Skip to main content
SpringerLink
Log in

Mutually unbiased maximally entangled bases in \(C^{d}\otimes C^{d}\) with d an odd prime power

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

We study mutually unbiased maximally entangled bases (MUMEBs) in bipartite system \(C^{d}\otimes C^{d}\) with \(d\ge 3\) a power of an odd prime number. By using the theory of finite fields, we provide a new and intuitive method to construct MUMEBs in \(C^{d}\otimes C^{d}\). And we construct \(d(d-1)\) MUMEBs in bipartite system \(C^{d}\otimes C^{d}\) explicitly.

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

Data availability

No new data were generated or analyzed in support of this research.

References

  1. D’Ariano, G.M., Paris, M.G.A., Sacchi, M.F.: Quantum tomography. Adv. Imaging Electron Phys. 128, 206–309 (2003)

    Google Scholar 

  2. Adamson, R.B.A., Steinberg, A.M.: Experimental quantum state estimation with mutually unbiased base. Phys. Rev. Lett. 105, 030406 (2010)

    Article  ADS  Google Scholar 

  3. Cerf, N.J., Bourennane, M., Karlsson, A., Gisin, N.: Secuity of quantum key distribution using \(d\)-level systems. Phys. Rev. Lett. 88, 127902 (2002)

    Article  ADS  Google Scholar 

  4. Bennett, C.H., Brassard, G.: Quantum cryptography: public key distribution and coin tossing. In: Proceeding of the IEEE International Conference on Computers. Systems and Signal Processing, pp. 175–179. Bangalore, India (IEEE, New York) (1984)

  5. Brub, D.: Optimal eavesdropping in quantum cryptography with six states. Phys. Rev. Lett. 81, 3018 (1998)

    Article  ADS  Google Scholar 

  6. Caruso, F., Bechmanm-Pasquinucci, H., Macchiavello, C.: Robustness of a quantum key distribution with two and three mutually unbiased bases. Phys. Rev. A 72, 032340 (2005)

    Article  ADS  Google Scholar 

  7. Aharonov, Y., Englert, B.-G., Naturforsch, Z.: The mean king’s problem: prime degrees of freedom. Phys. Lett. A 284, 1–5 (2001)

    Article  MathSciNet  ADS  Google Scholar 

  8. Hayashi, A., Horibe, M., Hashimoto, T.: Mean king’s problem with mutually unbiased bases and orthogonal Latin squares. Phys. Rev. A 71, 052331 (2005)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  9. Kimura, G., Tanaka, H., Ozawa, M.: Solution to the mean king’s problem with mutually unbiased bases for arbitary level. Phys. Rev. A 73, 050301 (2006)

    Article  ADS  Google Scholar 

  10. Ivanovic, I.D.: Geometrical description of quantal state determination. J. Phys. A Math. Gen. 14, 3241–3245 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  11. Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191, 363–381 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  12. Bandyopadhyay, S., Boykin, P.O., Roychowdhury, V., Vatan, F.: A new proof for the existence of mutually unbiased bases. Algorithmica 34, 512–518 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chaturvedi, S.: Aspects of mutually unbiased bases in odd-prime-power dimensions. Phys. Rev. A. 65, 044301 (2002)

    Article  ADS  Google Scholar 

  14. Durt, T., Englert, B.-G., Bengtsson, I., Życzkowski, K.: On mutually unbiased bases. Int. J. Quantum Inf. 8(04), 535–640 (2010)

    Article  MATH  Google Scholar 

  15. Tao, Y.H., Nan, H., Zhang, J., Fei, S.M.: Mutually unbiased maximally entangled bases in \(C^{d}\otimes C^{kd}\). Quantum Inf. Process. 14, 2291–2300 (2015)

    Article  MATH  ADS  Google Scholar 

  16. Liu, J.Y., Yang, M.H., Feng, K.Q.: Mutually unbiased maximally entangled bases in \(C^{d}\otimes C^{d}\). Quantum Inf. Process. 16, 1–8 (2017)

    Article  MATH  ADS  Google Scholar 

  17. Xu, D.M.: Construction of mutually unbiased maximally entangled bases through permutation of Hardamard matrices. Quantum Inf. Process. 16, 1–11 (2017)

    Article  Google Scholar 

  18. Luo, L.Z., Li, X.Y., Tao, Y.H.: Two types of maximally entangled bases and their mutually unbiased property in \(C^{d}\otimes C^{d^{\prime }}\). Int. J. Theor. Phys. 55, 5069–5076 (2016)

    Article  MATH  Google Scholar 

  19. Xu, D.M.: Trace-2 excluded subsets of special linear groups over finite fields and mutually unbiased maximally entangled bases. Quantum Inf. Process. 18, 213 (2019)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  20. Nan, H., Tao, Y.H., Wang, T.J., Zhang, J.: Mutually unbiased maximally entangled bases for the bipartite system \(C^{d}\otimes C^{d^{k}}\). Int. J. Theor. Phys. 55, 4324–4330 (2016)

    Article  MATH  Google Scholar 

  21. Xu, L.S., Zhang, G.J., Song, Y.Y., Tao, Y.H.: Mutually unbiased property of maximally entangled bases and product bases in \(C^{d}\otimes C^{d}\). Int. J. Theor. Phys. 57(11), 3463–3472 (2018)

    Article  MATH  Google Scholar 

  22. Song, Y.Y., Zhang, G.J., Xu, L., Tao, Y.H.: Mutually unbiased unextendible maximally entangled bases in \(C^{d}\otimes C^{d+1}\). Int. J. Theor. Phys. 57, 3785–3794 (2018)

    Article  MATH  Google Scholar 

  23. Han, Y.F., Zhang, G.J., Yong, X.L., Xu, L.S., Tao, Y.H.: Mutually unbiased special entangled bases with Schmidt number 2 in \(C^{3} \otimes C^{4k}\). Quantum Inf. Process. 17, 1–13 (2018)

    Article  MathSciNet  Google Scholar 

  24. Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  25. Scott, A.J.: Optimizing quantum process tomography with unitary 2-designs. J. Phys. A Math. Theor. 41(5), 055308 (2008)

    Article  MathSciNet  MATH  ADS  Google Scholar 

Download references

Acknowledgements

The work is supported by the NSFC under number 1227010065, Natural Science Foundation of Heilongjiang Province (LH2022F032).

Author information

Authors and Affiliations

  1. College of Science, Harbin University of Science and Technology, Harbin, 150080, China

    Lai-Zhen Luo, Yu Xia & Gui-Jun Zhang

Authors
  1. Lai-Zhen Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gui-Jun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gui-Jun Zhang.

Ethics declarations

Conflict of interest

The authors declare no competing financial interests.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luo, LZ., Xia, Y. & Zhang, GJ. Mutually unbiased maximally entangled bases in \(C^{d}\otimes C^{d}\) with d an odd prime power. Quantum Inf Process 22, 415 (2023). https://doi.org/10.1007/s11128-023-04168-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-023-04168-x

Keywords

Search

Navigation