The Compositional Structure of Multipartite Quantum Entanglement

While multipartite quantum states constitute a (if not the) key resource for quantum computations and protocols, obtaining a high-level, structural understanding of entanglement involving arbitrarily many qubits is a longstanding open problem in quantum computer science. In this paper we expose the algebraic and graphical structure of the GHZ-state and the W-state, as well as a purely graphical distinction that characterises the behaviours of these states. In turn, this structure yields a compositional graphical model for expressing general multipartite states. We identify those states, named Frobenius states, which canonically induce an algebraic structure, namely the structure of a commutative Frobenius algebra (CFA). We show that all SLOCC-maximal tripartite qubit states are locally equivalent to Frobenius states. Those that are SLOCC-equivalent to the GHZ-state induce special commutative Frobenius algebras, while those that are SLOCC-equivalent to the W-state induce what we call anti-special commutative Frobenius algebras. From the SLOCC-classification of tripartite qubit states follows a representation theorem for two dimensional CFAs. Together, a GHZ and a W Frobenius state form the primitives of a graphical calculus. This calculus is expressive enough to generate and reason about arbitrary multipartite states, which are obtained by “composing” the GHZand W-states, giving rise to a rich graphical paradigm for general multipartite entanglement. ? This work is supported by EPSRC Advanced Research Fellowship EP/D072786/1, by a Clarendon Studentship, by US Office of Naval Research Grant N00014-09-1-0248 and by EU FP6 STREP QICS. We thank Michael Hermann and Jamie Vicary for useful feedback. ar X iv :1 00 2. 25 40 v2 [ qu an tph ] 1 7 A ug 2 01 0 2 Bob Coecke and Aleks Kissinger