Quantifying entanglement with witness operators

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 72, Issue 2, 2005, Pages

  Brando, Fernando G S L ^a  

^a FEDERAL UNIVERSITY OF MINAS GERAIS   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

ENTANGLEMENT; ENTANGLEMENT QUANTIFIERS; GAUSSIAN STATES; WITNESS OPERATORS;

APPROXIMATION THEORY; ELEMENTARY PARTICLES; MATHEMATICAL OPERATORS; QUANTUM OPTICS; THERMODYNAMIC PROPERTIES;

ATOMIC PHYSICS;

EID: 27144465234     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.72.022310     Document Type: Article

References (93)

1. C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.53.2046 53, 2046 (1996).
2. V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.78.2275 78, 2275 (1997).
3. M. A. Nielsen, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.83. 436 83, 436 (1999).
4. G. Vidal, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.83.1046 83, 1046 (1999).
5. D. Jonathan and M. B. Plenio, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.83.1455 83, 1455 (1999).
6. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.60.1888 60, 1888 (1999).
7. M. Curty, M. Lewenstein, and N. Lbctkenhaus, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.92.217903 92, 217903 (2004).
8. D. Bruss, G. M. D\'e2\'80\'99Ariano, M. Lewenstein, C. Macchiavello, A. Sen De, and U. Sen, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.93. 210501 93, 210501 (2004).
9. W. Dbcr, G. Vidal, and J. I. Cirac, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.62.062314 62, 062314 (2000).
10. G. Vidal, J. Mod. Opt. JMOPEW 0950-0340 10.1080/095003400148268 47, 355 (2000).
11. L. Gurvits, in Proceedings of the 35th ACM Symposium on the Theory of Computing (ACM Press, New York, 2003), pp. 10- 19.
12. A. Peres, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.77.1413 77, 1413 (1996).
13. K. Chen, L. Wu, and L. Yang, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.69.022312 69, 022312 (2004).
14. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A PYLAAG 0375-9601 10.1016/S0375-9601(96)00706-2 223, 1 (1996).
15. B. M. Terhal, Phys. Lett. A PYLAAG 0375-9601 10.1016/S0375-9601(00)00401- 1 271, 319 (2000);
16. B. M. Terhal, Linear Algebr. Appl. LAAPAW 0024-3795 10.1016/S0024- 3795(00)00251-2 323, 61 (2000).
17. M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.62.052310 62, 052310 (2000).
18. B. M. Terhal, Theor. Comput. Sci. TCSCDI 0304-3975 10.1016/S0304-3975(02) 00139-1 287, 313 (2002).
19. G. Toth, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.69.052327 69, 052327 (2004).
20. C. Brukner, V. Vedral, and A. Zeilinger, e-print quant-ph/0410138;
21. C. Brukner and V. Vedral, e-print quant-ph /0406040.
22. L. A. Wu, S. Bandyopadhyay, M. S. Sarandy, and D. A. Lidar, e-print quant-ph/0412099.
23. G. Toth and O. Gbchne, AIP Conf. Proc. APCPCS 0094-243X 234, 234 (2004);
24. G. Toth and O. Gbchne, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.94.060501 94, 060501 (2005).
25. O. Gbchne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.66.062305 66, 062305 (2002);
26. O. Gbchne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, J. Mod. Opt. JMOPEW 0950-0340 10.1080/0950034021000057932 50, 1079 (2003).
27. M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gbchne, P. Hyllus, D. Bruss, M. Lewenstein, and Anna Sanpera, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.92.087902 92, 087902 (2004).
28. F. G. S. L. Brandao and R. O. Vianna, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.93.220503 93, 220503 (2004);
29. F. G. S. L. Brandao and R. O. Vianna, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.70.062309 70, 062309 (2004).
30. A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.88.187904 88, 187904 (2002);
31. A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.71.032333 71, 032333 (2005).
32. J. Eisert, P. Hyllus, O. Gbchne, and M. Curty, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.70.062317 70, 062317 (2004).
33. R. A. Bertlmann, H. Narnhofer, and W. Thirring, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.66.032319 66, 032319 (2002).
34. F. G. S. L. Brandao and R. O. Vianna, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.70.062309 70, 062309 (2004).
35. W. Dbcr and J. I. Cirac, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.61.042314 61, 042314 (2000).
36. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A PYLAAG 0375-9601 223, 1 (1996).
37. M. Lewenstein and A. Sanpera, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.80.2261 80, 2261 (1998).
38. A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.69.022308 69, 022308 (2004).
39. P. Hayden, B. M. Terhal, and A. Uhlmann, e-print quant-ph/0011095.
40. G. Vidal and R. F. Werner, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.65.032314 65, 032314 (2002).
41. J. Eisert, Ph.D. thesis, University of Potsdam, 2001 (unpublished).
42. K. Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.58.883 58, 883 (1998).
43. E. M. Rains, IEEE Trans. Inf. Theory IETTAW 0018-9448 10.1109/18.959270 47, 2921 (2001).
44. F. Verstraete, Ph.D. thesis Katholieke Universiteit Leuven, 2002 (unpublished).
45. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, England, 2000).
