Hierarchy of graph-diagonal states based on quantum discord and entanglement classification

  • Mohammad Ali Jafarizadeh
  • Naser KarimiEmail author
  • Davood Amidi Sahlan
  • Ahmad Heshmati
  • Marziyeh Yahyavi
Regular Article


For the relative entropy-based measure of quantum discord the key idea is to find the closest classical state (CCS) for a given state ρ, which is in general a more complicated problem. In this work, we study three and four qubit graph-diagonal states and give the explicit expressions of CCS for these states. Using the CCS, we compute the quantum discord of graph-diagonal states of three and four qubit systems and show that there is a hierarchy for the quantum discord of graph-diagonal states of any three and four qubit systems. Then we classify the entanglement of graph-diagonal states of three and four qubit systems and draw the hierarchy of entanglement of these graph-diagonal states. Finally, we compare the hierarchy of quantum discord and quantum entanglement of the these graph-diagonal states and show that the hierarchy of quantum entanglement is at least in equivalence of quantum discord.

Graphical abstract


Quantum Information 


  1. 1.
    R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys. 81, 865 (2009)ADSCrossRefGoogle Scholar
  2. 2.
    O. Guhne, G. Toth, Phys. Rep. 474, 1 (2009)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    K. Modi, A. Brodutch, H. Cable, T. Paterek, V. Vedral, Rev. Mod. Phys. 84, 1655 (2012)ADSCrossRefGoogle Scholar
  4. 4.
    M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)Google Scholar
  5. 5.
    C.H. Bennett, D.P. DiVincenzo, Nature 404, 247 (2000)ADSCrossRefGoogle Scholar
  6. 6.
    C.H. Bennett, D.P. DiVincenzo, C.A. Fuchs, T. Mor, E. Rains, P.W. Shor, J.A. Smolin, W.K. Wootters, Phys. Rev. A 59, 1070 (1999)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    M. Horodecki, P. Horodecki, R. Horodecki, J. Oppenheim, A. Sen, U. Sen, B. Synak-Radtke, Phys. Rev. A 71, 062307 (2005)ADSCrossRefGoogle Scholar
  8. 8.
    J. Niset, N.J. Cerf, Phys. Rev. A 74, 052103 (2006)ADSCrossRefGoogle Scholar
  9. 9.
    H. Ollivier, W.H. Zurek, Phys. Rev. Lett. 017901, 88 (2001)Google Scholar
  10. 10.
    L. Henderson, V. Vedral, J. Phys. A 34, 6899 (2001)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    S. Luo, Phys. Rev. A 77, 042303 (2008)ADSCrossRefGoogle Scholar
  12. 12.
    L. Mazzola, J. Piilo, S. Maniscalco, Phys. Rev. Lett. 104, 200401 (2010)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    M.D. Lang, C.M. Caves, Phys. Rev. Lett. 105, 150501 (2010)ADSCrossRefGoogle Scholar
  14. 14.
    J. Maziero, L.C. Celeri, R.M. Serra, V. Vedral, Phys. Rev. A 80, 044102 (2009)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    F.F. Fanchini, T. Werlang, C.A. Brasil, L.G.E. Arruda, A.O. Caldeira, Phys. Rev. A 81, 052107 (2010)ADSCrossRefGoogle Scholar
  16. 16.
    A. Datta, Phys. Rev. A 80, 052304 (2009)ADSCrossRefGoogle Scholar
  17. 17.
    T. Werlang, G. Rigolin, Phys. Rev. A 81, 044101 (2010)ADSCrossRefGoogle Scholar
  18. 18.
    M. Ali, A.R.P. Rau, G. Alber, Phys. Rev. A 81, 042105 (2010)ADSCrossRefGoogle Scholar
  19. 19.
    G. Adesso, A. Datta, Phys. Rev. Lett. 105, 030501 (2010)ADSCrossRefGoogle Scholar
  20. 20.
    M.A. Jafarizadeh et al., Eur. Phys. J. D 68, 136 (2014)ADSCrossRefGoogle Scholar
  21. 21.
    Z. Ma, Z. Chen, F.F. Fanchini, S. Fei, Sci. Rep. 5, 10262 (2015)ADSCrossRefGoogle Scholar
  22. 22.
    K. Modi, T. Paterek, W. Son, V. Vedral, M. Williamson, Phys. Rev. Lett. 104, 080501 (2010)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    H. Kim, M.-R. Hwang, E. Jung, D.K. Park, Phys. Rev. A 81, 052325 (2010)ADSCrossRefGoogle Scholar
  24. 24.
    P. Parashar, S. Rana, Phys. Rev. A 83, 032301 (2011)ADSCrossRefGoogle Scholar
  25. 25.
    J. Zhang, A. Chen, Quantum Phys. Lett. 1, 2 (2012)Google Scholar
  26. 26.
    M. Hein et al., in Quantum Computers, Algorithms and Chaos, edited by G. Casati, D.L. Shepelyansky, P. Zoller, G. Benenti (IOS, Amsterdam, 2006)Google Scholar
  27. 27.
    R. Raussendorf, H.J. Briegel, Phys. Rev. Lett. 86, 5188 (2001)ADSCrossRefGoogle Scholar
  28. 28.
    M.A. Jafarizadeh et al., Int. J. Theor. Phys. 55, 1543 (2015)CrossRefGoogle Scholar
  29. 29.
    R.V. Buniy, T.W. Kephart, J. Phys. A: Math. Theor. 45, 185304 (2012)ADSCrossRefGoogle Scholar
  30. 30.
    O. Gühne, B. Jungnitsch, T. Moroder, Y.S. Weinstein, Phys. Rev. A 84, 052319 (2011)ADSCrossRefGoogle Scholar
  31. 31.
    B. Jungnitsch, T. Moroder, O. Gühne, Phys. Rev. A 84, 032310 (2011)ADSCrossRefGoogle Scholar
  32. 32.
    R.V. Buniy, T.W. Kephart, J. Phys. A: Math. Theor. 45, 182001 (2012)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Mohammad Ali Jafarizadeh
    • 1
  • Naser Karimi
    • 2
    Email author
  • Davood Amidi Sahlan
    • 1
  • Ahmad Heshmati
    • 3
  • Marziyeh Yahyavi
    • 1
  1. 1.Department of Theoretical Physics and AstrophysicsTabriz UniversityTabrizIran
  2. 2.Department of scienceFarhangian UniversityTehranIran
  3. 3.Department of PhysicsShabestar Branch, Islamic Azad UniversityShabestarIran

Personalised recommendations