Skip to main content
SpringerLink
Log in

Entanglement criterion via general symmetric informationally complete measurements

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

We study the quantum separability problem by using general symmetric informationally complete measurements and present a separability criterion for arbitrary dimensional bipartite systems. We show by detailed examples that our criterion is more powerful than the existing ones in entanglement detection.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

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

    MATH  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  4. Gurvits, L.: In Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing. ACM Press, New York, Vol. 10 (2003)

  5. Gharibian, S.: Strong NP-hardness of the quantum separability problem. Quant. Inf. Comput. 10, 343 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Gurvits, L.: Classical complexity and quantum entanglement. J. Comput. Syst. Sci. 69, 448 (2003)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  10. Rudolph, O.: Some properties of the computable cross-norm criterion for separability. Phys. Rev. A 67, 032312 (2003)

    Article  ADS  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed quantum states: linear contractions and permutation criteria. Open Syst. Inf. Dyn. 13, 103 (2006)

    Article  MathSciNet  Google Scholar 

  13. Chen, K., Wu, L.A.: The generalized partial transposition criterion for separability of multipartite quantum states. Phys. Lett. A 306, 14 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  14. Chen, K., Wu, L.A.: Test for entanglement using physically observable witness operators and positive maps. Phys. Rev. A 69, 022312 (2004)

    Article  ADS  Google Scholar 

  15. Wocjan, P., Horodecki, M.: Characterization of combinatorially independent permutation separability criteria. Open Syst. Inf. Dyn. 12, 331 (2005)

    Article  MathSciNet  Google Scholar 

  16. Albeverio, S., Chen, K., Fei, S.M.: Generalized reduction criterion for separability of quantum states. Phys. Rev. A 68, 062313 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  17. Gühne, O., Hyllus, P., Gittsovich, O., Eisert, J.: Covariance matrices and the separability problem. Phys. Rev. Lett. 99, 130504 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  18. Vicente, J.D.: Separability criteria based on the Bloch representation of density matrices. Quant. Inf. Comput. 7, 624 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Vicente, J.D.: Further results on entanglement detection and quantification from the correlation matrix criterion. J. Phys. A Math. Theor. 41, 065309 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  20. 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 

  21. Gisin, N.: Bell’s inequality holds for all non-product states. Phys. Lett. A 154, 201 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  22. Yu, S., Pan, J.W., Chen, Z.B., Zhang, Y.D.: Comprehensive test of entanglement for two-level systems via the indeterminacy relationship. Phys. Rev. Lett. 91, 217903 (2003)

    Article  ADS  Google Scholar 

  23. Li, M., Fei, S.M.: Gisin’s theorem for arbitrary dimensional multipartite states. Phys. Rev. Lett. 104, 240502 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  24. Zhao, M.J., Ma, T., Fei, S.M., Wang, Z.X.: Inequalities detecting quantum entanglement for \(2\otimes d\) systems. Phys. Rev. A 83, 052120 (2011)

    Article  ADS  Google Scholar 

  25. Spengler, C., Huber, M., Brierley, S., Adaktylos, T., Hiesmayr, B.C.: Entanglement detection via mutually unbiased bases. Phys. Rev. A 86, 022311 (2012)

    Article  ADS  Google Scholar 

  26. Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. (N. Y. ) 191, 363 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  27. Chen, B., Ma, T., Fei, S.M.: Entanglement detection using mutually unbiased measurements. Phys. Rev. A 89, 064302 (2014)

    Article  ADS  Google Scholar 

  28. Liu, L., Gao, T., Yan, F.: Separability criteria via sets of mutually unbiased measurements. Sci. Rep. 5, 13138 (2015)

    Article  ADS  Google Scholar 

  29. Kalev, A., Gour, G.: Mutually unbiased measurements in finite dimensions. N. J. Phys. 16, 053038 (2014)

    Article  Google Scholar 

  30. Fuchs, C.A., Hoang, M.C., Stacey, B.C.: The SIC question: history and state of play. Axioms 6, 21 (2017)

    Article  Google Scholar 

  31. Appleby, D.M.: Symmetric informationally complete measurements of arbitrary rank. Opt. Spectrosc. 103, 416 (2007)

    Article  ADS  Google Scholar 

  32. Gour, G., Kalev, A.: Construction of all general symmetric informationally complete measurements. J. Phys. A Math. Theor. 47, 335302 (2014)

    Article  MathSciNet  Google Scholar 

  33. Chen, B., Li, T., Fei, S.M.: General SIC measurement-based entanglement detection. Quant. Inf. Process. 14, 2281 (2015)

    Article  ADS  Google Scholar 

  34. Xi, Y., Zheng, Z.J., Zhu, C.J.: Entanglement detection via general SIC-POVMs. Quant. Inf. Process. 15, 5119 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  35. Bae, J., Hiesmayr, B. C., McNulty, D.: Linking entanglement detection and state tomography via quantum 2-designs. arXiv:1803.02708

  36. Czartowski, J., Goyeneche, D., Życzkowski, K.: Entanglement properties of multipartite informationally complete quantum measurements. J. Phys. A Math. Theor 51, 305302 (2018)

    Article  MathSciNet  Google Scholar 

  37. Shen, S.Q., Li, M., Li-Jost, X., Fei, S.M.: Improved separability criteria via some classes of measurements. Quant. Inf. Process. 17, 111 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  38. Shang, J.W., Asadian, A, Zhu, H. J.: Enhanced entanglement criterion via symmetric informationally complete measurements. Phys. Rev. A 98, 022309 (2018)

  39. 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

This work is supported by the NSF of China under Grant No.11675113, the Research Foundation for Youth Scholars of BTBU QNJJ2017-03, Beijing Municipal Commission of Education under Grant Nos.KM201810011009 and KZ201810028042.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    Le-Min Lai, Shao-Ming Fei & Zhi-Xi Wang

  2. School of Science, Beijing Technology and Business University, Beijing, 100048, China

    Tao Li

  3. Max Planck Institute for Mathematics in the Sciences, 04103, Leipzig, Germany

    Shao-Ming Fei

Authors
  1. Le-Min Lai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shao-Ming Fei
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhi-Xi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhi-Xi Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lai, LM., Li, T., Fei, SM. et al. Entanglement criterion via general symmetric informationally complete measurements. Quantum Inf Process 17, 314 (2018). https://doi.org/10.1007/s11128-018-2084-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-018-2084-6

Keywords

Search

Navigation