We address local quantum estimation of bilinear Hamiltonians probed by
Gaussian states. We evaluate the relevant quantum Fisher information (QFI) and
derive the ultimate bound on precision. Upon maximizing the QFI we found that
single- and two-mode squeezed vacuum represent an optimal and universal class
of probe states, achieving the so-called Heisenberg limit to precision in terms
of the overall energy of the probe. We explicitly obtain the optimal observable
based on the symmetric logarithmic derivative and also found that homodyne
detection assisted by Bayesian analysis may achieve estimation of squeezing
with near-optimal sensitivity in any working regime. Besides, by comparison of
our results with those coming from global optimization of the measurement we
found that Gaussian states are effective resources, which allow to achieve the
ultimate bound on precision imposed by quantum mechanics using measurement
schemes feasible with current technology.Comment: revised version, 2 figure