The compositional structure of multipartite quantum entanglement

Journal article

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 long-standing 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 GHZ- and W-states, giving rise to a rich graphical paradigm for general multipartite entanglement.

Authors: Coecke, B; Kissinger, A

• Publication status: Published
• Publisher: Springer
• Journal: ICALP (2)
• Volume: 6199
• Issue: PART 2
• Pages: 297-308
• Publication date: 2010-02-12
• DOI: 10.1007/978-3-642-14162-1_25
• EISSN: 1611-3349
• ISSN: 0302-9743
• Language: English
• Keywords: math.QA; math.CT; quant-ph