A diagrammatic axiomatisation of the GHZ and W quantum states

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 under stochastic local operations and classical communication (SLOCC). 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. Through an explicit normalisation procedure, the ZW calculus was shown to be complete for the category of free abelian groups on a power of two generators (“qubits with integer coefficients”).