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Quantum Discord and Entropic Measures of Quantum Correlations: Optimization and Behavior in Finite XY Spin Chains

  • N. Canosa
  • M. Cerezo
  • N. Gigena
  • R. RossignoliEmail author
Chapter
Part of the Quantum Science and Technology book series (QST)

Abstract

We discuss a generalization of the conditional entropy and one-way information deficit in quantum systems, based on general entropic forms. The formalism allows to consider simple entropic forms for which a closed evaluation of the associated optimization problem in qudit-qubit systems is shown to become feasible, allowing to approximate that of the quantum discord. As application, we examine quantum correlations of spin pairs in the exact ground state of finite XY spin chains in a magnetic field through the quantum discord and information deficit. While these quantities show a similar behavior, their optimizing measurements exhibit significant differences, which can be understood and predicted through the previous approximations. The remarkable behavior of these quantities in the vicinity of transverse and non-transverse factorizing fields is also discussed.

Notes

Acknowledgements

The authors acknowledge support from CONICET (NG,NC,MC) and CIC (RR) of Argentina, and of CONICET grant PIP 11220150100732.

References

  1. 1.
    K. Modi et al., Rev. Mod. Phys. 84, 1655 (2012)ADSCrossRefGoogle Scholar
  2. 2.
    G. Adesso, T.R. Bromley, M. Cianciaruso, J. Phys. A 49, 473001 (2016)Google Scholar
  3. 3.
    B. Schumacher, Phys. Rev. A 51, 2738 (1995)Google Scholar
  4. 4.
    C.H. Bennett, H.J. Bernstein, S. Popescu, B. Schumacher, Phys. Rev. A 53, 2046 (1996)ADSCrossRefGoogle Scholar
  5. 5.
    M.A. Nielsen, I. Chuang, Quantum Computation and Quantum Information (Cambridge University, Press, 2000)zbMATHGoogle Scholar
  6. 6.
    S. Haroche, J.M. Raimond, Exploring the Quantum (Oxford University Press, Oxford, 2007)zbMATHGoogle Scholar
  7. 7.
    C.H. Bennett et al., Phys. Rev. Lett. 70, 1895 (1993); Phys. Rev. Lett. 76, 722 (1996)Google Scholar
  8. 8.
    R. Josza, N. Linden, Proc. R. Soc. A459, 2011 (2003)ADSGoogle Scholar
  9. 9.
    G. Vidal, Phys. Rev. Lett. 91, 147902 (2003)ADSCrossRefGoogle Scholar
  10. 10.
    R.F. Werner, Phys. Rev. A 40, 4277 (1989)ADSCrossRefGoogle Scholar
  11. 11.
    H. Ollivier, W.H. Zurek, Phys. Rev. Lett. 88, 017901 (2001)ADSCrossRefGoogle Scholar
  12. 12.
    L. Henderson, V. Vedral, J. Phys. A 34, 6899 (2001)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    V. Vedral, Phys. Rev. Lett. 90, 050401 (2003)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    W.H. Zurek, Phys. Rev. A 67, 012320 (2003)ADSCrossRefGoogle Scholar
  15. 15.
    E. Knill, R. Laflamme, Phys. Rev. Lett. 81, 5672 (1998)ADSCrossRefGoogle Scholar
  16. 16.
    A. Datta, A. Shaji, C.M. Caves, Phys. Rev. Lett. 100, 050502 (2008)ADSCrossRefGoogle Scholar
  17. 17.
    A. Datta, S.T. Flammia, C.M. Caves, Phys. Rev. A 72, 042316 (2005)ADSCrossRefGoogle Scholar
  18. 18.
    M. Koashi, A. Winter, Phys. Rev. A 69, 022309 (2004)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    A. Datta, S. Gharibian, Phys. Rev. A 79, 042325 (2009)ADSCrossRefGoogle Scholar
  20. 20.
