  In the paper, we give a characterization of arbitrary $n$-mode Gaussian
coherence breaking channels (GCBCs) and show the tensor product of a GCBC and
arbitrary a Gaussian channel maps all input states into product states. The
inclusion relation among GCBCs, Gaussian positive partial transpose channels
(GPPTC), entanglement breaking channels (GEBC) and Gaussian classical-quantum
channels (GCQC) is displayed. Finally, we prove that Gaussian classical
($\chi$-) capacity for the tensor products of GCBCs and arbitrary Gaussian
channels is additive.
