  Diagrammatic representation and manipulation of tensor networks has proven to
be a useful tool in mathematics, physics, and computer science. Here we present
several important and mostly well-known theorems regarding the dualities
between linear maps and bipartite pure quantum states, and the dualities
between quantum channels and bipartite mixed quantum states, in diagrammatic
form. The graphical presentation makes the proofs very compact and in some
cases even intuitive.
