Skip to main content
SpringerLink
Log in

Separable States with Unique Decompositions

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We search for faces of the convex set consisting of all separable states, which are affinely isomorphic to simplices, to get separable states with unique decompositions. In the two-qutrit case, we found that six product vectors spanning a five dimensional space give rise to a face isomorphic to the 5-dimensional simplex with six vertices, under a suitable linear independence assumption. If the partial conjugates of six product vectors also span a 5-dimensional space, then this face is inscribed in the face for PPT states whose boundary shares the fifteen 3-simplices on the boundary of the 5-simplex. The remaining boundary points consist of PPT entangled edge states of rank four. We also show that every edge state of rank four arises in this way. If the partial conjugates of the above six product vectors span a 6-dimensional space then we have a face isomorphic to 5-simplex, whose interior consists of separable states with unique decompositions, but with non-symmetric ranks. We also construct a face isomorphic to the 9-simplex. As applications, we give answers to questions in the literature Chen and Djoković (J Math Phys 54:022201, 2013) and Chen and Djoković (Commun Math Phys 323:241–284, 2013), and construct 3 ⊗ 3PPT states of type (9,5). For the qubit-qudit cases with d ≥ 3, we also show that (d + 1)-dimensional subspaces give rise to faces isomorphic to the d-simplices, in most cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Alfsen E., Shultz F.: Unique decompositions, faces, and automorphisms of separable states. J. Math. Phys. 51, 052201 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  2. Alfsen E., Shultz F.: Finding decompositions of a class of separable states. Linear Algebra Appl. 437, 2613–2629 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Augusiak, R., Bae, J., Tura, J., Lewenstein, M.: Checking the optimality of entanglement witnesses: an application to structural physical approximations. J. Phys. A. Math. Theor. 47, 065301 (2014)

    Google Scholar 

  4. Augusiak R., Tura J., Lewenstein M.: A note on the optimality of decomposable entanglement witnesses and completely entangled subspaces. J. Phys. A 44, 212001 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  5. Bhat B.V.R.: A completely entangled subspace of maximal dimension. Int. J. Quant. Inf. 4, 325–330 (2006)

    Article  MATH  Google Scholar 

  6. Chen L., Djoković D.Ž.: Description of rank four entangled states of two qutrits having positive partial transpose. J. Math. Phys. 52, 122203 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  7. Chen L., Djoković D.Ž.: Qubit-qudit states with positive partial transpose. Phys. Rev. A 86, 062332 (2012)

    Article  ADS  Google Scholar 

  8. Chen L., Djoković D.Ž.: Dimensions, lengths and separability in finite-dimensional quantum systems. J. Math. Phys. 54, 022201 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  9. Chen L., Djoković D.Ž.: Properties and construction of extreme bipartite states having positive partial transpose. Commun. Math. Phys. 323, 241–284 (2013)

    Article  ADS  MATH  Google Scholar 

  10. Cho S.-J., Kye S.-H., Lee S.G.: Generalized Choi maps in 3-dimensional matrix algebras. Linear Algebra Appl. 171, 213–224 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  11. Choi H.-S., Kye S.-H.: Facial structures for separable states. J. Korean Math. Soc. 49, 623–639 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Choi, M.-D.: Positive linear maps, Operator Algebras and Applications (Kingston, 1980), pp. 583–590. In: Proceedings of Symposia in Pure Mathematics, vol. 38. Part 2, American Mathematical Society (1982)

  13. Cohen, S.M.: Sums of product operators that remain product operators. preprint. arXiv:1210.0644

  14. DiVincenzo D.P., Terhal B.M., Thapliyal A.V.: Optimal decompositions of barely separable states. J. Mod. Opt. 47, 277–385 (2000)

    Article  MathSciNet  Google Scholar 

  15. Eom M.-H., Kye S.-H.: Duality for positive linear maps in matrix algebras. Math. Scand. 86, 130–142 (2000)

    MATH  MathSciNet  Google Scholar 

  16. Ha K.-C., Kye S.-H.: Construction of entangled states with positive partial transposes based on indecomposable positive linear maps. Phys. Lett. A 325, 315–323 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Ha K.-C., Kye S.-H.: Entanglement witnesses arising from exposed positive linear maps. Open Syst. Inf. Dyn. 18, 323–337 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ha K.-C., Kye S.-H.: Geometry of the faces for separble states arising from generalized Choi maps. Open Syst. Inf. Dyn. 19, 1250009 (2012)

