  Non-deterministic quantum noiseless linear amplifiers are a new technology
with interest in both fundamental understanding and new applications. With a
noiseless linear amplifier it is possible to perform tasks such as improving
the performance of quantum key distribution and purifying lossy channels.
Previous designs for noiseless linear amplifiers involving linear optics and
photon counting are non-optimal because they have a probability of success
lower than the theoretical bound given by the theory of generalised quantum
measurement. This paper develops a theoretical model which reaches this limit.
We calculate the fidelity and probability of success of this new model for
coherent states and Einstein-Podolsky-Rosen (EPR) entangled states.
