Estimating the coupling strength of a Hamiltonian with four different measurement strategies

In this paper, we compare different measurement strategies for estimating coupling strength of a Hamiltonian. In particular, we estimate the coupling strength coefficient when the Hamiltonian couples the pointer (measuring device) momentum to an observable of the system. We compare the performance of the estimations via four different measurement strategies: measuring the position or the momentum of the pointer with or without postselection of the system state. Without postselection, it turns out that all the information of the coupling strength can be extracted by measuring the position of the pointer. As a contrast, no information of the coupling strength can be extracted if the momentum is measured without postselection. Hence, measuring the position is a better choice if postselection is not allowed. However, when postselection is employed, we find that no matter which observable is measured, one can always achieve a great amplification of the measurement outcome, which can be larger than the initial standard deviation of the pointer observable under certain circumstances, at an acceptable cost of losing a little accuracy for the estimation of the coupling strength. With postselection, the better choice of the observable to be measured depends on the characteristic of the measurement devices.