  In contrast to classical systems, actual implementation of non-Hermitian
Hamiltonian dynamics for quantum systems is a challenge because the processes
of energy gain and dissipation are based on the underlying Hermitian
system-environment dynamics that is trace preserving. Recently, a scheme for
engineering non-Hermitian Hamiltonians as a result of repetitive measurements
on an anicillary qubit has been proposed. The induced conditional dynamics of
the main system is described by the effective non-Hermitian Hamiltonian arisng
from the procedure. In this paper we demonstrate the effectiveness of such a
protocol by applying it to physically relevant multi-spin models, showing that
the effective non-Hermitian Hamiltonian drives the system to a maximally
entangled stationary state. In addition, we report a new recipe to construct a
physical scenario where the quantum dynamics of a physical system represented
by a given non-Hermitian Hamiltonian model may be simulated. The physical
implications and the broad scope potential applications of such a scheme are
highlighted.
