Optimal input states and feedback for interferometric phase estimation - art. no. 053804

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 bq pre vious authors. We prove, that it is not possible to realize the canonical m easurement by counting photons in the outputs of the interferometer, even i f an adjustable auxiliary phase shift is allowed in the interferometer. How ever. we introduce a feedback algorithm based on Bayesian inference to cont rol this auxiliary phase shift. This makes the measurement close to a canon ical one, with a phase variance scaling slightly above N-2. With no feedbac k, the best result (given that the phase to be measured is completely unkno wn) 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 numbe rs more complicated schemes perform marginally better.