A diagrammatic approach to map-state and channel-state dualities

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.