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.
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.
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.
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.
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.
A classical simulation scheme of quantum computation given a restricted set of states and measurements may be---occasionally, but only occasionally---interpreted naturally as a statistical simulation of positive quasi-probability distributions on a phase space. In this dissertation, we explore phase space representations for finite-dimensional quantum systems and their negativities beyond the usual analogues of the Wigner function. The first line of study focuses on a characterization tool for valid quasi-probability distributions of (possibly mixed) quantum states. A quantum generalization of Bochner's theorem from classical probability theory simultaneously characterizes both the set of valid Wigner functions and the subset of positive ones. We extend this theorem to discrete phase spaces based on projective representations of abelian groups, including the discrete Wigner functions of Gross (J. Math. Phys. 47, 122107 (2006)) and Gibbons et al. (Phys. Rev. A 70, 062101 (2004)). The rest of the dissertation is then dedicated to phase space representations covariant with respect to a general Lie group. This means, in particular, that operators that lie in the same orbit of the group have the same amount of negativity in their quasi-probabilities---a natural requirement if one would like to interpret negativity as non-classicality, and the group action as a classical dynamics. We show that the construction due to Brif and Mann (Phys. Rev. A 59, 971 (1999)) of group covariant phase space representations satisfying the so-called Stratonovich-Weyl correspondence is essentially unique, given the group and an input state in the relevant representation. As an application, we construct quasi-probability representations on a compact phase space of fermionic Gaussian states tailored to a classically simulatable problem of fermionic linear optics and find that, in the first non-trivial case of four fermionic modes, the quasi-probabilities of the Gaussian states exhibit a considerable amount of negativity.
While negativity in phase space is a well-known signature of nonclassicality, a wide variety of nonclassical states require their characterization beyond negativity. We establish a framework of nonclassicality in phase space that addresses nonclassical states comprehensively with a direct experimental evidence. This includes the negativity of phase-space distribution as a special case and further analyzes quantum states with positive distributions effectively. We prove that it detects all nonclassical Gaussian states and all non-Gaussian states of arbitrary dimension remarkably by examining three phase-space points only. Our formalism also provides an experimentally accessible lower bound for a nonclassicality measure based on trace distance. Importantly, this foundational approach can be further adapted to constitute practical tests in two directions looking into particle and wave nature of bosonic systems, via an array of nonideal on-off detectors and coarse-grained homodyne measurement, respectively. All these tests are practically powerful in characterizing nonclasssical states reliably against noise, making a versatile tool for a broad range of quantum systems in quantum technologies.
Abstract We develop a phase-space description of tunneling in the quasi-exact sextic double well VQES(x)=21(x6+2x4−2(2λ+1)x2) V QES ( x ) = 1 2 ( x 6 + 2 x 4 - 2 ( 2 λ + 1 ) x 2 ) based on the Wigner quasiprobability. From high-accuracy wavefunctions, we introduce the complex Shannon entropy SW S W of the Wigner function W ( x , p ), whose imaginary part isolates coherence and negativity, and the entropy S∣W∣ S | W | of its modulus, a strictly real indicator of phase-space spreading. Together with the mutual information IW I W , these measures yield sharp signatures of the transition into the double-well regime ( λ>−21 λ > - 1 2 ) and the emergence of tunneling-induced parity doublets. High-precision spectral data are supported by a smooth Gaussian-process interpolant anchored in asymptotic scaling, ensuring analytic control of level behavior. The resulting phase-space framework provides a compact and sensitive characterization of nonclassical structure beyond probability-density approaches.