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

TL;DR

This work unifies all discrete $d\times d$ Wigner functions for a qudit through a stencil theorem: every valid DWF arises from cross-correlating a $2d\times 2d$ parent function with a stencil $M$, and a projector reduces the doubled space to the physical phase-space grid. It introduces a family of valid stencils, including the reduction stencil for odd $d$, the coarse-grain stencil for even $d$, and a Dirichlet-kernel stencil for odd $d$, yielding both known and novel DWFs (notably a new even-$d$ DWF). The authors prove the Stencil Theorem, construct invertible stencil-dependent maps between PPO frames and between DWFs, and show how all DWFs for the same dimension are representation-equivalent. This framework clarifies when features like negativity depend on representation, and it provides a systematic toolkit for comparing phase-space representations and guiding extensions to discrete Kirkwood–Dirac distributions and Cohen’s class. The approach has potential implications for resource theories, classical simulation, and dimension-agnostic studies of quantum-phase-space methods.

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.

Figures (2)

• Figure 1: Stencil plots of $M(\bm{m})$ (bright) and its projection $\bar{M}(\bm{m})$ (bright+faded) for the reduction stencil $M^{\textsc{rs}}$ (${d=3}$, left) and the coarse-grain stencil $M^{\textsc{cgs}}$ (${d=4}$, right). White, red, and cyan (of any saturation) indicate $0$, $+c$, $-c$, respectively, with a different constant $c$ for each stencil: $c^\textsc{rs} = 2$ and $c^\textsc{cgs}= 1$; for the projected stencils, $\bar{c}^\textsc{rs} = \tfrac{1}{2}$ and $\bar{c}^\textsc{cgs} = \tfrac{1}{4}$.
• Figure 2: Left: A function on phase space, $W:\mathcal{P}_d \to \mathbb{C}$, can represent different operators $[t]{\hat{O}}_i$ by varying the choice of $M$-PPO $[t]{\hat{A}}^{M_i}$ used to construct it. These are all related by a stencil-dependent linear map on operators, $\mathop{\mathrm{\mathcal{E}}}\nolimits$. Right: Analogously, each valid DWF can represent a given operator $[t]{\hat{O}}$ as a distinct phase-space function, all of which are related by a stencil-dependent linear map on phase-space functions, $\mathop{\mathrm{\mathsf E}}\nolimits$. By Theorem \ref{['thm:E_existence']}, these maps exist for all valid PPO frames.

Theorems & Definitions (14)

• Definition : discrete phase space
• Definition : faithful phase-space representation
• Definition : WHDO
• Definition : discrete PPO
• Definition : valid PPO frame
• Definition : valid DWF
• Definition : PPO marginals, marginalisation
• Definition : stencil
• Definition : $M$-DWF, $M$-PPO
• Definition : Valid stencil and $\mathcal{M}$