  The quantum switch is a higher-order operation that takes as an input two
quantum processes and combines them in a coherent superposition of two
alternative orders. Here we provide an approach to the quantum switch based on
the methods of categorical quantum mechanics. Specifically, we represent the
quantum switch as a sum of diagrams in the category of finite dimensional
Hilbert spaces, or, equivalently, as a sum of diagrams built from Selinger's
CPM construction. The sum-of-diagrams picture provides intuition for the
activation of classical capacity of completely depolarising channels (CDPCs)
and allows for generalisation to N-channel switches. We demonstrate the use of
these partially diagrammatic methods by deriving a permutation condition for
computing the output of any N-channel switch of CDPCs, we then use that
condition to prove that amongst all possible terms, the interference terms
associated to cyclic permutations of the N channels are the
information-transmitting terms with maximum normalisation
