Skip to main content
SpringerLink
Log in

Mutually Unbiased Measurement Based Entanglement Witnesses

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

Abstract

We study entanglement witness and present a construction of entanglement witnesses in terms of the mutually unbiased measurements (MUMs). These witnesses include the entanglement witnesses constructed from mutually unbiased bases (MUBs) as a special case. Comparing with the dimension dependence of MUBs, the witnesses can be always constructed from a complete set of d + 1 MUMs for any dimension d. We show that our witness can detect entanglement better than previous separability criterion given also by MUMs. And the approach can be experimentally implemented.

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.: . Phys. Rev. Lett. 67, 661 (1991)

    Article  ADS  MathSciNet  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. Peres, A.: . Phys. Rev. Lett. 77, 1413 (1996)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  7. Horodecki, P.: . Phys. Lett. A 232, 333 (1997)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  9. Chen, K., Wu, L.-A.: . Quantum Inf. Comput. 3, 193 (2003)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  13. Wocjan, P., Horodecki, M.: . Open Syst. Inf. Dyn. 12, 331 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  15. Guhne, O., Hyllus, P., Gittsovich, O., Eisert, J.: . Phys. Rev. Lett. 99, 130504 (2007)

    Article  ADS  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  22. Bengtsson, I., Żczkowski: Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  23. Bertlmann, R.A., Durstberger, K., Hiesmayr, B.C., Krammer, P.: . Phys. Rev. A 72, 052331 (2005)

    Article  ADS  Google Scholar 

  24. Chen, B., Ma, T., Fei, S.M.: . Phys. Rev. A 89, 064302 (2014)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

Download references

Acknowledgments

This work is supported by the NSF of China under Grant No. 11675113, the Research Foundation for Youth Scholars of Beijing Technology and Business University QNJJ2017-03, the Research Foundation for Educational Reform of Beijing Technology and Business University (jg185101), Scientific Research General Program of Beijing Municipal Commission of Education (Grant No.KM201810011009), NSF of Beijing under No. KZ201810028042 and Beijing Natural Science Foundation (Grant No. Z190005).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Technology and Business University, Beijing, 100048, China

    Tao Li

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

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

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

    Shao-Ming Fei

Authors
  1. Tao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Le-Min Lai
    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 Tao Li.

Additional information

Publisher’s Note

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

Appendix:

Appendix:

In order to show the advantage of the MUM based criterion (9), we consider the following 6 ⊗ 6 bipartite state,

$$ \begin{array}{@{}rcl@{}} \rho=\left( \begin{array}{cccccc} A & B & B & B & B & \alpha \\ B & A & B & B & B & \alpha \\ B & B & A & B & B & \alpha \\ B & B & B & A & B & \alpha \\ B & B & B & B & A & \alpha \\ \alpha^t & \alpha^t & \alpha^t & \alpha^t & \alpha^t & a \end{array} \right), \end{array} $$
(16)

where

$$ \begin{array}{@{}rcl@{}} A&=&\left( \begin{array}{ccccccc} 0.058 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0.022 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0.022 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.022 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.022 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0.022 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0.022 \end{array} \right),\ \\B&=&\left( \begin{array}{ccccccc} 0.036 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right), \ \alpha=\left( \begin{array}{c} 0.036\\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array} \right),\ \end{array} $$
(17)

a = 0.058 and αt is the transpose of α.

For d = 6, up to now one has only three MUBs [25,26,27,28]:

$$ \begin{array}{@{}rcl@{}} \left( \begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right), \end{array} $$
$$ \begin{array}{@{}rcl@{}} \left( \begin{array}{cccccc} 0.408 & 0.408 & 0.408 & 0.408 & 0.408 & 0.408 \\ 0.408 & 0.408 w & 0.408 w^{2} & 0.408 & 0.408 w & 0.408 w^{2} \\ 0.408 & 0.408 w^{2} & 0.408 w & 0.408 & 0.408 w^{2} & 0.408 w \\ 0.408 & 0.408 & 0.408 & -0.408 & -0.408 & -0.408 \\ 0.408 & 0.408 w & 0.408 w^{2} & -0.408 & -0.408 w & -0.408 w^{2} \\ 0.408 & 0.408 w^{2} & 0.408 w & -0.408 & -0.408 w^{2} & -0.408 w \end{array} \right), \end{array} $$
$$ \begin{array}{@{}rcl@{}} \left( \begin{array}{cccccc} 0.408 & 0.408 & 0.408 & -0.408 i & -0.408 i & -0.408 i \\ 0.408 w & 0.408 w^{2} & 0.408 & -0.408 i w & -0.408 i w^{2} & -0.408 i \\ 0.408 w & 0.408 & 0.408 w^{2} & -0.408 i w & -0.408 i & -0.408 i w^{2} \\ 0.408 i & 0.408 i & 0.408 i & -0.408 & 0.408 & 0.408 \\ 0.408 i w & 0.408 i w^{2} & 0.408 i & 0.408 w & 0.408 w^{2} & 0.408 \\ 0.408 i w & 0.408 i & 0.408 i w^{2} & 0.408 w & 0.408 & 0.408 w^{2} \end{array} \right), \end{array} $$

