Wigner-Positive States in Quantum Systems

• Wigner-Positive States are quantum states with nonnegative Wigner functions across phase space, defining a convex subset with unique structural properties.
• They are characterized by strict conditions in continuous, discrete, and infinite dimensions, with pure instances limited to Gaussian or stabilizer forms as per Hudson’s theorem.
• WPS underlie efficient classical simulation and resource theories, serving as a bridge between classical and quantum computational regimes.

A Wigner-Positive State (WPS) is a quantum state whose Wigner function is non-negative everywhere in phase space. This property singles out a convex subset of quantum states—across continuous, discrete, finite, and infinite-dimensional systems—with deep operational, structural, and computational implications in quantum information and the theory of quantum nonclassicality.

1. General Definition and Structural Properties

Let $\mathcal{H}$ be a Hilbert space (e.g., $L^2(\mathbb{R})$ for a single mode). For any trace-class density operator $\rho$ on $\mathcal{H}$, the Wigner function $W_\rho:\mathbb{R}^2\rightarrow\mathbb{R}$ (for the continuous-variable case) is

$$W_\rho(q,p) = \frac{1}{2\pi}\int_{\mathbb{R}} e^{ip y} \langle q - y/2| \rho|q + y/2\rangle dy,$$

with generalizations to finite-dimensional, spin, lattice, or angular momentum systems. $\rho$ is Wigner-positive (WPS) if $W_\rho(z)\ge 0$ for all phase-space points $z$.

The set $\mathcal{D}_+$ of Wigner-positive states is a closed, convex subset of the state space $\mathcal{D}$. In finite dimensions, $\mathcal{D}_+$ is compact and its (relative) interior consists of full-rank WPS whose Wigner functions are strictly positive everywhere. On $L^2(\mathbb{R})$, $\mathcal{D}_+$ is closed but has empty interior: every WPS can be arbitrarily well-approximated by non-WPS in the trace norm topology (Cerf et al., 16 Dec 2025).

The characterization of extreme points of $\mathcal{D}_+$ is essential for their convex-geometric structure. In finite dimensions, every extreme WPS (except for the Gaussian vacuum) has a Wigner function that vanishes somewhere; the boundary is stratified by the nodal sets of the Wigner function.

2. Pure-State Wigner-Positivity and Hudson’s Theorem

For pure states in continuous-variable systems, Hudson’s theorem states that $|\psi\rangle$ has $W_\psi(z)\ge 0$ if and only if $|\psi\rangle$ is a pure Gaussian state (displaced/squeezed vacuum). Non-Gaussian pure states necessarily display Wigner-function negativity, so the only extreme pure WPS are Gaussian (Corney et al., 2014, Herstraeten et al., 16 Dec 2025).

Discrete systems exhibit analogous constraints. For odd-prime $d$ qudit systems, the only pure WPS are stabilizer states (Mari et al., 2012). In the three-generator Wigner formalism for qubits, the pure WPS are precisely the stabilizer states; Clifford gates map WPS to WPS, realizing a non-contextual hidden variable model for Clifford circuits (Kocia et al., 2017).

On discrete, infinite lattices, the only pure WPS are position eigenstates; all other pure states, including discrete Gaussian wave packets, display regions of negativity (Hinarejos et al., 2012). On the quantum circle, only angular momentum eigenstates yield everywhere nonnegative Wigner functions (Skagerstam et al., 28 Apr 2025).

In relativistic quantum mechanics, Hudson’s theorem fails: explicit non-Gaussian Dirac spinors can exhibit strictly positive relativistic Wigner functions due to coherent superposition of particle and antiparticle components (Campos et al., 2014).

