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

arXiv:2503.09353 · Physical Review A

Grand unification of all discrete Wigner functions on d × d phase space

Lucky K. Antonopoulos, Dominic G. Lewis, J. Davis, Nicholas Funai, Nicolas C. Menicucci

Published Sep 24, 2025• 3 citations
arXivHTMLPDF

Abstract

Wigner functions help visualize quantum states and dynamics while supporting quantitative analysis in quantum information. In the discrete setting, many inequivalent constructions coexist for each Hilbert-space dimension. This fragmentation obscures which features are fundamental and which are artifacts of representation. We introduce a stencil-based framework that exhausts all possible $d\ifmmode\times\else\texttimes\fi{}d$ discrete Wigner functions for a single $d$-dimensional quantum system (including a novel one for even $d$), subsuming known forms. We also give explicit invertible linear maps between definitions within the same $d$, enabling direct comparison of operational properties and exposing representation dependence.

DOI
10.1103/s5wn-mysr

Related papers

Complete parameterization and parameter-space topology of discrete Wigner representations on $d \times d$ phase space

Lucky K. Antonopoulos, Nicolas C. Menicucci

Toroidal discrete Wigner functions (DWFs) for finite-dimensional systems are non-unique. We completely parameterize and topologically classify the labeled single-qudit ${d\times d}$ representations whose phase-point operators are Hermitian, unit-trace, Hilbert--Schmidt orthogonal, and Weyl--Heisenberg covariant, determining the independent parameter freedom in every dimension. Our previously established stencil theorem expresses this family as descents of a doubled ${2d\times2d}$ parent Wigner representation. Here, we solve its projected-stencil admissibility conditions explicitly to obtain the full parameter space of this family. In the symplectic-Fourier representation, admissibility fixes the modulus, leaving phase data on a ${d\times d}$ base cell satisfying a parity-dependent twisted-oddness relation. This gives the parameter space ${(S^1)^{(d^2-1)/2}}$ for odd ${d}$ and ${(S^1)^{(d^2-4)/2}\times\mathbb{Z}_{2}^{3}}$ for even ${d}$. Canonical horizontal and vertical marginals reduce these spaces to ${(S^1)^{(d-1)^2/2}}$ and ${(S^1)^{d\,(d-2)/2}\times\mathbb{Z}_{2}}$, respectively. The same phase data parameterize validity-preserving rephasings of the doubled Weyl--Heisenberg displacement operators. Requiring these operators to have order dividing the Hilbert-space dimension ${d}$ replaces each ${S^1}$ factor by a discrete ${\mathbb{Z}_{d}}$ factor. The associated symplectic-Fourier characteristic functions encode the same operator information, with pointwise magnitudes independent of the valid stencil convention and hence providing convention-independent information about the represented operator within this family. This classification separates the freedom intrinsic to DWF validity from that selected by marginal and displacement-algebra requirements and makes explicit the structural distinction between odd and even dimensions.

quant-pharXiv:2609.39119•Oct 1, 2026•0 citations

Representations of two-qubit and ququart states via discrete Wigner functions

Marcelo A. Marchiolli, D. Galetti

By means of a well-grounded mapping scheme linking Schwinger unitary operators and generators of the special unitary group $\mathrm{SU(N)}$, it is possible to establish a self-consistent theoretical framework for finite-dimensional discrete phase spaces which has the discrete $\mathrm{SU(N)}$ Wigner function as a legitimate by-product. In this paper, we apply these results with the aim of putting forth a detailed study on the discrete $\mathrm{SU(2)} \otimes \mathrm{SU(2)}$ and $\mathrm{SU(4)}$ Wigner functions, in straight connection with experiments involving, among other things, the tomographic reconstruction of density matrices related to the two-qubit and ququart states. Next, we establish a formal correspondence between both the descriptions that allows us to visualize the quantum correlation effects of these states in finite-dimensional discrete phase spaces. Moreover, we perform a theoretical investigation on the two-qubit X-states, which combines discrete Wigner functions and their respective marginal distributions in order to obtain a new function responsible for describing qualitatively the quantum correlation effects. To conclude, we also discuss possible extensions to the discrete Husimi and Glauber-Sudarshan distribution functions, as well as future applications on spin chains.

