  We study the quantum entropy of systems that are described by general
non-Hermitian Hamiltonians, including those which can model the effects of
sinks or sources. We generalize the von Neumann entropy to the non- Hermitian
case and find that one needs both the normalized and non-normalized density
operators in order to properly describe irreversible processes. It turns out
that such a generalization monitors the onset of disorder in quantum
dissipative systems. We give arguments for why one can consider the generalized
entropy as the informational entropy describing the flow of information between
the system and the bath. We illustrate the theory by explicitly studying few
simple models, including tunneling systems with two energy levels and
non-Hermitian detuning.
