  The quantum capacity of a memoryless channel is often used as a single figure
of merit to characterize its ability to transmit quantum information
coherently. The capacity determines the maximal rate at which we can code
reliably over asymptotically many uses of the channel. We argue that this
asymptotic treatment is insufficient to the point of being irrelevant in the
quantum setting where decoherence severely limits our ability to manipulate
large quantum systems in the encoder and decoder. For all practical purposes we
should instead focus on the trade-off between three parameters: the rate of the
code, the number of coherent uses of the channel, and the fidelity of the
transmission. The aim is then to specify the region determined by allowed
combinations of these parameters. Towards this goal, we find approximate and
exact characterizations of the region of allowed triplets for the qubit
dephasing channel and for the erasure channel with classical post-processing
assistance. In each case the region is parametrized by a second channel
parameter, the quantum channel dispersion. In the process we also develop
several general inner (achievable) and outer (converse) bounds on the coding
region that are valid for all finite-dimensional quantum channels and can be
computed efficiently. Applied to the depolarizing channel, this allows us to
determine a lower bound on the number of coherent uses of the channel necessary
to witness super-additivity of the coherent information.
