The European Physical Journal D

, Volume 47, Issue 2, pp 283–293 | Cite as

Two-qutrit entanglement witnesses and Gell-Mann matrices

  • M. A. JafarizadehEmail author
  • Y. Akbari
  • N. Behzadi
Quantum Optics and Quantum Information


The Gell-Mann λ matrices for Lie algebra su(3) are the natural basis for the Hilbert space of Hermitian operators acting on the states of a three-level system(qutrit). So the construction of EWs for two-qutrit states by using these matrices may be an interesting problem. In this paper, several two-qutrit EWs are constructed based on the Gell-Mann matrices by using the linear programming (LP) method exactly or approximately. The decomposability and non-decomposability of constructed EWs are also discussed and it is shown that the λ-diagonal EWs presented in this paper are all decomposable but by adding λ-non-diagonal terms, one can obtain various non-decomposable EWs.


03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.) 


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  1. M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000) Google Scholar
  2. The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation and Quantum Computation, edited by D. Bouwmeester, A. Ekert, A. Zeilinger (Springer, New York, 2000) Google Scholar
  3. J. Preskill, The Theory of Quantum Information and Quantum Computation (California Institute of Technology, Pasadena, CA, 2000) ph229/ Google Scholar
  4. B. Baumgartner, B.C. Hiesmayr, H. Narnhofer, Phys. Rev. A 74, 032327 (2006) CrossRefADSGoogle Scholar
  5. B. Baumgartner, B.C. Hiesmayr, H. Narnhofer, Phys. Lett. A (2007) DOI: 10.1016/j.physleta.2007.11.028 Google Scholar
  6. R.A. Bertlmann, K. Durstberger, B.C. Hiesmayr, P. Krammer, Phys. Rev. A 72, 052331 (2005) CrossRefADSGoogle Scholar
  7. A.B. Klimov, L.L. Sánchez-Soto, H. de Guise, G. Björk, J. Phys. A 37, 4097 (2004) zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. S.L. Woronowicz, Rep. Math. Phys. 10, 165 (1976) zbMATHCrossRefMathSciNetGoogle Scholar
  9. M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 223, 1 (1996) zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. A.C. Doherty, P.A. Parrilo, F.M. Spedalieri, Phys. Rev. A 71, 032333 (2005) CrossRefADSMathSciNetGoogle Scholar
  11. R.O. Vianna, A.C. Doherty, Phys. Rev. A 74, 052306 (2006) CrossRefADSGoogle Scholar
  12. M.A. Jafarizadeh, M. Rezaee, S.K.A. Seyed Yagoobi, Phys. Rev. A 72, 062106 (2005) CrossRefADSMathSciNetGoogle Scholar
  13. M.A. Jafarizadeh, M. Rezaee, S. Ahadpour, Phys. Rev. A 74, 042335 (2006) CrossRefADSMathSciNetGoogle Scholar
  14. M.A. Jafarizadeh, G. Najarbashi, H. Habibian, Phys. Rev. A 75, 052326 (2007) CrossRefADSGoogle Scholar
  15. M.A. Jafarizadeh, G. Najarbashi, Y. Akbari, H. Habibian, Eur. Phys. J. D (2008) DOI: 10.1140/epjd/e2008-00028-0 Google Scholar
  16. M.A. Jafarizadeh, R. Sufiani, Phys. Rev. A 77, 012105 (2008) CrossRefADSGoogle Scholar
  17. S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004) Google Scholar
  18. E.K.P. Chong, S.H. Żak, An Introduction to Optimization (John Wiley, NY, 2001) Google Scholar
  19. W. Pfeifer, The Lie Algebras su(N) An Introduction (Birkhäuser Verlag, 2003) Google Scholar
  20. W. Rudin, Functional Analysis (McGraw-Hill, Singapore, 1991) Google Scholar
  21. M. Lewenstein, D. Bruss, J.I. Cirac, B. Kraus, M. Kus, J. Samsonowicz, A. Sanpera, R. Tarrach, J. Mod. Opt. 47, 2841 (2000) MathSciNetGoogle Scholar
  22. R.A. Bertlmann, P. Krammer, e-print: arXiv:quant-ph/0706.1743v1 (2007) Google Scholar
  23. M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 82, 1056 (1999) zbMATHCrossRefADSMathSciNetGoogle Scholar
  24. A.C. Doherty, P.A. Parrilo, F.M. Spedalieri, Phys. Rev. Lett. 88, 187904 (2002) CrossRefADSGoogle Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Theoretical Physics and AstrophysicsTabriz UniversityTabrizIran
  2. 2.Institute for Studies in Theoretical Physics and MathematicsTehranIran
  3. 3.Research Institute for Fundamental SciencesTabrizIran

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