The effects of path absorption on phase sensitivity with entangled states

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