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Physica Scripta

Local Wigner-weight maps and integrated negativity as measures of nonclassicality in quantum chaotic billiards

Kyu-Won Park, Soojoon Lee, Kabgyun Jeong

Published May 21, 2026• 0 citations
Publisher

Abstract

Abstract The Wigner function is a phase-space quasiprobability distribution whose negative regions provide a direct local signature of nonclassicality. To identify where phase-sensitive structure concentrates, we define local positive and negative Wigner-weight maps and use the integrated Wigner negativity as a compact scalar diagnostic of nonclassical phase-space structure. A decomposition of the density operator reveals that off-diagonal coherences between hybridizing components generate oscillatory sign-alternating patterns, with the negative contribution maximized when component weights are comparable. Non-Gaussian chaotic eigenmodes exhibit a baseline negativity that is further amplified by such hybridization. We demonstrate the robustness of these diagnostics across two billiard geometries and suggest that the same sign-resolved analysis can be applied to other wave-chaotic platforms, where it may aid mode engineering and coherence control.

DOI
10.1088/1402-4896/ae719c

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Abstract The Wigner function is a phase-space quasiprobability distribution whose negative regions provide a direct local signature of nonclassicality. To identify where phase-sensitive structure concentrates, we define local positive and negative Wigner-weight maps and use the integrated Wigner negativity as a compact scalar diagnostic of nonclassical phase-space structure. A decomposition of the density operator reveals that off-diagonal coherences between hybridizing components generate oscillatory sign-alternating patterns, with the negative contribution maximized when component weights are comparable. Non-Gaussian chaotic eigenmodes exhibit a baseline negativity that is further amplified by such hybridization. We demonstrate the robustness of these diagnostics across two billiard geometries and suggest that the same sign-resolved analysis can be applied to other wave-chaotic platforms, where it may aid mode engineering and coherence control.
The Wigner function is a phase space quasi-probability distribution whose negative regions provide a direct, local signature of nonclassicality. To identify where phase-sensitive structure concentrates, we introduce local positive- and negative Wigner-mass maps and adopt the integrated Wigner negativity as a compact scalar measure of nonclassical phase space structure. A decomposition of the density operator reveals that off-diagonal coherences between hybridizing components generate oscillatory, sign-alternating patterns, with the negative contribution maximized when component weights are comparable. Non-Gaussian chaotic eigenmodes exhibit a baseline negativity that is further amplified by such hybridization. We validate these diagnostics across two billiard geometries and argue that the framework is transferable to other wave-chaotic platforms, where it can aid mode engineering and coherence control.
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.
A quantum state can be written in phase space, but the resulting object is not generally the probability density of a positive stochastic process on ordinary phase space. We spell this out for Wigner dynamics. If a positive phase-space process is required only to reproduce the Born density after integrating over momentum, the requirement fixes only an integrated current; the local drift and diffusion remain underdetermined. If one instead requires all Weyl-ordered expectation values, the phase-space object is fixed to be the Wigner function. For non-quadratic potentials the Wigner--Moyal generator contains higher-order, signed momentum-transfer terms, so it is not the Fokker--Planck generator of a positive Brownian diffusion. The exact Wigner function must therefore be reconstructed, in a stochastic representation, as a weighted empirical measure \[ \FW(\z,t)=\E_{\Pp}[W_tδ(\z-\z_t)], \qquad \z=(q,p), \] rather than the unweighted density of sampled carrier trajectories. With classical Hamiltonian flow as the carrier, all nonclassical correction beyond classical transport sits in the Moyal residual and can be represented by signed weights or branching events. The same split defines a residual diagnostic that vanishes for quadratic Hamiltonians and measures what classical carrier transport misses in anharmonic dynamics. The formulation also gives a forward--reverse relation for signed Wigner path measures. The ratio of forward and reversed contributions separates into a positive magnitude factor and a sign factor. This sign records the parity of the Wigner interference contribution; it is not a thermodynamic entropy production.
We propose a hierachy of nonclassicality criteria in phase space. Our formalism covers the negativity in phase space as a special case and further adresses nonclassicality for quantum states with positive phase-space distributions. Remarkably, it enables us to detect every nonclassical Gaussian state and every finite dimensional state in Fock basis by looking into only three phase-space points. Furthermore, our approach provides an experimentally accessible lower bound for the nonclassicality measure based on trace distance. We also extend our method to detecting genuine quantum non-Gaussianity of a state with a non-negative Wigner function. We finally establish our formalism by employing generalized quasiprobability distributions to demonstrate its power for a practical test using an on-off detector array.
Providing an operational characterization of the Wigner-positive states (WPS), i.e., the set of quantum states with non-negative Wigner function, is a longstanding open problem. For pure states, the only WPS are Gaussian states, but the situation is considerably more subtle for mixed states. Here, we approach the problem using convex geometry, reducing the question to the characterization of the extreme points of the set of WPS. We give a constructive method to generate a large class of such extreme WPS, which combines the following steps: (i) we characterize the phase-invariant extreme points of the superset of Wigner-positive quasi-states (WPQS); (ii) we introduce a new quantum map, named Vertigo map, which maps extreme WPQS to extreme WPS while preserving phase invariance; (iii) we identify families of extremality-preserving maps and use them to obtain extreme WPS while relaxing phase invariance. Our construction generates all extreme WPS of low dimension, starting from a specific kind of WPS known as beam-splitter states. Our results build upon new mathematical properties of the set of WPS derived in a companion paper and unveil the remarkable structure of mixed states with non-negative Wigner functions.