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.
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