Skip to main content
SpringerLink
Log in

Bell violation versus geometric measure of quantum discord and their dynamical behavior

  • Regular Article
  • Published:
The European Physical Journal D Aims and scope Submit manuscript

Abstract

Motivated by recent numerous works on the interplay among various measures of quantum correlations, we aim to investigate the relationship between the violation of Clauser-Horne-Shimony-Holt (CHSH) Bell inequality and geometric measure of quantum discord for two-qubit systems. Exact lower and upper bounds of Bell violation versus geometric discord are obtained for a specific and significant class of states, Bell diagonal states, and the respective states which suffice those bounds are also characterized. The dynamical behavior of these two quantifiers is carefully analyzed in the presence of decoherence, including Markovian, non-Markovian, and non-back-action quantum environments. The results suggest that Bell violation is closely related to geometric discord, like its relationship with other entanglement monotones.

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

Similar content being viewed by others

References

  1. J.S. Bell, Physics 1, 195 (1964)

    Google Scholar 

  2. J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, Phys. Rev. Lett. 23, 880 (1969)

    Article  ADS  Google Scholar 

  3. S. Pironio et al., Nature 464, 1021 (2010)

    Article  ADS  Google Scholar 

  4. J. Barrett, A. Acín, N. Gisin, Phys. Rev. Lett. 95, 010503 (2005)

    Article  ADS  Google Scholar 

  5. A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, V. Scarani, Phys. Rev. Lett. 98, 230501 (2007)

    Article  ADS  Google Scholar 

  6. S. Pironio, A. Acín, N. Brunner, N. Gisin, S. Massar, V. Scarani, New J. Phys. 11, 045021 (2009)

    Article  ADS  Google Scholar 

  7. F. Verstraete, M.M. Wolf, Phys. Rev. Lett. 89, 170401 (2002)

    Article  ADS  Google Scholar 

  8. Ł. Derkacz, L. Jakóbczyk, Phys. Lett. A 328, 26 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Ł. Derkacz, L. Jakóbczyk, Phys. Rev. A 72, 042321 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  10. L. Mazzola, B. Bellomo, R. Lo Franco, G. Compagno, Phys. Rev. A 81, 052116 (2010)

    Article  ADS  Google Scholar 

  11. N. Gisin, Phys. Lett. A 154, 201 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  12. R.F. Werner, Phys. Rev. A 40, 4277 (1989)

    Article  ADS  Google Scholar 

  13. W.J. Munro, K. Nemoto, A.G. White, J. Mod. Opt. 48, 1239 (2001)

    MathSciNet  ADS  MATH  Google Scholar 

  14. S. Ghosh, G. Kar, A. Sen(De), U. Sen, Phys. Rev. A 64, 044301 (2001)

    Article  ADS  Google Scholar 

  15. A. Miranowicz, A. Grudka, J. Opt. B 6, 542 (2004)

    Article  ADS  Google Scholar 

  16. H. Ollivier, W.H. Zurek, Phys. Rev. Lett. 88, 017901 (2001)

    Article  ADS  Google Scholar 

  17. L. Henderson, V. Vedral, J. Phys. A 34, 6899 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. S. Luo, Phys. Rev. A 77, 022301 (2008)

    Article  ADS  Google Scholar 

  19. K. Modi, T. Paterek, W. Son, V. Vedral, M. Williamson, Phys. Rev. Lett. 104, 080501 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  20. M.D. Lang, C.M. Caves, Phys. Rev. Lett. 105, 150501 (2010)

    Article  ADS  Google Scholar 

  21. S. Luo, S. Fu, Phys. Rev. A 82, 034302 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  22. A. Ferraro, L. Aolita, D. Cavalcanti, F.M. Cucchietti, A. Acín, Phys. Rev. A 81, 052318 (2010)

    Article  ADS  Google Scholar 

  23. J.-S. Xu, C.-F. Li, C.-J. Zhang, X.-Y. Xu, Y.-S. Zhang, G.-C. Guo, Phys. Rev. A 82, 042328 (2010)

    Article  ADS  Google Scholar 

  24. J.-S. Xu et al., Nat. Commun. 1, 7 (2010)

    Google Scholar 

  25. A. Datta, A. Shaji, C.M. Caves, Phys. Rev. Lett. 100, 050502 (2008)

    Article  ADS  Google Scholar 

  26. M. Piani, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 100, 090502 (2008)

    Article  ADS  Google Scholar 

  27. B.P. Lanyon, M. Barbieri, M.P. Almeida, A.G. White, Phys. Rev. Lett. 101, 200501 (2008)

    Article  ADS  Google Scholar 

  28. S. Luo, Phys. Rev. A 77, 042303 (2008)

    Article  ADS  Google Scholar 

  29. M. Ali, A.R.P. Rau, G. Alber, Phys. Rev. A 81, 042105 (2010)

    Article  ADS  Google Scholar 

  30. Q. Chen, C.J. Zhang, S.X. Yu, X.X. Yi, C.H. Oh, Phys. Rev. A 84, 042313 (2011)

    Article  ADS  Google Scholar 

  31. D. Girolami, G. Adesso, Phys. Rev. A 83, 052108 (2011)

    Article  ADS  Google Scholar 

  32. B. Dakić, V. Vedral, C. Brukner, Phys. Rev. Lett. 105, 190502 (2010)

    Article  ADS  Google Scholar 

  33. J. Batle, A. Plastino, A.R. Plastino, M. Casas, J. Phys. A: Math. Theor. 44, 505304 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  34. J. Batle, M. Casas, J. Phys. A: Math. Theor. 44, 445304 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  35. R. Horodecki, P. Horodecki, M. Horodecki, Phys. Lett. A 200, 340 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. R. Horodecki, Phys. Lett. A 210, 223 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, W.K. Wootters, Phys. Rev. Lett. 76, 722 (1996)

