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.

Figure 1

A pictorial comparison of the nonclassicality-breaking, entanglement-breaking, and complete positivity conditions in the channel parameter space $(a,b)$ for fixed $\det X$. Curves (1), (2), and (3) correspond to saturation of these conditions in that order. Curve (3) thus corresponds to quantum-limited channels. Frame (a) refers to the first canonical form $(\kappa 11,\mathrm{diag}(a,b))$, frame (c) to the second canonical form $(\kappa {\sigma }_3,\mathrm{diag}(a,b))$, and frame (d) to the third canonical form, singular $X$. Frame (b) refers to the limiting case $\kappa =1$, classical noise channel. In all four frames, the region to the right of (above) curve (1) corresponds to nonclassicality-breaking channels; the region to the right of (above) curve (2) corresponds to entanglement-breaking channels; curve (3) depicts the CP condition, so the region to the right of (above) it alone corresponds to physical channels. The region to the left (below) curve (3) is unphysical as channels. In frames (c) and (d), curves (2) and (3) coincide. In frame (b), curve (3) of (a) reduces to the $a$ and $b$ axis shown in bold. In frames (a) and (c), curves (1) and (2) meet at the point $(1+{\kappa }^2,1+{\kappa }^2)$, in frame (b) they meet at $(2,2)$, and in frame (d) at $(1,1)$. The region between (2) and (3) corresponds to the set of channels which are not entanglement breaking. That in frame (c) and (d) the two curves coincide proves that this set is vacuous for the second and third canonical forms. That in every frame the nonclassicality-breaking region is properly contained in the entanglement-breaking region proves that a nonclassicality-breaking channel is certainly an entanglement-breaking channel. The dotted curve in each frame indicates the orbit of a generic entanglement-breaking Gaussian channel under the action of a local unitary squeezing after the channel action. That the orbit of every entanglement-breaking channel passes through the nonclassicality-breaking region proves that the nonclassicality in all the output states of an entanglement-breaking channel can be removed by a fixed unitary squeezing, thus showing that every entanglement-breaking channel is “essentially” a nonclassicality-breaking channel.