Canonical Phase Measurements in the Presence of Photon Loss

We analyze the optimal state, as given by Berry and Wiseman [Phys. Rev. Lett {\bf 85}, 5098, (2000)], under the canonical phase measurement in the presence of photon loss. The model of photon loss is a generic fictitious beam splitter, and we present the full density matrix calculations, which are more direct and do not involve any approximations. We find for a given amount of loss the upper bound for the input photon number that yields a sub-shot noise estimate.