  We address the issue of precisely estimating small parameters encoded in a
general linear transformation of the modes of a bosonic quantum field. Such
Bogoliubov transformations frequently appear in the context of quantum optics.
We provide a set of instructions for computing the quantum Fisher information
for arbitrary pure initial states. We show that the maximally achievable
precision of estimation is inversely proportional to the squared average
particle number and that such Heisenberg scaling requires non-classical, but
not necessarily entangled states. Our method further allows us to quantify
losses in precision arising from being able to monitor only finitely many
modes, for which we identify a lower bound.
