Phase-matching condition for enhancement of phase sensitivity in quantum metrology

We find a phase-matching condition for enhancement of sensitivity in a Mach-Zehnder interferometer illuminated by an arbitrary state in one input port and an odd (even) state in the other port. Under this condition, the Fisher information becomes maximal with respect to the relative phase of two modes, and the phase sensitivity is enhanced. For the case with photon losses, we further find that the phase-matching condition remains unchanged with a coherent state and a coherent superposition state as the input states.