Classification of multiqubit mixed states: Separability and distillability properties

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 61, Issue 4, 2000, Pages 423141-4231411

  Dur W ^a   Cirac, J I ^a  

^a UNIVERSITY OF INNSBRUCK   (Austria)

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EID: 0345847394     PISSN: 10502947     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (35)

1. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76, 722 (1996); C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 54, 3824 (1996).
2. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76, 722 (1996); C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 54, 3824 (1996).
3. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).
4. H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 81, 5932 (1998); W. Dür, H.-J. Briegel, J. I. Cirac, and P. Zoller, Phys. Rev. A 59, 169 (1999); S. J. van Enk, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 78, 4293 (1997); S. J. van Enk, J. I. Cirac, and P. Zoller. Science 279, 205 (1998).
5. H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 81, 5932 (1998); W. Dür, H.-J. Briegel, J. I. Cirac, and P. Zoller, Phys. Rev. A 59, 169 (1999); S. J. van Enk, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 78, 4293 (1997); S. J. van Enk, J. I. Cirac, and P. Zoller. Science 279, 205 (1998).
6. H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 81, 5932 (1998); W. Dür, H.-J. Briegel, J. I. Cirac, and P. Zoller, Phys. Rev. A 59, 169 (1999); S. J. van Enk, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 78, 4293 (1997); S. J. van Enk, J. I. Cirac, and P. Zoller. Science 279, 205 (1998).
7. H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 81, 5932 (1998); W. Dür, H.-J. Briegel, J. I. Cirac, and P. Zoller, Phys. Rev. A 59, 169 (1999); S. J. van Enk, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 78, 4293 (1997); S. J. van Enk, J. I. Cirac, and P. Zoller. Science 279, 205 (1998).
8. D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Phys. Rev. Lett. 77, 2818 (1996).
9. R. F. Werner, Phys. Rev. A 40, 4277 (1989).
10. W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
11. A. Peres, Phys. Rev. Lett. 77, 1413 (1996).
12. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 223, 8 (1996).
13. P. Horodecki, Phys. Lett. A 232, 333 (1997).
14. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 78, 574 (1997).
15. M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, Phys. Rev. A 57, 4075 (1998).
16. J. Kempe, Phys. Rev. A 60, 910 (1999); A. V. Thapliyal, ibid. 59, 3336 (1998); G. Vidal, Phys. Rev. Lett. 83, 1048 (1999); N. Linden and S. Popescu, Fortschr. Phys. 46, 567 (1998); N. Linden, S. Popescu, and A. Sudbery, Phys. Rev. Lett. 83, 243 (1999); C. H. Bennett, S. Popescu, D. Rohrlich, J. A. Smolin, A. V. Thapliyal, e-print quant-ph/9908073.
17. J. Kempe, Phys. Rev. A 60, 910 (1999); A. V. Thapliyal, ibid. 59, 3336 (1998); G. Vidal, Phys. Rev. Lett. 83, 1048 (1999); N. Linden and S. Popescu, Fortschr. Phys. 46, 567 (1998); N. Linden, S. Popescu, and A. Sudbery, Phys. Rev. Lett. 83, 243 (1999); C. H. Bennett, S. Popescu, D. Rohrlich, J. A. Smolin, A. V. Thapliyal, e-print quant-ph/9908073.
