  Quantum metrology is being gradually studied for weak measurement systems.
For weak measurement systems with thermal state pointer, we find that in the
displacement space corresponding to imaginary weak values, the maximal QFI
after successful postselection can attain the level of thermal fluctuations,
without surpassing total QFI, and that QFI which increases with increasing
temperature can constantly improve the measurement precision. These results are
much better than that of weak measurement with pure state (i.e., Gaussian
state) pointer. On the other hand, in Kerr nonlinear interaction systems with
weak measurement, and by using thermal state pointer, we obtain in the phase
space successful postselection and postselected measurements both achieve the
Heisenberg limit of quantum metrology, and show weak measurement with thermal
states only obtain classical Fisher information (CFI) which increases with
increasing temperature and achieves classical enhanced scaling of N^2.
Moreover, weak measurement with thermal states has an advantage over that with
coherent states or mixed states of the light because generating these states
with more large uncertainty are limited under the current technology, but
thermal states with more large uncertainly are very easy to achieve with
increasing temperature in nature, regardless of thermal states of the light or
the matter.
