Three-qubit separable states of length ten with unique decompositions

  • Seung-Hyeok KyeEmail author


We construct one-parameter families of three-qubit separable states with length ten, which is strictly greater than the whole dimension eight. These states are located on the boundary of the convex set of all separable states, but they are in the interior of the convex set of all states with positive partial transposes. They are also decomposed into the convex sum of ten pure product states in a unique way.


Lengths of separable states Three-qubit states Unique decomposition Positive multi-linear maps Dual faces 

Mathematics Subject Classification

81P15 15A30 46L07 



The author is grateful to Kyung Hoon Han for valuable comments and discussion.


  1. 1.
    Alfsen, E., Shultz, F.: Unique decompositions, faces, and automorphisms of separable states. J. Math. Phys. 51, 052201 (2010)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Alfsen, E., Shultz, F.: Finding decompositions of a class of separable states. Linear Alg. Appl. 437, 2613–2629 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chen, L., Djoković, D.Ž.: Qubit-qudit states with positive partial transpose. Phys. Rev. A 86, 062332 (2012)ADSCrossRefGoogle Scholar
  4. 4.
    Chen, L., Djoković, D.Ž.: Dimensions, lengths and separability in finite-dimensional quantum systems. J. Math. Phys. 54, 022201 (2013)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, L., Djoković, D.Ž.: Properties and construction of extreme bipartite states having positive partial transpose. Commun. Math. Phys. 323, 241–284 (2013)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, L., Djoković, D.Ž.: Boundary of the set of separable states. Proc. R. Soc. A 471, 20150102 (2015)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, L., Djoković, D.Ž.: Dimension formula for induced maximal faces of separable states and genuine entanglement. Quantum Inf. Process. 14, 3335–3350 (2015)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen, L., Djoković, D.Ž.: Length filtration of the separable states. Proc. R. Soc. A (2016). CrossRefGoogle Scholar
  9. 9.
    Cohen, S. M.: Sums of product operators that remain product operators, preprint. arXiv:1210.0644
  10. 10.
    DiVincenzo, D.P., Terhal, B.M., Thapliyal, A.V.: Optimal decompositions of barely separable states. J. Mod. Opt. 47, 277–385 (2000)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ha, K.-C., Han, K. H., Kye, S.-H.: Separability of multi-qubit states in terms of diagonal and anti-diagonal entries, preprint. arXiv:1803.00175
  12. 12.
    Ha, K.-C., Kye, S.-H.: Exposedness of Choi type entanglement witnesses and applications to lengths of separable states. Open Syst. Inf. Dyn. 20, 1350012 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ha, K.-C., Kye, S.-H.: Geometry for separable states and construction of entangled states with positive partial transposes. Phys. Rev. A 88, 024302 (2013)ADSCrossRefGoogle Scholar
  14. 14.
    Ha, K.-C., Kye, S.-H.: Separable states with unique decompositions. Commun. Math. Phys. 328, 131–153 (2014)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Ha, K.-C., Kye, S.-H.: Multi-partite separable states with unique decompositions and construction of three qubit entanglement with positive partial transpose. J. Phys. A Math. Theor. 48, 045303 (2015)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Han, K.H., Kye, S.-H.: Construction of multi-qubit optimal genuine entanglement witnesses. J. Phys. A Math. Theor. 49, 175303 (2016)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Han, K.H., Kye, S.-H.: The role of phases in detecting three qubit entanglement. J. Math. Phys. 58, 102201 (2017)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Horodecki, P.: Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A 232, 333–339 (1997)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Kirkpatrick, K.A.: Uniqueness of a convex sum of products of projectors. J. Math. Phys. 43, 684–686 (2002)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Kiem, Y.-H., Kye, S.-H., Na, J.: Product vectors in the ranges of multi-partite states with positive partial transposes and permanents of matrices. Commun. Math. Phys. 338, 621–639 (2015)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Kye, S.-H.: Three-qubit entanglement witnesses with the full spanning properties. J. Phys. A Math. Theor. 48, 235303 (2015)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Kye, S.-H.: Indecomposable exposed positive bi-linear maps between two by two matrices. Acta Math. Viet. 43, 619–627 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lockhart, R.: Optomal ensemble length of mixed separable states. J. Math. Phys. 41, 6766–6771 (2000)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Sanpera, A., Tarrach, R., Vidal, G.: Local description of quantum inseparability. Phys. Lett. A 58, 826–830 (1998)MathSciNetGoogle Scholar
  25. 25.
    Skowronek, L.: Dualities and positivity in the study of quantum entanglement. Int. J. Quantum Inf. 8, 721–754 (2010)CrossRefGoogle Scholar
  26. 26.
    Uhlmann, A.: Entropy and optimal decompositions of states relative to a maximal commutative subalgebra. Open Sys. Inf. Dyn. 5, 209–227 (1998)CrossRefGoogle Scholar
  27. 27.
    Walgate, J., Scott, A.J.: Generic local distinguishability and completely entangled subspaces. J. Phys. A 41, 375305 (2008)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wootters, W.K.: entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

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

Personalised recommendations