Towards Normal Forms for GHZ/W Calculus

TL;DR

This paper introduces initial normal forms for a restricted class of graphs in the GHZ/W calculus, aiming to better identify quantum states through graphical properties, thus advancing the understanding of quantum state classification.

Contribution

It provides the first attempt at defining normal forms for simple, regular graphs in the GHZ/W calculus, facilitating state identification.

Findings

Normal forms established for simple, regular graphs.

Potential to extend to more complex graph classes.

Enhanced ability to classify quantum states graphically.

Abstract

Recently, a novel GHZ/W graphical calculus has been established to study and reason more intuitively about interacting quantum systems. The compositional structure of this calculus was shown to be well-equipped to sufficiently express arbitrary mutlipartite quantum states equivalent under stochastic local operations and classical communication (SLOCC). However, it is still not clear how to explicitly identify which graphical properties lead to what states. This can be achieved if we have well-behaved normal forms for arbitrary graphs within this calculus. This article lays down a first attempt at realizing such normal forms for a restricted class of such graphs, namely simple and regular graphs. These results should pave the way for the most general cases as part of future work.