Multipartite quantum states constitute a (if not the) key resource for quantum computations and protocols. However obtaining a generic, structural understanding of entanglement in N-qubit systems is still largely an open problem. Here we show that multipartite quantum entanglement admits a compositional structure. The two SLOCC-classes of genuinely entangled 3-qubit states, the GHZ-class and the W-class, exactly correspond with the two kinds of commutative Frobenius algebras on C^2, namely `special' ones and `anti-special' ones. Within the graphical language of symmetric monoidal categories, the distinction between `special' and `anti-special' is purely topological, in terms of `connected' vs.~`disconnected'. These GHZ and W Frobenius algebras form the primitives of a graphical calculus which is expressive enough to generate and reason about representatives of arbitrary N-qubit states.
This calculus induces a generalised graph state paradigm for measurement-based quantum computing, and refines the graphical calculus of complementary observables due to Duncan and one of the authors [ICALP'08], which has already shown itself to have many applications and admit automation.
References: Bob Coecke and Aleks Kissinger, http://arxiv.org/abs/1002.2540