Nonclassicality breaking is the same as entanglement breaking for bosonic Gaussian channels

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 behavior 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 term 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.