标题： 稳定子态的波函数和Wehrl猜想 显示英文标题

标题： The wave function of stabilizer states and the Wehrl conjecture

摘要： 我们关注由希尔伯特空间$L^2(A)$表示的量子系统，其中$A$是一个包含紧开子群的局部紧致阿贝尔群。 我们研究与威勒-海森堡算子相关的两个相互关联的问题。 首先，我们提供了一个完整而优雅的解决方案，用于描述稳定子态的波函数，这是一个在量子信息理论中出现的问题。 随后，我们证明了稳定子态正是魏尔熵泛函的最小化者，从而解决了对于任何此类群的魏尔猜想的类比问题。 此外，我们为稳定子态的集合构造了一个模空间，即对该集合的参数化，使其具有自然的代数结构，并推导出当$A$为有限时稳定子态数量的公式。 值得注意的是，这些结果甚至对于有限阿贝尔群来说也是新颖的。 显示英文摘要

摘要： 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.