Recovering the ideal results of a perturbed quantum emulator

We consider a quantum emulator which is a model of an ideal Hamiltonian of interest $H_S$. However, inevitably the system is perturbed by coupling to additional degrees of freedom. We study the case where we are interested in extracted a correlator from the emulated system in equilibrium. We show that it is possible to extract the ideal correlator from a perturbed system under certain conditions. The ideal correlator can be reconstructed if any n-time correlator of the ideal system can be written as a product of two-time correlators.