The entanglement and phase measurement performance of the damped NOON state

The evolution of the optical NOON state is investigated. The relative entropy of entanglement is exactly obtained for the NOON state simultaneously damped by symmetric amplitude damping and phase damping. The distillable entanglement of the phase-damped state is also obtained. The phase measurement performance of the damped state is studied according to the full measurement results of the quantum interferometry output. The phase sensitivity does not rely on the phase under estimate for the symmetrically amplitude-damped state.