  Mach-Zehnder interferometer is a common device in quantum phase estimation
and the photon losses in it are an important issue for achieving a high phase
accuracy. Here we thoroughly discuss the precision limit of the phase in the
Mach-Zehnder interferometer with a coherent state and a superposition of
coherent states as input states. By providing a general analytical expression
of quantum Fisher information, the phase-matching condition and optimal initial
parity are given. Especially, in the photon loss scenario, the sensitivity
behaviors are analyzed and specific strategies are provided to restore the
phase accuracies for symmetric and asymmetric losses.
