Degenerate quantum codes and the quantum channel capacity problem

A striking feature of quantum error correcting codes is that they can sometimes be used to correct more errors than they can uniquely identify. Indeed, such degenerate codes are known to outperform all non-degenerate codes for very noisy quantum channels. As a result, rather than being chosen randomly according to some i.i.d. distribution, capacity achieving quantum codes must be chosen in a highly structured fashion. While there is no systematic understanding of how to design such codes, I will describe our best understanding of the problem. I will also briefly discuss a single-letter upper bound on the quantum capacity and its relation to the capacitypsilas possible additivity.