Optimization of entanglement witnesses

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 62, Issue 5, 2000, Pages 052310-052311

  Lewenstein, M ^a   Kraus, B ^b   Cirac, J I ^b   Horodecki, P ^c  

^a LEIBNIZ UNIVERSITY HANNOVER   (Germany)

^b UNIVERSITY OF INNSBRUCK   (Austria)

^c GDANSK UNIVERSITY OF TECHNOLOGY   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; CORRELATION METHODS; DATA PROCESSING; ELECTRON ENERGY LEVELS; MATHEMATICAL OPERATORS; OPTIMIZATION;

ENTANGLEMENT WITNESSES (EW); QUANTUM-MECHANICAL EFFECTS;

QUANTUM THEORY;

EID: 0141782139     PISSN: 10502947     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevA.62.052310     Document Type: Article

References (54)

1. A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).
2. E. Schrödinger, Proc. Cambridge Philos. Soc. 31, 555 (1935).
3. A. Ekert, Phys. Rev. Lett. 67, 661 (1991).
4. C.H. Bennett and S.J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992).
5. C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W.K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).
6. See A. Ekert and R. Jozsa, Rev. Mod. Phys. 68, 733 (1996); A. Steane, Rep. Prog. Phys. 61, 117 (1998); A. Barenco, Contemp. Physics 37, 375 (1996).
7. See A. Ekert and R. Jozsa, Rev. Mod. Phys. 68, 733 (1996); A. Steane, Rep. Prog. Phys. 61, 117 (1998); A. Barenco, Contemp. Physics 37, 375 (1996).
8. See A. Ekert and R. Jozsa, Rev. Mod. Phys. 68, 733 (1996); A. Steane, Rep. Prog. Phys. 61, 117 (1998); A. Barenco, Contemp. Physics 37, 375 (1996).
9. R.F. Werner, Phys. Rev. A 40, 4277 (1989).
10. An excellent introduction to the problematics of quantum correlation and entanglement is provided by A. Peres, Quantum Theory: Concepts and Methods (Kluwer, Dordrecht, 1995).
11. See P. Horodecki, M. Horodecki, and R. Horodecki, Phys. Rev. Lett. 82, 1056 (1999).
12. A. Peres, Phys. Rev. Lett. 77, 1413 (1996).
13. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 223, 8 (1996).
14. T A ≥ 0. The relation between separable and positive partial transpose operators was established in Refs. [10, 11]. All separable operators have a positive partial transposition. However, the converse is not true in general. That is, there are positive partial transpose operators which are nonseparable.
15. P. Horodecki, Phys. Lett. A 232, 333 (1997).
16. C.H. Bennett, D.P. DiVincenzo, T. Mor, P.W. Shor, J.A. Smolin, and B.M. Terhal, Phys. Rev. Lett. 82, 5385 (1999); D.P. DiVincenzo, T. Mor, P.W. Shor, J.A. Smolin, and B.M. Terhal, quant-ph/9908070; C.H. Bennett, D.P. DiVincenzo, Ch.A. Fuchs, T. Mor, E. Rains, P.W. Shor, J.A. Smolin, and W.K. Wootters, Phys. Rev. A 59, 1070 (1999); See also R. Horodecki, M. Horodecki, and P. Horodecki, ibid. 60, 4144 (1999).
17. C.H. Bennett, D.P. DiVincenzo, T. Mor, P.W. Shor, J.A. Smolin, and B.M. Terhal, Phys. Rev. Lett. 82, 5385 (1999); D.P. DiVincenzo, T. Mor, P.W. Shor, J.A. Smolin, and B.M. Terhal, quant-ph/9908070; C.H. Bennett, D.P. DiVincenzo, Ch.A. Fuchs, T. Mor, E. Rains, P.W. Shor, J.A. Smolin, and W.K. Wootters, Phys. Rev. A 59, 1070 (1999); See also R. Horodecki, M. Horodecki, and P. Horodecki, ibid. 60, 4144 (1999).
18. C.H. Bennett, D.P. DiVincenzo, T. Mor, P.W. Shor, J.A. Smolin, and B.M. Terhal, Phys. Rev. Lett. 82, 5385 (1999); D.P. DiVincenzo, T. Mor, P.W. Shor, J.A. Smolin, and B.M. Terhal, quant-ph/9908070; C.H. Bennett, D.P. DiVincenzo, Ch.A. Fuchs, T. Mor, E. Rains, P.W. Shor, J.A. Smolin, and W.K. Wootters, Phys. Rev. A 59, 1070 (1999); See also R. Horodecki, M. Horodecki, and P. Horodecki, ibid. 60, 4144 (1999).
19. C.H. Bennett, D.P. DiVincenzo, T. Mor, P.W. Shor, J.A. Smolin, and B.M. Terhal, Phys. Rev. Lett. 82, 5385 (1999); D.P. DiVincenzo, T. Mor, P.W. Shor, J.A. Smolin, and B.M. Terhal, quant-ph/9908070; C.H. Bennett, D.P. DiVincenzo, Ch.A. Fuchs, T. Mor, E. Rains, P.W. Shor, J.A. Smolin, and W.K. Wootters, Phys. Rev. A 59, 1070 (1999); See also R. Horodecki, M. Horodecki, and P. Horodecki, ibid. 60, 4144 (1999).