arXiv:1908.02410•Aug 7, 2019•0 citations

On the discrete Wigner function for S U ( N )

Marcelo A Marchiolli, Diógenes Galetti

Abstract We present a self-consistent theoretical framework for finite-dimensional discrete phase spaces that leads us to establish a well-grounded mapping scheme between Schwinger unitary operators and generators of the special unitary group . This general mathematical construction provides a sound pathway to the formulation of a genuinely discrete Wigner function for arbitrary quantum systems described by finite-dimensional state vector spaces. To illustrate our results, we obtain a general discrete Wigner function for the group and apply this to the study of a particular three-level system. Moreover, we also discuss possible extensions to the discrete Husimi and Glauber–Sudarshan functions, as well as future investigations on multipartite quantum states.

arXiv:1908.01096•Journal of Physics A Mathematical and Theoretical•Aug 15, 2019•10 citations

Tomographic Universality of the Discrete Wigner Function

Isabel Sainz, Ernesto Camacho, Andrés García +1 more

We observe that the discrete Wigner functions (DWFs) of n-partite systems with odd local dimensions are tomographically universal, as reflected in the delta function form of the DWF for any stabilizer. However, in the n-qubit case, this property does not hold due to the non-factorization of the mapping kernel, the explicit form of which depends on a particular partition of the discrete phase space. Nonetheless, it turns out that the DWF for some specific stabilizers, not included in the set used for the construction of the Wigner map, takes on the form of a delta function. This implies that the possibility of classical simulations of Pauli measurements in a given stabilizer state for qubit systems is closely tied to the experimental setup.

Quantum Reports•Jan 19, 2024•1 citation

Optimized discrete Wigner representations and non-stabilizerness in qubit systems

A. B. Klimov, C. Muñoz, I. Sainz

The apparent absence of a stabilizer-specific phase-space description for qubit systems is not fundamental, but arises from fixing a state-independent discrete Wigner map. Allowing the phase-space representation to adapt to the state through a simple optimization procedure, we prove that every qubit stabilizer state admits an exact delta-function discrete Wigner representation for any fixed phase-space partition. This optimization can be implemented directly at the level of measurement statistics by selecting maximal probabilities across mutually unbiased bases forming the partition. This construction naturally leads to a non-stabilizer witness defined by the optimized value of the Wigner function at the origin. Stabilizer states saturate the universal lower bound of this witness for pure states, whereas a value strictly larger than the bound certifies non-stabilizerness. Although the witness is not a finite-size resource monotone, it becomes sharply concentrated and effectively invariant under random Clifford transformations in the large-qubit limit, suggesting an asymptotically linear growth law for broad families of non-stabilizer states. Our results restore a meaningful geometric phase-space characterization of qubit stabilizer structure and provide an operational diagnostic of non-stabilizerness.

quant-pharXiv:2610.10471•Oct 8, 2026•0 citations

Path integral formulation of finite-dimensional quantum mechanics in discrete phase space

Leonardo A. Pachon, Andres D. Gomez

We develop a path integral representation for the dynamics of quantum systems with a finite-dimensional Hilbert space, formulated entirely within a discrete phase space. Starting from the discrete Wigner function on ${\mathbb{Z}_d} \times {\mathbb{Z}_d}$ ($d$ an odd prime) and the associated Weyl transform built from generalized displacement operators, we derive an exact kernel that propagates the discrete Wigner function in time and, by iterating its composition law through a short-time approximation, obtain a sum-over-paths expression weighted by a discrete phase-space action -- the natural finite-dimensional counterpart of Marinov's functional. For Hamiltonians linear in the phase-space coordinates and at times strictly commensurate with the lattice, the fluctuation sum collapses to a deterministic shift, realizing the discrete analog of classical Hamiltonian flow. Applying the formalism to one and to two interacting qutrits ($d=3$), we show that the full entanglement dynamics -- captured by a closed-form linear entropy valid for all times -- requires the coherent contribution of all fluctuation sectors; the boundary-term (mean-field) sector alone fails to reproduce it. For a non-stabilizer Hamiltonian, where the short-time kernel is only approximate, the time-sliced path integral converges to the exact dynamics, including the dynamical generation of Wigner negativity. We discuss implications for the semiclassical simulation of many-body spin systems and for the characterization of non-classicality through Wigner negativity.

