Constructions of Mutually Unbiased Bases

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


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.


Quantum cryptography quantum state estimation Weil sums finite fields Galois rings 


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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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