  We determine both the quantum and the private capacities of low-noise quantum
channels to leading orders in the channel's distance to the perfect channel. It
has been an open problem for more than 20 years to determine the capacities of
some of these low-noise channels such as the depolarizing channel. We also show
that both capacities are equal to the single-letter coherent information of the
channel, again to leading orders. We thus find that, in the low noise regime,
super-additivity and degenerate codes have negligible benefit for the quantum
capacity, and shielding does not improve the private capacity beyond the
quantum capacity, in stark contrast to the situation when noisier channels are
considered.
