Cyclic mutually unbiased bases, Fibonacci polynomials and Wiedemann's conjecture
Ulrich Seyfarth, Kedar S. Ranade · Journal of Mathematical Physics · 2012
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 \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.