On the Moduli Space of Wigner Quasiprobability Distributions for N-Dimensional Quantum Systems

A mapping between operators on the Hilbert space of an N-dimensional quantum system and Wigner quasiprobability distributions defined on the symplectic flag manifold is discussed. The Wigner quasiprobability distribution is constructed as a dual pairing between the density matrix and the Stratonovich–Weyl kernel. It is shown that the moduli space of Stratonovich–Weyl kernels is given by the intersection of the coadjoint orbit space of the group SU(N) and a unit (N − 2)-dimensional sphere. The general considerations are exemplified by a detailed description of the moduli space of 2, 3, and 4-dimensional systems.