  We analyze precision bounds for a local phase estimation in the presence of
general, non-Markovian phase noise. We demonstrate that the metrological
equivalence of product and maximally entangled states that holds under strictly
Markovian dephasing fails in the non-Markovian case. Using an exactly solvable
model of a physically realistic finite bandwidth dephasing environment, we
demonstrate that the ensuing non-Markovian dynamics enables quantum correlated
states to outperform metrological strategies based on uncorrelated states using
otherwise identical resources. We show that this conclusion is a direct result
of the coherent dynamics of the global state of the system and environment and
therefore the obtained scaling with the number of particles, which surpasses
the standard quantum limit but does not achieve Heisenberg resolution,
possesses general validity that goes beyond specific models. This is in marked
contrast with the situation encountered under general Markovian noise, where an
arbitrarily small amount of noise is enough to restore the scaling dictated by
the standard quantum limit.
