Extreme non-negative Wigner functions

TL;DR

The paper develops a convex-geometry framework to characterize Wigner-positive quantum states by focusing on extreme points and introduces the Vertigo map, which converts extreme WPQS into extreme WPS while preserving phase invariance. By combining this map with extremality-preserving channels (e.g., Fock-bounded displacements and Gaussian unitaries), the authors construct large classes of extreme WPS, with beam-splitter states playing a central role as attractors and extremal seeds. In low-dimensional settings, they achieve a complete parametrization of extreme WPS, revealing a rich, structured landscape of mixed states with non-negative Wigner functions. The work connects phase-space polynomials to state-space extremality and sets the stage for further exploration of higher-dimensional extremal WPS and their operational implications in quantum information processing.

Abstract

Providing an operational characterization of the Wigner-positive states (WPS), i.e., the set of quantum states with non-negative Wigner function, is a longstanding open problem. For pure states, the only WPS are Gaussian states, but the situation is considerably more subtle for mixed states. Here, we approach the problem using convex geometry, reducing the question to the characterization of the extreme points of the set of WPS. We give a constructive method to generate a large class of such extreme WPS, which combines the following steps: (i) we characterize the phase-invariant extreme points of the superset of Wigner-positive quasi-states (WPQS); (ii) we introduce a new quantum map, named Vertigo map, which maps extreme WPQS to extreme WPS while preserving phase invariance; (iii) we identify families of extremality-preserving maps and use them to obtain extreme WPS while relaxing phase invariance. Our construction generates all extreme WPS of low dimension, starting from a specific kind of WPS known as beam-splitter states. Our results build upon new mathematical properties of the set of WPS derived in a companion paper and unveil the remarkable structure of mixed states with non-negative Wigner functions.

Figures (8)

• Figure 1: Mapping extreme WPQS onto extreme WPS. Within the cut of operators with Fock-bounded support (i.e., a bounded particle number $n$), the set of Wigner-positive states $\mathcal{D}^{n}_{+}$ (blue shaded region) appears as a convex subset of the Wigner-positive quasi-states $\mathcal{A}^{n}_{+}$. The Vertigo map $\mathcal{V}_{t}$ (orange arrows) sends extreme points of $\mathcal{A}^{n}_{+}$ to extreme points of $\mathcal{D}^{n}_{+}$ as soon as the parameter $t$ exceeds a threshold value $t_{0}>0$ (with $t_{0}$ generally depending on the initial point). The fixed points of the Vertigo map $\mathcal{V}_{t}$ (for $t\to\infty$) are binomial states. The last step of the construction of WPS (pictured in \ref{['fig:extreme_state_generation']}) relaxes Fock-boundedness while preserving extremality.
• Figure 2: The Vertigo map. Physical implementation of the Vertigo map $\mathcal{V}_t$ for $t\geq 1$, as the concatenation of a noiseless linear amplifier (NLA) $\mathcal{N}$ and a pure loss channel (PLC) $\mathcal{E}$. We have $\mathcal{V}_{t}=\mathcal{E}_{\eta}\circ\mathcal{N}_{g}$ with $g=2t-1$ and $\eta=t/(2t-1)$.
• Figure 3: Beam-splitter state. A balanced beam-splitter ($\eta=1/2$) acts on a separable state $\hat{\rho}_{\mathrm{sep}}$. After tracing over the second output mode, the beam-splitter state $\hat{\sigma}$ is obtained. Beam-splitter states are always Wigner positive.
• Figure 4: Pictorial depiction of our method for generating extreme WPS. We refer to Table \ref{['table:notations_convex_sets']} for notations. The set of WPS $\mathcal{D}_{+}$ (blue-shaded disc) is contained in the set of WPQS $\mathcal{A}_{+}$ (outer solid-line black ellipse). The subset of Fock-bounded phase-invariant WPQS $\mathcal{A}^{n}_{\oplus}$ (inner dashed ellipse) intersects with the set of Fock-bounded WPS $\mathcal{D}^{n}_{+}$ (dotted curve). The Vertigo map $\mathcal{V}$ (orange arrows) maps extreme points of $\mathcal{A}^{n}_{\oplus}$ to extreme points of $\mathcal{D}^{n}_\oplus$. The Fock-bounded displacement-like operator $\bar{D}$ (green arrows) sends extreme states of $\mathcal{D}^{n}_{\oplus}=\mathcal{A}^{n}_{\oplus}\cap\mathcal{D}^{n}_{+}$ to extreme states of $\mathcal{D}^{n}_{+}$. Finally, Gaussian unitary operations $G$ (yellow arrows) send extreme states of $\mathcal{D}^{n}_{+}$ to extreme states of $\mathcal{D}_{+}$.
• Figure 5: Chain of results for constructing extreme WPS. Any quasi-state $\hat{A} \in \mathrm{Extr}(\mathcal{A}^{n}_{\oplus})$, as characterized by Theorem \ref{['th:extrAo+n']}, is also an extreme point of $\mathcal{A}^{n}_{+}$ (Lemma \ref{['lemma:extreme-fock-diagonal-quasi-states-are-extreme']}). Under Vertigo evolution, $\hat{A}$ enters the set $\mathcal{D}$ at some finite $t_0$ (Lemma \ref{['lemma:vertigo_quasi-states_to_states']}), while remaining extreme in $\mathcal{A}^{n}_{+}$ (Lemma \ref{['lemma:vertigo_extreme_trajectories']}). Then, for $t\geq t_0$, Lemma \ref{['lemma:extreme_superset']} ensures that $\mathcal{V}^{\mathrm{norm}}_{t}[\hat{A}]$ belongs to $\mathrm{Extr}(\mathcal{D}^{n}_{+})$, and Lemma \ref{['lemma:extremality_preserving_map_v2']} guarantees that extremality is preserved under any $\bar{D}$-orbit. Finally, Lemma \ref{['lemma:extreme_subset_projector']} shows that the resulting state is also extreme in $\mathcal{D}_{+}$, and remains so under any Gaussian orbit (Lemma \ref{['lemma:extremality_preserving_map_v2']}).

Theorems & Definitions (32)

• Definition 1: Beam-splitter state
• Definition 2: Extreme point
• Theorem 1: Krein--Milman theorem for WPS, informal statement of Theorem 40 in mathpaper
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 2: Extreme points of $\mathcal{A}^{n}_{\oplus}$
• Lemma 4: Extreme Vertigo trajectories
• Lemma 5: $\mathcal{V}$ maps quasi-states to states
• Theorem 3: $\mathcal{V}$ maps extreme WPQS to extreme WPS