  The complementarity of two observables is often captured in uncertainty
relations, which quantify an inevitable tradeoff in knowledge. Here we study
complementarity in the context of an information processing task: we link the
complementarity of two observables to their usefulness for superdense coding
(SDC). In SDC, Alice sends two classical dits of information to Bob by sending
a single qudit. However, we show that encoding with commuting unitaries
prevents Alice from sending more than one dit per qudit, implying that
complementarity is necessary for SDC to be advantagous over a classical
strategy for information transmission. When Alice encodes with products of
Pauli operators for the $X$ and $Z$ bases, we quantify the complementarity of
these encodings in terms of the overlap of the $X$ and $Z$ basis elements. Our
main result explicitly solves for the SDC capacity as a function of the
complementarity, showing that the entropy of the overlap matrix gives the
capacity, when the preshared state is maximally entangled. We generalise this
equation to resources with symmetric noise such as a preshared Werner state. In
the most general case of arbitrary noisy resources, we obtain an analogous
lower bound on the SDC capacity. Our results shed light on the role of
complementarity in determining the quantum advantage in SDC and also seem
fundamentally interesting since they bear a striking resemblance to uncertainty
relations.
