  In this paper we show how \emph{the metric theory of tensor products}
developed by Grothendieck perfectly fits in the study of channel capacities, a
central topic in \emph{Shannon's information theory}. Furthermore, in the last
years Shannon's theory has been generalized to the quantum setting to let the
\emph{quantum information theory} step in. In this paper we consider the
classical capacity of quantum channels with restricted assisted entanglement.
In particular these capacities include the classical capacity and the unlimited
entanglement-assisted classical capacity of a quantum channel. To deal with the
quantum case we will use the noncommutative version of $p$-summing maps. More
precisely, we prove that the (product state) classical capacity of a quantum
channel with restricted assisted entanglement can be expressed as the
derivative of a completely $p$-summing norm.
