Skip to main content
SpringerLink
Log in

Quantum Fisher information for periodic and quasiperiodic anisotropic XY chains in a transverse field

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

In this work, the concept of quantum Fisher information (QFI) is used to characterize the quantum transitions and factorization transitions in one-dimensional anisotropic XY models with periodic coupling interaction and quasiperiodic one. For the periodic-two model, it is found that the Ising transition and anisotropic transition can be distinctively illustrated by the evolution of QFI and its first-order derivatives, confirmed additionally by the scaling behavior. For the quasiperiodic Fibonacci chain, the number of quantum phase transitions increases from one to the lth Fibonacci number \(F_{l}\) when the anisotropic parameter \(\gamma \) approaches zero. The phase diagram for the approximant \(F_{l}=8 \) is derived as an example. In addition, the factorization transition in the two models can be marked by the correlation quantity defined from the QFI. The present work demonstrates the implication of the QFI as a general fingerprint to characterize the quantum transitions and factorization transitions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Helstrom, C.W.: Quantum Detection and Estimation Theory. Academic Press, New York (1976)

    MATH  Google Scholar 

  2. Holevo, A.S.: Statistical Structure of Quantum Theory. North-Holland, Amsterdam (1982)

    MATH  Google Scholar 

  3. Fisher, R.A.: Theory of statistical estimation. Proc. Camb. Philos. Soc. 22, 700 (1925)

    Article  ADS  MATH  Google Scholar 

  4. Facchi, P., Kulkarni, R., Man’ko, V.I., Marmo, G., Sudarshan, E.C.G., Ventriglia, F.: Classical and quantum Fisher information in the geometrical formulation of quantum mechanics. Phys. Lett. A 374, 4801 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Braunstein, S.L., Caves, C.M.: Statistical distance and the geometry of quantum states. Phys. Rev. Lett. 72, 3439 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Braunstein, S.L., Caves, C.M., Milburn, G.J.: Generalized uncertainty relations: theory, examples, and Lorentz invariance. Ann. Phys. 247, 135 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Hübner, M.: Explicit computation of the Bures distance for density-matrices. Phys. Lett. A 163, 239 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  8. Hübner, M.: Computation of Uhlmann parallel transport for density-matrices and the Bures metric on 3-dimensional Hilbert-space. Phys. Lett. A 179, 226 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  9. Boixo, S., Flammia, S.T., Caves, C.M., Geremia, J.M.: Generalized limits for single-parameter quantum estimation. Phy. Rev. Lett. 98, 090401 (2007)

    Article  ADS  Google Scholar 

  10. Roy, S.M., Braunstein, S.L.: Exponentially enhanced quantum metrology. Phy. Rev. Lett. 100, 220501 (2008)

    Article  ADS  Google Scholar 

  11. Boixo, S., Datta, A., Davis, M.J., Flammia, S.T., Shaji, A., Caves, C.M.: Quantum metrology: dynamics versus entanglement. Phy. Rev. Lett. 101, 040403 (2008)

    Article  ADS  Google Scholar 

  12. Jin, G.R., Kim, S.W.: Storage of spin squeezing in a two-component Bose–Einstein condensate. Phy. Rev. Lett. 99, 170405 (2007)

    Article  ADS  Google Scholar 

  13. Pezzé, L., Smeri, A.: Entanglement, nonlinear dynamics, and the Heisenberg limit. Phy. Rev. Lett. 102, 100401 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  14. Monras, A., Paris, M.G.A.: Optimal quantum estimation of loss in bosonic channels. Phys. Rev. Lett. 98, 160401 (2007)

    Article  ADS  Google Scholar 

  15. Ma, J., Huang, Y.X., Wang, X.G., Sun, C.P.: Quantum Fisher information of the Greenberger–Horne–Zeilinger state in decoherance channels. Phys. Rev. A 84, 022302 (2011)

    Article  ADS  Google Scholar 

  16. Watanabe, Y., Sagawa, T., Ueda, M.: Optimal measurement on noisy quantum systems. Phys. Rev. Lett. 104, 020401 (2010)

    Article  ADS  Google Scholar 

  17. Luo, S.L.: Quantum Fisher information and uncertainty relations. Lett. Math. Phys. 53, 243 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Luo, S.L.: Wigner–Yanase skew information vs.quantum Fisher information. Proc. Am. Math. Soc. 132, 885 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, N., Luo, S.L.: Entanglement detection via quantum Fisher information. Phys. Rev. A. 88, 014301 (2013)

    Article  ADS  Google Scholar 

  20. Boixo, S., Monras, A.: Operational interpretation for global multipartite entanglement. Phys. Rev. Lett. 100, 100503 (2008)

    Article  ADS  Google Scholar 

  21. Hyllus, P., Laskowski, W., Kridchek, R., Schwemmer, C., Wieczorek, W., Weinfurter, H., Pezze, L., Smerzi, A.: Fisher information and multiparticle entanglement. Phys. Rev. A 85, 022321 (2012)

