Characterising the sets of quantum states with non-negative Wigner function

• URL: http://arxiv.org/abs/2512.14820v2
• Date: Thu, 18 Dec 2025 09:02:26 GMT
• Title: Characterising the sets of quantum states with non-negative Wigner function
• Abstract summary: We consider the convex sets $mathcal D_+(mathcal H)$ of Wigner-positive states (WPS) over $mathcal H$ with non-negative Wigner function.<n>We construct minimal sets of states that generate $mathcal D_+(mathcal H)$ through convex combinations.
• Score: 0.15393457051344295
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: For Hilbert spaces $\mathcal H\subseteq L^2(\mathbb R)$ we consider the convex sets $\mathcal D_+(\mathcal H)$ of Wigner-positive states (WPS), i.e.~density matrices over $\mathcal H$ with non-negative Wigner function. We investigate the topological structure of these sets, namely concerning closure, compactness, interior and boundary (in a relative topology induced by the trace norm). We also study their geometric structure and construct minimal sets of states that generate $\mathcal D_+(\mathcal H)$ through convex combinations. If $\mathcal H$ is finite-dimensional, the existence of such sets follows from a central result in convex analysis, namely the Krein-Milman theorem. In the infinite-dimensional case $\mathcal H=L^2(\mathbb R)$ this is not so, due to lack of compactness of the set $\mathcal D_+(\mathcal H)$. Nevertheless, we prove that a Krein-Milman theorem holds in this case, which allows us to extend most of the results concerning the sets of generators to the infinite-dimensional setting. Finally, we study the relation between the finite and infinite-dimensional sets of WPS, and prove that the former provide a hierarchy of closed subsets, which are also proper faces of the latter. These results provide a basis for an operational characterisation of the extreme points of the sets of WPS, which we undertake in a companion paper. Our work offers a unified perspective on the topological and geometric properties of the sets of WPS in finite and infinite dimensions, along with explicit constructions of minimal sets of generators.

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