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arXiv:2609.39119

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

Lucky K. Antonopoulos, Nicolas C. Menicucci

quant-phSubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations• Revised Oct 3, 2026 (v2)
v1 · Sep 30, 2026v2 · Oct 3, 2026
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Abstract

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.

Comments
(v2) Revised abstract, introduction and discussion for clearer scope, literature context and statement of contributions; clarified author contributions. 21 pages, 6 figures. (v1) 21 pages, 6 figures

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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.
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 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.
Schwinger's finite (D) dimensional periodic Hilbert space representations are studied on the toroidal lattice eeZDtimeseeZD{\\ee Z}_{D} \\times {\\ee Z}_{D}eeZD​timeseeZD​ with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the Wigner function. These subalgebras are shown to be admissible endowed with the non-negative norm of Hilbert space vectors. Hence, they provide the desired canonical basis for the algebraic formulation of the quantum phase problem. Certain equivalence classes in the space of labels are identified within each subalgebra, and connections with area-preserving canonical transformations are examined. The generalized representations of the Wigner function are examined in the finite-dimensional cyclic Schwinger basis. These representations are shown to conform to all fundamental conditions of the generalized phase space Wigner distribution. As a specific application of the Schwinger basis, the number-phase unitary operator pair in ${\\ee Z}_{D} \\times q-oscillator subalgebra, an {\\it algebraic} approach to the unitary quantum phase operator is established. This being the focus of this work, connections with the Susskind-Glogower- Carruthers-Nieto phase operator formalism as well as standard action-angle Wigner function formalisms are examined in the infinite-period limit. The concept of continuously shifted Fock basis is introduced to facilitate the Fock space representations of the Wigner function.
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.
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.