    Article  MathSciNet  Google Scholar 

  19. Ha K.-C., Kye S.-H., Park Y.S.: Entangled states with positive partial transposes arising from indecomposable positive linear maps. Phys. Lett. A 313, 163–174 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Hansen L.O., Hauge A., Myrheim J., Sollid P.Ø.: Low-rank positive-partial-transpose states and their relation to product vectors. Phys. Rev. A 85, 022309 (2012)

    Article  ADS  Google Scholar 

  21. Horodecki M., Horodecki P., Horodecki R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 1–8 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. Horodecki P.: Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A 232, 333–339 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. Horodecki P., Lewenstein M., Vidal G., Cirac I.: Operational criterion and constructive checks for the separability of low-rank density matrices. Phys. Rev. A 62(3), 032310 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  24. Kiem, Y.-H.: Algebraic geometry and matrix theory, private communication (2003)

  25. Kiem Y.-H., Kye S.-H., Lee J.: Existence of product vectors and their partial conjugates in a pair of spaces. J. Math. Phys. 52, 122201 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  26. Kirkpatrick K.A.: Uniqueness of a convex sum of products of projectors. J. Math. Phys. 43, 684–686 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Kraus B., Cirac J.I., Karnas S., Lewenstein M.: Separability in 2xN composite quantum systems. Phys. Rev. A 61, 062302 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  28. Kye S.-H.: Necessary conditions for optimality of decomposable entanglement witness. Rep. Math. Phys 69, 419–426 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  29. Kye S.-H., Osaka H.: Classification of bi-qutrit positive partial transpose entangled edge states by their ranks. J. Math. Phys. 53, 052201 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  30. Leinaas J.M., Myrheim J., Sollid P.Ø.: Numerical studies of entangled PPT states in composite quantum systems. Phys. Rev. A 81, 062329 (2010)

    Article  ADS  Google Scholar 

  31. Lewenstein M., Kraus B., Cirac J., Horodecki P.: Optimization of entanglement witness. Phys. Rev. A 62, 052310 (2000)

    Article  ADS  Google Scholar 

  32. Parthasarathy K.R.: On the maximal dimension of a completely entangled subspace for finite level quantum systems. Proc. Indian Acad. Sci. Math. Sci. 114, 365–374 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  33. Peres A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413–1415 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  34. Pittenger A.O., Rubin M.H.: Convexity and the separability problem of quantum mechanical density matrices. Linear Algebra Appl. 346, 47–71 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  35. Skowronek L.: Three-by-three bound entanglement with general unextendible product bases. J. Math. Phys. 52, 122202 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  36. Uhlmann A.: Entropy and optimal decompositions of states relative to a maximal commutative subalgebra. Open Syst. Inf. Dyn. 5, 209–227 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  37. Walgate J., Scott A.J.: Generic local distinguishability and completely entangled subspaces. J. Phys. A 41, 375305 (2008)

    Article  MathSciNet  Google Scholar 

  38. Wallach N.R.: An unentangled Gleason’s theorem. Contemp. Math. 305, 291–298 (2002)

    Article  MathSciNet  Google Scholar 

  39. Werner R.F.: Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden variable model. Phys. Rev. A 40, 4277–4281 (1989)

    Article  ADS  Google Scholar 

  40. Wootters W.K.: entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998)

    Article  ADS  Google Scholar 

  41. Woronowicz S.L.: Positive maps of low dimensional matrix algebras. Rep. Math. Phys. 10, 165–183 (1976)

    Article  ADS  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Applied Statistics, Sejong University, Seoul, 143-747, Korea

    Kil-Chan Ha

  2. Department of Mathematics and Institute of Mathematics, Seoul National University, Seoul, 151-742, Korea

    Seung-Hyeok Kye

Authors
  1. Kil-Chan Ha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Seung-Hyeok Kye
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Seung-Hyeok Kye.

Additional information

Communicated by M. B. Ruskai

Partially supported by NRFK 2012-0002600 and NRFK 2012-0000939.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ha, KC., Kye, SH. Separable States with Unique Decompositions. Commun. Math. Phys. 328, 131–153 (2014). https://doi.org/10.1007/s00220-014-1980-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-1980-6

Keywords

Search

Navigation