quant-pharXiv:2604.20776•Apr 23, 2026•0 citations
Wigner functions help visualize quantum states and dynamics while supporting quantitative analysis in quantum information. In the discrete setting, many inequivalent constructions coexist for each Hilbert-space dimension. This fragmentation obscures which features are fundamental and which are artifacts of representation. We introduce a stencil-based framework that exhausts all possible d\ifmmode×\else\texttimes\fidd\ifmmode\times\else\texttimes\fi{}dd\ifmmode×\else\texttimes\fid discrete Wigner functions for a single ddd-dimensional quantum system (including a novel one for even ddd), subsuming known forms. We also give explicit invertible linear maps between definitions within the same ddd, enabling direct comparison of operational properties and exposing representation dependence.
Complete parameterization and parameter-space topology of discrete Wigner representations on d×dd \times dd×d phase space
Toroidal discrete Wigner functions (DWFs) for finite-dimensional systems are non-unique. We completely parameterize and topologically classify the labeled single-qudit d×d{d\times d}d×d representations whose phase-point operators are Hermitian, unit-trace, Hilbert--Schmidt orthogonal, and Weyl--Heisenberg covariant, determining the independent parameter freedom in every dimension. Our previously established stencil theorem expresses this family as descents of a doubled 2d×2d{2d\times2d}2d×2d parent Wigner representation. Here, we solve its projected-stencil admissibility conditions explicitly to obtain the full parameter space of this family. In the symplectic-Fourier representation, admissibility fixes the modulus, leaving phase data on a d×d{d\times d}d×d base cell satisfying a parity-dependent twisted-oddness relation. This gives the parameter space (S1)(d2−1)/2{(S^1)^{(d^2-1)/2}}(S1)(d2−1)/2 for odd d{d}d and (S1)(d2−4)/2×Z23{(S^1)^{(d^2-4)/2}\times\mathbb{Z}_{2}^{3}}(S1)(d2−4)/2×Z23​ for even d{d}d. Canonical horizontal and vertical marginals reduce these spaces to (S1)(d−1)2/2{(S^1)^{(d-1)^2/2}}(S1)(d−1)2/2 and (S1)d (d−2)/2×Z2{(S^1)^{d\,(d-2)/2}\times\mathbb{Z}_{2}}(S1)d(d−2)/2×Z2​, respectively. The same phase data parameterize validity-preserving rephasings of the doubled Weyl--Heisenberg displacement operators. Requiring these operators to have order dividing the Hilbert-space dimension d{d}d replaces each S1{S^1}S1 factor by a discrete Zd{\mathbb{Z}_{d}}Zd​ factor. The associated symplectic-Fourier characteristic functions encode the same operator information, with pointwise magnitudes independent of the valid stencil convention and hence providing convention-independent information about the represented operator within this family. This classification separates the freedom intrinsic to DWF validity from that selected by marginal and displacement-algebra requirements and makes explicit the structural distinction between odd and even dimensions.
By means of a well-grounded mapping scheme linking Schwinger unitary operators and generators of the special unitary group SU(N)\mathrm{SU(N)}SU(N), it is possible to establish a self-consistent theoretical framework for finite-dimensional discrete phase spaces which has the discrete SU(N)\mathrm{SU(N)}SU(N) Wigner function as a legitimate by-product. In this paper, we apply these results with the aim of putting forth a detailed study on the discrete SU(2)⊗SU(2)\mathrm{SU(2)} \otimes \mathrm{SU(2)}SU(2)⊗SU(2) and SU(4)\mathrm{SU(4)}SU(4) Wigner functions, in straight connection with experiments involving, among other things, the tomographic reconstruction of density matrices related to the two-qubit and ququart states. Next, we establish a formal correspondence between both the descriptions that allows us to visualize the quantum correlation effects of these states in finite-dimensional discrete phase spaces. Moreover, we perform a theoretical investigation on the two-qubit X-states, which combines discrete Wigner functions and their respective marginal distributions in order to obtain a new function responsible for describing qualitatively the quantum correlation effects. To conclude, we also discuss possible extensions to the discrete Husimi and Glauber-Sudarshan distribution functions, as well as future applications on spin chains.
