Iwama found a complete set of identities for circuits solely consisting of Toffoli gates. I present a complete set of identities for the symmetric monoidal category generated by the Toffoli gate \emph{and ancillary bits}. I also provide a normal form for these circuits and prove an equivalence of categories into a subcategory of

Title: Structures for decoherence
Abstract:
This talk will introduce and develop the structure required for studying
decoherence in certain monoidal categories. Our driving example is a
decoherence structure in
![Rendered by QuickLaTeX.com \mathsf{CP}^*[\mathsf{FHilb}]](https://quicklatex.com/cache3/7a/ql_138bbffd2408ee74ee31a67f8eb4317a_l3.png)
Our goal is to move towards understanding the following:
Theorem:
Let C be a dagger compact closed category and

be a subcategory of C that inherits the dagger and compact closed structure. Suppose

has a decoherence structure with purification, then there exists an invertible dagger functor
![Rendered by QuickLaTeX.com \mathsf{CP}^*[C_\mathsf{pure}] \to C](https://quicklatex.com/cache3/6c/ql_8b75932db1aa8bc54cc019032ebd1a6c_l3.png)
such that

.