  We show that in presence of a local and uncorrelated dephasing noise, quantum
advantage can be obtained in the Fisher information-based lower bound of the
minimum uncertainty in estimating parameters of the system Hamiltonian. The
quantum advantage refers here to the benefit of initiating with a maximally
entangled state instead of a product one. This quantum advantage was known to
vanish in the same noisy scenario for a frequency estimation protocol.
Restoration of the better precision in frequency estimation with maximally
entangled probes can be obtained by incorporating an interaction between the
system particles. The interaction examined here is Ising in nature, and is
considered with or without a transverse magnetic field. There are instances,
e.g. where frequency estimation in presence of a transverse field is considered
and quantum advantage is not restored. A quantum advantage can also be obtained
while estimating the strength of the introduced magnetic field along the
transverse direction, whereas for the instances considered, using uncorrelated
probes is better in measuring the coupling parameter of the Ising interaction.
We also investigate the dependence of measurement precision on the entanglement
content, which is not necessarily maximal, of the initial state. The precision
in estimation of coupling constant decreases monotonically with the increase of
entanglement content of the initial state, while the same for frequency
estimation is independent of the entanglement content of the inputs.
