Characterising the sets of quantum states with non-negative Wigner function
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914211665281024 |
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| author | Cerf, Nicolas J. Chabaud, Ulysse Davis, Jack Dias, Nuno C. Prata, João N. Van Herstraeten, Zacharie |
| author_facet | Cerf, Nicolas J. Chabaud, Ulysse Davis, Jack Dias, Nuno C. Prata, João N. Van Herstraeten, Zacharie |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_14820 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterising the sets of quantum states with non-negative Wigner function Cerf, Nicolas J. Chabaud, Ulysse Davis, Jack Dias, Nuno C. Prata, João N. Van Herstraeten, Zacharie Mathematical Physics Functional Analysis Quantum Physics 52A07, 46B28, 46B22, 81P16, 81S30 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. |
| title | Characterising the sets of quantum states with non-negative Wigner function |
| topic | Mathematical Physics Functional Analysis Quantum Physics 52A07, 46B28, 46B22, 81P16, 81S30 |
| url | https://arxiv.org/abs/2512.14820 |