Cones of positive maps and their duality relations

Journal of Mathematical Physics

Volumn 50, Issue 6, 2009, Pages

  Skowronek, Łukasz ^a   Størmer, Erling ^b   Zyczkowski, Karol ^a,c  

^a JAGIELLONIAN UNIVERSITY   (Poland)

^b UNIVERSITY OF OSLO   (Norway)

^c CENTER FOR THEORETICAL PHYSICS   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 67650494727     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.3155378     Document Type: Article

References (59)

1. W. F. Stinespring, Proc. Am. Math. Soc. 0002-9939 6, 211 (1955). 10.2307/2032342
2. K. Kraus, Ann. Phys. 0003-3804 64, 311 (1971). 10.1016/0003-4916(71) 90108-4
3. E. Størmer, J. Funct. Anal. 0022-1236 66, 235 (1986). 10.1016/0022-1236(86)90072-8
4. T. Ando, Linear Algebr. Appl. 0024-3795 379, 3 (2004). 10.1016/j.laa.2003.06.005
5. A. S. Holevo, M. E. Shirokov, and R. F. Werner, Russ. Math. Surveys 0036-0279 60, 153 (2005).
6. M. Horodecki, P. W. Shor, and M. B. Ruskai, Rev. Math. Phys. 0129-055X 15, 629 (2003). 10.1142/S0129055X03001709
7. A. Jamiokowski, Rep. Math. Phys. 0034-4877 3, 275 (1972). 10.1016/0034-4877(72)90011-0
8. K. yczkowski and I. Bengtsson, Open Syst. Inf. Dyn. 1230-1612 11, 3 (2004). 10.1023/B:OPSY.0000024753.05661.c2
9. M. Asorey, A. Kossakowski, G. Marmo, E. C. Sudarshan, Open Syst. Inf. Dyn. 1230-1612 12, 319 (2005). 10.1007/s11080-005-4482-3
10. K. Takasaki and J. Tomiyama, Math. Z. 0025-5874 184, 101 (1983). 10.1007/BF01162009
11. K. S. Ranade and M. Ali, Open Syst. Inf. Dyn. 1230-1612 14, 371 (2007). 10.1007/s11080-007-9062-2
12. M. -D. Choi, Linear Algebr. Appl. 0024-3795 10, 285 (1975). 10.1016/0024-3795(75)90075-0
13. E. C. G. Sudarshan, P. M. Mathews, and J. Rau, Phys. Rev. 0096-8250 121, 920 (1961). 10.1103/PhysRev.121.920
14. E. Størmer, Acta Math. 0001-5962 110, 233 (1963). 10.1007/BF02391860
15. J. de Pillis, Pac. J. Math. 0030-8730 23, 129 (1967).
16. W. Arverson, Acta Math. 0001-5962 123, 141 (1969). 10.1007/BF02392388
17. M. -D. Choi, Can. J. Math. 0008-414X 24, 520 (1972).
18. M. -D. Choi, Linear Algebr. Appl. 0024-3795 12, 95 (1975). 10.1016/0024-3795(75)90058-0
19. S. L. Woronowicz, Commun. Math. Phys. 0010-3616 51, 243 (1976). 10.1007/BF01617922
20. S. L. Woronowicz, Rep. Math. Phys. 0034-4877 10, 165 (1976). 10.1016/0034-4877(76)90038-0
21. E. Størmer, Proc. Am. Math. Soc. 0002-9939 86, 402 (1982). 10.2307/2044436
22. A. G. Robertson, J. Lond. Math. Soc. 0024-6107 s2-32, 133 (1985). 10.1112/jlms/s2-32.1.133
23. H. Osaka, Linear Algebr. Appl. 0024-3795 153, 73 (1991). 10.1016/0024-3795(91)90211-E
24. S. -J. Cho, S. -H. Kye, and S. -G. Lee, Linear Algebr. Appl. 0024-3795 171, 213 (1992). 10.1016/0024-3795(92)90260-H
25. K. -C. Ha, Publ. Res. Inst. Math. Sci. 0034-5318 34, 591 (1998). 10.2977/prims/1195144425
26. M. -H. Eom and S. -H. Kye, Math. Scand. 0025-5521 86, 130 (2000).
27. W. A. Majewski and M. Marciniak, J. Phys. A 0305-4470 34, 5863 (2001). 10.1088/0305-4470/34/29/308
28. A. Kossakowski, Open Syst. Inf. Dyn. 1230-1612 10, 213 (2003). 10.1023/A:1025101606680
29. K. -C. Ha, Linear Algebr. Appl. 0024-3795 359, 277 (2003). 10.1016/S0024-3795(02)00415-9