Abstract We present a self-consistent theoretical framework for finite-dimensional discrete phase spaces that leads us to establish a well-grounded mapping scheme between Schwinger unitary operators and generators of the special unitary group . This general mathematical construction provides a sound pathway to the formulation of a genuinely discrete Wigner function for arbitrary quantum systems described by finite-dimensional state vector spaces. To illustrate our results, we obtain a general discrete Wigner function for the group and apply this to the study of a particular three-level system. Moreover, we also discuss possible extensions to the discrete Husimi and Glauber–Sudarshan functions, as well as future investigations on multipartite quantum states.
We observe that the discrete Wigner functions (DWFs) of n-partite systems with odd local dimensions are tomographically universal, as reflected in the delta function form of the DWF for any stabilizer. However, in the n-qubit case, this property does not hold due to the non-factorization of the mapping kernel, the explicit form of which depends on a particular partition of the discrete phase space. Nonetheless, it turns out that the DWF for some specific stabilizers, not included in the set used for the construction of the Wigner map, takes on the form of a delta function. This implies that the possibility of classical simulations of Pauli measurements in a given stabilizer state for qubit systems is closely tied to the experimental setup.
The apparent absence of a stabilizer-specific phase-space description for qubit systems is not fundamental, but arises from fixing a state-independent discrete Wigner map. Allowing the phase-space representation to adapt to the state through a simple optimization procedure, we prove that every qubit stabilizer state admits an exact delta-function discrete Wigner representation for any fixed phase-space partition. This optimization can be implemented directly at the level of measurement statistics by selecting maximal probabilities across mutually unbiased bases forming the partition. This construction naturally leads to a non-stabilizer witness defined by the optimized value of the Wigner function at the origin. Stabilizer states saturate the universal lower bound of this witness for pure states, whereas a value strictly larger than the bound certifies non-stabilizerness. Although the witness is not a finite-size resource monotone, it becomes sharply concentrated and effectively invariant under random Clifford transformations in the large-qubit limit, suggesting an asymptotically linear growth law for broad families of non-stabilizer states. Our results restore a meaningful geometric phase-space characterization of qubit stabilizer structure and provide an operational diagnostic of non-stabilizerness.
We develop a path integral representation for the dynamics of quantum systems with a finite-dimensional Hilbert space, formulated entirely within a discrete phase space. Starting from the discrete Wigner function on Zd×Zd{\mathbb{Z}_d} \times {\mathbb{Z}_d}Zd​×Zd​ (ddd an odd prime) and the associated Weyl transform built from generalized displacement operators, we derive an exact kernel that propagates the discrete Wigner function in time and, by iterating its composition law through a short-time approximation, obtain a sum-over-paths expression weighted by a discrete phase-space action -- the natural finite-dimensional counterpart of Marinov's functional. For Hamiltonians linear in the phase-space coordinates and at times strictly commensurate with the lattice, the fluctuation sum collapses to a deterministic shift, realizing the discrete analog of classical Hamiltonian flow. Applying the formalism to one and to two interacting qutrits (d=3d=3d=3), we show that the full entanglement dynamics -- captured by a closed-form linear entropy valid for all times -- requires the coherent contribution of all fluctuation sectors; the boundary-term (mean-field) sector alone fails to reproduce it. For a non-stabilizer Hamiltonian, where the short-time kernel is only approximate, the time-sliced path integral converges to the exact dynamics, including the dynamical generation of Wigner negativity. We discuss implications for the semiclassical simulation of many-body spin systems and for the characterization of non-classicality through Wigner negativity.