Description
Quantum states and operators can be represented in many different ways. One useful representation assigns values to points of a finite phase-space grid, giving a discrete analogue of the Wigner function. These discrete Wigner functions are useful in quantum information because they can reveal nonclassical behaviour, support calculations with finite-dimensional systems, and connect naturally to displacement-operator structures. However, unlike in the standard continuous setting, the finite-dimensional version is not unique: several different toroidal grid-based representations can satisfy the same basic requirements.
This work studies the freedom that remains once those requirements are imposed. Building on a stencil-based construction of discrete Wigner functions, I show that the problem of choosing a valid representation can be reduced to choosing phase data on a single d×d grid. The validity conditions impose simple consistency rules on these phases. Most grid points occur in opposite pairs, giving continuous choices, while special self-paired points give discrete choices. In this way, the remaining freedom is not just counted, but organised into a topological parameter space.
This produces an explicit description of the allowed family of representations. In odd dimensions, the remaining freedom forms a continuous torus. In even dimensions, additional disconnected discrete sectors appear. Imposing standard line-marginal conditions further reduces the family by fixing selected grid directions, but does not usually yield a unique representation. The same phase data also determines the associated displacement operators, giving an operator-level meaning to the classification.
The result is a global map of the convention-dependence in finite-dimensional Wigner representations. It clarifies which features are fixed by Hermiticity, normalisation, orthogonality, covariance, and marginal constraints, and which remain choices of convention.
| I am the presenting author | Yes |
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