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

Lucky K. Antonopoulos, Dominic G. Lewis, Jack Davis, Nicholas Funai, Nicolas C. Menicucci (Mar 13 2025).

Abstract: We present a universal framework for defining, visualising, and transforming between all possible $d\times d$ discrete Wigner functions for a single $d$-dimensional qudit. This framework constructs valid functions by cross-correlating a naturally emerging $2d\times 2d$ discrete Wigner function with an information-reorganising function called a stencil. Validity criteria for the Wigner function translate into stencil conditions, ensuring consistency. Additionally, we identify one stencil that is valid for all even dimensions (and which coincides with Wootters' function for $d=2$) and another valid for all odd dimensions, reproducing Gross' function (as well as Wootters' for odd prime dimensions). Furthermore, we establish invertible linear transformations between all valid functions within the same dimension, enabling a systematic comparison of their structural and operational properties, such as negativity and marginalisation.

Arxiv: https://arxiv.org/abs/2503.09353