Finite-dimensional Quantum Observables are the Special Symmetric Dagger-Frobenius Algebras of CP Maps

We use purity, a principle borrowed from the foundations of quantum information, to show that all special symmetric $\dagger$-Frobenius algebras in $\operatorname{CPM}\left(\operatorname{fHilb}\right)$ -- and, in particular, all classical structures -- are canonical, i.e. that they arise by doubling of special symmetric $\dagger$-Frobenius algebras in $\operatorname{fHilb}$. This provides an exact classification of finite-dimensional quantum observables.