  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.
