The wave function of stabilizer states and the Wehrl conjecture

We focus on quantum systems represented by a Hilbert space $L^2\left(A\right)$, where A is a locally compact Abelian group that contains a compact open subgroup. We examine two interconnected issues related to Weyl-Heisenberg operators. First, we provide a complete and elegant solution to the problem of describing the stabilizer states in terms of their wave functions — an issue that arises in quantum information theory. Subsequently, we demonstrate that the stabilizer states are exactly the minimizers of the Wehrl entropy, thereby solving the Wehrl-type entropy conjecture for any such group (in particular, for finite-dimensional vector spaces over non-Archimedean local fields). Additionally, we construct a moduli space for the set of stabilizer states, that is, a parametrization of this set, that endows it with a natural algebraic structure, and we derive a formula for the number of stabilizer states when A is finite. Indeed, these results are novel even for finite Abelian groups.