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Acta Mathematica Vietnamica

, Volume 43, Issue 4, pp 619–627 | Cite as

Indecomposable Exposed Positive Bi-linear Maps Between Two by Two Matrices

  • Seung-Hyeok Kye
Article
  • 34 Downloads

Abstract

Positive bi-linear maps between matrix algebras play important roles to detect tri-partite entanglement by the duality between bi-linear maps and tri-tensor products. We exhibit indecomposable positive bi-linear maps between 2 × 2 matrices which generate extreme rays in the cone of all positive bi-linear maps. In fact, they are exposed, and so detect entanglement of positive partial transpose (PPT) whose volume is nonzero.

Keywords

Positive multi-linear maps Exposed Indecomposable Separable Partial transpose 

Mathematics Subject Classification (2010)

46L07 15A30 81P15 

Notes

Acknowledgements

The author is grateful to Kyung-Hoon Han for useful discussion and comments. The author is partially supported by NRF-2017R1A2B4006655.

References

  1. 1.
    Chen, L., Han, K.H., Kye, S.-H.: Separability criterion for three-qubit states with a four dimensional norm. J. Phys. A 50(34), 27 pp (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cho, S.-J., Kye, S.-H., Lee, S.G.: Generalized Choi maps in three-dimensional matrix algebra. Linear Algebra Appl. 171, 213–224 (1992)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Choi, M.D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285–290 (1975)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Choi, M.D.: Positive semidefinite biquadratic forms. Linear Algebra Appl. 12 (2), 95–100 (1975)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Choi, M.D., Lam, T.T.: Extremal positive semidefinite forms. Math. Ann. 231(1), 1–18 (1977)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chruściński, D., Kossakowski, A.: Spectral conditions for positive maps. Commun. Math. Phys. 290(3), 1051–1064 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chruściński, D., Sarbicki, G.: Exposed positive maps in M 4(C). Open. Syst. Inf. Dyn. 19(3), 8 (2012)zbMATHGoogle Scholar
  8. 8.
    Eom, M. -H., Kye, S. -H.: Duality for positive linear maps in matrix algebras. Math. Scand. 86(1), 130–142 (2000)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ha, K.-C., Kye, S.-H.: Exposedness of Choi-type entanglement witnesses and applications to lengths of separable states. Open. Syst. Inf. Dyn. 20(4), 17 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ha, K. -C., Kye, S. -H.: Separable states with unique decompositions. Commun. Math. Phys. 328(1), 131–153 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ha, K. -C., Kye, S. -H.: Construction of exposed indecomposable positive linear maps between matrix algebras. Linear Multilinear Algebra 64(11), 2188–2198 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Han, K.H., Kye, S.-H.: Various notions of positivity for bi-linear maps and applications to tri-partite entanglement. J. Math. Phys. 57(1), 22 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Han, K.H., Kye, S.-H.: Construction of multi-qubit optimal genuine entanglement witnesses. J. Phys. A 49(17), 16 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Han, K.H., Kye, S.-H.: Separability of three qubit Greenberger-Horne-Zeilinger diagonal states. J. Phys. A 50(14), 17 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Han, K.H., Kye, S.-H.: The role of phases in detecting three-qubit entanglement. J. Math. Phys. 58(10), 19 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223(1-2), 1–8 (1996)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jamiokowski, A.: An effective method of investigation of positive maps on the set of positive definite operators. Rep. Math. Phys. 5, 415–424 (1974)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kye, S.-H.: Facial structures for various notions of positivity and applications to the theory of entanglement. Rev. Math. Phys. 25(2), 52 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kye, S.-H.: Three-qubit entanglement witnesses with the full spanning properties. J. Phys. A 48(23), 11 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Marciniak, M.: Rank properties of exposed positive maps. Linear Multilinear Algebra 61(7), 970–975 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Robertson, A.G.: Automorphisms of spin factors and the decomposition of positive maps. Quart. J. Math. Oxford Ser. (2) 34(133), 87–96 (1983)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Robertson, A.G.: Positive projections on C -algebras and extremal positive maps. J. London Math. Soc. (2) 32(1), 133–140 (1985)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefGoogle Scholar
  24. 24.
    Sarbicki, G., Chruściński, D.: A class of exposed indecomposable positive maps. J. Phys. A 46(1), 15 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Strmer, E.: Positive linear maps of operator algebras. Acta Math. 110, 233–278 (1963)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Woronowicz, S.L.: Positive maps of low dimensional matrix algebras. Rep. Math. Phys. 10(2), 165–183 (1976)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Woronowicz, S.L.: Nonextendible positive maps. Commun. Math. Phys. 51(3), 243–282 (1976)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Yopp, D.A., Hill, R.D.: Extremals and exposed faces of the cone of positive maps. Linear Multilinear Algebra 53(3), 167–174 (2005)MathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Institute of MathematicsSeoul National UniversitySeoulKorea

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