Skip to main content
SpringerLink
Log in

Constructing Entanglement Witnesses from Two Mutually Unbiased Bases

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

We study entanglement witnesses (EWs) and construct a linear map by taking two groups of mutually unbiased bases (MUBs) of two Hilbert spaces. We provide two new relations about matrices (may not be square matrices) with properties similar to unitary matrix. Using these two new relations, we obtain a positive map between two different linear spaces of complex matrices. Finally, we calculate a special case by applying our map to get an EW on \(\mathbb {C}^{3}\otimes \mathbb {C}^{4}\). We also find three entangled states detected by this EW.

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

Similar content being viewed by others

References

  1. Ekert, A. K.: Rev, Phys. Lett. 67, 661 (1991)

    Article  Google Scholar 

  2. Fuchs, C. A., Gisin, N., Griffiths, R. B., Niu, C.-S., Peres, A.: Phys. Rev. A 56, 1163 (1997)

    Article  ADS  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  5. Gurvis, L.: J. Comput. Syst. Sci. 69, 448 (2003)

    Article  Google Scholar 

  6. Peres, A.: Phys. Rev. Lett. 77, 1413 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  7. Horodecki, M., Horodecki, P., Horodecki, R.: Phys. Lett. A 223, 1 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  8. Rudolph, O.: Phys. Rev. A 67, 032312 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  9. Horodecki, M., Horodecki, P., Horodecki, R.: Open Syst. Inf. Dyn. 13, 103 (2006)

    Article  MathSciNet  Google Scholar 

  10. Chen, K., Wu, L.A.: Phys. Lett A 306, 14 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  11. Chen, K., Wu, L. A.: Phys. Rev. A 69, 022312 (2004)

    Article  ADS  Google Scholar 

  12. Albeverio, S., Chen, K., Fei, S.-M.: Phys. Rev. A 68, 062313 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  13. de Vicente, J.: Quantum Inf. Comput. 7, 624 (2007)

    MathSciNet  Google Scholar 

  14. de Vicente, J.: J. Phys. A Math. Theor. 41, 065309 (2008)

    Article  ADS  Google Scholar 

  15. Hofmann, H. F., Takeuchi, S.: Phys. Rev. A 68, 032103 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  16. Cerf, N. J., Adami, C., Gingrich, R. M.: Phys. Rev. A 60, 898 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  17. Terhal, B. M.: Phys. Lett. A 271, 319 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  18. Wootters, W. K., Fields, B. D.: Ann. Phys. (NY) 191, 363 (1989)

    Article  ADS  Google Scholar 

  19. Chruściński, D., Sarbicki, G., Wudarski, F.: Phys. Rev. A 97, 032318 (2018)

    Article  ADS  Google Scholar 

  20. Brierley, S., Weigert, S.: Phys. Rev. A 79, 052316 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  21. Raynal, P., Lü, X., Englert, B.-G.: Phys. Rev. A 83, 062303 (2011)

    Article  ADS  Google Scholar 

  22. McNulty, D., Weigert, S.: J. Phys A: Math. Theor. 45, 102001 (2012)

    Article  ADS  Google Scholar 

  23. Spengler, C., Huber, M., Brierley, S., Adaktylos, T., Hiesmayr, B. C.: Phys. Rev. A 86, 022311 (2012)

    Article  ADS  Google Scholar 

  24. Wootters, W. K., Fields, B. D.: Ann. Phys. 191, 363 (1988)

    Article  ADS  Google Scholar 

  25. Bandyopadhyay, S., Boykin, P., Roychowdhury, V., Vatan, .: Algoritheoremica 34, 512 (2002)

    Article  Google Scholar 

  26. Grassl, M.: On SIC-POVMs and MUBs in dimension 6 arxiv: quant-ph/0406175 (2004)

  27. Kalev, A., Gour, G.: New J. Phys. 16, 053038 (2014)

    Article  ADS  Google Scholar 

  28. Li, T., Lai, L.-M., Fei, S.-M., Wang, Z.-X.: Int. J. Theor. Phys. 58, 3973 (2019)

    Article  Google Scholar 

  29. Durt, T., Englert, B.-G., Bengtsson, I., Życzkowski, K.: Int. J. Quantum Inf. 08, 535 (2010)

    Article  Google Scholar 

  30. Terhal, B. M.: Linear Algebr. Appl. 323, 61 (1998)

    Article  MathSciNet  Google Scholar 

  31. Choi, M.-D.: Linear Algebr. Appl. 10, 285 (1975)

    Article  Google Scholar 

  32. Jamiołkowski, A.: Rep. Math. Phys. 3, 275 (1972)

    Article  ADS  Google Scholar 

  33. Brierley, S., Weigert, S.: Phys. Rev. A 78, 042312 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  34. Chruściński, D., Sarbicki, G.: J. Phys. A: Math. Theor. 47, 483001 (2014)

    Article  Google Scholar 

  35. Bengtsson, I., Życzkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

Download references

Acknowledgements

This work is supported by Key Research and Development Project of Guangdong Province under Grant No.2020B0303300001 and the Guangdong Basic and Applied Basic Research Foundation under Grant No.2020B1515310016.

Author information

Authors and Affiliations

  1. School of Mathematics, South China University of Technology, Guangzhou, 510641, China

    Kun Wang & Zhu-Jun Zheng

  2. Laboratory of Quantum Science and Engineering, South China University of Technology, Guangzhou, 510641, China

    Kun Wang & Zhu-Jun Zheng

Authors
  1. Kun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhu-Jun Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhu-Jun Zheng.

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

Wang, K., Zheng, ZJ. Constructing Entanglement Witnesses from Two Mutually Unbiased Bases. Int J Theor Phys 60, 274–283 (2021). https://doi.org/10.1007/s10773-020-04689-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-020-04689-6

Keywords

Search

Navigation