Optimal phase measurements with pure Gaussian states

We analyze the Heisenberg limit on phase estimation for Gaussian states without reference to a phase operator. We prove that the most sensitive states to phase measurements with a given energy are the squeezed vacuum states. We show that at least two kinds of measurements exist that asymptotically attain such limit. One of them is described in terms of POVM measurements and its efficiency is exhaustively explored. We also study Gaussian measurements where phase quadrature measurements are performed. We show that this type of measurements can also be optimal in the large sample limit