A Diagrammatic Axiomatisation for Qubit Entanglement

TL;DR

This paper introduces the ZW calculus, a complete diagrammatic axiomatisation for representing and reasoning about GHZ and W states in qubit entanglement, enhancing the graphical tools for quantum information.

Contribution

It presents a new graphical calculus, the ZW calculus, refining the ZX calculus to better classify multipartite qubit entanglement with symmetry and algebraic structure.

Findings

ZW calculus is complete for free abelian groups on powers of two generators

Provides an explicit normalisation procedure for the calculus

Refines the ZX calculus with enhanced symmetry and algebraic properties

Abstract

Diagrammatic techniques for reasoning about monoidal categories provide an intuitive understanding of the symmetries and connections of interacting computational processes. In the context of categorical quantum mechanics, Coecke and Kissinger suggested that two 3-qubit states, GHZ and W, may be used as the building blocks of a new graphical calculus, aimed at a diagrammatic classification of multipartite qubit entanglement that would highlight the communicational properties of quantum states, and their potential uses in cryptographic schemes. In this paper, we present a full graphical axiomatisation of the relations between GHZ and W: the ZW calculus. This refines a version of the preexisting ZX calculus, while keeping its most desirable characteristics: undirectedness, a large degree of symmetry, and an algebraic underpinning. We prove that the ZW calculus is complete for the category of free abelian groups on a power of two generators - "qubits with integer coefficients" - and provide an explicit normalisation procedure.