  A complete analysis of multi-mode bosonic Gaussian channels is proposed. We
clarify the structure of unitary dilations of general Gaussian channels
involving any number of bosonic modes and present a normal form. The maximum
number of auxiliary modes that is needed is identified, including all rank
deficient cases, and the specific role of additive classical noise is
highlighted. By using this analysis, we derive a canonical matrix form of the
noisy evolution of n-mode bosonic Gaussian channels and of their weak
complementary counterparts, based on a recent generalization of the normal mode
decomposition for non-symmetric or locality constrained situations. It allows
us to simplify the weak-degradability classification. Moreover, we investigate
the structure of some singular multi-mode channels, like the additive classical
noise channel that can be used to decompose a noisy channel in terms of a less
noisy one in order to find new sets of maps with zero quantum capacity.
Finally, the two-mode case is analyzed in detail. By exploiting the composition
rules of two-mode maps and the fact that anti-degradable channels cannot be
used to transfer quantum information, we identify sets of two-mode bosonic
channels with zero capacity.
