An analytic function approach to weak mutually unbiased bases

Quantum systems with variables in ${\mathbb Z}(d)$ are considered, and three different structures are studied. The first is weak mutually unbiased bases, for which the absolute value of the overlap of any two vectors in two different bases is $1/\sqrt{k}$ (where $k|d$) or $0$. The second is maximal lines through the origin in the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ phase space. The third is an analytic representation in the complex plane based on Theta functions, and their zeros. It is shown that there is a correspondence (triality) that links strongly these three apparently different structures. For simplicity, the case where $d=p_1\times p_2$, where $p_1,p_2$ are odd prime numbers different from each other, is considered.