On Families of Wigner Functions for N-Level Quantum Systems

A method for constructing all admissible unitary non-equivalent Wigner quasiprobability distributions providing the Stratonovic-h-Weyl correspondence for an arbitrary N-level quantum system is proposed. The method is based on the reformulation of the Stratonovich-Weyl correspondence in the form of algebraic "master equations" for the spectrum of the Stratonovich-Weyl kernel. The later implements a map between the operators in the Hilbert space and the functions in the phase space identified by the complex flag manifold. The non-uniqueness of the solutions to the master equations leads to diversity among the Wigner quasiprobability distributions. It is shown that among all possible Stratonovich-Weyl kernels for a N=(2j+1)-level system, one can always identify the representative that realizes the so-called SU(2)-symmetric spin-j symbol correspondence. The method is exemplified by considering the Wigner functions of a single qubit and a single qutrit.