Optimal observables to determine entanglement of a two qubit state

Regular Article

Abstract

Experimental determination of entanglement is important not only to characterize the state and use it in quantum information, but also in understanding complicated phenomena such as phase transitions. In this paper we show that in many cases, it is possible to determine entanglement of a two qubit state, as represented by concurrence, with a few observables, most of which are local. In particular, rank 1 and rank 2 states need exclusively measurement of local observables while rank 3 states need measurement of just one correlation observable in addition to local observables. Only the rank 4 states are shown to require a more detailed tomography. The analysis also sheds light on the other measure, non separability since it is a lower bound on concurrence.

Graphical abstract

Keywords

Quantum Information 

References

  1. 1.
    S. Filipp, P. Maurer, P.J. Leek, M. Baur, R. Bianchetti, J.M. Fink, M. Goppl, L. Steffen, J.M. Gambetta, A. Blais, A. Wallraff, Phys. Rev. Lett. 102, 200402 (2009) ADSCrossRefGoogle Scholar
  2. 2.
    Steffen et al., Science 313, 1423 (2006)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    M. Vasilyev, S.-K. Choi, P. Kumar, G. Mauro D’Ariano, Phys. Rev. Lett. 84, 2354 (2000)ADSCrossRefGoogle Scholar
  4. 4.
    K Banaszek, B Cramer, D Gross, New J. Phys. 15, 125020 (2013) ADSCrossRefGoogle Scholar
  5. 5.
    M. Agnew, J. Leach, M. McLaren, F.S. Roux, R.W. Boyd, Phys. Rev. A 84, 062101 (2011) ADSCrossRefGoogle Scholar
  6. 6.
    A. Chiuri, L. Mazzola, M. Paternostro, P. Mataloni, New J. Phys. 14, 085006 (2012) ADSCrossRefGoogle Scholar
  7. 7.
    L. Peng et al., New J. Phys. 15, 125027 (2013) CrossRefGoogle Scholar
  8. 8.
    S.A. Babichev, J. Appel, A.I. Lvovsky, Phys. Rev. Lett. 92, 193601 (2004) ADSCrossRefGoogle Scholar
  9. 9.
    Roos et al., Science 304, 1478 (2004)ADSCrossRefGoogle Scholar
  10. 10.
    D. Gross, IEEE Trans. Inform. Theory 57, 1548 (2011) MathSciNetCrossRefGoogle Scholar
  11. 11.
    J. Řeháčke, B.G. Englert, D. Kaszlikowski, Phys. Rev. A 70, 052321 (2004) ADSCrossRefGoogle Scholar
  12. 12.
    C. Cinelli, G. Di Nepi, F. De Martini, M. Barbieri, P. Mataloni, Phys. Rev. A 70, 022321 (2004) ADSCrossRefGoogle Scholar
  13. 13.
    A.M. Souza, M.S. Reis, D.O. Soares-Pinto, I.S. Oliveira, R.S. Sarthour, Phys. Rev. B 77, 104402 (2008) ADSCrossRefGoogle Scholar
  14. 14.
    J.B. Altepeter, E.R. Jeffrey, P.G. Kwiat, S. Tanzilli, N. Gisin, A. Acin, Phys. Rev. Lett. 95, 033601 (2005) ADSCrossRefGoogle Scholar
  15. 15.
    G. Tóth, O. Gühne, Phys. Rev. Lett. 94, 060501 (2005) ADSCrossRefGoogle Scholar
  16. 16.
    S.M. Fei, M.J. Zhao, K. Chen, Z.X. Wang, Phys. Rev. A 80, 032320 (2009) ADSCrossRefGoogle Scholar
  17. 17.
    Walborn et al., Nature 440, 1022 (2006)ADSCrossRefGoogle Scholar
  18. 18.
    S.P. Walborn, P.H. Souto Ribeiro, L. Davidovich, F. Mintert, A. Buchleitner, Phys. Rev. A 75, 032338 (2007) ADSCrossRefGoogle Scholar
