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

TL;DR

This paper explores the construction of cyclic mutually unbiased bases in power-of-two dimensions using Fibonacci polynomials, linking it to Wiedemann's conjecture and finite field theory, and discusses their equivalence.

Contribution

It establishes a connection between cyclic mutually unbiased bases and Fibonacci polynomials, providing solutions for dimensions 2^(2^k) and relating to Wiedemann's conjecture.

Findings

Constructed cyclic mutually unbiased bases in specific dimensions.

Linked the problem to Fibonacci polynomials and finite field theory.

Discussed the equivalence of different mutually unbiased bases.

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 F_2 with an irreducible characteristic polynomial that has a given Fibonacci index. For dimensions of the form 2^(2^k) 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.