where \(w=e^{\frac {2 \pi i}{3}}\). Taking \(\mathcal {O}^{(\alpha )}=\mathbb {I}\), we obtain Tr(Wρ) = 0.68 > 0 by numerical calculation. Therefore, the entanglement of the state ρ is not detected.

Now we use our MUM based criterion in terms of the 7 MUMs constructed as follows:

$$ \begin{array}{@{}rcl@{}} P^{(1)}_{1}=\left( \begin{array}{cccccc} 0.167 & 0.102 i & -0.014 i & -0.014 i & -0.014 i & -0.014 i \\ 0.102 i & 0.167 & 0 & 0 & 0 & 0 \\ 0.014 i & 0 & 0.167 & 0 & 0 & 0 \\ 0.014 i & 0 & 0 & 0.167 & 0 & 0 \\ 0.014 i & 0 & 0 & 0 & 0.167 & 0 \\ 0.014 i & 0 & 0 & 0 & 0 & 0.167 \end{array} \right); P^{(1)}_{2}=\left( \begin{array}{cccccc} 0.167 & -0.014 i & 0.102 i & -0.014 i & -0.014 i & -0.014 i \\ 0.014 i & 0.167 & 0 & 0 & 0 & 0 \\ 0.102 i & 0 & 0.167 & 0 & 0 & 0 \\ 0.014 i & 0 & 0 & 0.167 & 0 & 0 \\ 0.014 i & 0 & 0 & 0 & 0.167 & 0 \\ 0.014 i & 0 & 0 & 0 & 0 & 0.167 \end{array} \right); \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(1)}_{3}=\left( \begin{array}{cccccc} 0.167 & -0.014 i & -0.014 i & 0.102 i & -0.014 i & -0.014 i \\ 0.014 i & 0.167 & 0 & 0 & 0 & 0 \\ 0.014 i & 0 & 0.167 & 0 & 0 & 0 \\ 0.102 i & 0 & 0 & 0.167 & 0 & 0 \\ 0.014 i & 0 & 0 & 0 & 0.167 & 0 \\ 0.014 i & 0 & 0 & 0 & 0 & 0.167 \end{array} \right); P^{(1)}_{4}=\left( \begin{array}{cccccc} 0.167 & -0.014 i & -0.014 i & -0.014i &0.102 i & -0.014 i \\ 0.014 i & 0.167 & 0 & 0 & 0 & 0 \\ 0.014 i & 0 & 0.167 & 0 & 0 & 0 \\ 0.014 i & 0 & 0 & 0.167 & 0 & 0 \\ 0.102 i & 0 & 0 & 0 & 0.167 & 0 \\ 0.014 i & 0 & 0 & 0 & 0 & 0.167 \end{array} \right); \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(1)}_{5}=\left( \begin{array}{cccccc} 0.167 & -0.014 i & -0.014 i & -0.014i & -0.014 i & 0.102 i \\ 0.014 i & 0.167 & 0 & 0 & 0 & 0 \\ 0.014 i & 0 & 0.167 & 0 & 0 & 0 \\ 0.014 i & 0 & 0 & 0.167 & 0 & 0 \\ 0.014 i & 0 & 0 & 0 & 0.167 & 0 \\ 0.102 i & 0 & 0 & 0 & 0 & 0.167 \end{array} \right); P^{(1)}_{6} = \left( \begin{array}{cccccc} 0.167 & -0.047 i & -0.047 i & -0.047i & -0.047 i & -0.047 i \\ 0.047 i & 0.167 & 0 & 0 & 0 & 0 \\ 0.047 i & 0 & 0.167 & 0 & 0 & 0 \\ 0.047 i & 0 & 0 & 0.167 & 0 & 0 \\ 0.047 i & 0 & 0 & 0 & 0.167 & 0 \\ 0.047 i & 0 & 0 & 0 & 0 & 0.167 \end{array} \right). \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(2)}_{1}=\left( \begin{array}{cccccc} 0.167 & -0.102 & 0 & 0 & 0 & 0 \\ -0.102 & 0.167 & -0.014 i & -0.014 i & -0.014 i & -0.014 i \\ 0 & 0.014 i & 0.167 & 0 & 0 & 0 \\ 0 & 0.014 i & 0 & 0.167 & 0 & 0 \\ 0 & 0.014 i & 0 & 0 & 0.167 & 0 \\ 0 & 0.014 i & 0 & 0 & 0 & 0.167 \end{array} \right); P^{(2)}_{2}=\left( \begin{array}{cccccc} 0.167 & 0.014 & 0 & 0 & 0 & 0 \\ 0.014 & 0.167 & 0.102 i & -0.014 i & -0.014 i & -0.014 i \\ 0 & -0.102 i & 0.167 & 0 & 0 & 0 \\ 0 & 0.014 i & 0 & 0.167 & 0 & 0 \\ 0 & 0.014 i & 0 & 0 & 0.167 & 0 \\ 0 & 0.014 i & 0 & 0 & 0 & 0.167 \end{array} \right); \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(2)}_{3}=\left( \begin{array}{cccccc} 