Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The European Physical Journal Plus

Wigner-entropy and negativity signatures of tunneling in a sextic double well

A. M. Escobar-Ruiz, Angelina N. Mendoza Tavera, Robin P. Sagar, Christian Mateo Frausto-Avila, Mario Alan Quiroz-Juarez

Published Mar 6, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1140/epjp/s13360-026-07462-6

Related papers

Entropic Characterization of Tunneling and State Pairing in a Quasi-exactly Solvable Sextic Potential

Angelina N. Mendoza Tavera, A. M. Escobar-Ruiz, Robin P. Sagar

Abstract We study the (de)localization properties of a quasi-exactly solvable (QES) sextic potential $$V_{\text {QES}}(x) = \tfrac{1}{2}(x^6 + 2x^4 - 2(2\lambda + 1)x^2)$$ as a function of the tunable parameter $$\lambda \in [-\tfrac{3}{4},6]$$ . For the lowest states $$n=0,1,2,3$$ , variational wavefunctions are constructed that respect parity, have correct asymptotics, and admit analytical Fourier transforms. Energies agree with Lagrange-mesh and exact QES results with relative errors of order $$10^{-8}$$ for $$n=0,1,2$$ and $$10^{-6}$$ for $$n=3$$ . We show that information-theoretic indicators (Shannon entropy, Kullback–Leibler and Cumulative Residual Jeffreys divergences) are more sensitive than variance-based measures in detecting tunneling transitions, symmetry breaking, and level pairing. The Beckner–Białynicki-Birula–Mycielski entropic uncertainty relation is verified across all $$\lambda$$ , and trial densities are validated by divergences as small as $$10^{-10}$$ from exact solutions. Importantly, while QES solvability allows for analytic benchmarking, many of the informational signatures identified here—such as entropy sensitivity and divergence-based detection of quasi-degeneracy—are generic to symmetric double-well systems and not restricted to the QES framework.

arXiv:2506.22684•International Journal of Theoretical Physics•Nov 17, 2025•2 citations

Wigner–Husimi phase-space structure of quasi-exactly solvable sextic potential

Angelina N. Mendoza Tavera, A. M. Escobar-Ruiz, Robin P. Sagar

Abstract We present a comparative phase-space analysis of the Wigner function W , the Husimi distribution H , and a normalized auxiliary diagnostic derived from the Wigner function, | W | , for a one-dimensional quantum system undergoing a transition from a single-well to a symmetric double-well potential. Using the quasi-exactly solvable sextic oscillator as a benchmark, we analyze the two-dimensional phase-space structure of these quantities, together with their marginals and associated information-theoretic measures. We show that W uniquely captures phase-space interference, negativity, and the strongest information-theoretic signatures associated with the emergence of the double-well structure, whereas | W | and H primarily highlight geometric splitting and coarse-grained delocalization. In particular, | W | provides a sign-suppressed structural diagnostic that lies between the fully oscillatory Wigner description and the Gaussian-smoothed Husimi picture. These results clarify the distinct information retained by standard and derived phase-space characterizations of the state under continuous deformation of the confining potential.

Journal of Physics A Mathematical and Theoretical•Jun 22, 2026•0 citations

Wigner-Husimi phase-space structure of quasi-exactly solvable sextic potential

Angelina N. Mendoza Tavera, Adrian M. Escobar Ruiz, Robin P. Sagar

In this study, we compare the Wigner function $W$, its modulus, and the Husimi distribution $H$ in a one-dimensional quantum system exhibiting a transition from a single-well to a double-well configuration, using the quasi-exactly solvable sextic oscillator as a representative example. High-accuracy variational wavefunctions for the lowest states are used to compute two-dimensional phase-space structures, one-dimensional marginals, and the corresponding Shannon entropies, mutual information, and Cumulative Residual Jeffreys divergences. The analysis shows that the Wigner representation is uniquely responsive to interference effects and displays clear, nonmonotonic entropic behavior as the wells separate, whereas the modulus-Wigner and Husimi distributions account only for geometric splitting or coarse-grained delocalization. These findings establish a quantitative hierarchy in the ability of $W$, $|W|$, and $H$ to resolve structural changes in a quantum state and provide a general framework for assessing the descriptive power of different phase-space representations in systems with emerging bimodality or tunneling.

quant-pharXiv:2512.06295•Dec 9, 2025•0 citations

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

Kyu-Won Park, Jongin Jeong

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.

