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We investigate the performance of entangled coherent state for quantumenhanced phase estimation. An exact analytical expression of quantum Fisherinformation is derived to show the role of photon losses on the ultimate phasesensitivity. We find a transition of the sensitivity from the Heisenbergscaling to the classical scaling due to quantum decoherence of the photonstate. This quantum-classical transition is uniquely determined by the numberof photons being lost, instead of the number of incident photons or the photonloss rate alone. Our results also reveal that a crossover of the sensitivitybetween the entangled coherent state and the NOON state can occur even for verysmall photon loss rate.

Quantum Fisher information of entangled coherent state in the presence of photon losses: exact solution