Conclusive Precision Bounds for SU(1,1) Interferometers*

We revisit the quantum Fisher information (QFI) calculation in SU(1,1) interferometer considering different phase configurations. Firstly, when one of the input modes is a vacuum state, we show by using phase averaging, different phase configurations give same QFI. In this case, the QFI is linearly proportional to the average photon number of the second input state, and quadratically proportional to the average photon number generated by the OPA. This suggests that when fixing the squeezing strength of the OPA, to achieve higher sensitivity, one simply needs to inject a state with higher average photon number. Secondly, we compared the results of the phase-averaging method and the quantum Fisher information matrix method, and then we argued that for a SU(1,1) interferometer, phase averaging or quantum Fisher information matrix method is generally required, and they are essentially equivalent. Finally, we used the quantum Fisher information matrix method to calculate the precision limit for other common input states, such as two coherent state inputs or coherent state with squeezed vacuum inputs.