    A. Shabani, D.A. Lidar, Phys. Rev. Lett. 102, 100402 (2009)ADSCrossRefGoogle Scholar
  21. 21.
    K. Modi et al., Phys. Rev. Lett. 104, 080501 (2010)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    A. Ferraro et al., Phys. Rev. A 81, 052318 (2010)ADSCrossRefGoogle Scholar
  23. 23.
    F.F. Fanchini et al., Phys. Rev. A 84, 012313 (2011)ADSCrossRefGoogle Scholar
  24. 24.
    M. Horodecki et al., Phys. Rev. A 71, 062307 (2005)ADSCrossRefGoogle Scholar
  25. 25.
    J. Oppenheim et al., Phys. Rev. Lett. 89, 180402 (2002)ADSCrossRefGoogle Scholar
  26. 26.
    A. Streltsov, H. Kampermann, D. Bruß, Phys. Rev. Lett. 106, 160401 (2011)ADSCrossRefGoogle Scholar
  27. 27.
    B. Dakić, V. Vedral, C. Brukner, Phys. Rev. Lett. 105, 190502 (2010)ADSCrossRefGoogle Scholar
  28. 28.
    R. Rossignoli, N. Canosa, L. Ciliberti, Phys. Rev. A 82, 052342 (2010)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    R. Rossignoli, N. Canosa, L. Ciliberti, Phys. Rev. A 84, 052329 (2011)ADSCrossRefGoogle Scholar
  30. 30.
    D. Girolami, T. Tuffarelli, G. Adesso, Phys. Rev. Lett. 110, 240402 (2013)ADSCrossRefGoogle Scholar
  31. 31.
    S. Luo, S. Fu, C.H. Oh, Phys. Rev. A 85, 032117 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    M. Paula, T.R. de Olivera, M.S. Sarandi, Phys. Rev. A 87, 064101 (2013)ADSCrossRefGoogle Scholar
  33. 33.
    H. Hu, H. Fan, D.L. Zhou, W.M. Liu, Phys. Rev. A 87, 032340 (2013)Google Scholar
  34. 34.
    T. Nakano, M. Piani, G. Adesso, Phys. Rev. A 88, 012117 (2013)ADSCrossRefGoogle Scholar
  35. 35.
    T. Baumgratz, M. Cramer, M.B. Plenio, Phys. Rev. Lett. 113, 140401 (2014)ADSCrossRefGoogle Scholar
  36. 36.
    I. Marvian, R.W. Spekkens, Phys. Rev. A 90, 062110 (2014)Google Scholar
  37. 37.
    V. Madhok, A. Datta, Phys. Rev. A 83, 032323 (2011)ADSCrossRefGoogle Scholar
  38. 38.
    D. Cavalcanti et al., Phys. Rev. A 83, 032324 (2011)ADSCrossRefGoogle Scholar
  39. 39.
    M. Piani et al., Phys. Rev. Lett. 106, 220403 (2011)ADSCrossRefGoogle Scholar
  40. 40.
    D. Girolami, G. Adesso, Phys. Rev. Lett. 108, 150403 (2012)ADSCrossRefGoogle Scholar
  41. 41.
    T. Tufarelli et al., Phys. Rev. A 86, 052326 (2012)ADSCrossRefGoogle Scholar
  42. 42.
    Y. Huang, New. J. Phys. 16, 033027 (2014)Google Scholar
  43. 43.
    R. Dillenschneider, Phys. Rev. B 78, 224413 (2008)ADSCrossRefGoogle Scholar
  44. 44.
    M.S. Sarandy, Phys. Rev. A 80, 022108 (2009)ADSCrossRefGoogle Scholar
  45. 45.
    J. Maziero et al., Phys. Rev. A 82, 012106 (2010)ADSCrossRefGoogle Scholar
  46. 46.
    T. Werlang, G. Rigolin, Phys. Rev. A 81, 044101 (2010)ADSCrossRefGoogle Scholar
  47. 47.
    L. Ciliberti, R. Rossignoli, N. Canosa, Phys. Rev. A 82, 042316 (2010)ADSCrossRefGoogle Scholar
  48. 48.
    T. Werlang et al., Phys. Rev. Lett. 105, 095702 (2010)ADSCrossRefGoogle Scholar
  49. 49.
    T. Werlang, G.A.P. Ribeiro, G. Rigolin, Phys. Rev. A 83, 062334 (2011)ADSCrossRefGoogle Scholar
  50. 50.
    B.Q. Liu et al., Phys. Rev. A 83, 052112 (2011)ADSCrossRefGoogle Scholar
  51. 51.
    Y.C. Li, H.Q. Lin, Phys. Rev. A 83, 052323 (2011)ADSCrossRefGoogle Scholar
  52. 52.
    N. Canosa, L. Ciliberti, R. Rossignoli, Int. J. Mod. Phys. B 27, 1345033 (2012)ADSCrossRefGoogle Scholar
  53. 53.
