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, which are trace preserving. Recently, a scheme for engineering non-Hermitian Hamiltonians as a result of repetitive measurements on an ancillary qubit has been proposed. The induced conditional dynamics of the main system is described by the effective non-Hermitian Hamiltonian arising 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.
We reveals a connection between non-Hermitian and Hermitian systems by studying the connection between a family of non-Hermitian and Hermitian Hamiltonians based on exact solutions. In general, for a dynamic process in a non-Hermitian system
H, there always exists a parallel dynamic process governed by the corresponding Hermitian conjugate system ${\mathcal{H}}^{\ifmmode\dagger\else\textdagger\fi{}}$. We show that a linear superposition of the two parallel dynamics is exactly equivalent to the time evolution of a state under a Hermitian Hamiltonian
H, and we present the relations between ${H,\mathcal{H},{\mathcal{H}}^{\ifmmode\dagger\else\textdagger\fi{}}}$.