Product Weyl-Heisenberg covariant MUBs and Maximizers of Magick

In this work we investigate discrete structures in product Hilbert spaces. For monopartite systems of size d one relies on the Weyl-Heisenberg group WH(d), while in the case of composite Hilbert spaces we identify designs covariant with respect to the product group, [WH(p)]^{otimes n}. In analogy with magic -a quantity attaining its maximum for states fiducial with respect to WH(d) -we introduce a similar notion of magick, defined with respect to the product group. The maximum of this quantity over all equimodular vectors yields fiducial states that generate d textit{a priori} isoentangled mutually unbiased bases (MUBs), which, when supplemented by the identity, form their complete set. Such fiducial states are explicitly constructed in all prime-power dimensions pⁿ with pge 3. The result for pge 5 extends the construction of Klappenecker and Rötteler, whereas for p=3 it is mathematically distinct and is based on Galois rings. The global maximum of magick for d=2³ yields fiducial states corresponding to the symmetric informationally complete (SIC) generalized measurement of Hoggar. Our approach feeds into a unifying perspective in which highly symmetric quantum designs emerge from fiducial states with extremal properties via structured group-orbit constructions.