Semidefinite Programming Converse Bounds for Quantum Communication


We study the one-shot and asymptotic quantum communication assisted with the positive-partial-transpose-preserving (PPT) and no-signalling (NS) codes. We first show improved general semidefinite programming (SDP) finite blocklength converse bounds for quantum communication with a given infidelity tolerance and utilize them to study the depolarizing channel and amplitude damping channel in a small blocklength. Based on the one-shot bounds, we then derive a general SDP strong converse bound for the quantum capacity of an arbitrary quantum channel. In particular, we prove that the SDP strong converse bound is always smaller than or equal to the partial transposition bound introduced by Holevo and Werner, and the inequality could be strict. Furthermore, we show that the SDP strong converse bound can be refined as the max-Rains information, which is an analog to the Rains information introduced in [Tomamichel/Wilde/Winter, IEEE Trans. Inf. Theory 63:715, 2017]. This also implies that it is always no smaller than the Rains information. Finally, we establish an inequality relationship among some of these known strong converse bounds on quantum capacity.

IEEE Transactions on Information Theory