Complex-Entropy Framework for Wigner Negativity

• The paper presents a framework that extends classical entropy concepts to quantify Wigner negativity through complex-valued measures.
• It unifies classical uncertainty and quantum nonclassicality, remaining invariant under Gaussian unitaries while using the imaginary part to track negativity.
• The approach offers practical metrics for non-Gaussian state synthesis and dynamic phase-space analysis in quantum control applications.

A complex-entropy framework for Wigner negativity provides a systematic method to quantify and analyze quantum nonclassicality directly in phase space. By extending classical information-theoretic functionals, such as the Gibbs–Shannon entropy and relative entropy, to the field of sign-indefinite quasiprobability distributions (notably the Wigner function), the framework yields complex-valued entropic quantities whose imaginary part precisely isolates and quantifies the Wigner-negative regions. This unifies classical uncertainty (spread) and nonclassicality (negativity) in a single analytic structure, enabling rigorous tracking of their evolution under Gaussian unitaries, noise, and external control.

1. Complex Entropy for Real Wigner Functions

The Wigner function $W(x, p)$, normalized as $\iint W(x,p)\,dx\,dp=1$, typically takes both positive and negative values. Extending the Shannon differential entropy to a real but sign-changing $W$, the complex Wigner entropy is defined by analytic continuation of the logarithm: $$S_c[W] \equiv -\iint W(x,p)\ln W(x,p)\,dx\,dp = S_R[W] + i\,S_I[W]$$ with

$S_R[W] = -\iint W(x,p)\ln|W(x,p)|\,dx\,dp$

$S_I[W] = -\iint W(x,p)\arg W(x,p)\,dx\,dp$

Given that $W$ is real, $\arg W(x,p)$ is $0$ in regions where $W>0$ and $\pi$ where $W<0$: $$S_I[W] = \pi\int_{W<0}|W(x,p)|\,dx\,dp = \pi\,{\rm Vol}_{-}(W)$$ The imaginary part, $S_I[W]$, is thus exactly proportional to the total negative volume of the Wigner function; this quantity is widely recognized as the Kenfack–Życzkowski negativity and serves as a precise measure of phase-space nonclassicality (Cerf et al., 2023, Park et al., 3 Dec 2025, Pizzimenti et al., 2023).

2. Invariance under Gaussian Unitaries and Channels

Both real and imaginary parts of the complex Wigner entropy are invariant under Gaussian unitaries—combinations of phase-space displacements, rotations, and squeezing transformations. Consider a phase-space transformation $\mathbf{X}' = \mathcal{S}\mathbf{X} + \mathbf{d}$ with $\det\mathcal S=1$, the entropy satisfies: $S_R[W'] = S_R[W], \quad S_I[W'] = S_I[W]$ This invariance ensures the complex entropy fundamentally characterizes intrinsic phase-space features, independent of specific Gaussian transformations (Cerf et al., 2023, Pizzimenti et al., 2023).

Under Gaussian noise (additive convolution of the Wigner function with a Gaussian kernel), the evolution is governed by the diffusion equation $\partial_t W = \frac12\Delta W$. The complex Wigner entropy evolves according to the extended de Bruijn identity: $\frac{d}{dt} S_c[W] = \frac12 I_c[W]$ where $I_c[W]$ is the complex Fisher information. Importantly, the imaginary part, and thus the total negative volume, decays under Gaussian noise ($I_I \le 0$), reflecting monotonic classicalization (Cerf et al., 2023).

3. Sign-Resolved Channel Decomposition and Fisher Information

A refined analysis decomposes the Wigner function into positive and negative "lobes": $$W_+(r,p)=\max\{W(r,p),0\},\quad W_-(r,p)=\max\{-W(r,p),0\}$$ Defining $Z_+ = \iint W_+\,dr\,dp$ and $Z_- = \iint W_-\,dr\,dp = N$, and normalized distributions $P_{\pm} = W_{\pm}/Z_{\pm}$, the negative-lobe statistics $N = Z_-$ and $P_-$ respectively measure the total negative volume and its phase-space distribution (Park et al., 3 Dec 2025).

The Fisher information associated with the negative lobe, $I_-(\vartheta)$, quantifies the sensitivity of $P_-$ to an external parameter $\vartheta$: $$I_-(\vartheta) = \iint P_-(r,p;\vartheta) \left[ \partial_{\vartheta}\ln P_-(r,p;\vartheta) \right]^2\,dr\,dp$$ A Cauchy–Schwarz–type bound constrains the rate at which the imaginary entropy, and hence Wigner negativity, may vary with $\vartheta$: $$\left|\frac{dh_i}{d\vartheta}\right| \le \pi N \sqrt{\widetilde I_-}$$ where $\widetilde I_-$ is the non-centered second moment of the sensitivity score. This gives an operational limit on the reconfigurability of negativity in phase space (Park et al., 3 Dec 2025).

