  We formalize the correspondence between quantum states and quantum operations
isometrically, and harness its consequences. This correspondence was already
implicit in the various proofs of the operator sum representation of Completely
Positive-preserving linear maps; we go further and show that all of the
important theorems concerning quantum operations can be derived directly from
those concerning quantum states. As we do so the discussion first provides an
elegant and original review of the main features of quantum operations. Next
(in the second half of the paper) we find more results stemming from our
formulation of the correspondence. Thus we provide a factorizability condition
for quantum operations, and give two novel Schmidt-type decompositions of
bipartite pure states. By translating the composition law of quantum
operations, we define a group structure upon the set of totally entangled
states. The question whether the correspondence is merely mathematical or can
be given a physical interpretation is addressed throughout the text: we provide
formulae which suggest quantum states inherently define a quantum operation
between two of their subsystems, and which turn out to have applications in
quantum cryptography. Keywords: Kraus, CP-maps, superoperators, extremality,
trace-preserving, factorizable, triangular.
