We extend shapelets for the analysis of astronomical images to be available in a phase space, introducing \textit{Wigner function shapelets}(WFSs). Whereas conventional shapelets expand images separately in configuration or Fourier space using Hermite-Gaussian or Laguerre-Gaussian modes, WFSs represent images directly in the four-dimensional phase space with symplectic group $\mathrm{Sp}(4,\mathbb{R})$ , which is quantized by a phase-space cell $2π\lambdabar$ that determines a resolution limit of a telescope. WFSs consist of a bilinear form of the cross-Wigner function of the Laguerre-Gaussian modes as an orthogonal and complete basis for the Wigner function of an image, carrying out $\mathrm{SU}(2)$ irreducible representations of the phase space with the Hopf tori. We introduce a scalar function $\mathcal{W}_{k\ell} (Q_0,Q_2)$ from the $\mathrm{U}(1)\times \mathrm{U}(1)$ - covariant tori to a two-dimensional space of constants of motion $(Q_0,Q_2)$ -- the harmonic energy and axial angular momentum -- thereby yielding a natural phase-space ``band structure'', given a pair of winding number $(k,\ell) \in \mathbb{Z}^2$ . % The WFSs leverage key properties of the Wigner function for image analysis: (i) it encodes full information of an image in a symmetry-preserving way; (ii) its trasport equation naturally involves with a Liouville equation at $\lambdabar \rightarrow 0$; (iii) it admits positive/negative oscillatory patterns on $(Q_0,Q_2)$ plane that can be sensitive spatial coherent structure of galaxy morphology and cosmological imprints; and (iv) systematics and noise can be manipulated as a quantum channel operation. This paper aims to bring together all the formulae related to the Wigner function in the context of astrophysics and cosmology, formally organizing them in terminologies of both astronomy and of quantum information theory.We extend shapelets for the analysis of astronomical images to be available in a phase space, introducing \textit{Wigner function shapelets}(WFSs). Whereas conventional shapelets expand images separately in configuration or Fourier space using Hermite-Gaussian or Laguerre-Gaussian modes, WFSs represent images directly in the four-dimensional phase space with symplectic group $\mathrm{Sp}(4,\mathbb{R})$, which is quantized by a phase-space cell $2π\lambdabar$ that determines a resolution limit of a telescope. WFSs consist of a bilinear form of the cross-Wigner function of the Laguerre-Gaussian modes as an orthogonal and complete basis for the Wigner function of an image, carrying out $\mathrm{SU}(2)$ irreducible representations of the phase space with the Hopf tori. We introduce a scalar function $\mathcal{W}_{k\ell} (Q_0,Q_2)$ from the $\mathrm{U}(1)\times \mathrm{U}(1)$ - covariant tori to a two-dimensional space of constants of motion $(Q_0,Q_2)$ -- the harmonic energy and axial angular momentum -- thereby yielding a natural phase-space ``band structure'', given a pair of winding number $(k,\ell) \in \mathbb{Z}^2$. % The WFSs leverage key properties of the Wigner function for image analysis: (i) it encodes full information of an image in a symmetry-preserving way; (ii) its trasport equation naturally involves with a Liouville equation at $\lambdabar \rightarrow 0$; (iii) it admits positive/negative oscillatory patterns on $(Q_0,Q_2)$ plane that can be sensitive spatial coherent structure of galaxy morphology and cosmological imprints; and (iv) systematics and noise can be manipulated as a quantum channel operation. This paper aims to bring together all the formulae related to the Wigner function in the context of astrophysics and cosmology, formally organizing them in terminologies of both astronomy and of quantum information theory.

