  We study the estimation of a single parameter characterizing families of
unitary transformations acting on two systems. We consider the situation with
the presence of bottleneck, i.e. only one of the systems can be measured to
gather information. The estimation capabilities are related to unitaries'
generators. In particular, we establish continuity of quantum Fisher
information with respect to generators. Furthermore, we find conditions on the
generators to achieve the same maximum quantum Fisher information we would have
in the absence of bottleneck. We also discuss the usefulness of initial
entanglement across the two systems as well as across multiple estimation
instances.
