Quantum-channel capacity of very noisy channels

We present a family of additive quantum error-correcting codes whose capacities exceed those of quantum random coding (hashing) for very noisy channels. These codes provide nonzero capacity in a depolarizing channel for fidelity parameters $f$ when $f>\mathrm{}0.809\mathrm{}44$. Random coding has nonzero capacity only for $f>\mathrm{}0.810\mathrm{}71$; by analogy to the classical Shannon coding limit, this value had previously been conjectured to be a lower bound. We use the method introduced by Shor and Smolin of concatenating a nonrandom repetition (cat) code within a random code to obtain good codes. The cat code with block size five is shown to be optimal for single concatenation. The best known multiple-concatenated code we found has a block size of 25. We derive a general relation between the capacity attainable by these concatenation schemes and the coherent information of the inner code states.