  We study the time evolution of a PT-symmetric, non-Hermitian quantum system
for which the associated phase space is compact. We focus on the simplest
non-trivial example of such a Hamiltonian, which is linear in the angular
momentum operators. In order to describe the evolution of the system, we use a
particular disentangling decomposition of the evolution operator, which remains
numerically accurate even in the vicinity of the Exceptional Point. We then
analyze how the non-Hermitian part of the Hamiltonian affects the time
evolution of two archetypical quantum states, coherent and Dicke states. For
that purpose we calculate the Husimi distribution or Q function and study its
evolution in phase space. For coherent states, the characteristics of the
evolution equation of the Husimi function agree with the trajectories of the
corresponding angular momentum expectation values. This allows to consider
these curves as the trajectories of a classical system. For other types of
quantum states, e.g. Dicke states, the equivalence of characteristics and
trajectories of expectation values is lost.
