  Amplification of quantum states is inevitably accompanied with the
introduction of noise at the output. For protocols that are probabilistic with
heralded success, noiseless linear amplification in theory may still possible.
When the protocol is successful, it can lead to an output that is a noiselessly
amplified copy of the input. When the protocol is unsuccessful, the output
state is degraded and is usually discarded. Probabilistic protocols may improve
the performance of some quantum information protocols, but not for metrology if
the whole statistics is taken into consideration. We calculate the precision
limits on estimating the phase of coherent states using a noiseless linear
amplifier by computing its quantum Fisher information and we show that on
average, the noiseless linear amplifier does not improve the phase estimate. We
also discuss the case where abstention from measurement can reduce the cost for
estimation.
