Finite Fields and Applications

Volume 2948 of the series Lecture Notes in Computer Science pp 137-144

Constructions of Mutually Unbiased Bases

  • Andreas KlappeneckerAffiliated withDepartment of Computer Science, Texas A&M University
  • , Martin RöttelerAffiliated withDepartment of Combinatorics and Optimization, University of Waterloo

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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 ℂ d 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