Product Weyl--Heisenberg covariant mutually unbiased bases and extremal non-stabilizerness

Abstract

Discrete structures in product Hilbert spaces are investigated. For monopartite systems of size $d$ one relies on the Weyl--Heisenberg group $WH(d)$, while in the case of composite Hilbert spaces with the local dimensions $d_i$ we identify designs covariant with respect to the product group, ${⨂}_{i}WH(d_i)$. In analogy with magic -- a quantity attaining its maximum for states fiducial with respect to $WH(d)$ -- we introduce a similar quantifier of {\sl product magic}, defined with respect to the product group. The maximum of this quantity over all equimodular vectors yields fiducial states that generate $d={\prod }_i{d}_i$ \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 $d=p^n$ with $p\ge 3$. The result for $p\ge 5$ extends the construction of Klappenecker and R{ö}tteler, whereas for $p=3$ it is mathematically distinct and is based on Galois rings. The global maximum of the quantifier of product magic for $d=2^3$ 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.