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 $\iint W(x,p)\,dx\,dp=1$0 where $\iint W(x,p)\,dx\,dp=1$1: $\iint W(x,p)\,dx\,dp=1$2 The imaginary part, $\iint W(x,p)\,dx\,dp=1$3, 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 $\iint W(x,p)\,dx\,dp=1$4 with $\iint W(x,p)\,dx\,dp=1$5, the entropy satisfies: $\iint W(x,p)\,dx\,dp=1$6 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 $\iint W(x,p)\,dx\,dp=1$7. The complex Wigner entropy evolves according to the extended de Bruijn identity: $\iint W(x,p)\,dx\,dp=1$8 where $\iint W(x,p)\,dx\,dp=1$9 is the complex Fisher information. Importantly, the imaginary part, and thus the total negative volume, decays under Gaussian noise ($W$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$1 Defining $W$2 and $W$3, and normalized distributions $W$4, the negative-lobe statistics $W$5 and $W$6 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, $W$7, quantifies the sensitivity of $W$8 to an external parameter $W$9: $$S_c[W] \equiv -\iint W(x,p)\ln W(x,p)\,dx\,dp = S_R[W] + i\,S_I[W]$$0 A Cauchy–Schwarz–type bound constrains the rate at which the imaginary entropy, and hence Wigner negativity, may vary with $$S_c[W] \equiv -\iint W(x,p)\ln W(x,p)\,dx\,dp = S_R[W] + i\,S_I[W]$$1: $$S_c[W] \equiv -\iint W(x,p)\ln W(x,p)\,dx\,dp = S_R[W] + i\,S_I[W]$$2 where $$S_c[W] \equiv -\iint W(x,p)\ln W(x,p)\,dx\,dp = S_R[W] + i\,S_I[W]$$3 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" $$S_c[W] \equiv -\iint W(x,p)\ln W(x,p)\,dx\,dp = S_R[W] + i\,S_I[W]$$4—the unique Gaussian with matching moments—is: $$S_c[W] \equiv -\iint W(x,p)\ln W(x,p)\,dx\,dp = S_R[W] + i\,S_I[W]$$5 This complex quantity has

$$S_c[W] \equiv -\iint W(x,p)\ln W(x,p)\,dx\,dp = S_R[W] + i\,S_I[W]$$6

exactly the Wigner negative volume, and

$$S_c[W] \equiv -\iint W(x,p)\ln W(x,p)\,dx\,dp = S_R[W] + i\,S_I[W]$$7

the difference in (real-part) entropy between $$S_c[W] \equiv -\iint W(x,p)\ln W(x,p)\,dx\,dp = S_R[W] + i\,S_I[W]$$8 and $$S_c[W] \equiv -\iint W(x,p)\ln W(x,p)\,dx\,dp = S_R[W] + i\,S_I[W]$$9.

$S_R[W] = -\iint W(x,p)\ln|W(x,p)|\,dx\,dp$0 is faithful ($S_R[W] = -\iint W(x,p)\ln|W(x,p)|\,dx\,dp$1 iff $S_R[W] = -\iint W(x,p)\ln|W(x,p)|\,dx\,dp$2 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 $S_R[W] = -\iint W(x,p)\ln|W(x,p)|\,dx\,dp$3 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: $S_R[W] = -\iint W(x,p)\ln|W(x,p)|\,dx\,dp$4 yields $S_R[W] = -\iint W(x,p)\ln|W(x,p)|\,dx\,dp$5, $S_R[W] = -\iint W(x,p)\ln|W(x,p)|\,dx\,dp$6.
• Fock state: The Wigner function of $S_R[W] = -\iint W(x,p)\ln|W(x,p)|\,dx\,dp$7 exhibits alternating positive/negative rings; $S_R[W] = -\iint W(x,p)\ln|W(x,p)|\,dx\,dp$8 increases with $S_R[W] = -\iint W(x,p)\ln|W(x,p)|\,dx\,dp$9 and tracks growing nonclassicality.
• Schrödinger cat states: For $S_I[W] = -\iint W(x,p)\arg W(x,p)\,dx\,dp$0 with large $S_I[W] = -\iint W(x,p)\arg W(x,p)\,dx\,dp$1, $S_I[W] = -\iint W(x,p)\arg W(x,p)\,dx\,dp$2 and $S_I[W] = -\iint W(x,p)\arg W(x,p)\,dx\,dp$3 discriminating two-component negativity.
• Oval quantum billiard: As a control parameter $S_I[W] = -\iint W(x,p)\arg W(x,p)\,dx\,dp$4 tunes the billiard boundary, both the negative volume $S_I[W] = -\iint W(x,p)\arg W(x,p)\,dx\,dp$5 and negative-channel Fisher information $S_I[W] = -\iint W(x,p)\arg W(x,p)\,dx\,dp$6 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_I[W] = -\iint W(x,p)\arg W(x,p)\,dx\,dp$7 (real/entropy)	$S_I[W] = -\iint W(x,p)\arg W(x,p)\,dx\,dp$8 (imaginary/negativity)
Thermal state	$S_I[W] = -\iint W(x,p)\arg W(x,p)\,dx\,dp$9	$W$0
Fock state $W$1	increases with $W$2	$W$3 increases
Cat state	$W$4	$W$5

6. Interpretation, Applications, and Limitations

The real part $W$6 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 $W$7 provides a faithful, operationally meaningful quantifier of nonclassicality via negativity.

The framework enables:

• Resource-theoretic monotones for Wigner negativity ($W$8, $W$9) and non-Gaussianity ($\arg W(x,p)$0, $\arg W(x,p)$1).
• 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).