  The use of distributed amplifiers may have some potential advantages for the
transmission of quantum information through optical fibers. In addition to the
quantum noise introduced by the amplifiers, entanglement between atoms in the
amplifying media and the optical field corresponds to which-path information
that can further reduce the coherence. Here we analyze the effects of
decoherence in a phase-insensitive distributed amplifier by using perturbation
theory to calculate the state of the entire system including the atomic media.
For an initial coherent state, tracing over the atomic states allows the
reduced density matrix of the field to be expressed as a mixture of squeezed
states with a reduced spread in photon number and an increased phase
uncertainty. The amplifier noise and decoherence can be interpreted as being
due to entanglement with the environment rather than the amplification of
vacuum fluctuation noise.