    Article  ADS  Google Scholar 

  22. Sun, Z., Ma, J., Lu, X.-M., Wang, X.G.: Fisher information in a quantum-critical environment. Phys. Rev. A 82, 022306 (2010)

    Article  ADS  Google Scholar 

  23. Invernizzi, C., Korbman, M., Venuti, L.C., Paris, M.G.A.: Optimal quantum estimation in spin systems at criticality. Phys. Rev. A 78, 042106 (2008)

    Article  ADS  Google Scholar 

  24. Zanardi, P., Paris, M.G.A., Venuti, L.C.: Quantum criticality as a resource for quantum estimation. Phys. Rev. A 78, 042105 (2008)

    Article  ADS  Google Scholar 

  25. Salvatori, G., Mandarino, A., Paris, M.G.A.: Quantum metrology in Lipkin–Meshkov–Glick critical systems. Phys. Rev. A 90, 022111 (2014)

    Article  ADS  Google Scholar 

  26. Wang, T.-L., Wu, L.-N., Yang, W., Jin, G.-R., Lambert, N., Nori, F.: Quantum Fisher information as a signature of the superradiant quantum phase transition. New J. Phys. 16, 063039 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  27. Liu, X.M., Du, Z.Z., Cheng, W.W., Liu, J.M.: Quantum Fisher information of localization transitions in one-dimensional systems. Int. J. Theor. Phys. 54, 3033 (2015)

    Article  MATH  Google Scholar 

  28. Dagotto, E., Rice, T.M.: Surprises on the way from one- to two-dimensional quantum magnets: the ladder materials. Science 271, 618 (1996)

    Article  ADS  Google Scholar 

  29. Nguyen, T.N., Lee, P.A., zurLoye, H.C.: Design of a random quantum spin chain paramagnet: \(Sr_3CuPt_0.5Ir_0.5o_6\). Science 271, 489 (1996)

    Article  ADS  Google Scholar 

  30. Gambardella, P., Dallmeyer, A., Maiti, K., Malogoli, M., Eberhardt, W., Kern, K., Carbone, C.: Ferromagnetism in one-dimensional monatomic metal chains. Nature 416, 301 (2002)

    Article  ADS  Google Scholar 

  31. Matsumoto, M., Normand, B., Rice, T.M., Sigrist, M.: Antiferromagnet–ferromagnet and structural phase transitions in \(La_{0.88}MnO_{x}\) manganites. Phys. Rev. B 69, 054432 (2004)

    Article  ADS  Google Scholar 

  32. Dzyaloshinsky, I.: A thermodynamic theory of weak ferromagnetism of antiferromagnetics. J. Phys. Chem. Solids 4, 241 (1958)

    Article  ADS  Google Scholar 

  33. Moriya, T.: Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev. 120, 91 (1960)

    Article  ADS  Google Scholar 

  34. Dender, D.C., Hammar, P.R., Reich, D.H., Broholm, C., Aeppli, G.: Direct observation of field-induced incommensurate fluctuations in a one-dimensional \(S=1/2\) antiferromagnet. Phys. Rev. Lett. 79, 1750 (1997)

    Article  ADS  Google Scholar 

  35. Kohgi, M., Iwasa, K., Mignot, J.M., Fak, B., Gegenwart, P., Lang, M., Ochiai, A., Aoki, H., Suzuki, T.: Staggered field effect on the one-dimensional \(S=1/2\) antiferromagnet \(Yb_{4 }As_{3}\). Phys. Rev. Lett. 86, 2439 (2001)

    Article  ADS  Google Scholar 

  36. Tsukada, I., Takeya, J.T., Masuda, T., Uchinokura, K.: Two-stage spin-flop transitions in the \(S=1/2\) antiferromagnetic spin chain \(BaCu_{2}Si_{2}O_{7}\). Phys. Rev. Lett. 87, 127203 (2001)

    Article  ADS  Google Scholar 

  37. Amico, L., Fazio, R., Osterloh, A., Vedral, V.: Entanglement in many-body systems. Rev. Mod. Phys. 80, 517 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Raoul, D.: Quantum discord and quantum phase transition in spin chains. Phys. Rev. B 78, 224413 (2008)

    Article  Google Scholar 

  39. Guo, J.L., Wei, J.L., Qin, W., Mu, Q.X.: Examining quantum correlations in the XY spin chain by local quantum uncertainty. Quantum Inf Process 14, 1429 (2015)

    Article  ADS  MATH  Google Scholar 

  40. Cheng, W.W., Li, J.X., Shan, C.J., Gong, L.Y., Zhao, S.M.: Criticality, factorization and Wigner–Yanase Skew information in quantum spin chains. Quantum Inf Process 14, 2535 (2015)

    Article  ADS  MATH  Google Scholar 

  41. Luo, S.L.: Fisher information, kinetic energy and uncertainty relation inequalities. J. Phys. A: Math. Gen. 35, 5181 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Gibbons, G.W.: Typical states and density matrics. J. Geom. Phys. 8, 147 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Frieden, B.R.: Physics from Fisher information: A Unification. Cambridge University Press, Cambridge (1998)

