  When estimating the phase of a single mode, the quantum Fisher information
for a pure probe state is proportional to the photon number variance of the
probe state. In this work, we point out particular states that offer photon
number distributions exhibiting a large variance, which would help to improve
the local estimation precision. These theoretical examples are expected to
stimulate the community to put more attention to those states that we found,
and to work towards their experimental realization and usage in quantum
metrology.
