The Wigner function provides a useful quasiprobability representation of quantum mechanics, with applications in various branches of physics. Many nice properties of the Wigner function are intimately connected with the high symmetry of the underlying operator basis composed of phase point operators: any pair of phase point operators can be transformed to any other pair by a unitary symmetry transformation. We prove that, in the discrete scenario, this permutation symmetry is equivalent to the symmetry group being a unitary 2 design. Such a highly symmetric representation can only appear in odd prime power dimensions besides dimensions 2 and 8. It suffices to single out a unique discrete Wigner function among all possible quasiprobability representations. In the course of our study, we show that this discrete Wigner function is uniquely determined by Clifford covariance, while no Wigner function is Clifford covariant in any even prime power dimension.
The original Wigner function provides a way of representing in phase space the quantum states of systems with continuous degrees of freedom. Wigner functions have also been developed for discrete quantum systems, one popular version being defined on a $2N\ifmmode\times\else\texttimes\fi{}2N$ discrete phase space for a system with
N orthogonal states. Here we investigate an alternative class of discrete Wigner functions, in which the field of real numbers that labels the axes of continuous phase space is replaced by a finite field having
N elements. There exists such a field if and only if
N is a power of a prime; so our formulation can be applied directly only to systems for which the state-space dimension takes such a value. Though this condition may seem limiting, we note that any quantum computer based on qubits meets the condition and can thus be accommodated within our scheme. The geometry of our $N\ifmmode\times\else\texttimes\fi{}N$ phase space also leads naturally to a method of constructing a complete set of
N+1 mutually unbiased bases for the state space.
We study the Wigner Transform (WT) in the case of an infinite-dimensional continuous Hilbert space and a discrete Hilbert space of finite prime-number dimensionality
N. In the discrete case we define a family of WT's as a function of a phase parameter. It is only for a specific value of the parameter that all the properties we have examined, including the matrix elements of the phase-space operator
P and the relation between Wigner's function and Kirkwood's distribution, have a parallel in both types of Hilbert space. This close parallel was found constructing the two WT's in terms of a complete set of operators having similar properties, in the sense that the position and momentum operators appear disentangled in the two situations. We give a geometric interpretation of the properties involving the phase-space operator in the discrete case.
Toroidal discrete Wigner functions (DWFs) for finite-dimensional systems are non-unique. We completely parameterize and topologically classify the labeled single-qudit
d×d representations whose phase-point operators are Hermitian, unit-trace, Hilbert--Schmidt orthogonal, and Weyl--Heisenberg covariant, determining the independent parameter freedom in every dimension. Our previously established stencil theorem expresses this family as descents of a doubled
2d×2d parent Wigner representation. Here, we solve its projected-stencil admissibility conditions explicitly to obtain the full parameter space of this family. In the symplectic-Fourier representation, admissibility fixes the modulus, leaving phase data on a
d×d base cell satisfying a parity-dependent twisted-oddness relation. This gives the parameter space
(S1)(d2−1)/2 for odd
d and
(S1)(d2−4)/2×Z23 for even
d. Canonical horizontal and vertical marginals reduce these spaces to
(S1)(d−1)2/2 and
(S1)d(d−2)/2×Z2, respectively. The same phase data parameterize validity-preserving rephasings of the doubled Weyl--Heisenberg displacement operators. Requiring these operators to have order dividing the Hilbert-space dimension
d replaces each
S1 factor by a discrete
Zd factor. The associated symplectic-Fourier characteristic functions encode the same operator information, with pointwise magnitudes independent of the valid stencil convention and hence providing convention-independent information about the represented operator within this family. This classification separates the freedom intrinsic to DWF validity from that selected by marginal and displacement-algebra requirements and makes explicit the structural distinction between odd and even dimensions.