Mutually unbiased projectors and duality between lines and bases in finite quantum systems

Quantum systems with variables in the ring Z(d)$\mathbb{Z}\left(d\right)$ are considered, and the concepts of weak mutually unbiased bases and mutually unbiased projectors are discussed. The lines through the origin in the Z(d)×Z(d)$\mathbb{Z}\left(d\right)\times \mathbb{Z}\left(d\right)$ phase space, are classified into maximal lines (sets of d$d$ points), and sublines (sets of d_i$d_i$ points where d_i|d$d_i|d$). The sublines are intersections of maximal lines. It is shown that there exists a duality between the properties of lines (resp., sublines), and the properties of weak mutually unbiased bases (resp., mutually unbiased projectors).