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

TL;DR

This paper uses diagrammatic tensor network representations to clarify and prove dualities between linear maps, quantum states, and channels, making complex proofs more intuitive and compact.

Contribution

It provides a diagrammatic framework for understanding and proving dualities in quantum information theory, enhancing intuition and simplifying proofs.

Findings

Diagrammatic representations clarify dualities in quantum theory.

Graphical proofs are more compact and intuitive.

The approach unifies various dualities in a visual framework.

Abstract

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.