Tensors, !-graphs, and non-commutative quantum structures (extended version)

-graphs provide a means of reasoning about infinite families of string diagrams and have proven useful in manipulation of (co)algebraic structures like Hopf algebras, Frobenius algebras, and compositions thereof. However, they have previously been limited by an inability to express families of diagrams involving non-commutative structures which play a central role in algebraic quantum information and the theory of quantum groups. In this paper, we fix this shortcoming by offering a new semantics for non-commutative !-graphs using an enriched version of Penrose’s abstract tensor notation.