# FACIAL STRUCTURES FOR SEPARABLE STATES

• Choi, Hyun-Suk ;
• Kye, Seung-Hyeok
• Received : 2010.12.16
• Published : 2012.05.01
• 56 6

#### Abstract

The convex cone $\mathbb{V}_1$ generated by separable states is contained in the cone $\mathbb{T}$ of all positive semi-definite block matrices whose block transposes are also positive semi-definite. We characterize faces of the cone $\mathbb{V}_1$ induced by faces of the cone $\mathbb{T}$ which are determined by pairs of subspaces of matrices.

#### Keywords

separable states;faces;entanglement

#### References

1. E. Alfsen and F. Shultz, Unique decompositions, faces, and automorphisms of separable states, J. Math. Phys. 51 (2010), 052201, 13 pp.
2. W. Arveson, Quantum channels that preserve entanglement, Math. Ann. 343 (2009), no. 4, 757-771. https://doi.org/10.1007/s00208-008-0288-2
3. R. Augusiak, J. Grabowski, M.Kus, and M. Lewenstein, Searching for extremal PPT entangled states, Optics Commun. 283 (2010), 805-813. https://doi.org/10.1016/j.optcom.2009.10.050
4. S. Bandyopadhyay, S. Ghosh, and V. Roychowdhury, Non-full rank bound entangled states satisfying the range criterion, Phys. Rev. A 71 (2005), 012316, 6 pp.
5. I. Bengtsson and K. Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, Cambridge University Press, 2006.
6. C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Unextendible Product Bases and Bound Entanglement, Phys. Rev. Lett. 82 (1999), no. 26, part 1, 5385-5388. https://doi.org/10.1103/PhysRevLett.82.5385
7. D. Bruss and A. Peres, Construction of quantum states with bound entanglement, Phys. Rev. A 61 (2000), no. 3, 030301, 2 pp.
8. E.-S. Byeon and S.-H. Kye, Facial structures for positive linear maps in two-dimensional matrix algebra, Positivity 6 (2002), no. 4, 369-380. https://doi.org/10.1023/A:1021397312586
9. S.-J. Cho, S.-H. Kye, and S. G. Lee, Generalized Choi maps in three-dimensional matrix algebra, Linear Algebra Appl. 171 (1992), 213-224. https://doi.org/10.1016/0024-3795(92)90260-H
10. M.-D. Choi, Some assorted inequalities for positive linear maps on $C^{*}$-algebras, J. Operator Theory. 4 (1980), no. 2, 271-285.
11. M.-D. Choi, Positive linear maps, Operator Algebras and Applications (Kingston, 1980), pp. 583-590, Proc. Sympos. Pure Math. Vol 38. Part 2, Amer. Math. Soc., 1982.
12. D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement, Comm. Math. Phys. 238 (2003), no. 3, 379-410. https://doi.org/10.1007/s00220-003-0877-6
13. M.-H. Eom and S.-H. Kye, Duality for positive linear maps in matrix algebras, Math. Scand. 86 (2000), no. 1, 130-142.
14. L. Gurvits, Classical complexity and quantum entanglement, J. Comput. System Sci. 69 (2004), no. 3, 448-484. https://doi.org/10.1016/j.jcss.2004.06.003
15. K.-C. Ha and S.-H. Kye, Construction of entangled states with positive partial transposes based on indecomposable positive linear maps, Phys. Lett. A 325 (2004), no. 5-6, 315- 323. https://doi.org/10.1016/j.physleta.2004.04.008
16. K.-C. Ha and S.-H. Kye, Construction of 3 $\bigotimes$ 3 entangled edge states with positive partial transpose, J. Phys. A 38 (2005), no. 41, 9039-9050. https://doi.org/10.1088/0305-4470/38/41/014
17. K.-C. Ha, S.-H. Kye, and Y. S. Park, Entangled states with positive partial transposes arising from indecomposable positive linear maps, Phys. Lett. A 313 (2003), no. 3, 163-174. https://doi.org/10.1016/S0375-9601(03)00733-3
18. M. Horodecki, P. Horodecki, and R. Horodecki, Separability of mixed states: necessary and sufficient conditions, Phys. Lett. A 223 (1996), no. 1-2, 1-8. https://doi.org/10.1016/S0375-9601(96)00706-2
19. P. Horodecki, Separability criterion and inseparable mixed states with positive partial transposition, Phys. Lett. A 232 (1997), no. 5, 333-339. https://doi.org/10.1016/S0375-9601(97)00416-7
20. P. Horodecki, M. Lewenstein, G. Vidal, and I. Cirac, Operational criterion and construc- tive checks for the separablity of low rank density matrices, Phys. Rev. A 62 (2000), 032310, 10 pp.
21. M. Junge, C. Palazuelos, D. Perez-Garcia, I. Villanueva, and M. Wolf, Operator space theory: a natural framework for Bell inequalities, Phys. Rev. Lett. 104 (2010), no. 17, 170405, 4 pp.
22. J. K. Korbicz, M. L. Almeida, J. Bae, M. Lewenstein, and A. Acin, Structural approxi- mations to positive maps and entanglement-breaking channels, Phys. Rev. A 78 (2008), 062105, 17 pp.
23. S.-H. Kye, Facial structures for unital positive linear maps in the two dimensional matrix algebra, Linear Algebra Appl. 362 (2003), 57-73. https://doi.org/10.1016/S0024-3795(02)00459-7
24. S.-H. Kye, Facial structures for decomposable positive linear maps in matrix algebras, Positivity 9 (2005), no. 1, 63-79.
25. M. Marciniak, Rank properties of exposed positive maps, preprint, arXiv:1103.3497.
26. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77 (1996), no. 8, 1413-1415. https://doi.org/10.1103/PhysRevLett.77.1413
27. W. F. Stinespring, Positive functions on $C^{*}$-algebras, Proc. Amer. Math. Soc. 6 (1955),211-216.
28. E. Strmer, Positive linear maps of operator algebras, Acta Math. 110 (1963), 233-278. https://doi.org/10.1007/BF02391860
29. E. Strmer, Decomposable positive maps on $C^{*}$-algebras, Proc. Amer. Math. Soc. 86 (1982), no. 3, 402-404.
30. E. Strmer, Separable states and positive maps, J. Funct. Anal. 254 (2008), no. 8, 2303- 2312. https://doi.org/10.1016/j.jfa.2007.12.017
31. S. L. Woronowicz, Positive maps of low dimensional matrix algebras, Rep. Math. Phys. 10 (1976), no. 2, 165-183. https://doi.org/10.1016/0034-4877(76)90038-0

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