Quantum Information Processing

, Volume 8, Issue 5, pp 415–429 | Cite as

Entanglement and Berry phase in a 9 × 9 Yang–Baxter system

  • Chunfang Sun
  • Kang Xue
  • Gangcheng Wang


A M-matrix which satisfies the Hecke algebraic relations is presented. Via the Yang–Baxterization approach, we obtain a unitary solution \({\breve{R}(\theta,\varphi_{1},\varphi_{2})}\) of Yang–Baxter equation. It is shown that any pure two-qutrit entangled states can be generated via the universal \({\breve{R}}\)-matrix assisted by local unitary transformations. A Hamiltonian is constructed from the \({\breve{R}}\)-matrix, and Berry phase of the Yang–Baxter system is investigated. Specifically, for \({\varphi_{1}\,{=}\,\varphi_{2}}\), the Hamiltonian can be represented based on three sets of SU(2) operators, and three oscillator Hamiltonians can be obtained. Under this framework, the Berry phase can be interpreted.


Quantum entanglement Berry phase Yang–Baxter equation 


03.67.Mn 02.40.-k 03.65.Vf 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berry M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. London. Ser. A 392, 45–57 (1984)CrossRefADSzbMATHGoogle Scholar
  2. 2.
    Simon B.: Holonomy, the quantum adiabatic theorem, and Berry’s Phase. Phys. Rev. Lett. 51, 2167–2170 (1983)CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Korepin V.E., Wu A.C.T.: Adiabatic transport properties and Berry’s phase in Heisenberg-Ising ring. Int. J. of Mod. Phys. B 5, 497–507 (1991)CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Aharonov Y., Anandan J.: Phase change during a cyclic quantum evolution. Phys. Rev. Lett. 58, 1593–1596 (1987)PubMedCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Sjövist E., Pati A.K., Ekert A., Anandan J.S., Ericsson M., Oi D.K.L., Vedral V.: Geometric phases for mixed states in interferometry. Phys. Rev. Lett. 85, 2845–2849 (2000)CrossRefADSGoogle Scholar
  6. 6.
    Samuel J., Bhandari R.: General setting for Berry’s phase. Phys. Rev. Lett. 60, 2339–2342 (1988)PubMedCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Tong D.M., Sjöqvist E., Kwek L.C, Oh C.H.: Kinematic approach to geometric phase of mixed states under nonunitary evolutions. Phys. Rev. Lett. 93, 080405 (2004)PubMedCrossRefADSGoogle Scholar
  8. 8.
    Wilczek F., Zee A.: Appearance of gauge structure in simple dynamical systems. Phys. Rev. Lett. 52, 2111–2114 (1984)CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Suter D., Chingas G., Harris R., Pines A.: Berry’s phase in magnetic resonance. Mol. Phys. 61, 1327–1340 (1987)CrossRefADSGoogle Scholar
  10. 10.
    Goldman M., Fleury V., Guéron M.: NMR frequency shift under sample spinning. J. Magn. Reson. A 118, 11–20 (1996)CrossRefGoogle Scholar
  11. 11.
    Tycko R.: Adiabatic rotational splittings and Berry’s phase in nuclear quadrupole resonance. Phys. Rev. Lett. 58, 2281–2284 (1987)PubMedCrossRefADSGoogle Scholar
  12. 12.
    Appelt S., \({\ddot{W}}\)ackerle G., Mehring M.: Deviation from Berry’s adiabatic geometric phase in a 131Xe nuclear gyroscope. Phys. Rev. Lett. 72, 3921–3924 (1994)Google Scholar
  13. 13.
    Jones J.A., Pines A.: Geometric dephasing in zero-field magnetic resonance. J. Chem. Phys. 106, 3007–3016 (1997)CrossRefADSGoogle Scholar
  14. 14.
    Chiao R.Y., Wu Y.S.: Manifestations of Berry’s topological phase for the photon. Phys. Rev. Lett. 57, 933–936 (1986)PubMedCrossRefADSGoogle Scholar
  15. 15.
    Bohm A., Mostafazadeh A., Koizumi H., Niu Q., Zwanziger J.: The geometric phase in quantum systems. J. Phys. A Math. Gen. 36, 12345 (2003)ADSGoogle Scholar
  16. 16.
    Jones J., Vedral V., Ekert A.K., Castagnoli C.: Geometric quantum computation using nuclear magnetic resonance. Nature 403, 869–871 (2000)PubMedGoogle Scholar
  17. 17.
    Duan L.M., Cirac J.I., Zoller P.: Geometric manipulation of trapped ions for quantum computation. Science 292, 1695–1697 (2001)PubMedCrossRefGoogle Scholar
  18. 18.
    Wootters W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998)CrossRefADSGoogle Scholar
  19. 19.
    Ekert A., Ericsson M., Hayden P., Inamori H., Jones J.A., Oi D.K.L., Vedral V.: Geometric quantum computation. J. Mod. Opt. 47, 2501–2513 (2000)MathSciNetGoogle Scholar
  20. 20.
    Bennett C.H., DiVincenzo D.P.: Quantum information and computation. Nature 404, 247–255 (2000)PubMedCrossRefADSGoogle Scholar
  21. 21.
    Bennett C.H., Brassard G., Crépeau C., Jozsa R., Peres A., Wootters W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1993)PubMedCrossRefADSMathSciNetzbMATHGoogle Scholar
  22. 22.
    Bennett C.H., Wiesner S.J.: Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69, 2881–2884 (1992)PubMedCrossRefADSMathSciNetzbMATHGoogle Scholar
  23. 23.
