Skip to main content

Partial separability/entanglement violates distributive rules

Abstract

We found three qubit Greenberger–Horne–Zeilinger diagonal states which tell us that the partial separability of three qubit states violates the distributive rules with respect to the two operations of convex sum and intersection. The gaps between the convex sets involving the distributive rules are of nonzero volume.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

References

  1. 1.

    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    Schrödinger, E.: Die gegenwärtige situation in der quantenmechanik. Naturwissenschaften 23, 823–828 (1935)

    ADS  Article  Google Scholar 

  3. 3.

    Schrödinger, E.: Discussion of probability relations between separated systems. Math. Proc. Camb. Philos. Soc. 31, 555–563 (1935)

    ADS  Article  Google Scholar 

  4. 4.

    Amico, L., Fazio, R., Osterloh, A., Vedral, Vlatko: Entanglement in many-body systems. Rev. Mod. Phys. 80, 517–576 (2008)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5.

    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  6. 6.

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

    ADS  Article  Google Scholar 

  7. 7.

    Bennett, C.H., DiVincenzo, D.P., Mor, T., Shor, P.W., Smolin, J.A., Terhal, B.M.: Unextendible product bases and bound entanglement. Phys. Rev. Lett. 82, 5385–5388 (1999)

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Dür, W., Cirac, J.I., Tarrach, R.: Separability and distillability of multiparticle quantum systems. Phys. Rev. Lett. 83, 3562–3565 (1999)

    ADS  Article  Google Scholar 

  9. 9.

    Dür, W., Cirac, J.I.: Classification of multi-qubit mixed states: separability and distillability properties. Phys. Rev. A 61, 042314 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  10. 10.

    Seevinck, M., Uffink, J.: Partial separability and etanglement criteria for multiqubit quantum states. Phys. Rev. A 78, 032101 (2008)

    ADS  MathSciNet  Article  Google Scholar 

  11. 11.

    Acin, A., Bruß, D., Lewenstein, M., Sanpera, A.: Classification of mixed three-qubit states. Phys. Rev. Lett. 87, 040401 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    Szalay, Sz, Kökényesi, Z.: Partial separability revisited: necessary and sufficient criteria. Phys. Rev. A 86, 032341 (2012)

    ADS  Article  Google Scholar 

  13. 13.

    Han, K.H., Kye, S.-H.: Construction of three-qubit biseparable states distinguishing kinds of entanglement in a partial separability classification. Phys. Rev. A 99, 032304 (2019)

    ADS  Article  Google Scholar 

  14. 14.

    Han, K.H., Kye, S.-H.: On the convex cones arising from classifications of partial entanglement in the three qubit system. J. Phys. A Math. Theor. 53, 015301 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  15. 15.

    Szalay, Sz: Multipartite entanglement measures. Phys. Rev. A 92, 042329 (2015)

    ADS  Article  Google Scholar 

  16. 16.

    Szalay, Sz: Separability criteria for mixed three-qubit states. Phys. Rev. A 83, 062337 (2011)

    ADS  Article  Google Scholar 

  17. 17.

    Szalay, Sz: The classification of multipartite quantum correlation. J. Phys. A Math. Theor. 51, 485302 (2018)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Szalay, Sz: \(k\)-Stretchability of entanglement, and the duality of \(k\)-separability and \(k\)-producibility. Quantum 3, 204 (2019)

    Article  Google Scholar 

  19. 19.

    Greenberger, D.M., Horne, M.A., Zeilinger, A.: Going beyond Bell’s theorem. In: Kafatos, M. (ed.) Bell’s Theorem, Quantum Theory and Conceptions of the Universe. Fundamental Theories of Physics, vol. 37. Springer, Dordrecht (1989)

    Google Scholar 

  20. 20.

    Gao, T., Hong, Y.: Separability criteria for several classes of \(n\)-partite quantum states. Eur. Phys. J. D 61, 765–771 (2011)

    ADS  Article  Google Scholar 

  21. 21.

    Rafsanjani, S.M.H., Huber, M., Broadbent, C.J., Eberly, J.H.: Genuinely multipartite concurrence of N-qubit X matrices. Phys. Rev. A 86, 062303 (2012)

    ADS  Article  Google Scholar 

  22. 22.

    Han, K.H., Kye, S.-H.: Construction of multi-qubit optimal genuine entanglement witnesses. J. Phys. A Math. Theor. 49, 175303 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  23. 23.

    Ha, K.-C., Han, K.H., Kye, S.-H.: Separability of multi-qubit states in terms of diagonal and anti-diagonal entries. Quantum Inf. Process. 18, 34 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  24. 24.

    Kye, S.-H.: Facial structures for the positive linear maps between matrix algebras. Can. Math. Bull. 39, 74–82 (1996)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Birkhoff, G.: Lattice Theory, vol. 25. Am. Math. Soc., Providence, RI (1940)

    MATH  Google Scholar 

  26. 26.

    Salii, V.N.: Lattices with Unique Complements, Trans. Math. Monog., vol. 69. Am. Math. Soc., Providence, RI (1988)

    Book  Google Scholar 

  27. 27.

    Kay, A.: Optimal detection of entanglement in Greenberger–Horne–Zeilinger states. Phys. Rev. A 83, 020303(R) (2011)

    ADS  Article  Google Scholar 

  28. 28.

    Gühne, O.: Entanglement criteria and full separability of multi-qubit quantum states. Phys. Lett. A 375, 406–410 (2011)

    ADS  MathSciNet  Article  Google Scholar 

  29. 29.

    Han, K.H., Kye, S.-H.: Separability of three qubit Greenberger–Horne–Zeilinger diagonal states. J. Phys. A Math. Theor. 50, 145303 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  30. 30.

    Chen, L., Han, K.H., Kye, S.-H.: Separability criterion for three-qubit states with a four dimensional norm. J. Phys. A Math. Theor. 50, 345303 (2017)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Freese, R., Jezek, J., Nation, J.B.: Free Lattice. Math. Surv Monog., vol. 42. Am. Math. Soc, Providence, RI (1995)

    Book  Google Scholar 

Download references

Acknowledgements

Both KHH and SHK were partially supported by the grant NRF-2017R1A2B4006655, Korea. SzSz was supported by the National Research, Development and Innovation Fund of Hungary within the Researcher-initiated Research Program (Project Nr: NKFIH-K120569) and within the Quantum Technology National Excellence Program (Project Nr: 2017-1.2.1-NKP-2017-00001), by the Ministry for Innovation and Technology within the ÚNKP-19-4 New National Excellence Program, and by the Hungarian Academy of Sciences within the János Bolyai Research Scholarship and the “Lendület” Program.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Kyung Hoon Han.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Han, K.H., Kye, SH. & Szalay, S. Partial separability/entanglement violates distributive rules. Quantum Inf Process 19, 202 (2020). https://doi.org/10.1007/s11128-020-02710-9

Download citation

Keywords

  • Partially separable
  • Partially entangled
  • Lattice
  • Distributive rule

Mathematics Subject Classification

  • 81P15
  • 15A30
  • 46L05
  • 46L07