  We investigate the performance of entangled coherent state for quantum
enhanced phase estimation. An exact analytical expression of quantum Fisher
information is derived to show the role of photon losses on the ultimate phase
sensitivity. We find a transition of the sensitivity from the Heisenberg
scaling to the classical scaling due to quantum decoherence of the photon
state. This quantum-classical transition is uniquely determined by the number
of photons being lost, instead of the number of incident photons or the photon
loss rate alone. Our results also reveal that a crossover of the sensitivity
between the entangled coherent state and the NOON state can occur even for very
small photon loss rate.