0.167 & 0.014 & 0 & 0 & 0 & 0 \\ 0.014 & 0.167 & -0.014 i & 0.102 i & -0.014 i & -0.014 i \\ 0 & 0.014 i & 0.167 & 0 & 0 & 0 \\ 0 & -0.102 i & 0 & 0.167 & 0 & 0 \\ 0 & 0.014 i & 0 & 0 & 0.167 & 0 \\ 0 & 0.014 i & 0 & 0 & 0 & 0.167 \end{array} \right); P^{(2)}_{4}=\left( \begin{array}{cccccc} 0.167 & 0.014 & 0 & 0 & 0 & 0 \\ 0.014 & 0.167 & -0.014 i & -0.014 i & 0.102 i & -0.014 i \\ 0 & 0.014 i & 0.167 & 0 & 0 & 0 \\ 0 & 0.014 i & 0 & 0.167 & 0 & 0 \\ 0 & -0.102 i & 0 & 0 & 0.167 & 0 \\ 0 & 0.014 i & 0 & 0 & 0 & 0.167 \end{array} \right); \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(2)}_{5}=\left( \begin{array}{cccccc} 0.167 & 0.014 & 0 & 0 & 0 & 0 \\ 0.014 & 0.167 & -0.014 i & -0.014 i & -0.014 i & 0.102 i \\ 0 & 0.014 i & 0.167 & 0 & 0 & 0 \\ 0 & 0.014 i & 0 & 0.167 & 0 & 0 \\ 0 & 0.014 i & 0 & 0 & 0.167 & 0 \\ 0 & -0.102 i & 0 & 0 & 0 & 0.167 \end{array} \right); P^{(2)}_{6}=\left( \begin{array}{cccccc} 0.167 & 0.047 & 0 & 0 & 0 & 0 \\ 0.047 & 0.167 & -0.047 i & -0.047 i & -0.047 i & -0.047 i \\ 0 & 0.047 i & 0.167 & 0 & 0 & 0 \\ 0 & 0.047 i & 0 & 0.167 & 0 & 0 \\ 0 & 0.047 i & 0 & 0 & 0.167 & 0 \\ 0 & 0.047 i & 0 & 0 & 0 & 0.167 \end{array} \right). \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(3)}_{1}=\left( \begin{array}{cccccc} 0.167 & 0 & -0.102 & 0 & 0 & 0 \\ 0 & 0.167 & 0.014 i & 0 & 0 & 0 \\ -0.102 & 0.014 & 0.167 & -0.014 i & -0.014 i & -0.014 i \\ 0 & 0 & 0.014 i & 0.167 & 0 & 0 \\ 0 & 0 & 0.014 i & 0 & 0.167 & 0 \\ 0 & 0 & 0.014 i & 0 & 0 & 0.167 \end{array} \right); P^{(3)}_{2}=\left( \begin{array}{cccccc} 0.167 & 0 & 0.014 & 0 & 0 & 0 \\ 0 & 0.167 & -0.102 i & 0 & 0 & 0 \\ 0.014 & -0.102 & 0.167 & -0.014 i & -0.014 i & -0.014 i \\ 0 & 0 & 0.014 i & 0.167 & 0 & 0 \\ 0 & 0 & 0.014 i & 0 & 0.167 & 0 \\ 0 & 0 & 0.014 i & 0 & 0 & 0.167 \end{array} \right); \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(3)}_{3}=\left( \begin{array}{cccccc} 0.167 & 0 & 0.014 & 0 & 0 & 0 \\ 0 & 0.167 & 0.014 & 0 & 0 & 0 \\ 0.014 & 0.014 & 0.167 & 0.102 i & -0.014 i & -0.014 i \\ 0 & 0 & -0.102 i & 0.167 & 0 & 0 \\ 0 & 0 & 0.014 i & 0 & 0.167 & 0 \\ 0 & 0 & 0.014 i & 0 & 0 & 0.167 \end{array} \right); P^{(3)}_{4}=\left( \begin{array}{cccccc} 0.167 & 0 & 0.014 & 0 & 0 & 0 \\ 0 & 0.167 & 0.014 & 0 & 0 & 0 \\ 0.014 & 0.014 & 0.167 & -0.014 i & 0.102 i & -0.014 i \\ 0 & 0 & 0.014 i & 0.167 & 0 & 0 \\ 0 & 0 & -0.102 i & 0 & 0.167 & 0 \\ 0 & 0 & 0.014 i & 0 & 0 & 0.167 \end{array} \right); \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(3)}_{5}=\left( \begin{array}{cccccc} 0.167 & 0 & 0.014 & 0 & 0 & 0 \\ 0 & 0.167 & 0.014 & 0 & 0 & 0 \\ 0.014 & 0.014 & 0.167 & -0.014 i & -0.014 i & 0.102 i \\ 0 & 0 & 0.014 i & 0.167 & 0 & 0 \\ 0 & 0 & 0.014 i & 0 & 0.167 & 0 \\ 0 & 0 & -0.102 i & 0 & 0 & 0.167 \end{array} \right); P^{(3)}_{6}=\left( \begin{array}{cccccc} 0.167 & 0 & 0.047 & 0 & 0 & 0 \\ 0 & 0.167 & 0.047 & 0 & 0 & 0 \\ 