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On the mathematical foundations of mutually unbiased bases

  • Koen Thas
Article
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Abstract

In order to describe a setting to handle Zauner’s conjecture on mutually unbiased bases (MUBs) (stating that in \(\mathbb {C}^d\), a set of MUBs of the theoretical maximal size \(d + 1\) exists only if d is a prime power), we pose some fundamental questions which naturally arise. Some of these questions have important consequences for the construction theory of (new) sets of maximal MUBs. Partial answers will be provided in particular cases; more specifically, we will analyze MUBs with associated operator groups that have nilpotence class 2, and consider MUBs of height 1. We will also confirm Zauner’s conjecture for MUBs with associated finite nilpotent operator groups.

Keywords

Mutually unbiased base Zauner’s conjecture Pauli group Operator group Nilpotence 

Notes

Acknowledgements

I want to thank Markus Grassl for many helpful communications on the subject of the present paper. In particular, he provided valuable comments on the definition of “height” in Sect. 5 in a previous version of this note. I also want to thank two anonymous referees for several useful suggestions.

References

  1. 1.
    Aschbacher, M., Childs, A.M., Wocjan, P.: The limitations of nice mutually unbiased bases. J. Algebr. Comb. 25, 111–123 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bandyopadhyay, S., Boykin, P.O., Roychowdhury, V., Vatan, F.: A new proof for the existence of mutually unbiased bases. Algorithmica 34, 512–528 (2002)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Calderbank, A.R., Cameron, P.J., Kantor, W.M., Seidel, J.J.: \(\mathbb{Z}_4\)-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets. Proc. Lond. Math. Soc. 75, 436–480 (1997)CrossRefMATHGoogle Scholar
  4. 4.
    Delsarte, P., Goethals, P.M., Seidel, J.J.: Bounds for systems of lines and Jacobi polynomials. Philips Res. Rep. 30, 91–105 (1975)MATHGoogle Scholar
  5. 5.
    Gorenstein, D.: Finite Groups, 2nd edn. Chelsea Publishing Co., New York (1980)MATHGoogle Scholar
  6. 6.
    Hirschfeld, J.W.P.: Projective Geometries over Finite Fields, Second edition. Oxford Mathematical Monographs. Oxford University Press, New York (1998)Google Scholar
  7. 7.
    Hoggar, S.G.: t-designs in projective spaces. Eur. J. Comb. 3, 233–254 (1982)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ivanovic, I.D.: Geometrical description of quantum state determination. J. Phys. A 14, 3241–3245 (1981)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Kabatiansky, G.A., Levenshtein, V.I.: On bounds for packings on a sphere and in space. Probl. Inf. Transm. 14, 1–17 (1978)MathSciNetGoogle Scholar
  10. 10.
    Klappenecker, A., Rötteler, M.: Proceedings 2005 IEEE international symposium on information theory, pp. 1740–1744 (Preprint quant-ph/0502031) (2005)Google Scholar
  11. 11.
    Knarr, N.: Translation Planes. Lecture Notes in Mathematics. Springer, Berlin, Heidelberg, New York (1995)CrossRefMATHGoogle Scholar
  12. 12.
    Knill, E.: Technical Report LAUR-96-2717, Los Alamos National Laboratory (1996)Google Scholar
  13. 13.
    Schur, I.: Neue Begründung der Theorie der Gruppencharaktere, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, pp. 406–432 (1905)Google Scholar
  14. 14.
    Schwinger, J.: Unitary operator bases. Proc. Natl. Acad. Sci. USA 46, 570 (1960)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Thas, K.: The geometry of generalized Pauli operators of N-qudit Hilbert space, and an application to MUBs. Europhys. Lett. (EPL) 86, 60005 (2009)ADSCrossRefGoogle Scholar
  16. 16.
    Thas, K.: Surveys in Combinatorics. London Mathematical Society Lecture Note Series, vol. 392, pp. 235–331. Cambridge University Press, Cambridge (2011)MATHGoogle Scholar
  17. 17.
    Thas, K.: A Course on Elation Quadrangles. EMS Series of Lectures in Mathematics. European Mathematical Society, Zürich (2012)CrossRefMATHGoogle Scholar
  18. 18.
    Thas, K.: Unextendible mutually unbiased bases (after Mandayam, Bandyopadhyay, Grassl and Wootters). Entropy 18, 395 (2016)ADSCrossRefGoogle Scholar
  19. 19.
    Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191, 363–381 (1989)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Wootters, W.K.: Quantum measurements and finite geometry. Found. Phys. 36, 112–126 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Zauner, G.: Quantendesigns: Grundzüge einer nichtkommutativen Designtheorie. Ph. D. Thesis, Universität Wien (1999)Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsGhent UniversityGhentBelgium

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