Optimal quantum-enhanced interferometry

We analyze the ultimate bounds on the phase sensitivity of an interferometer, given the constraint that the state input to the interferometer's initial 50:50 beam splitter B is a product state of the two input modes. Requiring a product state is a natural restriction: If one were allowed to input an arbitrary, entangled two-mode state |Ξ to the beam splitter, one could generally just as easily input the state B|Ξ directly into the two modes after the beam splitter, thus rendering the beam splitter unnecessary. We find optimal states for a fixed photon number and for a fixed mean photon number.