  It was recently shown that an entangled coherent state, which is a
superposition of two different coherent states, can surpass the performance of
noon state in estimating an unknown phase-shift. This may hint at further
enhancement in phase estimation by incorporating more component states in the
superposition of resource state. We here introduce a four-headed cat state
(4HCS), a superposition of four different coherent states, and propose its
application to quantum phase estimation. We demonstrate the enhanced
performance in phase estimation by employing an entangled state via the 4HCS,
which can surpass that of the two-headed cat state (2HCS), particularly in the
regime of small average photon numbers. Moreover, we show that an entangled
state modified from the 4HCS can further enhance the phase estimation, even in
the regime of large average photon number under a photon-loss channel. Our
investigation further extends to incorporate an increasingly large number of
component states in the resource superposition state and clearly show its merit
in phase estimation.
