  We theoretically investigate the phase sensitivity with parity detection on a
Mach-Zehnder interferometer with a coherent state combined with a photon-added
squeezed vacuum state. When the phase shift approaches zero, the squeezed
vacuum state is indeed the optimal state within a constraint on the average
number of photons. However, when the phase shift to be estimated slightly
deviates from zero, the optimal state is neither the squeezed vacuum state nor
the photon-subtracted squeezed vacuum state, but the photon-added squeezed
vacuum state when they carry many photons. Finally, we show that the quantum
Cram\'{e}r-Rao bound can be reached by parity detection.
