Classify general-line marginal constraints

Extend the classification of valid single-qudit discrete Wigner representations beyond axis-parallel marginal lines to more general phase-space lines, and determine the resulting restrictions on the qudit dimension and admissible parameter space.

Background

The paper classifies marginal constraints only for canonical horizontal and vertical, or axis-parallel, lines. It notes that more general line marginals have been studied in earlier work and may impose additional restrictions on the dimension of the qudit.

A complete treatment would extend the stencil-based parameterization and topology analysis to these more general marginal structures, thereby determining whether additional lines reduce, discretize, or eliminate the currently admissible families.

References

More general lines are considered in Refs., but the associated marginal requirements can impose additional restrictions on the qudit dimension. Such generalizations are left to future work.

— Complete parameterization and parameter-space topology of discrete Wigner representations on $d \times d$ phase space  (2609.39119 - Antonopoulos et al., 30 Sep 2026) in Section 4.3, Subsection “Subspace imposed by canonical horizontal and vertical marginals”

For example, which admissible convention minimizes the total negative DWF weight of a given state while retaining specified marginal or displacement-order requirements? The characteristic-function viewpoint turns this into a constrained choice of phases with fixed magnitudes. The classification separates the continuous phase choices from the remaining discrete choices, while imposing the order-${d}$ restriction makes the search finite. Investigating this selection problem is left to future work.

— Complete parameterization and parameter-space topology of discrete Wigner representations on $d \times d$ phase space  (2609.39119 - Antonopoulos et al., 30 Sep 2026) in Section 6, Discussion and conclusions