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

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.