Constructions of Mutually Unbiased Bases

@inproceedings{Klappenecker2003ConstructionsOM,
  title={Constructions of Mutually Unbiased Bases},
  author={Andreas Klappenecker and Martin R{\"o}tteler},
  booktitle={International Conference on Finite Fields and Applications},
  year={2003}
}
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… 
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A constructive proof of the existence of mutually biased bases for dimensions that are powers of primes is presented and it is proved that in any dimension d the number of mutually unbiased bases is at most d+1.
There is no generalization of known formulas for mutually unbiased bases
In a quantum system having a finite number N of orthogonal states, two orthonormal bases {ai} and {bj} are called mutually unbiased if all inner products ⟨ai∣bj⟩ have the same modulus 1∕N. This
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