Duration: 35:11

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We show a meaningful theory of classical communication over quantum channels when assisted by no-signalling (NS) and PPT-preserving (PPT) codes, for which both the optimal success probability of a given rate and one-shot $\epsilon$-error capacity are formalized as semidefinite programs (SDPs). Based on this, we not only obtain improved SDP converse bounds on entanglement assisted and unassisted codes, but also derive an SDP strong converse bound for the classical capacity. Furthermore, we show that the classical capacity of an amplitude damping channel with parameter $\gamma$ is upper bounded by $\log_2(1+\sqrt{1-\gamma})$ and establish the strong converse property for the classical and private capacities of a new class of channels. For quantum capacity, we use similar techniques to derive an SDP strong converse bound and show that it is always smaller than or equal to the ``Partial transposition bound''. We further demonstrate that our bound for quantum capacity is better than several previously known bounds for an explicit class of quantum channels.