2025/12/16 by Nicolas J. Cerf, Cerf, Nicolas J., Ulysse Chabaud +9 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Quantum Information and Cryptography #Spectral Theory in Mathematical Physics #Quantum many-body systems
paper · pdf · doi:10.48550/arxiv.2512.14820
For Hilbert spaces \mathcal H⊆ L2(\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=L2(\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.