Extracting the internal nonlocality from the dilated Hermiticity

Abstract

To effectively realize a $\mathcal{P}\mathcal{T}$-symmetric system, one can dilate a $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian to some global Hermitian one and simulate its evolution in the dilated Hermitian system. However, with only a global Hermitian Hamiltonian, how do we know whether it is a dilation and is useful for simulation? To answer this question, we consider the problem of how to extract the internal nonlocality in the Hermitian dilation. We unveil that the internal nonlocality brings nontrivial correlations between the subsystems. By evaluating the correlations with local measurements in three different pictures, the resulting different expectations of the Bell operator reveal the distinction of the internal nonlocality. When the simulated $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian approaches its exceptional point, such a distinction tends to be most significant. Our results clearly make a distinction between the Hermitian dilation and other global Hamiltonians without internal nonlocality. They also provide the figure of merit to test the reliability of the simulation, as well as to verify a $\mathcal{P}\mathcal{T}$-symmetric (sub)system.