  Most of the schemes for "noiseless" amplification of coherent states, which
have recently been attracting theoretical and experimental interest, share a
common trait: the amplification is not truly noiseless, or perfect, for
non-zero success probability. While this must hold true for all
phase-independent amplification schemes, in this work we point out that truly
noiseless amplification is indeed possible, provided that the states which we
wish to amplify come from a finite set. Perfect amplification with unlimited
average gain is then possible with finite success probability, for example
using techniques for unambiguously distinguishing between quantum states. Such
realizations require only linear optics, no single-photon sources, nor any
photon counting. We also investigate the optimal success probability of perfect
amplification of a symmetric set of coherent states. There are two regimes:
low-amplitude amplification, where the target amplitude is below one, and
general amplification. For the low-amplitude regime, analytic results for the
optimal amplification success probabilities can be obtained. In this case a
natural bound imposed by the ratio of success probabilities of optimal
unambiguous discrimination of the source and amplified states can always be
reached. We also show that for general amplification this bound cannot always
be satisfied.
