arXiv:2512.03505

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

Kyu-Won Park, Jongin Jeong

quant-phSubmitted Dec 3, 2025• Announced Dec 4, 2025• 0 citations

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.

Comments: 7 pages, 4 figures

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arXiv:2512.03505

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

Kyu-Won Park, Jongin Jeong

quant-phSubmitted Dec 3, 2025• Announced Dec 4, 2025• 0 citations

arXiv HTML PDF

Abstract

Comments: 7 pages, 4 figures

Related papers

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Abstract We develop a phase-space description of tunneling in the quasi-exact sextic double well $V^{\textrm{QES}}(x) = \tfrac{1}{2}\,(x^6 + 2x^4 - 2(2\lambda +1)\,x^{2})$ 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 $S_W$ 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 $I_W$ I W , these measures yield sharp signatures of the transition into the double-well regime ( $\lambda >-\frac{1}{2}$ 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.Abstract We develop a phase-space description of tunneling in the quasi-exact sextic double well $$V^{\textrm{QES}}(x) = \tfrac{1}{2}\,(x^6 + 2x^4 - 2(2\lambda +1)\,x^{2})$$ 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 $$S_W$$ 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 $$I_W$$ I W , these measures yield sharp signatures of the transition into the double-well regime ( $$\lambda >-\frac{1}{2}$$ 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.

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Axiomatization of Rényi Entropy on Quantum Phase Space

Adam M. Brandenburger, Pierfrancesco La Mura

Phase-space versions of quantum mechanics -- from Wigner's original distribution to modern discrete-qudit constructions -- represent some states with negative quasi-probabilities. Conventional Shannon and Rényi entropies become complex-valued in this setting and lose their operational meaning. Building on the axiomatic treatments of Rényi (1961) and Daróczy (1963), we develop a conservative extension that applies to signed finite phase spaces and identify a single admissible entropy family, which we call signed Rényi $α$ -entropy (for a free parameter $α\ge 0$ ). The obvious signed Shannon candidate is ruled out because it violates extensivity. We prove four results that bolster the usefulness of the new measure. (i) It serves as a witness of the presence of cancellation, detecting the coexistence of positive and negative weight in a signed measure. (ii) For $α> 1$ , it is Schur-concave, delivering the intuitive property that mixing increases entropy (iii) The same parametric family obeys a quantum H-theorem, namely, that under de-phasing dynamics entropy cannot decrease. (iv) The $2$ -entropy is conserved under discrete Moyal-bracket dynamics, mirroring conservation of von Neumann entropy under unitary evolution on Hilbert space. We also comment on interpreting the Rényi order parameter as an inverse temperature. Overall, we believe that our investigation provides good evidence that our axiomatically derived signed Rényi entropy may be a useful addition to existing entropy measures employed in quantum information, foundations, and thermodynamics.Phase-space versions of quantum mechanics -- from Wigner's original distribution to modern discrete-qudit constructions -- represent some states with negative quasi-probabilities. Conventional Shannon and Rényi entropies become complex-valued in this setting and lose their operational meaning. Building on the axiomatic treatments of Rényi (1961) and Daróczy (1963), we develop a conservative extension that applies to signed finite phase spaces and identify a single admissible entropy family, which we call signed Rényi $α$-entropy (for a free parameter $α\ge 0$). The obvious signed Shannon candidate is ruled out because it violates extensivity. We prove four results that bolster the usefulness of the new measure. (i) It serves as a witness of the presence of cancellation, detecting the coexistence of positive and negative weight in a signed measure. (ii) For $α> 1$, it is Schur-concave, delivering the intuitive property that mixing increases entropy (iii) The same parametric family obeys a quantum H-theorem, namely, that under de-phasing dynamics entropy cannot decrease. (iv) The $2$-entropy is conserved under discrete Moyal-bracket dynamics, mirroring conservation of von Neumann entropy under unitary evolution on Hilbert space. We also comment on interpreting the Rényi order parameter as an inverse temperature. Overall, we believe that our investigation provides good evidence that our axiomatically derived signed Rényi entropy may be a useful addition to existing entropy measures employed in quantum information, foundations, and thermodynamics.

arXiv:2410.15976Oct 21, 20240 citations

Abstract We develop a phase-space description of tunneling in the quasi-exact sextic double well

V^{\textrm{QES}}(x) = \tfrac{1}{2}\,(x^6 + 2x^4 - 2(2\lambda +1)\,x^{2})

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

S_W

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

I_W

I W , these measures yield sharp signatures of the transition into the double-well regime (

\lambda >-\frac{1}{2}

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.

2\times10^{-21}

preserves a rigorously established entropy decrease. Optimisation over all one-mode Wigner-nonnegative states gives the sharp low-energy scale

E^γ/[\ln(1/E)]^β

, with

γ\simeq0.7412

and

β\simeq0.5861

. Because

γ<1

, tensor products can have vanishing total energy and trace distance from vacuum while their entropy deficit diverges. We derive the exact half-transmission threshold for universal Shannon-entropy recovery and connect it to companion results showing residual non-Gaussian structure. The counterexamples reveal distinct controls on entropy: coherence sets the local descent, mode number amplifies it, and loss restores the vacuum bound.

α

-entropy (for a free parameter

α\ge 0

). The obvious signed Shannon candidate is ruled out because it violates extensivity. We prove four results that bolster the usefulness of the new measure. (i) It serves as a witness of the presence of cancellation, detecting the coexistence of positive and negative weight in a signed measure. (ii) For

α> 1

, it is Schur-concave, delivering the intuitive property that mixing increases entropy (iii) The same parametric family obeys a quantum H-theorem, namely, that under de-phasing dynamics entropy cannot decrease. (iv) The

2

-entropy is conserved under discrete Moyal-bracket dynamics, mirroring conservation of von Neumann entropy under unitary evolution on Hilbert space. We also comment on interpreting the Rényi order parameter as an inverse temperature. Overall, we believe that our investigation provides good evidence that our axiomatically derived signed Rényi entropy may be a useful addition to existing entropy measures employed in quantum information, foundations, and thermodynamics.