I'm looking at the paper Categorical Quantum Mechanics II: Classical-Quantum interaction by Coecke and Kissinger (arxiv link), and I'm having difficulty with one particular aspect.
Throughout the paper, quantum wires are defined as "doubled" wires, while the classical world is represented by single wires. In particular, quantum spiders are simply doubled classical spiders. My understanding of this is we use the canonical $\dagger$-Frobenius algebra on $A \otimes A^*$, given $\dagger$-Frobenius algebras on $A$ and $A^*$, to generate the quantum spiders on systems of the form $A \otimes A^*$. This seems like enough to treat classical and quantum operations "on the same footing", i.e. every object in our category can have an associated Frobenius algebra, given ones specified on the ground objects, and if a process is doubled, then it's quantum. The encoding/decoding maps allow for conversion between the two. I don't understand that there is any real difference between what happens at the classical and quantum levels, except that the quantum is doubled - so whether one interprets a particular morphism as a single "thick/doubled" quantum wire or as two classical ones is irrelevant.
However, Definition 3.20 states that there is a second canonical Frobenius algebra structure on any objects of the form $A \otimes A^*$, namely the "pants" algebra $M_n$. Further, the paragraph following this - and my reading on the $CP^*$ construction - seems to suggest that the correct embedding of completely positive quantum processes into the category $CP^*$ of mixed classical/quantum processes is in fact given by considering quantum processes as acting on objects of the form $A \otimes A^*$ with the pants algebra as the associated Frobenius algebra - whereas each component $A$ or $A^*$ may have associated to it a completely unrelated Frobenius algebra structure. If this is correct, does this imply there are actually two Frobenius algebra structures associated to $A \otimes A^*$ - the pants algebra, and the other canonical "doubled" Frobenius algebra, described above? Does this mean that the "doubled" Frobenius algebra on $A \otimes A^*$ actually represents classical communication, but the pants algebra on the same object is quantum communication?
I must have become quite confused here, as both of these "understandings" cannot be correct!
