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arXiv:2406.06173 · Journal de Mathématiques Pures et Appliquées

The wave function of stabilizer states and the Wehrl conjecture

Fabio Nicola

Published Jul 21, 2025• 0 citations
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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 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.

DOI
10.1016/j.matpur.2025.103778

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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 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.
A Wehrl entropy construction is proposed for an arbitrary locally compact abelian group GGG. It is proved that the Wehrl entropy is not less than a non-negative integer, which is an invariant of the group GGG. The minimum of the Wehrl entropy is achieved on coherent states.
A construction of the Wehrl entropy is proposed for an arbitrary locally compact abelian group GGG . It is proved that the Wehrl entropy is not less than a certain nonnegative integer, which is an invariant of the group GGG . The minimum of the Wehrl entropy is attained on coherent states.
The Wehrl entropy is the entropy of the probability distribution in phase space corresponding to the Husimi function in terms of coherent states. We explain the significance of the Wehrl entropy in quantum information theory, and present the theory behind the Lieb conjecture, which states that, in finite dimensions, the minimum Wehrl entropy occurs for Bloch coherent states. This was proven by Lieb and Solovej in 2012. We present the theory behind coherent states, with a particular emphasis on Bloch coherent states, and give a geometrical representation of quantum states as points on a sphere. Using this representation, we identify spherical arrangements of 2–9 points that maximize the Wehrl entropy locally. We conjecture that these maxima are in fact global. Furthermore, we investigate how the maximally entangled symmetric states are related to the states corresponding to maximal Wehrl entropy.
Abstract Stabilizer states serve as ‘classical objects’ in the stabilizer formalism of quantum theory, and play an important role in quantum error correction, fault-tolerant quantum computation, and quantum communication. They provide an efficient description of many basic features of quantum theory and exhibit a rich structure. For prime dimensional systems, they may be defined by two quite different yet equivalent ways: The first is via stabilizer groups (maximal Abelian subgroups of the discrete Heisenberg–Weyl group). The second is via the orbits of the Clifford group acting on any computational basis state. However, in a general dimensional system, this equivalence breaks down, and consequently, it is desirable to clarify the difference and relation between the above two approaches to stabilizer states. In this work, we show that these two approaches are equivalent if and only if the system dimension is square-free (i.e. has no square factor). Furthermore, we completely reveal the relation between the Clifford orbits and stabilizer states in an arbitrary dimensional system, derive the explicit expressions of the Clifford orbits and determine their cardinalities.
Let GGG be a compact connected semisimple Lie group, and let π:G→U(Vλ)π:G\to U(V_λ)π:G→U(Vλ​) be a finite-dimensional irreducible unitary representation associated with the highest weight λλλ. We prove that coherent projectors are the unique minimizers of the Wehrl entropy over all density matrices on VλV_λVλ​. Similar extremal properties hold for power functions in place of the entropy function. The proof uses a second variation at the extremizer by perturbation in directions associated with Killing fields, similar to the strategy of Frank and Lieb used in \cite{FrankLieb}. Then the problem reduces to the uncertainty principle characterizing coherent states. This solves the Wehrl-type entropy problem for arbitrary compact connected semisimple Lie groups.
In this paper, we determine the complete ground-state spaces for a broad class of generalized stabilizer quantum models defined over CW-decompositions of non-compact manifolds of arbitrary dimensions and topological type. We introduce a C∗C^*C∗-algebraic framework in which the quantum spin algebra $\A$ is realized as the twisted group C∗C^*C∗-algebra of an infinite discrete Pauli module PPP endowed with a symplectic structure. We establish a correspondence between stabilizer models and isotropic subgroups of PPP: Every Lagrangian extension of such a subgroup determines a C∗C^*C∗-diagonal of $\A$, whose Weyl groupoid encodes the dynamics through a continuous 1-cocycle. This realization yields an explicit family of extremal ground-states for every admissible Lagrangian model, together with their decomposition into sectors. When the sector space is T0T_0T0​, this gives a complete characterization of the ground-state spaces. We also prove that every Lagrangian stabilizer model admits a unique frustration-free ground state with a spectral gap. Our framework applies in arbitrary dimension, including higher abelian quantum doubles, homological codes, and a broad class of stabilizer models.