Complete Graphical Language for Hermiticity-Preserving Superoperators

TL;DR

This paper introduces a universal and complete graphical language for Hermiticity-preserving superoperators, enabling diagrammatic analysis of antilinear transformations in quantum physics.

Contribution

It extends the ZW-calculus to create a graphical language for Hermiticity-preserving superoperators, facilitating diagrammatic reasoning in quantum mechanics.

Findings

Provides a normal form for Hermitian matrices within the graphical language

Enables diagrammatic study of antilinear transformations like the Choi-Jamio{ }kowski isomorphism

Extends existing graphical languages to cover Hermiticity-preserving superoperators

Abstract

Universal and complete graphical languages have been successfully designed for pure state quantum mechanics, corresponding to linear maps between Hilbert spaces, and mixed states quantum mechanics, corresponding to completely positive superoperators. In this paper, we go one step further and present a universal and complete graphical language for Hermiticity-preserving superoperators. Such a language opens the possibility of diagrammatic compositional investigations of antilinear transformations featured in various physical situations, such as the Choi-Jamio{\l}kowski isomorphism, spin-flip, or entanglement witnesses. Our construction relies on an extension of the ZW-calculus exhibiting a normal form for Hermitian matrices.