(F’) to axiom (F).

Dagger Frobenius algebras in quantum mechanics
----------------------------------------------

Frobenius algebras provide a high-level algebraic way of talking about *orthonormal bases*, and hence can be seen as modeling quantum mechanical *observables*.

To put this in context, we recall the *no-cloning theorem* [@wootterszurek:nocloning], which says that there is no quantum evolution ([*i.e.* ]{}unitary operator) $f 
\colon H \to H {\ensuremath{\otimes}}H$ such that, for any ${{|} \phi\rangle} \in H$, $$f{{|} \phi\rangle} = {{|} \phi\rangle} {\ensuremath{\otimes}}{{|} \phi\rangle}.$$ A general form of no-cloning holds for structural reasons in categorical quantum mechanics [@abramsky:nocloning]. In particular, there is no *natural*, [*i.e.* ]{}uniform or basis-independent, family of diagonal morphisms in a compact closed category, unless the category collapses, so that endomorphisms are scalar multiples of the identity.

However, if we drop naturality, we *can* define such maps in ${\ensuremath{\mathbf{Hilb}}}$ in a basis-dependent fashion. Moreover, it turns out that such maps can be used to *uniquely determine* bases. Firstly, consider *copying maps*, which can be defined in arbitrary dimension: for a given basis $\{{{|} i\rangle}\}_{i \in I}$ of $H$, define $\delta \colon H \to H {\ensuremath{\otimes}}H$ by (continuous linear extension of) ${{|} i\rangle} \mapsto {{|} ii\rangle}$.

For example, consider the map $\delta_{\text{std}} \colon {\ensuremath{\mathbb{C}}}^2 \to
{\ensuremath{\mathbb{C}}}^2 