  We formulate, with full generality, the asymptotic estimation theory for
Gaussian states in terms of their first and second moments. By expressing the
quantum Fisher information (QFI) and the elusive symmetric logarithmic
derivative (SLD) in terms of the state's moments (and their derivatives) we are
able to obtain the noncommutative extension of the well known expression for
the Fisher information of a Gaussian probability distribution. Focusing on
models with fixed first moments and identical Williamson 'diagonal' states
--which include pure state models--, we obtain their SLD and QFI, and elucidate
what features of the Wigner function are fundamentally accessible, and at what
rates. In addition, we find the optimal homodyne detection scheme for all such
models, and show that for pure state models they attain the fundamental limit.
