  We investigate the quantum capacity of noisy quantum channels which can be
represented by coupling a system to an effectively small environment. A
capacity formula is derived for all cases where both system and environment are
two-dimensional--including all extremal qubit channels. Similarly, for channels
acting on higher dimensional systems we show that the capacity can be
determined if the channel arises from a sufficiently small coupling to a qubit
environment. Extensions to instances of channels with larger environment are
provided and it is shown that bounds on the capacity with unconstrained
environment can be obtained from decompositions into channels with small
environment.
