  Nonclassicality and entanglement are notions fundamental to quantum
information processes involving continuous variable systems. That these two
notions are intimately related has been intuitively appreciated for quite some
time. An aspect of considerable interest is the behaviour of these attributes
of a state under the action of a noisy channel. Inspired by the notion of
entanglement-breaking channels, we define the concept of
nonclassicality-breaking channels in a natural manner. We show that the notion
of nonclassicality-breaking is essentially equivalent---in a clearly defined
sense of the phrase `essentially'---to the notion of entanglement-breaking, as
far as bosonic Gaussian channels are concerned. This is notwithstanding the
fact that the very notion of entanglement-breaking requires reference to a
bipartite system, whereas the definition of nonclassicality-breaking makes no
such reference. Our analysis rests on our classification of channels into
nonclassicality-based, as against entanglement-based, types of canonical forms.
Our result takes ones intuitive understanding of the close relationship between
nonclassicality and entanglement a step closer.
