  We investigate the classical communication over quantum channels when
assisted by no-signaling (NS) and positive-partial-transpose-preserving (PPT)
codes, for which both the optimal success probability of a given transmission
rate and the one-shot $\epsilon$-error capacity are formalized as semidefinite
programs (SDPs). Based on this, we obtain improved SDP finite blocklength
converse bounds of general quantum channels for entanglement-assisted codes and
unassisted codes. Furthermore, we derive two SDP strong converse bounds for the
classical capacity of general quantum channels: for any code with a rate
exceeding either of the two bounds of the channel, the success probability
vanishes exponentially fast as the number of channel uses increases. In
particular, applying our efficiently computable bounds, we derive an improved
upper bound on the classical capacity of the amplitude damping channel. We also
establish the strong converse property for the classical and private capacities
of a new class of quantum channels. We finally study the zero-error setting and
provide efficiently computable upper bounds on the one-shot zero-error capacity
of a general quantum channel.
