Constructions of Mutually Unbiased Bases

  • Andreas Klappenecker
  • Martin Rötteler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2948)

Abstract

Two orthonormal bases B and B′ of a d-dimensional complex inner-product space are called mutually unbiased if and only if |〈b|b′ 〉|2 = 1/d holds for all b ∈ B and b′ ∈ B′. The size of any set containing pairwise mutually unbiased bases of ℂd cannot exceed d + 1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist. We give a simplified proof of this fact based on the estimation of exponential sums. We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.

Keywords

Quantum cryptography quantum state estimation Weil sums finite fields Galois rings 

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References

  1. 1.
    Alltop, W.O.: Complex sequences with low periodic correlations. IEEE Transactions on Information Theory 26(3), 350–354 (1980)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aravind, P.K.: Solution to the king’s problem in prime power dimensions. Z. Naturforschung 58a, 2212 (2003)Google Scholar
  3. 3.
    Bandyopadhyay, S., Boykin, P.O., Roychowdhury, V., Vatan, F.: A new proof of the existence of mutually unbiased bases. Algorithmica 34, 512–528 (2002)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bechmann-Pasquinucci, H., Tittel, W.: Quantum cryptography using larger alphabets. Phys. Rev. A 61(6), 062308 (2000)Google Scholar
  5. 5.
    Bennett, C.H., Brassard, G.: Quantum cryptography: Public key distribution and coin tossing. In: Proceedings of the IEEE Intl. Conf. Computers, Systems, and Signal Processing, pp. 175–179. IEEE, Los Alamitos (1984)Google Scholar
  6. 6.
    Beth, T., Jungnickel, D., Lenz, H.: Design Theory, 2nd edn., vol. 2. Cambridge University Press, Cambridge (1999)Google Scholar
  7. 7.
    Bollobás, B.: Random Graphs. Academic Press, London (1985)MATHGoogle Scholar
  8. 8.
    Carlet, C.: One-weight Z4-linear codes. In: Buchmann, J., Høholdt, T., Stichtenoth, H., Tapia-Recillas, H. (eds.) Coding Theory, Cryprography and Related Areas, pp. 57–72. Springer, Heidelberg (2000)Google Scholar
  9. 9.
    Chaturvedi, S.: Aspects of mutually unbiased bases in odd-prime-power dimensions. Phys. Rev. A 65, 044301 (2002)Google Scholar
  10. 10.
    Chowla, S., Erdös, P., Strauss, E.G.: On the maximal number of pairwise orthogonal latin squares of given order. Canadian J. Math. 12, 204–208 (1960)MATHCrossRefGoogle Scholar
  11. 11.
    Delsarte, P., Goethals, J.M., Seidel, J.J.: Bounds for systems of lines, and Jacobi polynomials. Philips Res. Repts., 91–105 (1975)Google Scholar
  12. 12.
    Englert, B.-G., Aharonov, Y.: The mean king’s problem: Prime degrees of freedom. Phys. Letters 284, 1–5 (2001)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Hoggar, S.G.: t-designs in projective spaces. Europ. J. Combin. 3, 233–254 (1982)MATHMathSciNetGoogle Scholar
  14. 14.
    Ivanović, I.D.: Geometrical description of quantal state determination. J. Phys. A 14, 3241–3245 (1981)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Kabatiansky, G.A., Levenshtein, V.I.: Bounds for packings on a sphere and in space. Problems of Information Transmission 14(1), 1–17 (1978)Google Scholar
  16. 16.
    Laywine, C.F., Mullen, G.L.: Discrete Mathematics Using Latin Squares. John Wiley, New York (1998)MATHGoogle Scholar
  17. 17.
    Lidl, R., Niederreiter, H.: Introduction to finite fields and their applications, 2nd edn. Cambridge University Press, Cambridge (1994)MATHGoogle Scholar
  18. 18.
    Schwinger, J.: Unitary operator bases. Proc. Nat. Acad. Sci. U.S.A. 46, 570–579 (1960)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Vaidman, L., Aharonov, Y., Albert, D.Z.: How to ascertain the values of σ x, σ y, and σ z. Phys. Rev. Lett. 58, 1385–1387 (1987)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Wan, Z.-X.: Quaternary Codes. World-Scientific, Singapore (1997)MATHGoogle Scholar
  21. 21.
    Wilson, R.M.: Concerning the number of mutually orthogonal Latin squares. Discr. Math. 9, 181–198 (1974)MATHCrossRefGoogle Scholar
  22. 22.
    Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Physics 191, 363–381 (1989)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Yang, K., Helleseth, T., Kumar, P.V., Shanbhag, A.G.: On the weight hierarchy of Kerdock codes over Z4. IEEE Transactions on Information Theory 42(5), 1587–1593 (1996)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Zauner, G.: Quantendesigns – Grundzüge einer nichtkommutativen Designtheorie. PhD thesis, Universität Wien (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Andreas Klappenecker
    • 1
  • Martin Rötteler
    • 2
  1. 1.Department of Computer ScienceTexas A&M UniversityCollege StationUSA
  2. 2.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

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