  We consider the general problem of estimating an unknown control parameter of
an open quantum system. We establish a direct relation between the evolution of
both system and environment and the precision with which the parameter can be
estimated. We show that when the open quantum system undergoes a first-order
dynamical phase transition the quantum Fisher information (QFI), which gives
the upper bound on the achievable precision of any measurement of the system
and environment, becomes quadratic in observation time (cf. "Heisenberg
scaling"). In fact, the QFI is identical to the variance of the dynamical
observable that characterises the phases that coexist at the transition, and
enhanced scaling is a consequence of the divergence of the variance of this
observable at the transition point. This identification allows to establish the
finite time scaling of the QFI. Near the transition the QFI is quadratic in
time for times shorter than the correlation time of the dynamics. In the regime
of enhanced scaling the optimal measurement whose precision is given by the QFI
involves measuring both system and output. As a particular realisation of these
ideas, we describe a theoretical scheme for quantum enhanced phase estimation
using the photons being emitted from a quantum system near the coexistence of
dynamical phases with distinct photon emission rates.