20. For a primer on separability, see M. Lewenstein, D. Bruß, J. I. Cirac, B. Kraus, M. Kuś, J. Samsonowicz, A. Sanpera, and R. Tarrach, in Proceedings of the Conference "Quantum Optics Kuhtai 2000," edited by F. Ehlotzky and P. L. Knight [J. Mod. Opt. (to be published)].
21. For an extensive review, see M. Horodecki, P. Horodecki, and R. Horodecki, in Quantum Information - Basic Concepts and Experiments, edited by A. Zeilinger, H. Weinfurter, R. Werner, and Th. Beth (Springer, Berlin, 2000).
22. R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A 230, 377 (1996).
23. K. Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Phys. Rev. A 58, 883 (1998).
24. G. Vidal and R. Tarrach, Phys. Rev. A 59, 141 (1999); S.L. Braunstein, C.M. Caves, R. Jozsa, N. Linden, S. Popescu, and R. Schack, Phys. Rev. Lett. 83, 1054 (1999).
25. G. Vidal and R. Tarrach, Phys. Rev. A 59, 141 (1999); S.L. Braunstein, C.M. Caves, R. Jozsa, N. Linden, S. Popescu, and R. Schack, Phys. Rev. Lett. 83, 1054 (1999).
26. B. Kraus, J.I. Cirac, S. Karnas, and M. Lewenstein, e-print quant-ph/9912010; Phys. Rev. A (to be published).
27. B. Kraus, J.I. Cirac, S. Karnas, and M. Lewenstein, e-print quant-ph/9912010; Phys. Rev. A (to be published).
28. P. Horodecki, M. Lewenstein, G. Vidal, and J.I. Cirac, e-print quant-ph/0002089; Phys. Rev. A (to be published).
29. P. Horodecki, M. Lewenstein, G. Vidal, and J.I. Cirac, e-print quant-ph/0002089; Phys. Rev. A (to be published).
30. M. Lewenstein and A. Sanpera, Phys. Rev. Lett. 80, 2261 (1998).
31. A. Sanpera, R. Tarrach, and G. Vidal, Phys. Rev. A 58, 826 (1998); G. Vidal, Ph.D. thesis, Universitat de Barcelona, 1999 (unpublished).
32. A. Sanpera, R. Tarrach, and G. Vidal, Phys. Rev. A 58, 826 (1998); G. Vidal, Ph.D. thesis, Universitat de Barcelona, 1999 (unpublished).
33. For recent progress on optimal decompositions see B.G. Englert and N. Metwally, quant-ph/9912989; quant-ph/0007053.
34. B.M. Terhal, e-print quant-ph/9810091; also see Phys. Lett. A 271, 319 (2000).
35. B.M. Terhal, e-print quant-ph/9810091; also see Phys. Lett. A 271, 319 (2000).
36. A. Jamiolkowski, Rep. Math. Phys. 3, 275 (1972).
37. E. Störmer, Acta Math. 110, 233 (1963).
38. S.L. Woronowicz, Rev. Mod. Phys. 10, 165 (1976).
39. S.L. Woronowicz, Commun. Math. Phys. 51, 243 (1976); P. Kryszyński and S.L. Woronowicz, Lett. Math. Phys. 3, 319 (1979); M.D. Choi, Proc. Sympos. Pure Math. 38, 583 (1982).
40. S.L. Woronowicz, Commun. Math. Phys. 51, 243 (1976); P. Kryszyński and S.L. Woronowicz, Lett. Math. Phys. 3, 319 (1979); M.D. Choi, Proc. Sympos. Pure Math. 38, 583 (1982).
41. S.L. Woronowicz, Commun. Math. Phys. 51, 243 (1976); P. Kryszyński and S.L. Woronowicz, Lett. Math. Phys. 3, 319 (1979); M.D. Choi, Proc. Sympos. Pure Math. 38, 583 (1982).
42. B.
43. T.
44. W. Rudin, Functional Analysis (McGraw Hill, New York, 1973).
45. -1 X is the projector in R(X). If X has no zero eigenvalue, the pseudoinverse coincides with the inverse operator.
46. For more examples of PPTES's also see (a) Ref. [9]; (b) D. Bruß and A. Peres, Phys. Rev. A 61, 030301(R) (2000); (c) P. Horodecki and M. Lewenstein, e-print quant-ph/0001035; Phys. Rev. Lett, (to be published).
47. For more examples of PPTES's also see (a) Ref. [9]; (b) D. Bruß and A. Peres, Phys. Rev. A 61, 030301(R) (2000); (c) P. Horodecki and M. Lewenstein, e-print quant-ph/0001035; Phys. Rev. Lett, (to be published).
48. For more examples of PPTES's also see (a) Ref. [9]; (b) D. Bruß and A. Peres, Phys. Rev. A 61, 030301(R) (2000); (c) P. Horodecki and M. Lewenstein, e-print quant-ph/0001035; Phys. Rev. Lett, (to be published).
49. M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 (1999).
50. N.J. Cerf, C. Adami, and R.M. Gingrich, Phys. Rev. A 60, 893 (1999).
51. b, where the partial transposition is taken in that basis.
52. These states can be prepared locally, and can be used to construct PPTES's by mixing them "weakly" with entangled states. This provides an interesting possibility of an experimental realization of PPTES's. This suggestion was formulated by A. Weinfurter.
53. A. Peres, Found. Phys. 29, 589 (1999).
54. R.F. Werner and M.M. Wolf, e-print quant-ph/9910063.