  We consider various forms of a process, which we call {\em gluing}, for
combining two or more complementary quantum channel pairs
$(\mathcal{B},\mathcal{C})$ to form a composite. One type of gluing combines a
perfect channel with a second channel to produce a \emph{generalized erasure
channel} pair $(\mathcal{B}_g,\mathcal{C}_g)$. We consider two cases in which
the second channel is (i) an amplitude-damping, or (ii) a phase-damping qubit
channel; (ii) is the \emph{dephrasure channel} of Leditzky et al. For both (i)
and (ii), $(\mathcal{B}_g,\mathcal{C}_g)$ depends on the damping parameter
$0\leq p\leq 1$ and a parameter $0 \leq \lambda \leq 1$ that characterizes the
gluing process. In both cases we study $Q^{(1)}(\mathcal{B}_g)$ and
$Q^{(1)}(\mathcal{C}_g)$, where $Q^{(1)}$ is the channel coherent information,
and determine the regions in the $(p,\lambda)$ plane where each is zero or
positive, confirming previous results for (ii). A somewhat surprising result
for which we lack any intuitive explanation is that $Q^{(1)}(\mathcal{C}_g)$ is
zero for $\lambda \leq 1/2$ when $p=0$, but is strictly positive (though
perhaps extremely small) for all values of $\lambda> 0$ when $p$ is positive by
even the smallest amount. In addition we study the nonadditivity of
$Q^{(1)}(\mathcal{B}_g)$ for two identical channels in parallel. It occurs in a
well-defined region of the $(p,\lambda)$ plane in case (i). In case (ii) we
have extended previous results for the dephrasure channel without, however,
identifying the full range of $(p,\lambda)$ values where nonadditivity occurs.
Again, an intuitive explanation is lacking.