46. G. Vidal and R. Tarrach, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.59.141 59, 141 (1999).
47. S. Karnas and M. Lewenstein, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/34/35/318 34, 6919 (2001).
48. R. Reemtsen, Semi-Infinite Programming (Nonconvex Optimization and Its Applications) (Kluwer Academic, Dordrecht, 1998).
49. M. Steiner, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.67.054305 67, 054305 (2003).
50. A. W. Harrow and M. A. Nielsen, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.68.012308 68, 012308 (2003).
51. M. Keyl, Phys. Rep. PRPLCM 0370-1573 10.1016/S0370-1573(02)00266-1 369, 431 (2002).
52. F. G. S. L. Brandao (unpublished).
53. F. Verstraete, M. Popp, and J. I. Cirac, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.92.027901 92, 027901 (2004).
54. J. B. Lassere, SIAM J. Optim. SJOPE8 1052-6234 11, 137 (2001).
55. A. Miyake and F. Verstraete, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.69.012101 69, 012101 (2004).
56. G. Giedke and J. I. Cirac, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.66.032316 66, 032316 (2002).
57. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.36.3868 36, 3868 (1987).
58. G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.70.022318 70, 022318 (2004).
59. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (Society for Industrial and Applied Mathematics, New York, 1994).
60. B. Tal, Conic and Robust Optimization, http://iew3.technion.ac.il/\'e2\ '80\rdblquote Home/Users/morbt/rom.pdf. See examples 18.p (p. 81), 11 (p. 55), and 9 (p. 54).
61. K. G. H. Vollbrecht and R. F. Werner, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.64.062307 64, 062307 (2001).
62. F. Verstraete and J. I. Cirac, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.91.010404 91, 010404 (2003).
63. S. D. Bartlett and H. M. Wiseman, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.91.097903 91, 097903 (2003).
64. B. M. Terhal, D. P. DiVincenzo, and D. W. Leung, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.86.5807 86, 5807 (2001).
65. N. Schuch, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.92.087904 92, 087904 (2004).
66. P. Zanardi, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.65.042101 65, 042101 (2002).
67. J. R. Gittings and A. J. Fisher, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.66.032305 66, 032305 (2002).
68. R. Paskauskas and L. You, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.64.042310 64, 042310 (2001).
69. J. Schliemann, J. I. Cirac, M. Kus, M. Lewenstein, and D. Loss, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.64.022303 64, 022303 (2001).
70. H. M. Wiseman and J. A. Vaccaro, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.91.097902 91, 097902 (2003).
71. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.84.2014 84, 2014 (2000).
72. C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.54.3824 54, 3824 (1996).
73. B. M. Terhal and KarlGerd H. Vollbrecht, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.85.2625 85, 2625 (2000).
74. W. K. Wootters, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.80. 2245 80, 2245 (1998).
75. P. Horodecki, Phys. Lett. A PYLAAG 0375-9601 10.1016/S0375-9601(97)00416- 7 232, 333 (1997).
76. K. Audenaert, F. Verstraete, and B. De Moor, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.64.052304 64, 052304 (2001).
77. M. A. Nielsen, Ph.D. thesis, University of Mexico, 1998, quant-ph/0011036.
78. M. C. Arnesen, S. Bose, and V. Vedral, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.87.017901 87, 017901 (2001).
79. X. Wang, H. Fu, and A. I. Solomon, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/34/50/312 34, 11307 (2001);
80. X. Wang and K. Molmer, Eur. Phys. J. D EPJDF6 1434-6060 10.1007/s10053-002-8844-x 18, 385 (2002).
81. X. Wang and P. Zanardi, Phys. Lett. A PYLAAG 0375-9601 10.1016/S0375-9601(02)00885-X 301, 1 (2002);
82. X. Wang, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.66.044305 66, 044305 (2002).
83. S. Bose and V. Vedral, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.61. 040101 61, 040101 (R) (2000).
84. K. M. O\'e2\'80\'99Connor and W. K. Wootters, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.63.052302 63, 052302 (2001).
85. T. J. Osborne and M. A. Nielsen, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.66.032110 66, 032110 (2002).
86. A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature (London) NATUAS 0028-0836 10.1038/416608a 416, 608 (2002).
87. G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.90.227902 90, 227902 (2003).
88. S. Ghose, T. F. Rosenbaum, G. Aeppli, and S. N. Coppersmith, Nature (London) NATUAS 0028-0836 10.1038/nature01888 425, 48 (2003).
89. T. Eggeling and R. F. Werner, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.63.042111 63, 042111 (2000).
90. H. Weyl, The Classical Groups (Princeton University, Princeton, NJ, 1946).
91. K. Sakai, M. Shiroishi, Y. Nishiyama, and M. Takahashi, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.67.065101 67, 065101 (R) (2003).
92. G. Toth, O. Gbchne, and J. I. Cirac, Poster at Gordon Research Conference on Quantum Information, Ventura, California, February 22- 27, 2004 (unpublished).
93. V. Vedral, New J. Phys. NJOPFM 1367-2630 10.1088/1367-2630/6/1/022 6, 22 (2004).