    Murao M., Jonathan D., Plenio M.B., Vedral V.: Quantum telecloning and multiparticle entanglement. Phys. Rev. A 59, 156–161 (1999)CrossRefADSGoogle Scholar
  24. 24.
    Yang C.N.: Some Exact results for the many-body problem in one dimension with repulsive delta–function interaction. Phys. Rev. Lett. 19, 1312–1315 (1967)CrossRefADSMathSciNetzbMATHGoogle Scholar
  25. 25.
    Yang C.N.: S matrix for the one-dimensional N-body problem with repulsive or attractive–function interaction. Phys. Rev. 168, 1920–1923 (1968)CrossRefADSGoogle Scholar
  26. 26.
    Baxter R.J.: Exactly solved models in statistical mechanics. Academic, New York (1982)zbMATHGoogle Scholar
  27. 27.
    Baxter R.J.: Partition function of the eight-vertex lattice model. Ann. Phys. 70, 193–228 (1972)CrossRefADSMathSciNetzbMATHGoogle Scholar
  28. 28.
    Drinfel’d V.G.: Hopf algebras and the quantum Yang–Baxter equation. Soviet Math. Dokl. 32, 254C258 (1985)Google Scholar
  29. 29.
    Drinfel’d V.G.: A new realization of Yangians and quantized affine algebras. Soviet Math. Dokl. 36, 212–216 (1988)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Drinfel’d, V.G.: Quantum groups. In: Proceedngs of International Congress on Mathematics, vol. 269, pp. 798–820. Academic, Berkeley (1986)Google Scholar
  31. 31.
    Kitaev A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003)CrossRefADSMathSciNetzbMATHGoogle Scholar
  32. 32.
    Kauffman L.H., Lomonaco S.J. Jr: Braiding operators are universal quantum gates. New J. Phys. 6, 134 (2004)CrossRefADSGoogle Scholar
  33. 33.
    Franko J.M., Rowell E.C., Wang Z.: Extraspecial 2-groups and images of braid group representations. J. Knot Theory Ramif. 15, 413–428 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Zhang Y., Kauffman L.H., Ge M.L.: Universal quantum gate, Yang-Baxterization and Hamiltonian. Int. J. Quant. Inf. 3, 669–678 (2005)CrossRefzbMATHGoogle Scholar
  35. 35.
    Zhang Y., Ge M.L.: GHZ states, almost-complex structure and Yang–Baxter equation. Quant. Inf. Proc. 6, 363–379 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Zhang, Y., Rowell, E.C., Wu, Y.S., Wang, Z.H., Ge, M.L.: From extraspecial two-Groups to GHZ states. arXiv:quant-ph/0706.1761v2Google Scholar
  37. 37.
    Chen J.L., Xue K., Ge M.L.: Braiding transformation, entanglement swapping, and Berry phase in entanglement space. Phys. Rev. A. 76, 042324 (2007)CrossRefADSGoogle Scholar
  38. 38.
    Chen J.L., Xue K., Ge M.L.: Berry phase and quantum criticality in Yang–Baxter systems. Ann. Phys. 323, 2614–2623 (2008)CrossRefADSMathSciNetzbMATHGoogle Scholar
  39. 39.
    Chen, J.L, Xue, K., Ge, M.L.: All pure two-qudit entangled states can be generated via a universal Yang–Baxter matrix assisted by local unitary transformations. arXiv:quantph/0809.2321v1Google Scholar
  40. 40.
    Hu S.W., Xue K., Ge M.L.: Optical simulation of the Yang–Baxter equation. Phys. Rev. A 78, 022319 (2008)CrossRefADSMathSciNetGoogle Scholar
  41. 41.
    Wang G.C., Xue K., Wu C.F., Liang H., Oh C.H.: Entanglement and the Berry phase in a new Yang–Baxter system. J. Phys. A Math. Theor. 42, 125207 (2009)CrossRefADSGoogle Scholar
  42. 42.
    Sun, C.F., Hu, T.T., Wang, G.C., Wu, C.F., Xue, K.: Thermal entanglement in the systems constructed from the Yang–Baxter \({\breve{R}}\)-matrix. Int. J. Quant. Inf. 7, 5 (2009)Google Scholar
  43. 43.
    Nielsen M.A., Chuang I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  44. 44.
    Ekert A.K.: Quantum cryptography based on Bells theorem. Phys. Rev. Lett. 67, 661–663 (1991)PubMedCrossRefADSMathSciNetzbMATHGoogle Scholar
  45. 45.
    Raussendorf R., Briegel H.J.: A one-way quantum computer. Phys. Rev. Lett. 86, 5188–5191 (2001)PubMedCrossRefADSGoogle Scholar
  46. 46.
    Prevedel R., Walther P., Tiefenbacher F., Bohi P., Kaltenbaek R., Jennewein T., Zeilinger A.: High-speed linear optics quantum computing using active feed-forward. Nature 445, 65–69 (2007)PubMedCrossRefADSGoogle Scholar
  47. 47.
    Zyczkowski K., Horodecki P., sanpera A., lewenstein M.: Volume of the set of separable states. Phys. Rev. A 58, 883–892 (1998)CrossRefADSMathSciNetGoogle Scholar
  48. 48.
    Vidal G., Werner R.F.: Computable measure of entanglement. Phys. Rev. A 65, 032314 (2002)CrossRefADSGoogle Scholar
  49. 49.
    Zhang W., Feng D., Gilmore R.: Coherent states: theory and some applications. Rev. Mod. Phys. 62, 867–927 (1990)CrossRefADSMathSciNetGoogle Scholar
  50. 50.
    Chaturvedi S., Sriram M.S., Srinivasan V.: Berry’s phase for coherent states. J. Phys. A Math. Gen 20, L1071–L1075 (1987)CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.School of PhysicsNortheast Normal UniversityChangchunPeople’s Republic of China

Personalised recommendations