  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.
