  Operator-sum or Kraus representations for single-mode Bosonic Gaussian
channels are developed, and several of their consequences explored. Kraus
operators are employed to bring out the manner in which the unphysical matrix
transposition map when accompanied by injection of a threshold classical noise
becomes a physical channel. The action of the quantum-limited attenuator and
amplifier channels as simply scaling maps on suitable quasi-probabilities in
phase space is examined in the Kraus picture. Consideration of cumulants is
used to examine the issue of fixed points. In the cases of
entanglement-breaking channels a description in terms of rank one Kraus
operators is shown to emerge quite simply. In contradistinction, it is shown
that there is not even one finite rank operator in the entire linear span of
Kraus operators of the quantum-limited amplifier or attenuator families, an
assertion far stronger than the statement that these are not entanglement
breaking channels. A characterization of extremality in terms of Kraus
operators, originally due to Choi, is employed to show that all quantum-limited
Gaussian channels are extremal. The fact that every noisy Gaussian channel can
be realised as product of a pair of quantum-limited channels is used to
construct a discrete set of linearly independent Kraus operators for noisy
Gaussian channels, including the classical noise channel, and these Kraus
operators have a particularly simple structure.
