  We analyze the Heisenberg limit on phase estimation for Gaussian states. In
the analysis, no reference to a phase operator is made. We prove that the
squeezed vacuum state is the most sensitive for a given average photon number.
We provide two adaptive local measurement schemes that attain the Heisenberg
limit asymptotically. One of them is described by a positive operator-valued
measure and its efficiency is exhaustively explored. We also study Gaussian
measurement schemes based on phase quadrature measurements. We show that
homodyne tomography of the appropriate quadrature attains the Heisenberg limit
for large samples. This proves that this limit can be attained with local
projective Von Neuman measurements.
