  Degradable quantum channels are an important class of completely positive
trace-preserving maps. Among other properties, they offer a single-letter
formula for the quantum and the private classical capacity and are
characterized by the fact that a complementary channel can be obtained from the
channel by applying a degrading channel. In this work we introduce the concept
of approximate degradable channels, which satisfy this condition up to some
finite $\varepsilon\geq0$. That is, there exists a degrading channel which upon
composition with the channel is $\varepsilon$-close in the diamond norm to the
complementary channel. We show that for any fixed channel the smallest such
$\varepsilon$ can be efficiently determined via a semidefinite program.
Moreover, these approximate degradable channels also approximately inherit all
other properties of degradable channels. As an application, we derive improved
upper bounds to the quantum and private classical capacity for certain channels
of interest in quantum communication.