    Book  MATH  Google Scholar 

  44. Shechtman, D., Blech, I., Gratias, D., Cahn, J.W.: Metallic Phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951 (1984)

    Article  ADS  Google Scholar 

  45. Merlin, R., Bajema, K., Clarke, R., Juang, F.Y., Bhattacharya, P.K.: Quasiperiodic GaAs–AlAs heterostructures. Phys. Rev. Lett. 55, 1768 (1985)

    Article  ADS  Google Scholar 

  46. Hida, K.: New universality class in spin-one-half Fibonacci Heisenberg chains. Phys. Rev. Lett. 93, 037205 (2004)

    Article  ADS  Google Scholar 

  47. Luck, J.M.: Critical-behavior of the aperiodic quantum Ising chain in a transverse magnetic-field. J. Stat. Phys. 72, 417 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Zhong, M., Tong, P.Q.: The Ising and anisotropy phase transition of the periodic \(XY\) model in a transverse field. J. Phys. A: Math. Theor. 43, 505302 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Tong, P., Zhong, M.: Quantum phase transitions of periodic anisotropic \(XY\) chain in a transverse field. Phys. B 304, 91 (2001)

    Article  ADS  Google Scholar 

  50. Campbell, S., Richens, J., Gullo, NLo: Criticality, factorization, and long-range correlations in the anisotropic \(XY\) model. Phys. Rev. A 88, 062305 (2013)

    Article  ADS  Google Scholar 

  51. Roscilde, T., Verrucchi, P., Fubini, A., Haas, S., Tognetti, V.: Entanglement and factorized ground states in two-dimensional quantum antiferromagnets. Phys. Rev. Lett. 94, 147208 (2005)

    Article  ADS  Google Scholar 

  52. Giampaplo, S.M., Adesso, G., Illuminati, F.: Theory of ground state factorization in quantum cooperative systems. Phys. Rev. Lett. 100, 197201 (2008)

    Article  ADS  Google Scholar 

  53. Giampaplo, S.M., Adesso, G., Illuminati, F.: Separability and ground-state factorization in quantum spin systems. Phys. Rev. B 79, 224434 (2009)

    Article  ADS  Google Scholar 

  54. Giampaplo, S.M., Adesso, G., Illuminati, F.: Probing quantum frustrated systems via factorization of the ground state. Phys. Rev. Lett. 104, 207202 (2010)

    Article  ADS  Google Scholar 

  55. Tomasello, B., Rossini, D., Hamma, A., Amico, L.: Ground-state factorization and correlations with broken symmetry. EPL 96, 27002 (2011)

    Article  ADS  Google Scholar 

  56. Luo, S.L., Fu, S.S., Oh, C.H.: Quantifying correlations via the Wigner–Yanase skew information. Phys. Rev. A 85, 032117 (2012)

    Article  ADS  Google Scholar 

  57. Lieb, E., Schultz, T., Mattis, D.: 2 Soluble models of an antiferromagnetic chain. Ann. Phys. 16, 407 (1961)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  58. Osenda, O., Huang, Z., Kais, S.: Entanglement localization by a single defect in a spin chain. Phys. Rev. A 74, 062316 (2003)

    Google Scholar 

  59. Huang, Z., Osenda, O., Kais, S.: Entanglement of formation for one-dimensional magnetic systems with defects. Phys. Lett. A 322, 137 (2004)

    Article  ADS  MATH  Google Scholar 

  60. Tong, P.Q., Liu, X.X.: Lee-Yang Zeros of periodic and quasiperiodic anisotropic \(XY\) chains in a transverse Field. Phys. Rev. Lett. 97, 017201 (2006)

    Article  ADS  Google Scholar 

  61. Dong, S., Liu, J.M., Cheong, S.W., Ren, Z.F.: Multiferroic materials and magnetoelectric physics: symmetry, entanglement, excitation, and topology. Adv. Phys. 64, 519 (2015)

Download references

Acknowledgments

This work was supported by the National 973 Projects of China (Grants No. 2015CB654602), the Natural Science Foundation of China (Grants Nos. 11234005, 11374147), and the Priority Academic Program Development of Jiangsu Higher Education Institutions, China.

Author information

Authors and Affiliations

  1. Laboratory of Solid State Microstructure and Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, 210093, China

    X. M. Liu, Z. Z. Du & J.-M. Liu

  2. Institute of Mathematical and Physical Sciences, Jiangsu University of Science and Technology, Zhenjiang, 212003, China

    X. M. Liu

Authors
  1. X. M. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Z. Z. Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J.-M. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to X. M. Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, X.M., Du, Z.Z. & Liu, JM. Quantum Fisher information for periodic and quasiperiodic anisotropic XY chains in a transverse field. Quantum Inf Process 15, 1793–1810 (2016). https://doi.org/10.1007/s11128-015-1237-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-015-1237-0

Keywords

Search

Navigation