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Classification of qubit entanglement: SL(2,ℂ) versus SU(2) invariance

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Abstract

The role of SU(2) invariants for the classification of multiparty entanglement is discussed and exemplified for the Kempe invariant I 5 of pure three-qubit states. It is found being an independent invariant only in presence of both W-type entanglement and three-tangle. In this case, constant I 5 allows for a wide range of both three-tangle and concurrences. This means that I 5 provides no information on the entanglement in the system in addition to that contained already in the tangles (concurrences and three-tangle) themselves. Furthermore, norm-preserving SL ⊗3 orbits of states with equal tangles but continuously varying I 5 are shown to exist. As a consequence, I 5 can be increased (and decreased) by general local operations. Nevertheless, numerical analysis of random SLOCC’s has not shown any violation of the monotone property of I 5. In case I 5 finally turned out to being an entanglement monotone, this would imply that both SU(2) invariance and the stronger monotone property are too weak requirements for the characterization and quantification of entanglement for systems of three qubits, and that SL(2,ℂ) invariance is required. This conclusion can be extended to general multipartite systems (including higher local dimension) because the entanglement classes of three-qubit systems appear as subclasses.

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References

  1. M.A. Nielsen, J. Kempe, Phys. Rev. Lett. 86, 5184 (2001)

    Article  ADS  Google Scholar 

  2. K. Zyczkowski, I. Bengtsson, Ann. Phys. 195, 115 (2002)

    MathSciNet  ADS  Google Scholar 

  3. G. Vidal, J. Mod. Opt. 47, 355 (2000)

    MathSciNet  ADS  Google Scholar 

  4. C.H. Bennett, S. Popescu, D. Rohrlich, J.A. Smolin, A.V. Thapliyal, Phys. Rev. A 63, 012307 (2001)

    Article  ADS  Google Scholar 

  5. A. Uhlmann, Phys. Rev. A 62, 032307 (2000)

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  8. F. Verstraete, J. Dehaene, B.D. Moor, Phys. Rev. A 68, 012103 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  9. W. Dür, G. Vidal, J.I. Cirac, Phys. Rev. A 62, 062314 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  10. V. Coffman, J. Kundu, W.K. Wootters, Phys. Rev. A 61, 052306 (2000)

    Article  ADS  Google Scholar 

  11. S. Ishizaka, M.B. Plenio, Phys. Rev. A 72, 042325 (2005)

    Article  ADS  Google Scholar 

  12. F. Verstraete, J. Dehaene, B. DeM̃oor, H. Verschelde, Phys. Rev. A 65, 052112 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  13. J.-G. Luque, J.-Y. Thibon, Phys. Rev. A 67, 042303 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  14. A. Miyake, Phys. Rev. A 67, 012108 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  15. A. Osterloh, J. Siewert, Phys. Rev. A 72, 012337 (2005). arXiv:quant-ph/0410102

    Article  ADS  Google Scholar 

  16. A. Osterloh, J. Siewert, Int. J. Quantum Inf. 4, 531 (2006). arXiv:quant-ph/0506073

    Article  MATH  Google Scholar 

  17. J.-G. Luque, J.-Y. Thibon, J. Phys. A 39, 371 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  18. O. Chterental, D.Ž. Đoković, Linear Algebra Research Advances (Nova Science, Hauppauge, 2007). arXiv:quant-ph/0612184

    Google Scholar 

  19. J.-G. Luque, J.-Y. Thibon, F. Toumazet, Math. Struct. Comput. Sci. 17, 1133 (2007)

    MathSciNet  MATH  Google Scholar 

  20. D.Ž. Đoković, A. Osterloh, J. Math. Phys. 50, 033509 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  21. P. Olver, Classical Invariant Theory (Cambridge University Press, Cambridge, 1999)

    Book  MATH  Google Scholar 

  22. E. Briand, J.-G. Luque, J.-Y. Thibon, J. Phys. A 36, 9915 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. A. Acín, A. Andrianov, L. Costa, E. Janè, J.I. Latorre, R. Tarrach, Phys. Rev. Lett. 85, 1560 (2000)

    Article  ADS  Google Scholar 

  24. H.A. Carteret, A. Higushi, A. Sudbery, J. Math. Phys. 41, 7932 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. A. Acín, A. Andrianov, E. Jane, R. Tarrach, J. Phys. A 34, 6725 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. R.M. Gingrich, Phys. Rev. A 65, 052302 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  27. A. Sudbery, J. Phys. A 34, 643 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. J. Kempe, Phys. Rev. A 60, 910 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  29. C.H. Bennett, S. Popescu, D. Rohrlich, J.A. Smolin, A.V. Thapliyal, Phys. Rev. A 63, 012307 (2001)

    Article  ADS  Google Scholar 

  30. R. Lohmayer, A. Osterloh, J. Siewert, A. Uhlmann, Phys. Rev. Lett. 97, 260502 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  31. C. Eltschka, A. Osterloh, J. Siewert, A. Uhlmann, New J. Phys. 10, 043014 (2008)

    Article  ADS  Google Scholar 

  32. W.-L. Yang, J.-L. Chen, Phys. Rev. A 76, 034301 (2007)

    Article  MathSciNet  ADS  Google Scholar 

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  1. Fachbereich Physik, Universität Duisburg-Essen, 47048, Duisburg, Germany

    A. Osterloh

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  1. A. Osterloh
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Osterloh, A. Classification of qubit entanglement: SL(2,ℂ) versus SU(2) invariance. Appl. Phys. B 98, 609–616 (2010). https://doi.org/10.1007/s00340-009-3859-3

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  • DOI: https://doi.org/10.1007/s00340-009-3859-3

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