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Cyclic mutually unbiased bases, Fibonacci polynomials and Wiedemann's conjecture

### Abstract

We relate the construction of a complete set of cyclic mutually unbiased bases, i.e., mutually unbiased bases generated by a single unitary operator, in power-of-two dimensions to the problem of finding a symmetric matrix over with an irreducible characteristic polynomial that has a given Fibonacci index. For dimensions of the form , we present a solution that shows an analogy to an open conjecture of Wiedemann in finite field theory. Finally, we discuss the equivalence of mutually unbiased bases.

© 2012 American Institute of Physics

Received 10 May 2012
Accepted 14 May 2012
Published online 05 June 2012

Acknowledgments:
The authors acknowledge funding by CASED and by BMBF/QuOReP.

Article outline:

I. INTRODUCTION
II. METHODS AND FORMALISM
A. Fibonacci polynomials
B. Finding a stabilizer matrix
III. EXPLICIT CONSTRUCTION OF THE REDUCED STABILIZER MATRIX FOR *m* = 2^{ k }
A. Irreducible self-reciprocal polynomials and the characteristic polynomial of
B. Fibonacci index of χ_{ B } and Wiedemann's conjecture
IV. EQUIVALENCE OF SETS OF MUTUALLY UNBIASED BASES
V. CONCLUSIONS

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