4. Complex-Valued Relative Entropy and Non-Gaussianity

For broader quantification of nonclassicality, the complex-valued relative entropy between a generic Wigner function and its Gaussian "associate" $G(r)$—the unique Gaussian with matching moments—is: $$\mu[W] \equiv D[W\Vert G] = \int W(r)\ln\frac{W(r)}{G(r)}\,dr$$ This complex quantity has

$\Im \mu[W] = \pi \int_{W<0}|W(r)|\,dr$

exactly the Wigner negative volume, and

$\Re\mu[W] = -h[W] + h[G]$

the difference in (real-part) entropy between $W$ and $G$.

$\mu[W]$ is faithful ($\mu[W]=0$ iff $W$ is Gaussian), invariant under Gaussian unitaries, and, under a sufficient monotonicity condition on the Fisher information, the real part decays monotonically under Gaussian channels. This structure enables $\mu[W]$ to serve as a bona fide non-Gaussianity measure unifying phase-space negativity and non-Gaussian shape (Pizzimenti et al., 2023).

5. Case Studies and Illustrative Examples

• Thermal (Gaussian) state: $$W(x,p) = \frac{1}{\pi\sigma^2} \exp(-\frac{x^2+p^2}{\sigma^2})$$ yields $S_I = 0$, $S_R = \ln(\pi \sigma^2) + 1$.
• Fock state: The Wigner function of $|n\rangle$ exhibits alternating positive/negative rings; $S_I=\pi\,{\rm Vol}_{-}$ increases with $n$ and tracks growing nonclassicality.
• Schrödinger cat states: For $$|\psi\rangle \propto |\alpha\rangle + |-\alpha\rangle$$ with large $|\alpha|$, $S_R \approx \ln\pi+1+\ln2$ and $S_I\approx1$ discriminating two-component negativity.
• Oval quantum billiard: As a control parameter $\vartheta$ tunes the billiard boundary, both the negative volume $N(\vartheta)$ and negative-channel Fisher information $I_-(\vartheta)$ peak near avoided crossings, signifying mode hybridization and heightened nonclassical interference. The Fisher bound is saturated at maxima, and phase-space slices exhibit finely structured negative regions (Park et al., 3 Dec 2025).

Example	$S_R$ (real/entropy)	$S_I$ (imaginary/negativity)
Thermal state	$\ln(\pi \sigma^2)+1$	$0$
Fock state $|n\rangle$	increases with $n$	$\pi\,{\rm Vol}_{-}(W)$ increases
Cat state	$\ln\pi+1+\ln2$	$\approx 1$

6. Interpretation, Applications, and Limitations

The real part $S_R$ of the complex Wigner entropy generalizes phase-space differential entropy for distributions that may change sign, but loses direct operational meaning in highly nonclassical (sign-indefinite) cases. The imaginary part $S_I$ provides a faithful, operationally meaningful quantifier of nonclassicality via negativity.

The framework enables:

• Resource-theoretic monotones for Wigner negativity ($S_I$, $\Im\mu$) and non-Gaussianity ($S_R$, $\Re\mu$).
• Figures of merit for non-Gaussian state synthesis (cat/GKP encoding, photon subtraction).
• Quantitative diagnostics for phase-space nonclassicality in mesoscopic and wave-chaotic systems with phase-space representations.

The construction is generic and extends to any real-valued phase-space representation, and to situations where symmetry-breaking or parameter control (as in quantum billiards) renders phase-space negativity and its reconfiguration experimentally or theoretically significant (Cerf et al., 2023, Park et al., 3 Dec 2025, Pizzimenti et al., 2023).

7. Broader Perspective and Outlook

The complex-entropy approach unifies the quantification of entropy and phase-space negativity, naturally connecting the geometry of the Wigner function’s support to information-theoretic measures. The sign-resolved decomposition and Fisher-channel analysis enable sensitivity diagnostics with operationally significant bounds. While the real part of the entropy loses some classical interpretation in general, the imaginary part supplies a robust, invariant, and monotone quantifier of quantum nonclassicality.

Numerical evidence shows robust monotonicity properties of the complex-valued non-Gaussianity measure under loss and noise, especially in quantum-optical resource states. A plausible implication is that these complex-entropy metrics are well-suited for optimizing and benchmarking non-Gaussian quantum protocols beyond what is possible with purely classical or negativity-only diagnostics.

Key results lay the groundwork for future studies extending these entropic and Fisher-information frameworks to multimode, high-dimensional, and dynamical quantum settings, and for investigating the operational role of quantum negativity across a broad class of wave-chaotic and quantum control systems (Cerf et al., 2023, Park et al., 3 Dec 2025, Pizzimenti et al., 2023).