Reduction criterion of separability and limits for a class of distillation protocols

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 59, Issue 6, 1999, Pages 4206-4216

  Horodecki, Michał ^a   Horodecki, Paweł ^b  

^a UNIVERSITY OF GDA SK   (Poland)

^b GDANSK UNIVERSITY OF TECHNOLOGY   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000525266     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.59.4206     Document Type: Article

References (59)

1. A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935)
2. E. Schrödinger, Naturwissenschaften 23, 807 (1935)
3. J. S. Bell, Physics (N.Y.) 1, 195 (1964).
4. D. Deutsch, Proc. R. Soc. London, Ser. A 425, 73 (1989)
5. P. Shor, SIAM J. Comput. 26, 1484 (1997).
6. A. Ekert, Phys. Rev. Lett. 67, 661 (1991).
7. C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992).
8. K. Mattle, H. Weinfurter, P. Kwiat, and A. Zeilinger, Phys. Rev. Lett. 76, 4656 (1996).
9. C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).
10. D. Bouwmeester, J.-W. Pan, K. Mattle, M. Elbl, H. Weinfurter, and A. Zeilinger, Nature (London) 390, 575 (1997)
11. D. Boschi, S. Brance, F. de Martini, L. Hardy, and S. Popescu, Phys. Rev. Lett. 80, 1121 (1998).
12. R. Cleve and H. Buhrman, Phys. Rev. A 56, 1201 (1997).
13. R. F. Werner, Phys. Rev. A 40, 4277 (1989).
14. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76, 722 (1996).
15. D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Phys. Rev. Lett. 77, 2818 (1996).
16. C. H. Bennett, D. P. Di Vincenzo, J. Smolin, and W. K. Wootters, Phys. Rev. A 54, 3824 (1996).
17. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 78, 574 (1997).
18. R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A 200, 340 (1995).
19. R. Horodecki and P. Horodecki, Phys. Lett. A 194, 147 (1994).
20. R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A 210, 377 (1996).
21. R. Horodecki and M. Horodecki, Phys. Rev. A 54, 1838 (1996).
22. A. Peres, Phys. Rev. Lett. 77, 1413 (1996).
23. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 223, 1 (1996).
24. Throughout this paper the (Formula presented) system means the system described by the Hilbert space (Formula presented) so that, e.g., a (Formula presented) system corresponds to the two-spin-(Formula presented) or, in general, two two-level (two-qubit) one.
25. N. Gisin, Phys. Lett. A 210, 151 (1996).
26. C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A 53, 2046 (1996).
27. This result can be immediately generalized to cover the (Formula presented) case.
28. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 80, 5239 (1998)
29. P. Horodecki, Phys. Lett. A 232, 333 (1997).
30. G. Lindblad, Commun. Math. Phys. 40, 147 (1975)
31. W. F. Stinespring, Proc. Am. Math. Soc. 26, 211 (1955).
32. K. Kraus, States, Effects and Operations: Fundamental Notions of Quantum Theory (Wiley, New York, 1991).
33. P. Busch and J. Lahti, Found. Phys. 20, 1429 (1990)
34. A. Sanpera, R. Tarrach, and G. Vidal, e-print quant-ph/9707041.
35. S. L. Woronowicz, Rep. Math. Phys. 10, 165 (1976).
36. A. Kossakowski (private communication).
37. After completion of the first version of this manuscript, we received information that this condition was independently derived and discussed in Ref. 31.
38. N. J. Cerf, C. Adami, and R. M. Gingrich, e-print quant-ph/9710001.
39. This is in agreement with the numerical evidence obtained by N. Cerf and R. Gingrich (private communication).
40. S. Popescu, Phys. Rev. Lett. 72, 797 (1994).
41. Quite recently, B. Terhal (e-print quant-ph/9810091) introduced a new positive map, which has a structure similar to our map but produces a criterion independent of the Peres one. Such criteria allow us to explore the region of bound entanglement 24.
42. E. Rains (private communication).
43. B. Schumacher and M. A. Nielsen, Phys. Rev. A 54, 2629 (1996).
44. R. Jozsa, J. Mod. Opt. 41, 2315 (1994).
45. P. W. Shor and J. A. Smolin, e-print quant-ph/9604006;
46. D. DiVincenzo, P. W. Shor, and J. A. Smolin, Phys. Rev. A 57, 830 (1998).
47. E. M. Rains, e-print quant-ph/9707002.
48. V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys. Rev. Lett. 78, 2275 (1997).
49. S. M. Barnett and S. J. D. Phoenix, Phys. Rev. A 44, 535 (1991).
50. R. Horodecki, M. Horodecki, and P. Horodecki, Phys. Lett. A 222, 21 (1996).
51. S. Popescu, Phys. Rev. Lett. 74, 2619 (1995).
52. N. D. Mermin (unpublished)
53. S. Popescu, in The Dilemma of Einstein, Podolsky and Rosen—60 Years After, edited by A. Mann and M. Revzen (IOP, Bristol, 1996)
54. S. Teufel, K. Brendl, D. Dürr, S. Goldstein, and N. Zanghí, Phys. Rev. A 56, 1217 (1997)
55. Phys. Rev. AM. Żukowski, R. Horodecki, M. Horodecki, and P. Horodecki, 58, 1694 (1998).
56. P. Shor and R. Laflamme, Phys. Rev. Lett. 78, 1600 (1997)
57. E. Rains, e-print quant-ph/9611001;IEEE Trans. Inf. Theory (to be published)
58. P. Shor and R. Laflamme, Phys. Rev. Lett. 44, 1388 (1998) (also available as e-print quant-ph/9612015).
59. A. Jamiołkowski, Rep. Math. Phys. 3, 275 (1972).