    Article  ADS  Google Scholar 

  38. R. Horodecki, M. Horodecki, Phys. Rev. A 54, 1838 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  39. C. Spengler, M. Huber, B.C. Hiesmayr, J. Phys. A: Math. Theor. 44, 065304 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  40. L. Mazzola, J. Piilo, S. Maniscalco, Phys. Rev. Lett. 104, 200401 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  41. J. Maziero, L.C. Celeri, R.M. Serra, V. Vedral, Phys. Rev. A 80, 044102 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  42. T. Werlang, S. Souza, F.F. Fanchini, C.J. Villas Boas, Phys. Rev. A 80, 024103 (2009)

    Article  ADS  Google Scholar 

  43. F.F. Fanchini, T. Werlang, C.A. Brasil, L.G.E. Arruda, A.O. Caldeira, Phys. Rev. A 80, 024103 (2009)

    Article  ADS  Google Scholar 

  44. B. Wang, Z.-Y. Xu, Z.-Q. Chen, M. Feng, Phys. Rev. A 81, 014101 (2010)

    Article  ADS  Google Scholar 

  45. X.M. Lu, Z.J. Xi, Z. Sun, X. Wang, Quant. Inform. Comput. 10, 0994 (2010)

    MathSciNet  Google Scholar 

  46. Z.Y. Xu, W.L. Yang, X. Xiao, M. Feng, J. Phys. A: Math. Theor. 44, 395304 (2011)

    Article  MathSciNet  Google Scholar 

  47. A. Miranowicz, Phys. Lett. A 327, 272 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. L. Jakóbczyk, A. Jamróz, Phys. Lett. A 318, 318 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. A.G. Kofman, A.N. Korotkov, Phys. Rev. A 77, 052329 (2008)

    Article  ADS  Google Scholar 

  50. Y. Yeo, J.H. An, C.H. Oh, Phys. Rev. A 82, 032340 (2010)

    Article  ADS  Google Scholar 

  51. B. Bellomo et al., Int. J. Quantum Inf. 9, 63 (2011)

    Article  MATH  Google Scholar 

  52. L. Mazzola, Phys. Scr. T140, 014055 (2010)

    Article  ADS  Google Scholar 

  53. K. Kraus, States, Effect, and Operations: Fundamental Notions in Quantum Theory (Springer-Verlag, Berlin, 1983)

  54. M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)

  55. S. Hill, W.K. Wootters, Phys. Rev. Lett. 78, 5022 (1997)

    Article  ADS  Google Scholar 

  56. W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)

    Article  ADS  Google Scholar 

  57. B. Bellomo, R. Lo Franco, G. Compagno, Phys. Rev. A 78, 062309 (2008)

    Article  ADS  Google Scholar 

  58. G. Jaeger, K. Ann, Phys. Lett. A 372, 2212 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  59. T. Yu, J.H. Eberly, Phys. Rev. Lett. 93, 140404 (2004)

    Article  ADS  Google Scholar 

  60. T. Yu, J.H. Eberly, Phys. Rev. Lett. 97, 140403 (2006)

    Article  ADS  Google Scholar 

  61. T. Yu, J.H. Eberly, Science 323, 598 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  62. B. Bellomo, R. Lo Franco, G. Compagno, Phys. Rev. Lett. 99, 160502 (2007)

    Article  ADS  Google Scholar 

  63. R. Lo Franco, B. Bellomo, E. Andersson, G. Compagno, Phys. Rev. A 85, 032318 (2012)

    Article  ADS  Google Scholar 

  64. K. Zyczkowski, P. Horodecki, M. Horodecki, R. Horodecki, Phys. Rev. A 65, 012101 (2001)

    Article  MathSciNet  ADS  Google Scholar 

  65. R. Alicki, K. Lendi, Quantum Dynamical Semigroups and Applications, Lect. Notes Phys. 717 (Springer, Berlin Heidelberg, 2007)

  66. A. Acin, S. Massar, S. Pironio, Phys. Rev. Lett. 108, 100402 (2012)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, P.R. China

    Y. Yao, H. W. Li, M. Li, Z. Q. Yin, W. Chen & Z. F. Han

Authors
  1. Y. Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. W. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Z. Q. Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. W. Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Z. F. Han
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Z. Q. Yin or Z. F. Han.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yao, Y., Li, H.W., Li, M. et al. Bell violation versus geometric measure of quantum discord and their dynamical behavior. Eur. Phys. J. D 66, 295 (2012). https://doi.org/10.1140/epjd/e2012-30259-1

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1140/epjd/e2012-30259-1

Keywords

Search

Navigation