0.047 & 0.047 & 0.167 & -0.047 i & -0.047 i & -0.047 i \\ 0 & 0 & 0.047 i & 0.167 & 0 & 0 \\ 0 & 0 & 0.047 i & 0 & 0.167 & 0 \\ 0 & 0 & 0.047 i & 0 & 0 & 0.167 \end{array} \right). \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(4)}_{1}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & -0.102 & 0 & 0 \\ 0 & 0.167 & 0 & 0.014 & 0 & 0 \\ 0 & 0 & 0.167 & 0.014 & 0 & 0 \\ -0.102 & 0.014 & 0.014 & 0.167 & -0.014 i & -0.014 i\\ 0 & 0 & 0 & 0.014 i & 0.167 & 0 \\ 0 & 0 & 0 & 0.014 i & 0 & 0.167 \end{array} \right); P^{(4)}_{2}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0.014 & 0 & 0 \\ 0 & 0.167 & 0 & -0.102 & 0 & 0 \\ 0 & 0 & 0.167 & 0.014 & 0 & 0 \\ 0.014 & -0.102 & 0.014 & 0.167 & -0.014 i & -0.014 i\\ 0 & 0 & 0 & 0.014 i & 0.167 & 0 \\ 0 & 0 & 0 & 0.014 i & 0 & 0.167 \end{array} \right); \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(4)}_{3}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0.014 & 0 & 0 \\ 0 & 0.167 & 0 & 0.014 & 0 & 0 \\ 0 & 0 & 0.167 & -0.102 & 0 & 0 \\ 0.014 &0.014 & -0.102 & 0.167 & -0.014 i & -0.014 i\\ 0 & 0 & 0 & 0.014 i & 0.167 & 0 \\ 0 & 0 & 0 & 0.014 i & 0 & 0.167 \end{array} \right); P^{(4)}_{4}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0.014 & 0 & 0 \\ 0 & 0.167 & 0 & 0.014 & 0 & 0 \\ 0 & 0 & 0.167 &0.014 & 0 & 0 \\ 0.014 &0.014 & 0.014 & 0.167 & 0.102 i & -0.014 i\\ 0 & 0 & 0 & -0.102 i & 0.167 & 0 \\ 0 & 0 & 0 & 0.014 i & 0 & 0.167 \end{array} \right); \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(4)}_{5}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0.014 & 0 & 0 \\ 0 & 0.167 & 0 & 0.014 & 0 & 0 \\ 0 & 0 & 0.167 &0.014 & 0 & 0 \\ 0.014 &0.014 & 0.014 & 0.167 & -0.014 i & 0.102 i\\ 0 & 0 & 0 & 0.014 i & 0.167 & 0 \\ 0 & 0 & 0 & -0.102 i & 0 & 0.167 \end{array} \right); P^{(4)}_{6}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0.047 & 0 & 0 \\ 0 & 0.167 & 0 & 0.047 & 0 & 0 \\ 0 & 0 & 0.167 &0.047 & 0 & 0 \\ 0.047 &0.047 & 0.047 & 0.167 & -0.047 i & -0.047 i\\ 0 & 0 & 0 & 0.047 i & 0.167 & 0 \\ 0 & 0 & 0 & 0.047 i & 0 & 0.167 \end{array} \right). \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(5)}_{1}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0 & -0.102 & 0 \\ 0 & 0.167 & 0 & 0 & 0.014 & 0 \\ 0 & 0 & 0.167 & 0 & 0.014 & 0 \\ 0 & 0 & 0 & 0.167 & 0.014 & 0 \\ -0.102 & 0.014 & 0.014 & 0.014 & 0.167 & -0.014 i \\ 0 & 0 & 0 & 0 & 0.014 i & 0.166667 \end{array} \right); P^{(5)}_{2}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0 & 0.014 & 0 \\ 0 & 0.167 & 0 & 0 & -0.102 & 0 \\ 0 & 0 & 0.167 & 0 & 0.014 & 0 \\ 0 & 0 & 0 & 0.167 & 0.014 & 0 \\ 0.014 & -0.102 & 0.014 & 0.014 & 0.167 & -0.014 i \\ 0 & 0 & 0 & 0 & 0.014 i & 0.167 \end{array} \right); \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(5)}_{3}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0 & 0.014 & 0 \\ 0 & 0.167 & 0 & 0 & 0.014 & 0 \\ 0 & 0 & 0.167 & 0 & -0.102 & 0 \\ 0 & 0 & 0 & 0.167 & 0.014 & 0 \\ 0.014 & 0.014 & -0.102 & 0.014 & 0.167 & -0.014 i \\ 0 & 0 & 0 & 0 & 0.014 i & 0.167 \end{array} \right); P^{(5)}_{4}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0 & 0.014 & 0 \\ 0 & 0.167 & 0 & 0 & 0.014 & 0 \\ 0 & 0 & 0.167 & 0 & 0.014 & 0 \\ 0 & 0 & 0 & 0.167 & -0.102 & 0 \\ 0.014 & 0.014 & 0.014 & -0.102 & 0.167 &-0.014 i \\ 0 & 0 & 0 & 0 &0.014 i & 0.167 \end{array} \right); \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(5)}_{5}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0 & 0.014 & 0 \\ 0 & 0.167 & 0 & 0 & 0.014 & 0 \\ 0 & 0 & 0.167 & 0 & 0.014 & 0 \\ 0 & 0 & 0 & 0.167 & 0.014 & 0 \\ 0.014 & 0.014 & 0.014 & 0.014 & 0.167 & 0.102 i \\ 0 & 0 & 0 & 0 & -0.102 i & 0.167 \end{array} \right); P^{(5)}_{6}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0 & 0.047 & 0 \\ 0 & 0.167 & 0 & 0 & 0.047 & 0 \\ 0 & 0 & 0.167 & 0 & 0.047 & 0 \\ 0 & 0 & 0 & 0.167 & 0.047 & 0 \\ 0.047 & 0.047 & 0.047 & 0.047 & 0.167 & -0.047 i \\ 0 & 0 & 0 & 0 & 0.047 i & 0.167 \end{array} \right). \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(6)}_{1}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0 & 0 & -0.102 \\ 0 & 0.167 & 0 & 0 & 0 & 0.014 \\ 0 & 0 & 0.167 & 0 & 0 & 0.014 \\ 0 & 0 & 0 & 0.167 & 0 & 0.014 \\ 0 & 0 & 0 & 0 & 0.167 & 0.014 \\ -0.102 & 0.014 & 0.014 & 0.014 & 0.014 & 0.167 \end{array} \right); P^{(6)}_{2}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0 & 0 & 0.014 \\ 0 & 0.167 & 0 & 0 & 0 & -0.102 \\ 0 & 0 & 0.167 & 0 & 0 & 0.014 \\ 0 & 0 & 0 & 0.167 & 0 & 0.014 \\ 0 & 0 & 0 & 0 & 0.167 & 0.014 \\ 0.014 & -0.102 & 0.014 & 0.014 & 0.014 & 0.167 \end{array} \right); \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(6)}_{3}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0 & 0 & 0.014 \\ 0 & 0.167 & 0 & 0 & 0 & 0.014 \\ 0 & 0 & 0.167 & 0 & 0 & -0.102 \\ 0 & 0 & 0 & 0.167 & 0 & 0.014 \\ 0 & 0 & 0 & 0 & 0.167 & 0.014 \\ 0.014 & 0.014 & -0.102 & 0.014 & 0.014 & 0.167 \end{array} \right); P^{(6)}_{4}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0 & 0 & 0.014 \\ 0 & 0.167 & 0 & 0 & 0 & 0.014 \\ 0 & 0 & 0.167 & 0 & 0 & 0.014 \\ 0 & 0 & 0 & 0.167 & 0 & -0.102 \\ 0 & 0 & 0 & 0 & 0.167 & 0.014 \\ 0.014 & 0.014 & 0.014 & -0.102 & 0.014 & 0.167 \end{array} \right); \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(6)}_{5}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0 & 0 & 0.014 \\ 0 & 0.167 & 0 & 0 & 0 & 0.014 \\ 0 & 0 & 0.167 & 0 & 0 & 0.014 \\ 0 & 0 & 0 & 0.167 & 0 & 0.014 \\ 0 & 0 & 0 & 0 & 0.167 & -0.102 \\ 0.014 & 0.014 & 0.014 & 0.014 & -0.102 & 0.167 \end{array} \right); P^{(6)}_{6}=\left( \begin{array}{cccccc} 0.167 & 0 & 0 & 0 & 0 & 0.047 \\ 0 & 0.167 & 0 & 0 & 0 & 0.047 \\ 0 & 0 & 0.167 & 0 & 0 & 0.047 \\ 0 & 0 & 0 & 0.167 & 0 & 0.047 \\ 0 & 0 & 0 & 0 & 0.167 & 0.047 \\ 0.047 & 0.047 & 0.047 & 0.047 & 0.047 & 0.167 \end{array} \right). \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(7)}_{1}=\left( \begin{array}{cccccc} 