  In the space of quantum channels, we establish the geometry that allows us to
make statistical predictions about relative volumes of entanglement breaking
channels among all the Gaussian quantum channels. The underlying metric is
constructed using the Choi-Jamio{\l}kowski isomorphism between the
continuous-variable Gaussian states and channels. This construction involves
the Hilbert-Schmidt distance in quantum state space. The volume element of the
one-mode Gaussian channels can be expressed in terms of local symplectic
invariants. We analytically compute the relative volumes of the one-mode
Gaussian entanglement breaking and incompatibility breaking channels. Finally,
we show that, when given the purities of the Choi-Jamio{\l}kowski state of the
channel, one can determine whether or not such channel is incompatibility
breaking.
