Grand Unification of All Discrete Wigner Functions on $d\times d$ Phase Space

Abstract

Wigner functions help visualise 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 artefacts of representation. We introduce a stencil-based framework that exhausts all possible $d\times d$ 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.

(v4) 8 Pages, 2 figures. Minor edits. Version published in PRA. (V3) 8 Pages, 2 figures. Rewritten for clarity, with the addition of new results. (v2) 8 pages, 2 figures. Theorem 5 clarified. Added a reference. Fixed typos. (v1) 8 pages, 2 figures

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