Idempotents in Dagger Categories: (Extended Abstract)

Dagger compact closed categories were studied by Abramsky and Coecke (under the name “strongly compact closed categories”) as an abstract presentation of the category of Hilbert spaces and linear maps, and as a framework in which to carry out the interpretation of quantum protocols. I subsequently showed that dagger compact closed categories can also describe mixed quantum computation, where the morphisms are completely positive maps. I introduced the CPM construction as a way to pass from the pure to the mixed setting. One technical detail of the CPM(C) construction is that it does not preserve biproducts. Therefore, to obtain an interpretation of classical types such as bit = I ⊕ I, one must work in the free biproduct completion CPM(C). In this paper, we show that there is another view of classical types, namely as splittings of self-adjoint idempotents on quantum types. We show that all the objects of CPM(C) arise as such splittings.