"Pretty strong" converse for the quantum capacity of degradable channels

Abstract

We exhibit a possible road towards a strong converse for the quantum capacity of degradable channels. In particular, we show that all degradable channels obey what we call a "pretty strong" converse: When the code rate increases above the quantum capacity, the fidelity makes a discontinuous jump from 1 to at most 0.707, asymptotically. A similar result can be shown for the private (classical) capacity. Furthermore, we can show that if the strong converse holds for symmetric channels (which have quantum capacity zero), then degradable channels obey the strong converse: The above-mentioned asymptotic jump of the fidelity at the quantum capacity is then from 1 down to 0.

17 pages, requires IEEEtran.cls. v2 has new title and updated references; several small errors are corrected and some explanations expanded. v3 is final version accepted by journal; improved presentation, more references and a few more typos corrected