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.

Moyal Bracket and Ehrenfest’s Theorem in Born–Jordan Quantization

The usual Poisson bracket { A , B } can be identified with the so-called Moyal bracket { A , B } M for larger classes of symbols than was previously thought, provided that one uses the Born–Jordan quantization rule instead of the better known Weyl correspondence. We apply our results to a generalized version of Ehrenfest’s theorem on the time evolution of averages of operators.