While traditional quantum mechanics focusses on systems conserving energy and probability, described by Hermitian Hamiltonians, in recent decades there has been ever growing interest in the use of non-Hermitian Hamiltonians. These can effectively describe loss and gain in a quantum system. In particular systems with a certain balance of loss and gain, PT-symmetric systems, have attracted considerable attention. The realisation of PT-symmetric quantum dynamics in optical systems has opened up a whole new field of investigations.

The properties of non-Hermitian quantum systems are often less intuitive than those of conventional Hermitian systems. Here we make use of the Husimi representation in phase space to analyse dynamical and spectral features. We consider the flow of the Husimi phase-space distribution in a semiclassical limit, leading to a first order partial differential equation, that helps illuminate the foundations of the full quantum evolution. Further, we demonstrate how ingredients of the dynamics can be used to construct approximate Husimi distributions of characteristic quantum states.