astro-ph.COCross-listedarXiv:2602.01141Feb 3, 2026

The geometry of Wigner's function

Nandor L. Balazs

Lecture notes in physicsJan 1, 19871 citation

Geometry of two-qubit system and hopf fibration

Wen Wei Wong

Complex Hopf fibration and quaternionic Hopf fibration provide distinct ways of describing the state space for two-qubit quantum states. In this research, we have studied the geometry of quantum states for a two-level quantum system, along with the correspondence between complex Hopf bundle and quaternionic Hopf bundle. In the first part of our study, we investigate the behaviour of local coordinates for both Hopf bundles under different degree of entanglement such as entangled states and non-entangled states. Fubini-Study metric for complex projective space is also obtained. Its form suggests that for the intermediate entangled states, complex projective space CP3 can be described as a set of flat three-tori parametrized by a three-sphere. The local inhomogeneous coordinate of CP3 (base space of complex Hopf fibration) is found to carry the description of both subsystems A and B, whereas in the case of maximally entangled state, basis elements of local coordinates CP3 is not linearly independent of each other. Next, we construct a base space map between CP3 and S4, which is denoted as η map. After the mapping, we obtained phases in the base space manifold, different sections and coordinate charts are related by transition functions. We found that there is an inherent symmetry of coordinate transformation corresponds to different sections of CP3, which is expressed in terms of transition functions having the U(1) group structure. Also under η map, phases and transition function in S4 is doubled over that of CP3, indicating subtle symmetric changes after the mapping. The base space coordinates of quaternionic Hopf bundle are consist of two parts, whereby the first part is invariant to the coordinate transformation in CP3 but sensitive to the coordinate transformation in S4.Complex Hopf fibration and quaternionic Hopf fibration provide distinct ways of describing the state space for two-qubit quantum states. In this research, we have studied the geometry of quantum states for a two-level quantum system, along with the correspondence between complex Hopf bundle and quaternionic Hopf bundle. In the first part of our study, we investigate the behaviour of local coordinates for both Hopf bundles under different degree of entanglement such as entangled states and non-entangled states. Fubini-Study metric for complex projective space is also obtained. Its form suggests that for the intermediate entangled states, complex projective space CP3 can be described as a set of flat three-tori parametrized by a three-sphere. The local inhomogeneous coordinate of CP3 (base space of complex Hopf fibration) is found to carry the description of both subsystems A and B, whereas in the case of maximally entangled state, basis elements of local coordinates CP3 is not linearly independent of each other. Next, we construct a base space map between CP3 and S4, which is denoted as η map. After the mapping, we obtained phases in the base space manifold, different sections and coordinate charts are related by transition functions. We found that there is an inherent symmetry of coordinate transformation corresponds to different sections of CP3, which is expressed in terms of transition functions having the U(1) group structure. Also under η map, phases and transition function in S4 is doubled over that of CP3, indicating subtle symmetric changes after the mapping. The base space coordinates of quaternionic Hopf bundle are consist of two parts, whereby the first part is invariant to the coordinate transformation in CP3 but sensitive to the coordinate transformation in S4.

Mar 1, 20130 citations

A tomographic approach to Wigner's function

J. Bertrand, P. Bertrand

Foundations of PhysicsApr 1, 1987339 citations

Partial revivals of wave packets: An action-angle phase-space description

Quanlin Jie, Shun-Jin Wang, Lian-Fu Wei

Partial revivals of wave packets are investigated by means of the action-angle Wigner function, which is formulated in terms of action quantum numbers and angle variables. In the eigenbasis of the Hamiltonian, the phase shifts of the expansion coefficients of the wave packet are approximated by a quadratic function of action quantum numbers. Under this approximation, the evolution of the action-angle Wigner function of the wave packet, similarly to the classical Liouville density, can be viewed as a result of each phase-space point moving along a classical orbit. The partial and full revivals of wave packets are direct consequences of two facts: (i) The action-angle Wigner function is distributed only on discrete tori of the phase space and (ii) phase-space points of nearby tori move against each other with a constant speed.Partial revivals of wave packets are investigated by means of the action-angle Wigner function, which is formulated in terms of action quantum numbers and angle variables. In the eigenbasis of the Hamiltonian, the phase shifts of the expansion coefficients of the wave packet are approximated by a quadratic function of action quantum numbers. Under this approximation, the evolution of the action-angle Wigner function of the wave packet, similarly to the classical Liouville density, can be viewed as a result of each phase-space point moving along a classical orbit. The partial and full revivals of wave packets are direct consequences of two facts: (i) The action-angle Wigner function is distributed only on discrete tori of the phase space and (ii) phase-space points of nearby tori move against each other with a constant speed.

Physical Review AMay 1, 199813 citations

\mathrm{Sp}(4,\mathbb{R})

, which is quantized by a phase-space cell $2π\lambdabar$ that determines a resolution limit of a telescope. WFSs consist of a bilinear form of the cross-Wigner function of the Laguerre-Gaussian modes as an orthogonal and complete basis for the Wigner function of an image, carrying out

\mathrm{SU}(2)

irreducible representations of the phase space with the Hopf tori. We introduce a scalar function

\mathcal{W}_{k\ell} (Q_0,Q_2)

from the

\mathrm{U}(1)\times \mathrm{U}(1)

- covariant tori to a two-dimensional space of constants of motion

(Q_0,Q_2)

-- the harmonic energy and axial angular momentum -- thereby yielding a natural phase-space ``band structure'', given a pair of winding number

(k,\ell) \in \mathbb{Z}^2

. % The WFSs leverage key properties of the Wigner function for image analysis: (i) it encodes full information of an image in a symmetry-preserving way; (ii) its trasport equation naturally involves with a Liouville equation at $\lambdabar \rightarrow 0$; (iii) it admits positive/negative oscillatory patterns on

(Q_0,Q_2)

plane that can be sensitive spatial coherent structure of galaxy morphology and cosmological imprints; and (iv) systematics and noise can be manipulated as a quantum channel operation. This paper aims to bring together all the formulae related to the Wigner function in the context of astrophysics and cosmology, formally organizing them in terminologies of both astronomy and of quantum information theory.