Phase estimation with photon number constraint

Many researches proposed the use of the noon state as the input state for phase estimation, which is one topic of quantum metrology. This is because the input noon state provides the maximum Fisher information at the specific point. However, the Fisher information does not necessarily give the attainable bound for estimation error. In this paper, we adopt the local asymptotic mini-max criterion as well as the mini-max criterion, and show that the maximum Fisher information does not give the attainable bound for estimation error under these criteria in the phase estimation. We also propose the optimal input state under the constraints for photon number of the input state instead of the noon state.