  The estimation of parameters characterizing dynamical processes is central to
science and technology. The estimation error changes with the number N of
resources employed in the experiment (which could quantify, for instance, the
number of probes or the probing energy). Typically, it scales as 1/N^(1/2).
Quantum strategies may improve the precision, for noiseless processes, by an
extra factor 1/N^(1/2). For noisy processes, it is not known in general if and
when this improvement can be achieved. Here we propose a general framework for
obtaining attainable and useful lower bounds for the ultimate limit of
precision in noisy systems. We apply this bound to lossy optical interferometry
and atomic spectroscopy in the presence of dephasing, showing that it captures
the main features of the transition from the 1/N to the 1/N^(1/2) behaviour as
N increases, independently of the initial state of the probes, and even with
use of adaptive feedback.
