  The quantum switch is a quantum process that creates a coherent control
between different unitary operations, which is often described as a quantum
process which transforms a pair of unitary operations $(U_1, U_2)$ into a
controlled unitary operation that coherently applies them in different orders
as ${\vert {0} \rangle\!\langle {0} \vert} \otimes U_1 U_2 + {\vert {1}
\rangle\!\langle {1} \vert} \otimes U_2 U_1$. This description, however, does
not directly define its action on non-unitary operations. The action of the
quantum switch on non-unitary operations is then chosen to be a ``natural''
extension of its action on unitary operations. In general, the action of a
process on non-unitary operations is not uniquely determined by its action on
unitary operations. It may be that there could be a set of inequivalent
extensions of the quantum switch for non-unitary operations. We prove, however,
that the natural extension is the only possibility for the quantum switch for
the 2-slot case. In other words, contrary to the general case, the action of
the quantum switch on non-unitary operations (as a linear and completely CP
preserving supermap) is completely determined by its action on unitary
operations. We also discuss the general problem of when the complete
description of a quantum process is uniquely determined by its action on
unitary operations and identify a set of single-slot processes which are
completely defined by their action on unitary operations.