quant-pharXiv:2512.03505•Dec 4, 2025•0 citations

Exact Cahill-Glauber Hierarchy for the Double-Morse Ground State

Firoz Chogle, Berihu Teklu, Mauro F. Pereira

Phase-space and quasiprobability methods now play operational roles in quantum technologies, characterizing localization, non-Gaussianity, nonclassical resources, and coarse-graining. We develop an exact, representation-consistent analysis of the lowest quasi-exact ground state of the symmetric double-Morse oscillator. In the double-Morse potential, the dimensionless parameter $A$ controls the separation of the minima and the central barrier, thereby changing the physical ground state. At fixed $A$, the Cahill--Glauber parameter $s$ labels the quasiprobability $W_A^{(s)}(q,p)$: $s=0$, $-1$, and $1$ give the Wigner, Husimi $Q$, and Glauber--Sudarshan $P$ representations, respectively. Although the potential is double-welled for $0<A<1$, the exact ground-state amplitude is single-peaked at the origin and lies above the barrier; as $A$ approaches unity, the merged well remains locally quartic rather than harmonic. Closed analytical expressions are obtained for the Wigner function and Weyl characteristic function. The Wigner function displays the $A$-dependent exchange between position and momentum localization and retains negative regions, certifying nonclassicality and, for this pure state, non-Gaussianity. The Weyl function is its Fourier dual, generates symmetrically ordered moments and cumulants, and yields the full $s$-ordered hierarchy. For $s<0$, isotropic Gaussian smoothing suppresses fine sign-changing structure while preserving the large-scale localization envelope. The Husimi endpoint is nonnegative without implying classicality, whereas the $P$ representation remains distributional. Thus, $A$ controls the physical phase-space geometry, while $s$ controls how the same non-Gaussian and nonclassical state is resolved across complementary representations

quant-pharXiv:2608.23316•APL Quantum•Aug 25, 2026•0 citations

Complex-valued Wigner entropy of a quantum state

Nicolas J. Cerf, Anaelle Hertz, Zacharie Van Herstraeten

Abstract It is common knowledge that the Wigner function of a quantum state may admit negative values, so that it cannot be viewed as a genuine probability density. Here, we examine the difficulty in finding an entropy-like functional in phase space that extends to negative Wigner functions and then advocate the merits of defining a complex-valued entropy associated with any Wigner function. This quantity, which we call thecomplex Wigner entropy, is defined via the analytic continuation of Shannon’s differential entropy of the Wigner function in the complex plane. We show that the complex Wigner entropy enjoys interesting properties, especially its real and imaginary parts are both invariant under Gaussian unitaries (displacements, rotations, and squeezing in phase space). Its real part is physically relevant when considering the evolution of the Wigner function under a Gaussian convolution, while its imaginary part is simply proportional to the negative volume of the Wigner function. Finally, we define the complex-valued Fisher information of any Wigner function, which is linked (via an extended de Bruijn’s identity) to the time derivative of the complex Wigner entropy when the state undergoes Gaussian additive noise. Overall, it is anticipated that the complex plane yields a proper framework for analyzing the entropic properties of quasiprobability distributions in phase space.

