  The quantum capacity of bosonic Gaussian quantum channels can be non-additive
in a particularly striking way: a pair of such optical-fiber type channels can
individually have zero quantum capacity but super-activate each other such that
the combined channel has strictly positive capacity. This has been shown in
[Nature Photonics 5, 624 (2011)] where it was conjectured that squeezing is a
necessary resource for this phenomenon. We provide a proof of this conjecture
by showing that for gauge covariant channels a Choi matrix with positive
partial transpose implies that the channel is entanglement-breaking. In
addition, we construct an example which shows that this implication fails to
hold for Gaussian channels which arise from passive interactions with a
squeezed environment.