    L. Ciliberti, N. Canosa, R. Rossignoli, Phys. Rev. A 88, 012119 (2013)ADSCrossRefGoogle Scholar
  54. 54.
    N. Gigena, R. Rossignoli, Phys. Rev. A 90, 042318 (2014)ADSCrossRefGoogle Scholar
  55. 55.
    N. Gigena, R. Rossignoli, J. Phys. A 47, 015302 (2014)ADSMathSciNetCrossRefGoogle Scholar
  56. 56.
    J. Kurmann, H. Thomas, G. Müller, Physica A 112, 235 (1982)ADSCrossRefGoogle Scholar
  57. 57.
    T. Roscilde et al., Phys. Rev. Lett. 93, 167203 (2004)ADSCrossRefGoogle Scholar
  58. 58.
    T. Roscilde et al., Phys. Rev. Lett. 94, 147208 (2005)ADSCrossRefGoogle Scholar
  59. 59.
    L. Amico et al., Phys. Rev. A 74, 022322 (2006)ADSCrossRefGoogle Scholar
  60. 60.
    F. Baroni et al., J. Phys. A 40, 9845 (2007)ADSMathSciNetCrossRefGoogle Scholar
  61. 61.
    R. Rossignoli, N. Canosa, J.M. Matera, Phys. Rev. A 77, 052322 (2008)ADSCrossRefGoogle Scholar
  62. 62.
    S.M. Giampaolo, G. Adesso, F. Illuminati, Phys. Rev. Lett. 100, 197201 (2008)ADSCrossRefGoogle Scholar
  63. 63.
    S.M. Giampaolo, G. Adesso, F. Illuminati, Phys. Rev. B 79, 224434 (2009)ADSCrossRefGoogle Scholar
  64. 64.
    R. Rossignoli, N. Canosa, J.M. Matera, Phys. Rev. A 80, 062325 (2009)ADSCrossRefGoogle Scholar
  65. 65.
    N. Canosa, R. Rossignoli, J.M. Matera, Phys. Rev. B 81, 054415 (2010)ADSCrossRefGoogle Scholar
  66. 66.
    S. Campbell, J. Richens, N. Lo Gullo, T. Busch, Phys. Rev. A 88, 062305 (2013)Google Scholar
  67. 67.
    M. Cerezo, R. Rossignoli, N. Canosa, Phys. Rev. B 92, 224422 (2015)Google Scholar
  68. 68.
    H. Wehrl, Rev. Mod. Phys. 50, 221 (1978)ADSMathSciNetCrossRefGoogle Scholar
  69. 69.
    N. Canosa, R. Rossignoli, Phys. Rev. Lett. 88, 170401 (2002)ADSMathSciNetCrossRefGoogle Scholar
  70. 70.
    C. Tsallis, J. Stat. Phys. 52, 479 (1988); C. Tsallis, Introduction to Non-Extensive Statistical Mechanics (Springer, New York, 2009)Google Scholar
  71. 71.
    R. Filip, Phys. Rev. A 65, 062320 (2002)ADSCrossRefGoogle Scholar
  72. 72.
    H. Nakazato et al., Phys. Rev. A 85, 042316 (2012)ADSCrossRefGoogle Scholar
  73. 73.
    T. Tanaka, G. Kimura, H. Nakazato, Phys. Rev. A 87, 012303 (2013)Google Scholar
  74. 74.
    V. Vedral, Rev. Mod. Phys. 74, 197 (2002)ADSCrossRefGoogle Scholar
  75. 75.
    R. Bhatia, Matrix Analysis (Springer, New York, USA, 1997)CrossRefzbMATHGoogle Scholar
  76. 76.
    N. Canosa, L. Ciliberti, R. Rossignoli, Entropy 17, 1634 (2015)ADSMathSciNetCrossRefGoogle Scholar
  77. 77.
    R. Rossignoli, J.M. Matera, N. Canosa, Phys. Rev. A 86, 022104 (2012)ADSCrossRefGoogle Scholar
  78. 78.
    S. Hill, W.K. Wootters, Phys. Rev. Lett. 78, 5022 (1997); W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)Google Scholar
  79. 79.
    P. Rungta, C.M. Caves, Phys. Rev. A 67, 012307 (2003); P. Rungta et al., Phys. Rev. A 64, 042315 (2001)Google Scholar
  80. 80.
    E. Lieb, T. Schultz, D. Mattis, Ann. Phys. (NY) 16, 407 (1961)ADSCrossRefGoogle Scholar
  81. 81.
    N. Canosa, R. Rossignoli, Phys. Rev. A 75, 032350 (2007)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • N. Canosa
    • 1
  • M. Cerezo
    • 1
  • N. Gigena
    • 1
  • R. Rossignoli
    • 1
    Email author
  1. 1.Departamento de Física-IFLPUniversidad Nacional de La PlataLa PlataArgentina

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