Limits in the quantum characteristic function description of open quantum systems

In this paper I investigate the usability of the quantum characteristic function for the description of the dynamics of open quantum systems, focussing on the non-Markovian regime. It has been shown that non-Markovian generalized master equations containing memory kernels may lead to non-physical time evolutions characterized by negative values of the density matrix diagonal elements [S.M. Barnett and S. Stenholm, Phys. Rev. A {\bf 64}, 033808 (2001)]. The main purpose of the paper is to show that there exist situations in which the quantum characteristic function of the non-Markovian open quantum system is perfectly well defined while the corresponding density matrix loses positivity. Therefore unphysical situations may not show up in a description of the dynamics via the quantum characteristic function. I conclude that, in general, the quantum characteristic function cannot be used for describing non-Markovian dynamics