3. Convex Geometry, Extremality, and Construction of Mixed WPS

The full set of WPS (mixed states) is significantly richer. By the Krein–Milman theorem (in finite dimensions), $$\mathcal{D}_+ = \text{conv}(\mathrm{Extr}\,\mathcal{D}_+)$$; in infinite dimensions,

$$\mathcal{D}_+ = \text{closure}_1\left (\text{conv}\,\mathrm{Extr}\,\mathcal{D}_+ \right )$$

where closure is in the trace norm (Cerf et al., 16 Dec 2025). The task of characterizing all extreme WPS is nontrivial and the subject of current research (Herstraeten et al., 16 Dec 2025).

Key methods include:

• Phase-invariant extreme WPQS: Diagonal in Fock basis with Wigner function $W_A(\alpha) = P(|\alpha|^2) e^{-2|\alpha|^2}$, with $P(t)$ a specific non-negative polynomial (Herstraeten et al., 16 Dec 2025).
• Vertigo map: An extremality-preserving, phase-space dilation composed with amplification and loss channels, mapping extreme WPQS into extreme WPS. Fixed points of the Vertigo map correspond to binomial states.
• Fock-bounded displacement and Gaussian unitaries: Applied to generate orbits of extreme WPS, possibly breaking phase invariance.

In all cases, one can approximate any WPS as a limit of convex combinations of finite-rank (Fock-bounded) extreme WPS; this provides an operational scheme for constructing arbitrary WPS.

4. Explicit Criteria, Witnesses, and Operational Aspects

For single-mode states, Chabaud et al. introduced an explicit set of linear inequalities that fully characterize $\mathcal{D}_+$ via Fock-state populations:

• For each $n\ge 1$, the $n$-Fock projection $\langle n|\rho|n\rangle$ must not exceed a threshold $f_n^*$. These thresholds are computed by infinite-dimensional linear programming and are optimized over all diagonal states, using sum-of-squares and semidefinite programming hierarchies (Chabaud et al., 2021).
• The set of witnesses $\{|n\rangle\langle n|\}$ (and their displacements) forms a complete separating family for the convex body of Wigner-positive states.

For Gaussian convex hulls, there are non-trivial lower bounds on the value of the Wigner function at the phase-space origin for all convex mixtures of Gaussian states: $$W_\rho(0) \geq \frac{2}{\pi} e^{-2\bar n(1+\bar n)},\qquad \bar n = \text{Tr}\left [\rho\, a^\dagger a \right ]$$ Violation of these bounds (even if $W_\rho(z)\ge 0$ everywhere) certifies quantum non-Gaussianity (Genoni et al., 2013). Extended criteria subject the state to optimal Gaussian operations (displacement, squeezing), leading to detection protocols that require only a small number of Wigner-value and energy measurements.

5. Wigner-Positive States under Operations, Quantum Simulation, and Resource Theory

Wigner-positivity is preserved under a wide range of operations:

• Gaussian unitaries (continuous-variable) and Clifford unitaries (discrete, odd-$d$) always map WPS to WPS (Mari et al., 2012).
• Positive-Wigner quantum channels correspond to Choi matrices with non-negative Wigner functions. The composition of such maps preserves WPS.
• Beam-splitter mixtures of WPS also yield WPS; in particular, all output single-mode states from tracing over arbitrary product-inputs after a 50:50 beam splitter are WPS, and the construction provides an explicit convex decomposition for extremal passive states (Herstraeten et al., 2021, Herstraeten et al., 8 Nov 2024).

The presence of only Wigner-positive states, channels, and measurements allows for efficient classical simulation of quantum circuits via classical Markov processes in phase-space. This generalizes the Gottesman–Knill theorem: the power of negativity as a quantum computational resource is underlined, as only when at least one element in the circuit has Wigner negativity can exponential quantum speedup be achieved (Mari et al., 2012).

