  Phase sensitive amplifiers (PSA), contrary to usual phase insensitive
amplifiers (PIA), are in principle capable to achieve noiseless amplification,
i.e. exhibit a quantum-limited noise figure (NF) of 0 dB. When implemented
using four-wave mixing (FWM) in a nonlinear fiber, extra waves can be generated
by undesired FWM processes, which may introduce extra input ports for vacuum
fluctuations, thus potentially degrading the NF. In this situation, we give
here a general analytical quantum derivation of the PSA NF, valid for an
arbitrary number of nonlinearly coupled modes. This expression is usable as
soon as a linear input-output relation can be found for the annihilation and
creation operators of the involved modes. It predicts that the noise level
depends on the number of interacting waves. We illustrate the usefulness of
this expression in the case of six waves, corresponding to four interacting
quantum modes. In this example the signal NF is degraded by 0.4 dB, compared to
10 dB obtained for PIA operation of the same scheme.
