Quantum capacities of channels with small environment

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.