Graph States and the Necessity of Euler Decomposition

Coecke and Duncan recently introduced a categorical formalisation of the interaction of complementary quantum observables. In this paper we use their diagrammatic language to study graph states, a computationally interesting class of quantum states. We give a graphical proof of the fixpoint property of graph states. We then introduce a new equation, for the Euler decomposition of the Hadamard gate, and demonstrate that Van den Nest’s theorem—locally equivalent graphs represent the same entanglement—is equivalent to this new axiom. Finally we prove that the Euler decomposition equation is not derivable from the existing axioms.