation’ Problem. Proc. Roy. Soc. Lond. A 322, 251–280 (1971)MathSciNetADSMATHCrossRefGoogle Scholar
  12. 12.
    Levy, D.: Structure of Temperley-Lieb algebras and its application to 2D statistical models. Phys. Rev. Lett. 64, 499–502 (1990)MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Rui, H., Xi, C.: The representation theory of cyclotomic Temperley-Lieb algebras. Commentar. Math. Helv. 79, 427–450 (2004)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Baxter, R.J.: The inversion relation method for some two-dimensional exactly solved models in lattice statistics J. Stat. Phys. 28, 1 (1982)Google Scholar
  15. 15.
    Jimbo, M.: A q-difference analogue of U(g) and the Yang-Baxter equation. Lett. Math. Phys. 10, 63–69 (1985)MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Kassel, C.: Quantum groups. Springer, Berlin (1995)MATHCrossRefGoogle Scholar
  17. 17.
    Martin, P.P.: Potts models and related problems in statistical mechanics. World Scientific, Singapore (1991)MATHCrossRefGoogle Scholar
  18. 18.
    Baxter, R.J.: Exactly solved models in statistical mechanics (Chap. 12). Academic Press, New York (1982)Google Scholar
  19. 19.
    Kauffman, L.H.: Knots and physics. World Scientific, Singapore (1991)MATHGoogle Scholar
  20. 20.
    Goodman, F.M., de la Harpe, P., Jones, V.F.R.: Coxeter graphs and towers of algebras, vol. 14. Math Sci Research Inst Publications, Springer, NewYork (1989)CrossRefGoogle Scholar
  21. 21.
    Savit, R.: Duality in field theory and statistical systems. Rev. Mod. Phys. 52, 453–487 (1980)MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    Alvarez, M., Martin, P.P.: A Temperley-Lieb category for 2-manifolds. arXiv:0711.4777v1 [math-ph]Google Scholar
  23. 23.
    Sun, C.F., Wang, G.C., Hu, T.T., Zhou, C.C., Wang, Q.Y., Xue, K.: The representations of Temperley-Lieb algebra and entanglement in a Yang-Baxter system. Int. J. Quantum Inf. 7, 1285–1293 (2009)MATHCrossRefGoogle Scholar
  24. 24.
    Wang, G.C., Xue, K., Sun, C.F., Hu, T.T.,Zhou, C.C., Du, G.J.: Quantum tunneling effect and quantum Zeno effect in a Topological system. arXiv:1012.1474v2Google Scholar
  25. 25.
    Sutherland, B., Shastry, B.S.: Adiabatic transport properties of an exactly soluble one-dimensional quantum many-body problem. Phys. Rev. Lett. 65, 1833–1837 (1990)MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    Korepin, V.E., Wu, A.C.T.: Adiabatic transport properties and Berry’s phase in Heisenberg-Ising ring. Int. J. Mod. Phys. B 5, 497–507 (1991)MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    Beyers, N., Yang, C.N.: Theoretical considerations concerning quantized magnetic flux in superconducting cylinders. Phys. Rev. Lett. 7, 46–49 (1961)ADSCrossRefGoogle Scholar
  28. 28.
    Fath, G., Solyom, J.: Isotropic spin-1 chain with twisted boundary condition. Phys. Rev. B 47, 872–881 (1993)ADSCrossRefGoogle Scholar
  29. 29.
    Barnum, H., Knill, E., Ortiz, G., Viola, L.: Generalizations of entanglement based on coherent states and convex sets. Phys. Rev. A 68, 032308 (2003)MathSciNetADSCrossRefGoogle Scholar
  30. 30.
    Meyer, D.A., Wallach, N.R.: Global entanglement in multiparticle systems. J. Math. Phys. 43, 4273 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  31. 31.
    Brennen, G.K.: An observable measure of entanglement for pure states of multi-qubit systems. J. Quantum Inf. Comput. 3, 619–626 (2003)MathSciNetMATHGoogle Scholar
  32. 32.
    Marx, R., Fahmy, A., Kauffman, L., Lomonaco, S., Sporl, A., Pomplun, N.S., Schulte-Herbruggen, T., Myers, J.M., Glaser, S.J.: Nuclear-magnetic-resonance quantum calculations of the Jones polynomial. Phys. Rev. A 81, 032319 (2009)ADSCrossRefGoogle Scholar
  33. 33.
    Kauffman, L.H., Lomonaco Jr, S.J.: Braiding operators are universal quantum gates. New J. Phys. 6, 134 (2004)ADSCrossRefGoogle Scholar
  34. 34.
    Jimbo, M.: Yang-Baxter Equations on Integrable Systems. World Scientific, Singapore (1990)CrossRefGoogle Scholar
  35. 35.
    Guijiao, Du, Xue, Kang, Sun, Chunfang, Wang, Gangcheng, Zhou, Chengcheng: The topological basis realization of four-qubit XXZ spin chain. Int. J. Quanum Inf. 10, 1250021 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Guijiao Du
    • 1
  • Kang Xue
    • 1
  • Chengcheng Zhou
    • 2
  • Chunfang Sun
    • 1
  • Gangcheng Wang
    • 1
  1. 1.School of PhysicsNortheast Normal UniversityChangchunPeople’s Republic of China
  2. 2.School of ScienceChangchun University of Science and TechnologyChangchunPeople’s Republic of China

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