Optimal input states and feedback for interferometric phase estimation

We derive optimal N-photon two-mode input states for interferometric phase measurements. Under canonical measurements the phase variance scales as $N^{-2}$ for these states, as compared to $N^{-1}$ or $N^{-1/2}$ for states considered by previous authors. We prove that it is not possible to realize the canonical measurement by counting photons in the outputs of the interferometer, even if an adjustable auxiliary phase shift is allowed in the interferometer. However, we introduce a feedback algorithm based on Bayesian inference to control this auxiliary phase shift. This makes the measurement close to a canonical one, with a phase variance scaling slightly above $N^{-2}.$ With no feedback, the best result (given that the phase to be measured is completely unknown) is a scaling of $N^{-1}.$ For optimal input states having up to four photons, our feedback scheme is the best possible one, but for higher photon numbers more complicated schemes perform marginally better.