  Quantum metrology utilizes quantum effects to reach higher precision
measurements of physical quantities compared with their classical counterparts.
However the ubiquitous decoherence obstructs its application. Recently,
non-Markovian effects are shown to be effective in performing quantum optical
metrology under locally dissipative environments\cite{PhysRevLett.123.040402}.
However, the mechanism is still rather hazy. Here, we uncover the reason why
forming a bound state can protect the quantumness against a dissipative ambient
via the quantum Fisher information of entangled coherent states. An exact
analytical expression of the quantum Fisher information in the
long-encoding-time condition is derived, which reveals that the dynamics of
precision can asymptotically reach the ideal-case-promised one easily when the
average photon number is small. Meanwhile, the scaling exhibits a transition
from the weak Heisenberg limit to the sub-classical limit with the increase of
average photon number. Our work provides a recipe to realize ultrasensitive
measurements in the presence of noise by utilizing non-Markovian effects.