  19. 19.
    D.F.V. James, P.G. Kwiat, W.J. Munro, A.G. White, Phys. Rev. A 64, 052312 (2001) ADSCrossRefGoogle Scholar
  20. 20.
    P. Horodecki, Phys. Rev. Lett. 90, 167901 (2003) ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    O. Gühne, P. Hyllus, D. Bruß, A. Ekert, M. Lewenstein, C. Macchiavello, A. Sanpera, Phys. Rev. A 66, 062305 (2002) ADSCrossRefGoogle Scholar
  22. 22.
    O. Gühne, P. Hyllus, Int. J. Theor. Phys. 42, 1001 (2003) CrossRefGoogle Scholar
  23. 23.
    I. Sargolzahi, S.Y. Mirafzali, M. Sarbishaei, Quantum Inform. Comput. 11, 0079 (2011) MathSciNetGoogle Scholar
  24. 24.
    Y.S. Teo et al., New J. Phys. 14, 105020 (2012) CrossRefGoogle Scholar
  25. 25.
    L.H. Zhang, Q. Yang, M. Yang, W. Song, Z.L. Cao, Phys. Rev. A 88, 062342 (2013) ADSCrossRefGoogle Scholar
  26. 26.
    K. Bartkiewicz, J. Beran, K. Lemr, M. Norek, A. Miranowicz, Phys. Rev. A 91, 022323 (2015) ADSCrossRefGoogle Scholar
  27. 27.
    K. Bartkiewicz, P. Horodecki, K. Lemr, A. Miranowicz, K. Życzkowski, Phys. Rev. A 91, 032315, (2015)ADSCrossRefGoogle Scholar
  28. 28.
    Wen-Long Yang, Jin-Ling Chen, Phys. Rev. A. 76, 034301 (2007) ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    R.F. Werner, Phys. Rev. A 40, 4277 (1989)ADSCrossRefGoogle Scholar
  30. 30.
    W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998) ADSCrossRefGoogle Scholar
  31. 31.
    M.A. Nielsen, Phys. Rev. Lett. 83, 436 (1999)ADSCrossRefGoogle Scholar
  32. 32.
    S. Bhardwaj, V. Ravishankar, Phys. Rev. A 77, 022322 (2008) ADSCrossRefGoogle Scholar
  33. 33.
    S. Hill, W.K. Wootters, Phys. Rev. Lett. 78, 5022 (1997) ADSCrossRefGoogle Scholar
  34. 34.
    V. Coffman, J. Kundu, W.K. Wootters, Phys. Rev. A 61, 052306 (2000) ADSCrossRefGoogle Scholar
  35. 35.
    I. Bose, E. Chattopadhyay, Phys. Rev. A 66, 062320 (2002) ADSCrossRefGoogle Scholar
  36. 36.
    L.A. Wu, M.S. Sarandy, D.A. Lidar, Phys. Rev. Lett. 93, 250404 (2004) ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    H. Wunderich, M.B. Plenio, J. Mod. Opt. 56, 2100 (2009) ADSCrossRefGoogle Scholar
  38. 38.
    K.M.R. Audenaert, M.B. Plenio, New J. Phys. 8, 266 (2006)ADSCrossRefGoogle Scholar
  39. 39.
    I. Affleck, T. Kennedy, E.H. Lieb, H. Tasaki, Phys. Rev. Lett. 59, 799 (1987)ADSCrossRefGoogle Scholar
  40. 40.
    S.J. Gu, H.Q. Lin, Y.Q. Li, Phys. Rev. A 68, 042330 (2003) ADSCrossRefGoogle Scholar
  41. 41.
    X. Wang, K. Mølmer, Eur. Phys. J. D 18, 385 (2002)ADSGoogle Scholar
  42. 42.
    K.M. O’Connor, W.K. Wootters, Phys. Rev. A 63, 052302 (2001) ADSCrossRefGoogle Scholar
  43. 43.
    M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, Phys. Rev. Lett. 92, 177901 (2004)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of PhysicsThe University of Texas at AustinAustinUSA
  2. 2.Department of PhysicsIndian Institute of TechnologyNew DelhiIndia
  3. 3.Department of PhysicsIndian Institute of TechnologyKanpurIndia

Personalised recommendations