  We give the optimal bounds on the phase-estimation precision for mixed
Gaussian states in the single-copy and many-copy regimes. Specifically, we
focus on displaced thermal and squeezed thermal states. We find that while for
displaced thermal states an increase in temperature reduces the estimation
fidelity, for squeezed thermal states a larger temperature can enhance the
estimation fidelity. The many-copy optimal bounds are compared with the minimum
variance achieved by three important single-shot measurement strategies. We
show that the single-copy canonical phase measurement does not always attain
the optimal bounds in the many-copy scenario. Adaptive homodyning schemes do
attain the bounds for displaced thermal states, but for squeezed states they
yield fidelities that are insensitive to temperature variations and are,
therefore, sub-optimal. Finally, we find that heterodyne measurements perform
very poorly for pure states but can attain the optimal bounds for sufficiently
mixed states. We apply our results to investigate the influence of losses in an
optical metrology experiment. In the presence of losses squeezed states cease
to provide Heisenberg limited precision and their performance is close to that
of coherent states with the same mean photon number.
