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# The compositional structure of multipartite quantum entanglement

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## Recording Details

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PIRSA Number:
10030114

## Abstract

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