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

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Abstract

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.

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References

  1. Berry M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. London. Ser. A 392, 45–57 (1984)

    Article  ADS  MATH  Google Scholar 

  2. Simon B.: Holonomy, the quantum adiabatic theorem, and Berry’s Phase. Phys. Rev. Lett. 51, 2167–2170 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  4. Aharonov Y., Anandan J.: Phase change during a cyclic quantum evolution. Phys. Rev. Lett. 58, 1593–1596 (1987)

    Article  PubMed  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  6. Samuel J., Bhandari R.: General setting for Berry’s phase. Phys. Rev. Lett. 60, 2339–2342 (1988)

    Article  PubMed  ADS  MathSciNet  Google Scholar 

  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)

    Article  PubMed  ADS  CAS  Google Scholar 

  8. Wilczek F., Zee A.: Appearance of gauge structure in simple dynamical systems. Phys. Rev. Lett. 52, 2111–2114 (1984)

    Article  ADS  CAS  MathSciNet  Google Scholar 

  9. Suter D., Chingas G., Harris R., Pines A.: Berry’s phase in magnetic resonance. Mol. Phys. 61, 1327–1340 (1987)

    Article  ADS  Google Scholar 

  10. Goldman M., Fleury V., Guéron M.: NMR frequency shift under sample spinning. J. Magn. Reson. A 118, 11–20 (1996)

    Article  CAS  Google Scholar 

  11. Tycko R.: Adiabatic rotational splittings and Berry’s phase in nuclear quadrupole resonance. Phys. Rev. Lett. 58, 2281–2284 (1987)

    Article  PubMed  ADS  CAS  Google Scholar 

  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. Jones J.A., Pines A.: Geometric dephasing in zero-field magnetic resonance. J. Chem. Phys. 106, 3007–3016 (1997)

    Article  ADS  CAS  Google Scholar 

  14. Chiao R.Y., Wu Y.S.: Manifestations of Berry’s topological phase for the photon. Phys. Rev. Lett. 57, 933–936 (1986)

    Article  PubMed  ADS  CAS  Google Scholar 

  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)

    ADS  Google Scholar 

  16. Jones J., Vedral V., Ekert A.K., Castagnoli C.: Geometric quantum computation using nuclear magnetic resonance. Nature 403, 869–871 (2000)

    PubMed  Google Scholar 

  17. Duan L.M., Cirac J.I., Zoller P.: Geometric manipulation of trapped ions for quantum computation. Science 292, 1695–1697 (2001)

    Article  PubMed  Google Scholar 

  18. Wootters W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998)

    Article  ADS  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  20. Bennett C.H., DiVincenzo D.P.: Quantum information and computation. Nature 404, 247–255 (2000)

    Article  PubMed  ADS  CAS  Google Scholar 

  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)

    Article  PubMed  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  PubMed  ADS  MathSciNet  MATH  Google Scholar 

  23. Murao M., Jonathan D., Plenio M.B., Vedral V.: Quantum telecloning and multiparticle entanglement. Phys. Rev. A 59, 156–161 (1999)

    Article  ADS  CAS  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  26. Baxter R.J.: Exactly solved models in statistical mechanics. Academic, New York (1982)

    MATH  Google Scholar 

  27. Baxter R.J.: Partition function of the eight-vertex lattice model. Ann. Phys. 70, 193–228 (1972)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Drinfel’d V.G.: Hopf algebras and the quantum Yang–Baxter equation. Soviet Math. Dokl. 32, 254C258 (1985)

    Google Scholar 

  29. Drinfel’d V.G.: A new realization of Yangians and quantized affine algebras. Soviet Math. Dokl. 36, 212–216 (1988)

    MathSciNet  MATH  Google Scholar 

  30. Drinfel’d, V.G.: Quantum groups. In: Proceedngs of International Congress on Mathematics, vol. 269, pp. 798–820. Academic, Berkeley (1986)

  31. Kitaev A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003)

    Article  ADS  CAS  MathSciNet  MATH  Google Scholar 

  32. Kauffman L.H., Lomonaco S.J. Jr: Braiding operators are universal quantum gates. New J. Phys. 6, 134 (2004)

    Article  ADS  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang Y., Kauffman L.H., Ge M.L.: Universal quantum gate, Yang-Baxterization and Hamiltonian. Int. J. Quant. Inf. 3, 669–678 (2005)

    Article  MATH  Google Scholar 

  35. Zhang Y., Ge M.L.: GHZ states, almost-complex structure and Yang–Baxter equation. Quant. Inf. Proc. 6, 363–379 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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/0706.1761v2

  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)

    Article  ADS  Google Scholar 

  38. Chen J.L., Xue K., Ge M.L.: Berry phase and quantum criticality in Yang–Baxter systems. Ann. Phys. 323, 2614–2623 (2008)

    Article  ADS  CAS  MathSciNet  MATH  Google Scholar 

  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/0809.2321v1

  40. Hu S.W., Xue K., Ge M.L.: Optical simulation of the Yang–Baxter equation. Phys. Rev. A 78, 022319 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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. Nielsen M.A., Chuang I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  44. Ekert A.K.: Quantum cryptography based on Bells theorem. Phys. Rev. Lett. 67, 661–663 (1991)

    Article  PubMed  ADS  MathSciNet  MATH  Google Scholar 

  45. Raussendorf R., Briegel H.J.: A one-way quantum computer. Phys. Rev. Lett. 86, 5188–5191 (2001)

    Article  PubMed  ADS  CAS  Google Scholar 

  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)

    Article  PubMed  ADS  CAS  Google Scholar 

  47. Zyczkowski K., Horodecki P., sanpera A., lewenstein M.: Volume of the set of separable states. Phys. Rev. A 58, 883–892 (1998)

    Article  ADS  CAS  MathSciNet  Google Scholar 

  48. Vidal G., Werner R.F.: Computable measure of entanglement. Phys. Rev. A 65, 032314 (2002)

    Article  ADS  Google Scholar 

  49. Zhang W., Feng D., Gilmore R.: Coherent states: theory and some applications. Rev. Mod. Phys. 62, 867–927 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  50. Chaturvedi S., Sriram M.S., Srinivasan V.: Berry’s phase for coherent states. J. Phys. A Math. Gen 20, L1071–L1075 (1987)

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Physics, Northeast Normal University, 130024, Changchun, People’s Republic of China

    Chunfang Sun, Kang Xue & Gangcheng Wang

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  1. Chunfang Sun
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  2. Kang Xue
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  3. Gangcheng Wang
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Correspondence to Kang Xue.

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Sun, C., Xue, K. & Wang, G. Entanglement and Berry phase in a 9 × 9 Yang–Baxter system. Quantum Inf Process 8, 415–429 (2009). https://doi.org/10.1007/s11128-009-0118-9

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  • DOI: https://doi.org/10.1007/s11128-009-0118-9

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