Amendable Gaussian channels: Restoring entanglement via a unitary filter

We show that there exist Gaussian channels which are amendable. A channel that is entanglement-breaking of order 2 [A. De Pasquale and V. Giovannetti, Phys. Rev. A 86, 052302 (2012)] is amendable if there exists an unitary filter that, once applied in between two actions of the channel, removes the entanglement-breaking property of the overall transformation. We find that, depending on the structure of the channel, the unitary filter can be a squeezing transformation or a phase-shift operation. We also propose two realistic quantum optics experiments where the amendability of Gaussian channels can be verified by exploiting the fact that it is sufficient to test the entanglement-breaking properties of two-mode Gaussian channels on input states with finite energy (which are not maximally entangled).