---
title: Wigner Logarithmic Negativity
url: https://www.emergentmind.com/topics/wigner-logarithmic-negativity
type: topic
---

# Wigner Logarithmic Negativity

Wigner logarithmic negativity (WLN) is a fundamental resource-theoretic quantifier of quantum state non-classicality, constructed from the phase-space Wigner function. By taking the logarithm of the $L^1$-norm of the Wigner function, WLN provides an additive and operationally meaningful measure of Wigner negativity for both continuous-variable (CV) and discrete-variable (DV) systems. It underpins resource theories of non-Gaussianity, magic states, and complexity, and serves as a primitive monotone controlling the cost and power of non-Gaussian state manipulation and classical simulation hardness [1804.05763][2506.02110].

## 1. Formal Definition

For an $n$-mode CV density operator $\rho$, the Wigner function is
\[
W_\rho(\mathbf r) = \frac1{(2\pi)^n}\int d^{2n}\mathbf s\, e^{i\,\mathbf s^T \Omega\, \mathbf r}\, \chi_\rho(-\mathbf s)
\]
where $\chi_\rho$ is the Weyl characteristic function and $\Omega$ the symplectic form. The Wigner logarithmic negativity is defined by
\[
\mathsf W(\rho) = \log\Bigl(\int d^{2n}\mathbf r\,|W_\rho(\mathbf r)|\Bigr) = \log\Vert W_\rho \Vert_1
\]
Here, $\|W_\rho\|_1$ is the $L^1$-norm, interpreted as the sum of absolute values of the Wigner function. For CV states with positive Wigner function, $\|W_\rho\|_1 = 1$, so WLN vanishes.

In DV systems, e.g. for a $D$-dimensional Hilbert space equipped with phase-point operators $A(q,p)$ as in Wootters’ construction, the discrete Wigner function
\[
W_\rho(q,p) = \frac{1}{D}\Tr[\rho\,A(q,p)]
\]
gives rise to discrete Wigner negativity
\[
\mathcal N(\rho) = \sum_{q,p} |W_\rho(q,p)|
\]
and mana or logarithmic Wigner negativity
\[
\mathcal M(\rho) = \log \mathcal N(\rho)
\]
which is directly analogous to the CV WLN [2506.02110].

## 2. Resource Theoretic Context and Operational Meaning

The WLN is embedded in resource theories that distinguish free (classically simulable) operations and states from genuine quantum resources:

- In the CV theory, free operations come from Gaussian protocols: linear optics, squeezers, Gaussian ancillas, homodyne/heterodyne detection, and classical feed-forward.
- Free states are either the Gaussian convex hull $\mathcal G$ or the set of positive-Wigner states $\mathcal W_+ = \{ \rho : W_\rho(\mathbf r) \geq 0\;\forall\mathbf r \}$.

Resource states—those with negative Wigner functions—cannot be generated from free states by free operations, and the degree of negativity quantifies "non-Gaussianity" or "magic." For DV (qudit) systems, the analogous resource theory is that of magic states (Clifford-stabilizer framework), wherein mana is likewise a monotone [1804.05763][2506.02110].

Operationally, WLN sets bounds on conversion rates in resource distillation: if $k$ copies of $\rho$ are used to probabilistically generate $m$ copies of $\sigma$ via Gaussian protocols with success probability $p$, additivity and monotonicity yield
\[
k\,\mathsf W(\rho) \ge p\,m\,\mathsf W(\sigma) \quad\implies\quad \mathbb E[n] \ge m\, \frac{\mathsf W(\sigma)}{\mathsf W(\rho)}
\]
which quantifies the minimal cost in resource consumption.

## 3. Mathematical Properties

WLN displays several key mathematical properties:

- **Faithfulness**: $\mathsf W(\rho) = 0$ if and only if $W_\rho(\mathbf r) \ge 0$ everywhere.
- **Monotonicity**: Non-increasing under deterministic Gaussian protocols, and, on average, under probabilistic (coarse-grained) Gaussian measurements:
  \[
  \sum_i p_i \mathsf W(\rho_i) \le \mathsf W(\rho)
  \]
- **Additivity**:
  \[
  \mathsf W(\rho \otimes \sigma) = \mathsf W(\rho) + \mathsf W(\sigma)
  \]
- **Non-convexity**: The logarithmic form destroys convexity; in general
  \[
  \mathsf W(p\rho + (1-p)\sigma) \not\le p\mathsf W(\rho) + (1-p)\mathsf W(\sigma)
  \]
- **Analogue to Mana**: For DV systems, the mana $\mathcal M$ shares analogous properties, including faithfulness and monotonicity under stabilizer operations [1804.05763][2506.02110].

