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

Understand

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.