The wave function of stabilizer states and the Wehrl conjecture

TL;DR

This paper characterizes stabilizer states in quantum systems over locally compact Abelian groups, showing they minimize Wehrl entropy and providing a parameterization of their set, extending known results to new group structures.

Contribution

It provides a complete description of stabilizer states via wave functions and proves they minimize Wehrl entropy for a broad class of groups, including finite Abelian groups.

Findings

Stabilizer states are the wave function minimizers of Wehrl entropy.

A moduli space for stabilizer states is constructed with an algebraic structure.

Explicit formulas for the number of stabilizer states when the group is finite.

Abstract

We focus on quantum systems represented by a Hilbert space $L^2(A)$, 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 precisely the minimizers of the Wehrl entropy functional, thereby resolving the analog of the Wehrl conjecture for any such group. Additionally, we construct a moduli space for the set of stabilizer states, that is, a parameterization 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. Notably, these results are novel even for finite Abelian groups.