Complex Wigner entropy and Fisher control of negativity in an oval quantum billiard

TL;DR

The paper tackles quantifying Wigner negativity in phase space by extending Gibbs–Shannon entropy to the real Wigner function, where the imaginary part of the complex entropy provides an entropy-like measure of nonclassicality proportional to the negativity volume $N( heta)$. It introduces sign-resolved positive/negative channels with shape probabilities $P_{ heta, pm}$ to separate the total amount of negativity from its spatial distribution. A Fisher-information analysis on the negative channel yields $F_-( heta)$ and the noncentered $ ilde{F}_-( heta)$, leading to the bound $|dh_i/d heta| \

Abstract

We develop a complex-entropy framework for Wigner negativity and apply it to avoided crossings in an oval quantum billiard. For a real Wigner function the Gibbs--Shannon functional becomes complex; its imaginary part, proportional to the Wigner-negative volume, serves as an entropy-like measure of phase-space nonclassicality. A sign-resolved decomposition separates the total negative weight from its phase-space distribution and defines a negative-channel Fisher information that quantifies how sensitively the negative lobe reshapes as a control parameter is varied. This structure yields a Cauchy--Schwarz bound that limits how rapidly the imaginary entropy, and hence the Wigner negativity, can change with the parameter. In the oval billiard, avoided crossings display enhanced negativity and an amplified negative-channel Fisher response, providing a clear phase-space signature of mode hybridization. The construction is generic and extends to other wave-chaotic and mesoscopic systems with phase-space representations.

Figures (4)

• Figure 1: (Top) Eigenvalue trajectories of two interacting modes near the avoided crossing, with sampling points $A\!-\!C$ (mode 1) and $D\!-\!F$ (mode 2). (Bottom) Corresponding intensity patterns $|\psi(x,y)|^{2}$, showing the exchange of spatial structure between the two modes across the avoided-crossing region.
• Figure 2: Representative Wigner sections corresponding to the eigenvalue trajectories in Fig. 1. Panels (a),(b) show mode 2 at $D$; (c),(d) at $E$; and (e),(f) at $F$. Panels (g),(h) show mode 1 at $A$; (i),(j) at $B$; and (k),(l) at $C$. Each pair displays the phase-space slices $W_\vartheta(x,y{=}0,p_x,p_y{=}0)$ (left) and $W_\vartheta(x{=}0,y,p_x{=}0,p_y)$ (right). The colormap is centered at zero (red: positive, blue: negative), so blue regions directly visualize Wigner negativity. The negative domains intensify and become more finely structured as the system approaches the avoided-crossing region, providing a clear phase-space signature of mode hybridization.
• Figure 3: Complex Wigner entropy for the two modes shown in Fig. 1. Panels (a) and (b) correspond to mode 2, displaying the real part $h_r(\vartheta)$ and the imaginary part $h_i(\vartheta)$. Panels (c) and (d) correspond to mode 1, showing the same quantities for the lower branch. Because $h_i(\vartheta) \propto N(\vartheta)$ (the Wigner negativity), its local maximum near the center of the avoided crossing reflects the enhancement of phase-space interference. The insets in (b) and (d) plot the magnitude $|dh_i/d\vartheta|$, whose nearly vanishing value marks the precise location of the peak of $h_i$, allowing clear identification of the center of the avoided crossing (AC).
• Figure 4: Positive- and negative-channel Fisher informations for the two modes in Fig. 1. Panels (a) and (b) correspond to mode 2, showing the positive-channel $F_{+}(\vartheta)$ and negative-channel $F_{-}(\vartheta)$ on a vertical scale of $10^{8}$. Panels (c) and (d) display the same quantities for mode 1. In both modes $F_{-}(\vartheta)$ is systematically larger, typically exceeding $F_{+}(\vartheta)$ by more than a factor of two, which indicates that the negative Wigner lobe undergoes a stronger parameter-induced reshaping. Both $F_{+}(\vartheta)$ and $F_{-}(\vartheta)$ exhibit clear peaks near the avoided-crossing region, providing a sharp phase-space signature of mode hybridization that is consistent with the maxima of $h_i(\vartheta)$ in Fig. \ref{['Figure-3']}.