6. Wigner Entropy, Entropic Bounds, and Additive Noise

For WPS, the Wigner function is a genuine probability distribution, so its Shannon differential entropy $S_W(\rho)$ (Wigner entropy) is well defined. This entropy is invariant under all Gaussian unitaries and, conjecturally, lower bounded: $S_W(\rho) \geq \ln\pi + 1,$ with equality for pure Gaussian states. This "Wigner-entropy conjecture" has been proven for passive states, and for a broad family—beam-splitter states—rigorously for all $\alpha\geq 1/2$ in the Rényi entropy family: $$h_\alpha(W_\rho) \geq \ln\pi + \frac{\ln\alpha}{\alpha-1}$$ (Herstraeten et al., 2021, Herstraeten et al., 8 Nov 2024). The entropy-power inequality, $N_{\rm out}\geq \eta N_A + (1-\eta)N_B$, takes a form for WPS mirroring the classical additive noise channel.

Because $S_W$ serves as a monotone under Gaussian and beam splitter operations, and tightens entropic uncertainty relations, it is anticipated to be central in the resource theory of Wigner negativity.

7. WPS in Discrete, Lattice, and Spin Systems

• Spin-$j$ systems: Wigner-positivity defines a convex polytope in the spectrum simplex, with geometric characterization via linear constraints of the eigenvalues with respect to the Stratonovich–Weyl kernels. This polytope sits strictly inside the simplex and shows nontrivial relations to symmetric absolute separability and the positivity of the Glauber–Sudarshan $P$-function (Denis et al., 2023).
• Two-qubit X-states: The intersection of Wigner-positivity and separability (double classicality) in the X-state subspace coincides with the set of absolutely separable X-states, characterized by explicit inequalities on the eigenvalues (Khvedelidze et al., 5 Oct 2025).
• Lattice systems: For a particle on an infinite lattice, the only pure Wigner-positive states are the sharply localized basis states; the regular Gaussian packets and their superpositions always show either "ghost" or genuine regular negativity (Hinarejos et al., 2012).
• Circle phase space: Pure angular momentum eigenstates are the sole Wigner-positive pure states; all other superpositions yield negative regions in Wigner-type functions (Skagerstam et al., 28 Apr 2025).

8. Nonclassicality, Contextuality, and Quantum Information Implications

While Wigner-positivity is often interpreted as "classicality," the situation is nuanced:

• Certain positive Wigner function states—Gaussian or mixtures thereof—are not only efficiently classically simulatable, but also serve as boundaries for the onset of nonclassicality in resource theories.
• Strong contextuality, as quantified by violation of non-contextuality Bell-type inequalities, can be achieved by Wigner-positive states when using appropriately chosen (pseudo-spin) observables. In particular, displaced squeezed vacuum states can achieve violations scaling as $0.842(\sqrt{2})^{N-1}$ for $N$ embedded pseudo-spins (Roy, 2018).
• In relativistic quantum theory, non-Gaussian Dirac spinors can be strictly Wigner-positive due to the interplay of particle–antiparticle degrees of freedom, breaking the nonrelativistic link between positivity and classical simulation (Campos et al., 2014).
• In the stochastic simulation of Gaussian states, the use of the positive Wigner representation does not guarantee compliance with Bell inequalities: the "measurement" mapping from the phase-space variable to observed data can introduce physically negative (e.g. normally ordered) outcomes, violating local realism even when the underlying phase-space density is positive (Lantz et al., 2020).

9. Hierarchy of Classicality Notions in Phase Space

A phase-space-based hierarchy emerges: $$\text{Pure Gaussian states} \subset \left( \text{Convex hull of pure Gaussian states} \right) \subset \mathcal{D}_+ \subset \mathcal{D}$$ (Genoni et al., 2013). Distinct from Wigner-positivity are more restrictive convex hulls (Gaussian mixtures), and more permissive notions (mixtures of arbitrary WPS, etc.).

In summary, Wigner-Positive States form a mathematically rich, physically crucial family, pivotal both for understanding classical simulability in quantum computation and for defining operational boundaries between classical and quantum resources in quantum information theory. Recent advances in convex geometry, entropic analysis, and operational criteria have deepened the understanding of this set, though its complete characterization—especially in infinite-dimensional systems—remains a central open challenge.