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

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 $2d\times 2d$ 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 $d\times d$ 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.

(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