  Quantum capacities are fundamental quantities that are notoriously hard to
compute and can exhibit surprising properties such as superadditivity. Thus, a
vast amount of literature is devoted to finding tight and computable bounds on
these capacities. We add a new viewpoint by giving operationally motivated
bounds on several capacities, including the quantum capacity and private
capacity of a quantum channel and the one-way distillable entanglement and
private key of a quantum state. These bounds are generally phrased in terms of
capacity quantities involving the complementary channel or state. As a tool to
obtain these bounds, we discuss partial orders on quantum channels and states,
such as the less noisy and the more capable order. Our bounds help to further
understand the interplay between different capacities, as they give operational
limitations on superadditivity and the difference between capacities in terms
of the information-theoretic properties of the complementary channel or state.
They can also be used as a new approach towards numerically bounding
capacities, as discussed with some examples.
