The compositional structure of multipartite quantum entanglement

Abstract

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.

31 pages, heavily revised from previous version. Notably, added a section on Frobenius states and removed the section on the Z/X-calculus. Full treatment of complementarity is relegated to future work

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• ZX-Calculus and Extended Hypergraph Rewriting Systems I: A Multiway Approach to Categorical Quantum Information Theory
• Higher order singular value decomposition and the reduced density matrices of three qubits
• Quantum entanglement, supersymmetry, and the generalized Yang-Baxter equation
• A Diagrammatic Axiomatisation for Qubit Entanglement
• Relating toy models of quantum computation: comprehension, complementarity and dagger mix autonomous categories