Semidefinite programming converse bounds for classical communication over quantum channels

We study the classical communication over quantum channels when assisted by no-signalling (NS) and PPT-preserving (PPT) codes. We first show that both the optimal success probability of a given transmission rate and one-shot c-error capacity can be formalized as semidefinite programs (SDPs) when assisted by NS or NSnPPT codes. Based on this, we derive SDP finite blocklength converse bounds for general quantum channels, which also reduce to the converse bound of Polyanskiy, Poor, and Verdu for classical channels. Furthermore, we derive an SDP strong converse bound for the classical capacity of a general quantum channel: for any code with a rate exceeding this bound, the optimal success probability vanishes exponentially fast as the number of channel uses increases. In particular, applying our efficiently computable bound, we derive improved upper bounds to the classical capacity of the amplitude damping channels and also establish the strong converse property for a new class of quantum channels.