Constructing the Wigner quasiprobability distributions of finite-dimensional quantum systems

The report is devoted to the formulation of the Wigner functions (WF) method for the description of finite-dimensional quantum systems. It is shown how the Stratonovich - Weyl correspondence rules can be reformulated as the “master algebraic equations” on the spectrum of a self dual Stratonovich-Weyl (SW) kernel of the Wigner - Weyl mapping between the Hilbert space operators and functions on a phase space corresponding to an N-level quantum system. Our generic approach in constructing the SW kernel doesn't use any specific symmetry of quantum mechanical system under consideration. The form of the master equations gives rise to an interesting duality between the space of the states of a quantum mechanical system and the space of SW kernels. Although, the ambiguity of the choice in the SW kernel, dictated by the master equations is going to be analyzed through the detailed description of the corresponding moduli space, the question of a physically motivated fixing of the SW kernel remains an open problem. A general scheme will be exemplified by constructing Wigner functions of a single qubit (N=2), qutrit (N=3) an quatrit (N=4).