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The compositional structure of multipartite quantum entanglement | | Bob Coecke
; Aleks Kissinger
; | | Date: |
12 Feb 2010 | | 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 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.
We consider N-qubit states to be equivalent as computational resources if
they can be inter-converted by stochastic local (quantum) operations and
classical communication (SLOCC). There are only two SLOCC-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
over qubits 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 due to Duncan and one of the
authors (arXiv:0906.4725), 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. | | Source: | arXiv, 1002.2540 | | Services: | Forum | Review | PDF | Favorites | |
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