  We demonstrate that quantum fluctuations can cause, under certain conditions,
the dynamical instability of pure states that can result in their evolution
into mixed states. It is shown that the degree and type of such an instability
are controlled by the environment-induced anti-Hermitian parts of Hamiltonians.
Using the quantum-statistical approach for non-Hermitian Hamiltonians and
related non-linear master equation, we derive the equations that are necessary
to study the stability properties of any model described by a non-Hermitian
Hamiltonian. It turns out that the instability of pure states is not
preassigned in the evolution equation but arises as the emergent phenomenon in
its solutions. In order to illustrate the general formalism and different types
of instability that may occur, we perform the local stability analysis of some
exactly solvable two-state models, which can be used in the theories of open
quantum-optical and spin systems.
