  Recently the power of positive partial transpose preserving (PPTp) and
no-signalling (NS) codes in quantum communication has been studied. We continue
with this line of research and show that the NS/PPTp/NS$\cap$PPTp codes
assisted zero-error quantum capacity depends only on the non-commutative
bipartite graph of the channel and the one-shot case can be computed
efficiently by semidefinite programming (SDP). As an example, the activated
PPTp codes assisted zero-error quantum capacity is carefully studied. We then
present a general SDP upper bound $Q_\Gamma$ of quantum capacity and show it is
always smaller than or equal to the "Partial transposition bound" introduced by
Holevo and Werner, and the inequality could be strict. This upper bound is
found to be additive, and thus is an upper bound of the potential PPTp assisted
quantum capacity as well. We further demonstrate that $Q_\Gamma$ is strictly
better than several previously known upper bounds for an explicit class of
quantum channels. Finally, we show that $Q_\Gamma$ can be used to bound the
super-activation of quantum capacity.
