Wigner Logarithmic Negativity

• Wigner Logarithmic Negativity is a quantifier of quantum non-classicality derived from the L1-norm of the Wigner function, applicable to both continuous-variable and discrete-variable systems.
• It serves as a robust monotone in resource theories by distinguishing non-Gaussian or magic states from free Gaussian states, ensuring additivity and monotonicity under Gaussian protocols.
• WLN provides operational insights into quantum resource distillation, simulation hardness, and dynamical complexity growth, with practical computation via analytic and numerical integration methods.

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 (Albarelli et al., 2018, Basu et al., 2 Jun 2025).

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 $n$0-dimensional Hilbert space equipped with phase-point operators $n$1 as in Wootters’ construction, the discrete Wigner function

$n$2

gives rise to discrete Wigner negativity

$n$3

and mana or logarithmic Wigner negativity

$n$4

which is directly analogous to the CV WLN (Basu et al., 2 Jun 2025).

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 $n$5 or the set of positive-Wigner states $n$6.

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 (Albarelli et al., 2018, Basu et al., 2 Jun 2025).

Operationally, WLN sets bounds on conversion rates in resource distillation: if $n$7 copies of $n$8 are used to probabilistically generate $n$9 copies of $\rho$0 via Gaussian protocols with success probability $\rho$1, additivity and monotonicity yield

$\rho$2

which quantifies the minimal cost in resource consumption.

3. Mathematical Properties

WLN displays several key mathematical properties:

• Faithfulness: $\rho$3 if and only if $\rho$4 everywhere.
• Monotonicity: Non-increasing under deterministic Gaussian protocols, and, on average, under probabilistic (coarse-grained) Gaussian measurements:

$\rho$5

• Additivity:

$\rho$6

• Non-convexity: The logarithmic form destroys convexity; in general

$\rho$7

• Analogue to Mana: For DV systems, the mana $\rho$8 shares analogous properties, including faithfulness and monotonicity under stabilizer operations (Albarelli et al., 2018, Basu et al., 2 Jun 2025).

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 $\rho$9.
• In the DV context, $$W_\rho(\mathbf r) = \frac1{(2\pi)^n}\int d^{2n}\mathbf s\, e^{i\,\mathbf s^T \Omega\, \mathbf r}\, \chi_\rho(-\mathbf s)$$0 is computed via straightforward summation over the discrete phase space.

Computational bottlenecks arise for large $$W_\rho(\mathbf r) = \frac1{(2\pi)^n}\int d^{2n}\mathbf s\, e^{i\,\mathbf s^T \Omega\, \mathbf r}\, \chi_\rho(-\mathbf s)$$1 or $$W_\rho(\mathbf r) = \frac1{(2\pi)^n}\int d^{2n}\mathbf s\, e^{i\,\mathbf s^T \Omega\, \mathbf r}\, \chi_\rho(-\mathbf s)$$2 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, $$W_\rho(\mathbf r) = \frac1{(2\pi)^n}\int d^{2n}\mathbf s\, e^{i\,\mathbf s^T \Omega\, \mathbf r}\, \chi_\rho(-\mathbf s)$$3 (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 $$W_\rho(\mathbf r) = \frac1{(2\pi)^n}\int d^{2n}\mathbf s\, e^{i\,\mathbf s^T \Omega\, \mathbf r}\, \chi_\rho(-\mathbf s)$$4:

• Generic basis: Wigner negativity (and thus WLN) grows rapidly under time evolution, reaching $$W_\rho(\mathbf r) = \frac1{(2\pi)^n}\int d^{2n}\mathbf s\, e^{i\,\mathbf s^T \Omega\, \mathbf r}\, \chi_\rho(-\mathbf s)$$5 on $$W_\rho(\mathbf r) = \frac1{(2\pi)^n}\int d^{2n}\mathbf s\, e^{i\,\mathbf s^T \Omega\, \mathbf r}\, \chi_\rho(-\mathbf s)$$6 timescales, i.e., exponential in $$W_\rho(\mathbf r) = \frac1{(2\pi)^n}\int d^{2n}\mathbf s\, e^{i\,\mathbf s^T \Omega\, \mathbf r}\, \chi_\rho(-\mathbf s)$$7 (Basu et al., 2 Jun 2025).
• Krylov basis: Early-time growth of Wigner negativity is only polynomial, as bounded by entropy measures. For $$W_\rho(\mathbf r) = \frac1{(2\pi)^n}\int d^{2n}\mathbf s\, e^{i\,\mathbf s^T \Omega\, \mathbf r}\, \chi_\rho(-\mathbf s)$$8, $$W_\rho(\mathbf r) = \frac1{(2\pi)^n}\int d^{2n}\mathbf s\, e^{i\,\mathbf s^T \Omega\, \mathbf r}\, \chi_\rho(-\mathbf s)$$9; 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., $\chi_\rho$0 limits of JT gravity). The Krylov basis enables an effective semiclassical description at sub-exponential times, whereas generic bases yield rapid complexity growth (Basu et al., 2 Jun 2025).

6. Representative Examples

WLN’s sensitivity to specific non-Gaussian features is evident in the following cases (Albarelli et al., 2018):

State Class	Notable WLN Behavior	Limiting Value/Dependency
Cubic-phase	$\chi_\rho$1 increases with nonlinearity $\chi_\rho$2; diverges logarithmically as $\chi_\rho$3	$\chi_\rho$4
Photon-added/-subtracted Gaussians	Maximized by single-photon Fock state $\chi_\rho$5	$\chi_\rho$6
Cat states	WLN saturates to finite value as $\chi_\rho$7 (odd cat: approaches $\chi_\rho$8)	$\chi_\rho$9 (for $\Omega$0)

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: $\Omega$1 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 $\Omega$2-integration, providing direct operational bounds for resource-theoretic and complexity-theoretic tasks (Albarelli et al., 2018, Basu et al., 2 Jun 2025).

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 (Basu et al., 2 Jun 2025).