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

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