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 poweroftwo dimensions to the problem of finding a symmetric matrix over \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}_2$\end{document}F2 with an irreducible characteristic polynomial that has a given Fibonacci index. For dimensions of the form \documentclass[12pt]{minimal}\begin{document}$2^{2^k}$\end{document}22k, 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.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 June 2012
 DOI:
 10.1063/1.4723825
 arXiv:
 arXiv:1104.0202
 Bibcode:
 2012JMP....53f2201S
 Keywords:

 03.67.a;
 02.10.De;
 02.30.Lt;
 Quantum information;
 Algebraic structures and number theory;
 Sequences series and summability;
 Quantum Physics
 EPrint:
 11 pages, added chapter on equivalence