## 4. Computation and Evaluation

To compute WLN:

- For analytic states (e.g., Fock, cat, or cubic-phase states), closed-form Wigner functions may be integrated numerically.
- In higher dimensions or for mixed/experimental states, numerical procedures such as adaptive quadrature, Monte Carlo phase-space sampling, or phase-space tomography are employed to estimate $\|W_\rho\|_1$.
- In the DV context, $\mathcal M(\rho)$ is computed via straightforward summation over the discrete phase space.

Computational bottlenecks arise for large $n$ or $D$ due to the exponential scaling of phase-space volume.

## 5. Physical Interpretation and Dynamical Growth

WLN measures irreducible quantum nonclassicality:

- In CV frameworks, it quantifies non-Gaussianity not achievable by free Gaussian operations.
- In the DV magic-state resource theory, $\mathcal M$ (mana) quantifies the "magic" resource necessary for operations beyond Clifford circuits, directly controlling classical simulation cost.

In dynamical settings under chaotic Hamiltonians and large Hilbert space dimension $D$:

- **Generic basis**: Wigner negativity (and thus WLN) grows rapidly under time evolution, reaching $O(\sqrt D)$ on $t=O(1)$ timescales, i.e., exponential in $\log D$ [2506.02110].
- **Krylov basis**: Early-time growth of Wigner negativity is only polynomial, as bounded by entropy measures. For $t \ll e^{O(D)}$, $\mathcal M(\psi_t) \lesssim \log t$; only at exponentially late times does WLN reach the generic-basis plateau.

This reveals a deep connection between basis choice, dynamical complexity generation, and emergent semiclassical phase-space pictures (e.g., $q\to0$ limits of JT gravity). The Krylov basis enables an effective semiclassical description at sub-exponential times, whereas generic bases yield rapid complexity growth [2506.02110].

## 6. Representative Examples

WLN’s sensitivity to specific non-Gaussian features is evident in the following cases [1804.05763]:

| State Class    | Notable WLN Behavior                                         | Limiting Value/Dependency           |
|----------------|--------------------------------------------------------------|-------------------------------------|
| Cubic-phase    | $\mathsf W$ increases with nonlinearity $\chi = \gamma e^{3r}$; diverges logarithmically as $\chi \to \infty$    | $\mathsf W(|\gamma,r\rangle) = f(\chi)$    |
| Photon-added/-subtracted Gaussians | Maximized by single-photon Fock state $|1\rangle$    | $\mathsf W(\text{state}) \leq \mathsf W(|1\rangle)$ |
| Cat states     | WLN saturates to finite value as $|\alpha| \to \infty$ (odd cat: approaches $\log 2$) | $\lim_{|\alpha|\to\infty}\mathsf W = \log(1+|\sin 2\phi|)$ (for $\theta=\pi$)  |

WLN therefore distinguishes subtle differences between states: e.g., for cat states, non-Gaussianity measures (relative entropy) diverge, but WLN saturates due to finite-volume negative regions in the Wigner function.

## 7. Comparison to Alternative Measures

- **Relative entropy of non-Gaussianity**: $\delta[\rho] = \inf_{\tau\in\mathcal G} S(\rho\Vert\tau)$ is not monotonic under Gaussian protocols unless convex-roof extended, which is computationally impractical.
- **Volume of negativity**: Fails to yield additive monotones or to satisfy convexity.
- **WLN/Mana**: Satisfies additivity, faithfulness, monotonicity, and efficient computability via $L^1$-integration, providing direct operational bounds for resource-theoretic and complexity-theoretic tasks [1804.05763][2506.02110].

WLN thus emerges as a primitive, robust, and operationally grounded quantifier—distinguishing Gaussian-simulable states (WLN zero) from non-Gaussian resources, dictating ancilla construction costs, and tracking complexity generation under quantum dynamics. Its role links quantum optics, resource theory, simulation complexity, and, via recent developments, semiclassical gravity duality [2506.02110].

Source: https://www.emergentmind.com/topics/wigner-logarithmic-negativity