I'm looking at the paper Categorical quantum mechanics II: classical-quantum interaction by Coecke and Kissinger (arxiv link), and I'm having trouble with one aspect in particular.
Throughout the document, quantum wires are defined as "doubled" wires, whereas the classical world is represented by single wires. In particular, quantum spiders are simply doubled spiders. My understanding of this is that we use the canonical $ dagger $-Algebra of Frobenius on $ A otimes A ^ * $given $ dagger $-Algebras of Frobenius on $ A $ and $ A ^ * $, to generate quantum spiders on systems of the form $ A otimes A ^ * $. This seems sufficient to treat the classical and quantum operations "on the same footing", ie each object in our category may have an associated Frobenius algebra, from those specified on the ground objects, and if a process is doubled, it is quantum. Coding / decoding cards allow conversion between the two. I do not understand that there is a real difference between what happens at the classical and quantum levels, if it is that the quantum is doubled – that we thus interpret a particular morphism as a simple "thick / doubled" quantum wire or two classic wires is out of place.
However, the definition 3.20 indicates that there is a second canonical algebraic structure of Frobenius on any object of the form. $ A otimes A ^ * $, namely the "pants" algebra $ M_n $. In addition, the following paragraph – and my readings on the $ CP ^ * $ construction – seems to suggest that the correct insertion of fully positive quantum processes into the category $ CP ^ * $ mixed classical / quantum processes is actually given by considering quantum processes as acting on objects of the form $ A otimes A ^ * $ with the algebra of the pants as the associated Frobenius algebra – while each component $ A $ or $ A ^ * $ may have associated a completely independent Frobenius algebra structure. If this is correct, does this imply that there are actually two Frobenius algebra structures associated with $ A otimes A ^ * $ – the algebra of the pants, and the other canonical "doubled" Frobenius algebra, described above? Does this mean that the algebra of Frobenius "doubled" on $ A otimes A ^ * $ actually represents classical communication, but the algebra of pants on the same object is quantum communication?
I must have been pretty confused here, because these two "understandings" can not be correct!