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

TL;DR

The paper develops a unified convex-analytic framework to characterize sets of quantum states with nonnegative Wigner functions, D_+(H), across finite and infinite dimensions. It thoroughly analyzes topology (closure, compactness, interiors, boundaries) and geometry (extreme points, faces, and minimal generator sets), proving finite-dimensional D_+(H) is closed and compact with interior characterized by full rank and empty nodal sets, and establishing a Krein–Milman-type decomposition in the infinite-dimensional setting via the weak*-topology. It constructs explicit minimal generator sets and demonstrates a hierarchical relation between finite- and infinite-dimensional Wigner-positive state sets, showing that finite cases form proper closed faces of the infinite case. The results lay groundwork for an operational characterization of extreme Wigner-positive states and connect nodal-set structure to convex-generator theory, with a companion paper detailing the extreme points further.

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.

Figures (5)

• Figure 1: The ambient space is the affine space $\mathcal{A}^1$. Each point $(q,{\rm Re}\, s,{\rm Im}\, s)$ represents a hermitian, trace-class operator $\rho =q|0\rangle\langle0| +(1-q) |1\rangle\langle1|+s|0\rangle\langle1| + \overline{s} |1\rangle\langle0|\in \mathcal{A}^1$. The outer ellipsoid represents the set $\mathcal{D}^1$, while the inner ellipsoid corresponds to $\mathcal{D}^1_+$. Moreover, the north pole is the state $|0\rangle\langle0|$ and the south pole is $|1\rangle\langle1|$. The Euclidean distance coincides with the trace distance.
• Figure 2: Illustration of \ref{['X_+']} for the case $\mathcal{H}=\mathcal{H}^1={\rm span}\{|0\rangle,|1\rangle\}$. Each point $(p,s)$ inside the larger ellipse represents a density matrix $\rho{(p,s)}= (1-p) \,\rho_0+ p \,|1\rangle\langle1| +s [|0\rangle\langle1| + |1\rangle\langle0| ],$ (cf. \ref{['DM']} below). The set of density matrices $\mathcal{D}(\mathcal{H})$ (grey region) and of WPS $\mathcal{D}_+(\mathcal{H})$ (blue region) are represented for the case of $s$ real. The origin corresponds to the state $\rho_0=|0\rangle\langle0|$. In this case, $Y(\mathcal{H})=\partial^i\mathcal{D}(\mathcal{H}) \backslash \{\rho_0\}$ and $X_+(\mathcal{H})\coloneqq{\rm Ran} \, F=\partial^i\mathcal{D}_+(\mathcal{H}) \backslash \{\rho_0\}$.
• Figure 3: Set of density matrices $\mathcal{D}(\mathcal{H})$ (grey area) and of WPS $\mathcal{D}_+(\mathcal{H})$ for the case of real $s$. Each point $(p,s)$ inside the larger ellipse represents a density matrix $\rho{(p,s)}= (1-p) \,\rho_0+ p \,|n\rangle\langle n| +s [|0\rangle\langle n| + |n\rangle\langle0| ].$ The blue line (without the point $(0,0)$) represents the set $Y(\mathcal{H})$. The origin is the state $\rho_0=|0\rangle\langle0|$. The coloured domains, in decreasing order of their area size, represent the sets of WPS for the cases of $n=1,\dots,4$, respectively. Their boundaries (without the point $(0,0)$) represent the sets $X_+(\mathcal{H})$ of the states $\rho_+$.
• Figure 4: Two views of the (affine) cones of density matrices $\mathcal{D}_{|0\rangle}$ (green domain) and of WPS $\mathcal{D}_+^2 \cap \mathcal{D}_{|0\rangle}$ (blue domain) for the case of real $s$. Each point $(p,q,s)$ in the larger domain represents a density matrix $\rho(p,q,s)= (1-p-q) \,\rho_0+ q \,|1\rangle\langle1| +p \, |2\rangle\langle2| + s \left(|1\rangle\langle2| + |2\rangle\langle1|\right).$ The vertex of the cone is the state $\rho_0=|0\rangle\langle0|$, and the base of the cone is the set $\mathcal{D}(\langle0\rangle^\perp) \subset Y^2$ (restricted to real $s$). A cross section of this figure (for s=0) is displayed in \ref{['fig:combined']} (left and right) for a=0 and c=0, respectively.
• Figure 5: Affine cones of density matrices $\mathcal{D}_v$ (grey domain) and of WPS $\mathcal{D}_+^2 \cap \mathcal{D}_v$ for the case $s=0$. Each point $(p,q)$ in the triangular domain represents a density matrix $\rho{(p,q)}= (1-p-q) \,\rho_0+ q \,|u\rangle\langle u| +p \, |w\rangle\langle w|$ for some $u,w\in \langle v\rangle^\perp$. The origin is the state $\rho_0=|0\rangle\langle0|$. The hypotenuse is the set $\mathcal{D}(\langle v\rangle^\perp)$ (restricted to $s=0$). The points inside the domains bounded by the coloured lines are the WPS. The figure on the left corresponds to the cases $v=|0\rangle-a|1\rangle, \,u=\tfrac{a|0\rangle+|1\rangle}{\sqrt{1+a^2}}$ and $w=|2\rangle$, for $a=0,1,3$. On the right, we have $v=|0\rangle-c|2\rangle,\, u=|1\rangle$ and $w=\tfrac{c|0\rangle+|2\rangle}{\sqrt{1+c^2}}$ for $c=0,1,3$.

Theorems & Definitions (65)

• Theorem 3: Krein--Milman
• Theorem 4
• Remark 5
• Theorem 6
• proof
• Theorem 7
• proof
• Remark 8
• Theorem 9
• proof