Title: The ZX& calculus
Abstract: Consider ZX&, the fragment of the ZX calculus generated by the copying/addition Frobenius algebras, the not gate and the and gate. I prove that this fragment is complete and universal for a prop of spans of sets, by freely adding units and counits to the inverse products of TOF (the category generated by the Toffoli gate and computational basis states/effects). To prove completeness, I first show show that adding a counit to TOF is the same as computing the classical channels, which is the the same as the discrete Cartesian completion. The completeness of ZX& is obtained via a two way translation between this extension of TOF with units and counits, and ZX&.