18. J. Kempe, Phys. Rev. A 60, 910 (1999); A. V. Thapliyal, ibid. 59, 3336 (1998); G. Vidal, Phys. Rev. Lett. 83, 1048 (1999); N. Linden and S. Popescu, Fortschr. Phys. 46, 567 (1998); N. Linden, S. Popescu, and A. Sudbery, Phys. Rev. Lett. 83, 243 (1999); C. H. Bennett, S. Popescu, D. Rohrlich, J. A. Smolin, A. V. Thapliyal, e-print quant-ph/9908073.
19. J. Kempe, Phys. Rev. A 60, 910 (1999); A. V. Thapliyal, ibid. 59, 3336 (1998); G. Vidal, Phys. Rev. Lett. 83, 1048 (1999); N. Linden and S. Popescu, Fortschr. Phys. 46, 567 (1998); N. Linden, S. Popescu, and A. Sudbery, Phys. Rev. Lett. 83, 243 (1999); C. H. Bennett, S. Popescu, D. Rohrlich, J. A. Smolin, A. V. Thapliyal, e-print quant-ph/9908073.
20. J. Kempe, Phys. Rev. A 60, 910 (1999); A. V. Thapliyal, ibid. 59, 3336 (1998); G. Vidal, Phys. Rev. Lett. 83, 1048 (1999); N. Linden and S. Popescu, Fortschr. Phys. 46, 567 (1998); N. Linden, S. Popescu, and A. Sudbery, Phys. Rev. Lett. 83, 243 (1999); C. H. Bennett, S. Popescu, D. Rohrlich, J. A. Smolin, A. V. Thapliyal, e-print quant-ph/9908073.
21. J. Kempe, Phys. Rev. A 60, 910 (1999); A. V. Thapliyal, ibid. 59, 3336 (1998); G. Vidal, Phys. Rev. Lett. 83, 1048 (1999); N. Linden and S. Popescu, Fortschr. Phys. 46, 567 (1998); N. Linden, S. Popescu, and A. Sudbery, Phys. Rev. Lett. 83, 243 (1999); C. H. Bennett, S. Popescu, D. Rohrlich, J. A. Smolin, A. V. Thapliyal, e-print quant-ph/9908073.
22. W. Dür, J. I. Cirac, and R. Tarrach, Phys. Rev. Lett. 83, 3562 (1999).
23. S. L. Braunstein, C. M. Caves, R. Jozsa, N. Linden, S. Popescu, and R. Schack, Phys. Rev. Lett. 83, 1054 (1999).
24. G. Vidai and R. Tarrach, Phys. Rev. A 59, 141 (1999).
25. W. Dür, J. I. Cirac, M. Lewenstein, and D. Bruß, e-print quant-ph/9910022.
26. D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, A. V. Thapliyal, e-print quant-ph/9910026.
27. M. Lewenstein, J. I. Cirac, and S. Karnas, e-print quant-ph/9903012.
28. G. E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and Its Applications (Addison-Wesley Publishing Company, Reading, MA, 1976).
29. D. M. Greenberger, M. Home, and A. Zeilinger, Bell's Theorem, Quantum Theory, and Conceptions of the Universe, edited by M. Kafatos (Kluwer, Dordrecht 69, 1989);
30. D. Bouwmeester et al., Phys. Rev. Lett. 82, 1345 (1999).
31. C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Phys. Rev. Lett. 82, 5385 (1999).
32. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 82, 1056 (1999).
33. M. Zukowski, A. Zeilinger, M. A. Home, and A. K. Ekert, Phys. Rev. Lett. 71, 4287 (1993).
34. For example, we have that the following combination is in principle allowed: The state ρ is 4-inseparable, 3-inseparable with respect to all possible 3-partite splits (3a)-(3f), but 2-separable with respect to all possible bipartite splits (2a)-(2g). On the other hand, we have that the following combination is not allowed: ρ is 4-inseparable, 3-separable with respect to all possible 3-partite splits (3a)-(3f) and 2-separable with respect to the bipartite splits (2b)-(2g), but 2-inseparable with respect to the bipartite split (2a). Since, e.g., the 3-separability with respect to the 3-partite split (3a) already implies the 2-separability with respecl to the bipartite split (2a) (see Fig. 2), it follows that this configuration is impossible and thus forbidden.
35. In R. Schack and C. M. Caves, e-print quant-ph/9904109, it was independently shown using different methods that for N = 3, the state ρ(x) is separable for x≤s 1/5. For larger N however, only weaker bounds were obtained.