pseudo-telepathy. First considerations along these lines will appear in a companion paper [@paper1b].

We now suggest a number of other possible applications of our categorical :

-   Quantum functions are closely related to quantum communication. Indeed, in Remark \[rem:teleportation\], we note that quantum bijections between a matrix algebra $\mathrm{Mat}_n$ and an $n^2$-element classical set can be understood as generalised teleportation protocols. More generally, it has been observed that quantum homomorphisms between quantum graphs admit an interpretation in terms of zero-error source-channel coding [@Stahlke2016].

-   Our classification in [@paper1b] is based on the study of Frobenius monads in $\QGraph$. As symmetric monoidal $2$-categories (see Remark \[rem:symmmon2cat\]), $\QSet$ and $\QGraph$ admit several other as yet unexplored algebraic structures.

-   Our quantisation framework may be extended to other combinatorial theories besides finite set theory and graph theory. In particular, in Section \[app:quantumrelation\], we show how Kuperberg and Weaver’s finite-dimensional quantum relations [@Kuperberg2012] fit into our framework; this suggests, for example, a $2$-category of quantum posets and quantum monotone functions.=-2

Outline of the paper
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In Section \[sec:background\], we recall the diagrammatic calculus for Hilbert spaces and linear maps as well as the correspondence between finite-dimensional $C^*$-algebras and certain Frobenius algebras. We express Gelfand 