"Peculiar quantum evolution in subsystems described by an effective non-Hermitian Hamiltonians":
When a discrete quantum subsystem is coupled to a continuum, the bound states become metastable/resonance states (e.g. Stark ionization). These resonances can be described by an effective non-Hermitian Hamiltonian. Without the Hermitian constraint new possibilities become available. For example, it is possible to discriminate between non-orthogonal states, to reduce entropy, or to have an evolution which is much faster than the energy difference in the system. After a short review on time-dependent non-Hermitian Hamiltonian dynamics, and non-Hermitian degeneracies (exceptional points), we will discuss how to quantify the cost of these non-Hermitian "magic tricks". As it turns out, studying the question of performance vs. resources reveals a lot about the proper tools needed for the study of non-Hermitian system.