Gaussian channels that are eventually entanglement breaking yet asymptotically nonclassicality saving

A complete classification of single-mode bosonic Gaussian channels on the basis of being quantum-limited or entanglement breaking under $n$-fold composition is obtained. Parametric forms for all single-mode bosonic Gaussian channels that remain quantum-limited under $n$-fold composition is obtained. It is shown that the obtained parametric forms of quantum-limited amplifier, attenuator, and singular noise channels are entanglement saving, i.e., they do not break entanglement under finite $n$-fold composition. All other single-mode bosonic Gaussian channels, quantum limited or not, are shown to be eventually entanglement breaking. Nonclassicality breaking under multiple composition of a single-mode bosonic Gaussian channel is also studied. We outline a family of single-mode bosonic Gaussian channels that are eventually entanglement breaking, but asymptotically nonclassicality saving. We illustrate examples of channels that are eventually nonclassicality breaking, asymptotically nonclassicality breaking, and asymptotically nonclassicality saving.