  When measuring phase of quantum states of light, the optimal single-shot
measurement implements projection on the un-physical phase states. If we want
to improve the precision further we need to accept a reduced probability of
success, either by implementing a probabilistic measurement or by
probabilistically manipulating the measured quantum state by means of noiseless
amplification. We analyze the limits of this approach by finding the optimal
probabilistic measurement which, for a given rate of success, maximizes the
precision with which the phase can be measured.
