Complete parameterization and parameter-space topology of discrete Wigner representations on d × 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× 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 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 base cell satisfying a parity-dependent twisted-oddness relation. This gives the parameter space (S1)(d2−1)/2 for odd d and (S1)(d2−4)/2×ℤ23 for even d. Canonical horizontal and vertical marginals reduce these spaces to (S1)(d−1)2/2 and (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 replaces each S1 factor by a discrete 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.
Related
- Minimal representations of topology-preserving quantum-like states
- Specimen design for material parameter identification using topology optimization
- Discrete Geometric Physics. Volume 1: Computational Topodynamics of Media. Volume 2: From Chaos to Infinity. — Complete Theory in a Single Document
- Optimized discrete Wigner representations and non-stabilizerness in qubit systems
- Discrete topology sensing in turbulence via signed pairwise graphlets of the velocity gradient tensor
BibTeX
@misc{antonopoulos2026complete,
title = {{Complete parameterization and parameter-space topology of discrete Wigner representations on $d \times d$ phase space}},
author = {Lucky K. Antonopoulos and Nicolas C. Menicucci},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.39119},
eprint = {2609.39119},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39119},
}
RIS
TY - UNPB
TI - Complete parameterization and parameter-space topology of discrete Wigner representations on d × d phase space
AU - Lucky K. Antonopoulos
AU - Nicolas C. Menicucci
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2609.39119
UR - https://arxiv.org/abs/2609.39119
AB - Toroidal discrete Wigner functions (DWFs) for finite-dimensional systems are non-unique. We completely parameterize and topologically classify the labeled single-qudit 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 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 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×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×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 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.
ER -