It is known that the phase measurement using lossless Mach-Zehnder interferometer with certain entangled
N-photon states can achieve a phase sensitivity of the order of 1/N, the Heisenberg limit. However, the output
observable used is different for different input states to achieve the Heisenberg limit. In this paper, we show
that it is possible to achieve this limit just by parity measurement for all the commonly proposed entangled
states. Taking account of path absorption effect, the phase shifts with parity measurement for these states are
obtained and indicate that the N00N state still remains to be the best candidate to perform phase detection if the
transmittance of the medium is not too small and the number of photons is not very large.