  We show a general relationship between a superposition of macroscopically
distinct states and sensitivity in quantum metrology. Generalized cat states
are defined by using an index which extracts the coherence between
macroscopically distinct states, and a wide variety of states, including a
classical mixture of an exponentially large number of states, has been
identified as the generalized cat state with this criterion. We find that, if
we use the generalized cat states for magnetic field sensing without noise, we
achieve the Heisenberg limited sensitivity. Moreover, we even show that
sensitivity of generalized cat states achieves the ultimate scaling sensitivity
beyond the standard quantum limit under the effect of dephasing. As an example,
we investigate the sensitivity of a generalized cat state that is attainable
through a single global manipulation on a thermal equilibrium state and find an
improvement of a few orders of magnitude from the previous sensors. Clarifying
a wide class that includes such a peculiar state as metrologically useful, our
results significantly broaden the potential of quantum metrology.
