The Compositional Structure of Multipartite Quantum Entanglement

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 a long-standing open problem in quantum computer science. Here we show that multipartite quantum entanglement admits a compositional structure, and hence is subject to modern computer science methods. Recall that two N-qubit states are SLOCC-equivalent if they can be inter-converted by stochastic local (quantum) operations and classical communication. There are only two SLOCC-equivalence classes of genuinely entangled 3-qubit states, the GHZ-class and the W-class, and we show that these exactly correspond with two kinds of internal commutative Frobenius algebras on C2 in the symmetric monoidal category of Hilbert spaces and linear maps, 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 refines the graphical calculus of complementary observables in [5, ICALP'08], which has already shown itself to have many applications and admit automation. Our result also induces a generalised graph state paradigm for measurement-based quantum computing.

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