  We investigate the ultimate precision limits for quantum phase estimation in
terms of the coherence, $C$, of the probe. For pure states, we give the minimum
estimation variance attainable, $V(C)$, and the optimal state, in the
asymptotic limit when the probe system size, $n$, is large. We prove that pure
states are optimal only if $C$ scales as $n$ with a sufficiently large
proportionality factor, and that the rank of the optimal state increases with
decreasing $C$, eventually becoming full-rank. We show that the variance
exhibits a Heisenberg-like scaling, $V(C) \sim a_n/C^2$, where $a_n$ decreases
to $\pi^2/3$ as $n$ increases, leading to a dimension-independent relation.