arXiv:2310.19296•Quantum Studies Mathematics and Foundations•May 16, 2024•11 citations
Abstract We develop a phase-space description of tunneling in the quasi-exact sextic double well VQES(x)=12 (x6+2x4−2(2λ+1) x2)V^{\textrm{QES}}(x) = \tfrac{1}{2}\,(x^6 + 2x^4 - 2(2\lambda +1)\,x^{2})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 SWS_WSW​ S W of the Wigner function W ( x , p ), whose imaginary part isolates coherence and negativity, and the entropy S∣W∣S_{|W|}S∣W∣​ S | W | of its modulus, a strictly real indicator of phase-space spreading. Together with the mutual information IWI_WIW​ I W , these measures yield sharp signatures of the transition into the double-well regime ( λ>−12\lambda >-\frac{1}{2}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.
Abstract We study the (de)localization properties of a quasi-exactly solvable (QES) sextic potential VQES(x)=12(x6+2x4−2(2λ+1)x2)V_{\text {QES}}(x) = \tfrac{1}{2}(x^6 + 2x^4 - 2(2\lambda + 1)x^2)VQES​(x)=21​(x6+2x4−2(2λ+1)x2) as a function of the tunable parameter λ∈[−34,6]\lambda \in [-\tfrac{3}{4},6]λ∈[−43​,6] . For the lowest states n=0,1,2,3n=0,1,2,3n=0,1,2,3 , variational wavefunctions are constructed that respect parity, have correct asymptotics, and admit analytical Fourier transforms. Energies agree with Lagrange-mesh and exact QES results with relative errors of order 10−810^{-8}10−8 for n=0,1,2n=0,1,2n=0,1,2 and 10−610^{-6}10−6 for n=3n=3n=3 . We show that information-theoretic indicators (Shannon entropy, Kullback–Leibler and Cumulative Residual Jeffreys divergences) are more sensitive than variance-based measures in detecting tunneling transitions, symmetry breaking, and level pairing. The Beckner–Białynicki-Birula–Mycielski entropic uncertainty relation is verified across all λ\lambdaλ , and trial densities are validated by divergences as small as 10−1010^{-10}10−10 from exact solutions. Importantly, while QES solvability allows for analytic benchmarking, many of the informational signatures identified here—such as entropy sensitivity and divergence-based detection of quasi-degeneracy—are generic to symmetric double-well systems and not restricted to the QES framework.
Abstract We present a comparative phase-space analysis of the Wigner function W , the Husimi distribution H , and a normalized auxiliary diagnostic derived from the Wigner function, | W | , for a one-dimensional quantum system undergoing a transition from a single-well to a symmetric double-well potential. Using the quasi-exactly solvable sextic oscillator as a benchmark, we analyze the two-dimensional phase-space structure of these quantities, together with their marginals and associated information-theoretic measures. We show that W uniquely captures phase-space interference, negativity, and the strongest information-theoretic signatures associated with the emergence of the double-well structure, whereas | W | and H primarily highlight geometric splitting and coarse-grained delocalization. In particular, | W | provides a sign-suppressed structural diagnostic that lies between the fully oscillatory Wigner description and the Gaussian-smoothed Husimi picture. These results clarify the distinct information retained by standard and derived phase-space characterizations of the state under continuous deformation of the confining potential.
In this study, we compare the Wigner function WWW, its modulus, and the Husimi distribution HHH in a one-dimensional quantum system exhibiting a transition from a single-well to a double-well configuration, using the quasi-exactly solvable sextic oscillator as a representative example. High-accuracy variational wavefunctions for the lowest states are used to compute two-dimensional phase-space structures, one-dimensional marginals, and the corresponding Shannon entropies, mutual information, and Cumulative Residual Jeffreys divergences. The analysis shows that the Wigner representation is uniquely responsive to interference effects and displays clear, nonmonotonic entropic behavior as the wells separate, whereas the modulus-Wigner and Husimi distributions account only for geometric splitting or coarse-grained delocalization. These findings establish a quantitative hierarchy in the ability of WWW, ∣W∣|W|∣W∣, and HHH to resolve structural changes in a quantum state and provide a general framework for assessing the descriptive power of different phase-space representations in systems with emerging bimodality or tunneling.
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.
Phase-space and quasiprobability methods now play operational roles in quantum technologies, characterizing localization, non-Gaussianity, nonclassical resources, and coarse-graining. We develop an exact, representation-consistent analysis of the lowest quasi-exact ground state of the symmetric double-Morse oscillator. In the double-Morse potential, the dimensionless parameter AAA controls the separation of the minima and the central barrier, thereby changing the physical ground state. At fixed AAA, the Cahill--Glauber parameter sss labels the quasiprobability WA(s)(q,p)W_A^{(s)}(q,p)WA(s)​(q,p): s=0s=0s=0, −1-1−1, and 111 give the Wigner, Husimi QQQ, and Glauber--Sudarshan PPP representations, respectively. Although the potential is double-welled for 0<A<10<A<10<A<1, the exact ground-state amplitude is single-peaked at the origin and lies above the barrier; as AAA approaches unity, the merged well remains locally quartic rather than harmonic. Closed analytical expressions are obtained for the Wigner function and Weyl characteristic function. The Wigner function displays the AAA-dependent exchange between position and momentum localization and retains negative regions, certifying nonclassicality and, for this pure state, non-Gaussianity. The Weyl function is its Fourier dual, generates symmetrically ordered moments and cumulants, and yields the full sss-ordered hierarchy. For s<0s<0s<0, isotropic Gaussian smoothing suppresses fine sign-changing structure while preserving the large-scale localization envelope. The Husimi endpoint is nonnegative without implying classicality, whereas the PPP representation remains distributional. Thus, AAA controls the physical phase-space geometry, while sss controls how the same non-Gaussian and nonclassical state is resolved across complementary representations
Abstract It is common knowledge that the Wigner function of a quantum state may admit negative values, so that it cannot be viewed as a genuine probability density. Here, we examine the difficulty in finding an entropy-like functional in phase space that extends to negative Wigner functions and then advocate the merits of defining a complex-valued entropy associated with any Wigner function. This quantity, which we call thecomplex Wigner entropy, is defined via the analytic continuation of Shannon’s differential entropy of the Wigner function in the complex plane. We show that the complex Wigner entropy enjoys interesting properties, especially its real and imaginary parts are both invariant under Gaussian unitaries (displacements, rotations, and squeezing in phase space). Its real part is physically relevant when considering the evolution of the Wigner function under a Gaussian convolution, while its imaginary part is simply proportional to the negative volume of the Wigner function. Finally, we define the complex-valued Fisher information of any Wigner function, which is linked (via an extended de Bruijn’s identity) to the time derivative of the complex Wigner entropy when the state undergoes Gaussian additive noise. Overall, it is anticipated that the complex plane yields a proper framework for analyzing the entropic properties of quasiprobability distributions in phase space.