  Quantum-enhanced metrology can be achieved by entangling a probe with an
auxiliary system, passing the probe through an interferometer, and subsequently
making measurements on both the probe and auxiliary system. Conceptually, this
corresponds to performing metrology with the purification of a (mixed) probe
state. We demonstrate via the quantum Fisher information how to design mixed
states whose purifications are an excellent metrological resource. In
particular, we give examples of mixed states with purifications that allow
(near) Heisenberg-limited metrology, and provide example entangling
Hamiltonians that can generate these states. Finally, we present the optimal
measurement and parameter-estimation procedure required to realize these
sensitivities (i.e. that saturate the quantum Cram\'er-Rao bound). Since pure
states of comparable metrological usefulness are typically challenging to
generate, it may prove easier to use this approach of entanglement and
measurement of an auxiliary system. An example where this may be the case is
atom interferometry, where entanglement with optical systems is potentially
easier to engineer than the atomic interactions required to produce
nonclassical atomic states.