0.086 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0.289 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0.164 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.156 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.153 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0.149 \end{array} \right); P^{(7)}_{2}=\left( \begin{array}{cccccc} 0.135 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0.108 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0.297 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.156 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.153 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0.149 \end{array} \right); \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(7)}_{3}=\left( \begin{array}{cccccc} 0.155 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0.127 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0.117 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.299 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.153 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0.149 \end{array} \right); P^{(7)}_{4}=\left( \begin{array}{cccccc} 0.165 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0.138 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0.128 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.121 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.299 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0.149 \end{array} \right); \end{array} $$
$$ \begin{array}{@{}rcl@{}} P^{(7)}_{5}=\left( \begin{array}{cccccc} 0.172 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0.145 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0.135 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.128 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.123 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0.298 \end{array} \right); P^{(7)}_{6}=\left( \begin{array}{cccccc} 0.287 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0.193 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0.159 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.136 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.119 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0.106 \end{array} \right). \end{array} $$

By calculation with \(\mathcal {O}^{(\alpha )}=\mathbb {I}\), we get Tr(Wρ) = − 0.0114 < 0. This is, the state ρ is entangled. The effectiveness of (9) is due to the fact that there always exists a complete set of mutually unbiased measurements, which is not the case for mutually unbiased bases.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, T., Lai, LM., Fei, SM. et al. Mutually Unbiased Measurement Based Entanglement Witnesses. Int J Theor Phys 58, 3973–3985 (2019). https://doi.org/10.1007/s10773-019-04263-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-019-04263-9

Keywords

Search

Navigation