Extracting the internal nonlocality from the dilated Hermiticity

TL;DR

This paper investigates how to identify and extract internal nonlocality in Hermitian dilations of $ ext{PT}$-symmetric Hamiltonians, providing a method to verify the simulation's reliability and distinguish it from other global Hamiltonians.

Contribution

It introduces a way to detect internal nonlocality in Hermitian dilations, revealing correlations that distinguish them from other Hamiltonians and aiding in verifying $ ext{PT}$-symmetric system simulations.

Findings

Internal nonlocality reveals nontrivial correlations between subsystems.

Distinction of internal nonlocality becomes more significant near exceptional points.

Provides a figure of merit for testing the reliability of $ ext{PT}$-symmetric system simulations.

Abstract

To effectively realize a $\cal PT$-symmetric system, one can dilate a $\cal PT$-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 $\cal PT$-symmetric Hamiltonian approaches its exceptional point, such a distinction tends to be most significant. Our results clearly make a difference 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 $\cal PT$-symmetric (sub)system.