30. S. -H. Kye, Linear Algebr. Appl. 0024-3795 362, 57 (2003). 10.1016/S0024-3795(02)00459-7
31. F. Benatti, R. Floreanini, and M. Piani, Open Syst. Inf. Dyn. 1230-1612 11, 325 (2004). 10.1007/s11080-004-6622-6
32. D. Chruściński and A. Kossakowski, Open Syst. Inf. Dyn. 1230-1612 14, 275 (2007). 10.1007/s11080-007-9052-4
33. E. Størmer, J. Funct. Anal. 0022-1236 254, 2303 (2008). 10.1016/j.jfa.2007.12.017
34. E. Størmer, " Duality of cones of positive maps.," e-print arXiv:0810.4253.
35. A. Peres, Phys. Rev. Lett. 0031-9007 77, 1413 (1996). 10.1103/PhysRevLett.77.1413
36. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 0375-9601 223, 1 (1996). 10.1016/S0375-9601(96)00706-2
37. M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 0031-9007 80, 5239 (1998). 10.1103/PhysRevLett.80.5239
38. P. Horodecki, M. Horodecki, and R. Horodecki, Phys. Rev. Lett. 0031-9007 82, 1056 (1999). 10.1103/PhysRevLett.82.1056
39. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, e-print arXiv:quant-ph/0702225.
40. B. Terhal, Phys. Lett. A 0375-9601 271, 319 (2000). 10.1016/S0375- 9601(00)00401-1
41. O. Gühne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, J. Mod. Opt. 0950-0340 50, 1079 (2003). 10.1080/09500340308234554
42. M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bruß, M. Lewenstein, and A. Sanpera, Phys. Rev. Lett. 0031-9007 92, 087902 (2004). 10.1103/PhysRevLett.92.087902
43. Y. Tokunaga, T. Yamamoto, M. Koashi, and N. Imoto, Phys. Rev. A 1050-2947 74, 020301 (R) (2006). 10.1103/PhysRevA.74.020301
44. Y. Tokunaga, S. Kuwashiro, T. Yamamoto, Phys. Rev. Lett. 0031-9007 100, 210501 (2008). 10.1103/PhysRevLett.100.210501
45. M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, Phys. Rev. A 1050-2947 62, 052310 (2000). 10.1103/PhysRevA.62.052310
46. G. Sarbicki, J. Phys. A 0305-4470 41, 375303 (2008). 10.1088/1751-8113/ 41/37/375303
47. G. Sarbicki, " General theory of detection and optimality.," e-print arXiv:0905.0778.
48. . Skowronek, " Quantum entanglement and certain problems in mathematics.," M.S. thesis, Kraków, 2008 (http://chaos.if.uj.edu.pl/ ~karol/kzstudent.htm).
49. D. Chruściński and A. Kossakowski, " Spectral conditions for positive maps.," e-print arXiv:0809.4909.
50. L. Clarisse, Phys. Rev. A 1050-2947 71, 032332 (2005). 10.1103/PhysRevA.71.032332
51. M. -D. Choi, J. Oper. Theory 0379-4024 4, 271 (1980).
52. K. Tanahashi and J. Tomiyama, Can. Math. Bull. 0008-4395 31, 308 (1988).
53. I. Bengtsson and K. yczkowski, Geometry of Quantum States (Cambridge University Press, Cambridge, 2006).
54. B. Terhal and P. Horodecki, Phys. Rev. A 1050-2947 61, 040301 (2000). 10.1103/PhysRevA.61.040301
55. A. Sanpera, D. Bruß, and M. Lewenstein, Phys. Rev. A 1050-2947 63, 050301 (R) (2001). 10.1103/PhysRevA.63.050301
56. F. Hulpke, D. Bruß, M. Lewenstein, and A. Sanpera, Quantum Inf. Comput. 1533-7146 4, 207 (2004).
57. R. T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1997).
58. St. J. Szarek, E. Werner, and K. yczkowski, J. Math. Phys. 0022-2488 49, 032113 (2008). 10.1063/1.2841325
59. J. K. Korbicz, M. L. Almeida, J. Bae, M. Lewenstein, and A. Acin, Phys. Rev. A 1050-2947 78, 062105 (2008). 10.1103/PhysRevA.78.062105