Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
SpringerLink
Log in

A hierarchy of entanglement criteria for four-qubit symmetric Greenberger–Horne–Zeilinger diagonal states

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

With a two-step method of optimizing entanglement witness, we propose a set of necessary and sufficient entanglement criteria for four-qubit symmetric Greenberger–Horne–Zeilinger (GHZ) diagonal states. The criterion set contains four criteria. Two of them are linear with density matrix elements. The other two criteria are nonlinear with density matrix elements. The criterion set has a nest structure. A proper subset of the criteria is necessary and sufficient for entanglement of the corresponding subset of states. We illustrate the nest structure of criterion set with highly symmetric GHZ diagonal state set, its subset the general Werner state set and its superset the symmetric GHZ diagonal state set.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

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

    Article  ADS  MathSciNet  Google Scholar 

  2. Horodecki, M., Horodecki, P., Horodecki, R.: Separablity of mixed stated states: necessary and sufficient conditions. Phys. Lett. A 223, 1 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  3. Rudolph, O.: Computable cross-norm criterion for separability. Lett. Math. Phys. 70, 57 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  4. Chen, K., Wu, L.A.: A matrix realignment method for recognizing entanglement. Quant. Inf. Comput. 3, 193 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Gühne, O.: Characterizing entanglement via uncertainty relations. Phys. Rev. Lett. 92, 117903 (2004)

    Article  ADS  Google Scholar 

  6. Doherty, A.C., Parrilo, P.A., Spedalieri, F.M.: Distinguishing separable and entangled states. Phys. Rev. Lett. 88, 187904 (2002)

    Article  ADS  Google Scholar 

  7. Li, M., Wang, J., Fei, S.-M., Li-Jost, X.: Quantum separability criteria for arbitrary-dimensional multipartite states. Phys. Rev. A 89, 022325 (2014)

    Article  ADS  Google Scholar 

  8. Chen, X.Y., Jiang, L.Z.: Quantum separable criterion based on characteristic function. Sci. Sin-Phys. Mech. Astron. 48, 020302 (2018). (in Chinese)

    Article  Google Scholar 

  9. Gühne, O., Tóth, G.: Entanglement detection. Phys. Rep. 474, 1 (2009)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  11. Horodecki, M., Horodecki, P., Horodecki, R.: Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps. Phys. Lett. A 283, 1 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  12. Gerke, S., Sperling, J., Vogel, W., Cai, Y., Roslund, J., Treps, N., Fabre, C.: Is genuine multipartite entanglement really genuine? Phys. Rev. Lett. 117, 110502 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  13. Shahandeh, F., Ringbauer, M., Loredo, J.C., Ralph, T.C.: Ultrafine entanglement witnessing. Phys. Rev. Lett. 118, 110502 (2017)

    Article  ADS  Google Scholar 

  14. Chen, X., Hu, X.Y., Zhou, D.L.: Entanglement witness game. Phys. Rev. A 95, 052326 (2017)

    Article  ADS  Google Scholar 

  15. Sperling, J., Vogel, W.: Multipartite entanglement witnesses. Phys. Rev. Lett. 111, 110503 (2013)

    Article  ADS  Google Scholar 

  16. Kay, A.: Optimal detection of entanglement in Greenberger–Horne–Zeilinger states. Phys. Rev. A 83, 020203(R) (2011)

    Article  Google Scholar 

  17. Chen, X.Y.: Graph-state basis for Pauli channels. Phys. Rev. A 83, 052316 (2011)

    Article  ADS  Google Scholar 

  18. Kaufmann, H., Ruster, T., Schmiegelow, C.T., Luda, M.A., Kaushal, V., Schulz, J., von Lindenfels, D., Schmidt-Kaler, F., Poschinger, U.G.: Scalable creation of long-lived multipartite entanglement. Phys. Rev. Lett. 119, 150503 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  19. Su, Z.-E., Tang, W.-D., Wu, D., Cai, X.-D., Yang, T., Li, L., Liu, N.-L., Lu, C.-Y., Zukowski, M., Pan, J.-W.: Experimental test of irreducible four-qubit Greenberger–Horne–Zeilinger paradox. Phys. Rev. A 95, 030103(R) (2017)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  21. Chen, X.Y., Jiang, L.Z., Yu, P., Tian, M.: Necessary and sufficient fully separable criterion and entanglement for three-qubit Greenberger–Horne–Zeilinger diagonal states. Quant. Inf. Process. 14, 2463–2476 (2015)

    Article  ADS  Google Scholar 

  22. Chen, X.Y., Jiang, L.Z., Xu, Z.A.: Matched witness for multipartite entanglement. Quant. Inf. Process. 16, 95 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  23. Chen, X.Y., Jiang, L.Z., Xu, Z.A.: Precise detection of multipartite entanglement in four qubit GreenbergerCHorneCZeilinger diagonal states. Front. Phys. 13, 130317 (2018)

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  25. Gühne, O., Seevinck, M.: Separability criteria for genuine multipartite entanglement. New J. Phys. 12, 053002 (2010)

    Article  ADS  Google Scholar 

  26. Pittenger, A.O., Rubin, M.H.: Note on separability of the werner states in arbitrary dimensions. Op. Commun. 179, 447 (2000)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

Download references

Acknowledgements

Support from the National Natural Science Foundation of China (Grant Nos: 61871347 and 11375152) is gratefully acknowledged.

Author information

Authors and Affiliations

  1. College of Information and Electronic Engineering, Zhejiang Gongshang University, Hangzhou, 310018, China

    Xiao-yu Chen & Li-zhen Jiang

Authors
  1. Xiao-yu Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Li-zhen Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiao-yu Chen.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Xy., Jiang, Lz. A hierarchy of entanglement criteria for four-qubit symmetric Greenberger–Horne–Zeilinger diagonal states. Quantum Inf Process 18, 262 (2019). https://doi.org/10.1007/s11128-019-2373-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-019-2373-8

Keywords

Search

Navigation