Categories and Quantum Informatics: Complementarity

This chapter studies what happens when we have two interacting Frobenius structures. Specifically, we are interested in when they are “maximally incompatible”, or complementary, and give a definition that makes sense in arbitrary monoidal dagger categories in Section 6.1. We will see that it comes down to the standard notion of mutually unbiased bases from quantum information theory in the category of Hilbert spaces, and classify the complementary groupoids in the category of sets and relations. We will also characterize complementarity in terms of a canonical morphism being isometric. This is exemplified by discussing the Deutsch–Jozsa algorithm in Section 6.2, where the canonical morphism plays the role of an oracle function. Section 6.3 links complementarity to the subject of Hopf algebra. It turns out that this well-studied notion gives rise to a stronger form of complementarity that we characterize. Finally, Section 6.4 discusses how many-qubit gates can be modeled in categorical quantum mechanics using only complementary Frobenius structures, such as controlled negation, controlled phase gates, and arbitrary single qubit gates. We have been using colours to distinguish between monoid multiplication and comonoid comultiplication . We have also been indicating that one is the dagger of the other by abbreviating = to just a single colour . From this chapter on, we will deal with two Frobenius structures, each carrying both a multiplication and a comultiplication. When this is the case we will specialize to dagger Frobenius structures, so we can distinguish them. By drawing the operations of a single Frobenius structure in a single colour, we can speak about two dagger Frobenius structures (A, , , , ) and (A, , , , ), in a way perfectly consistent with our conventions. Nevertheless, many results hold more generally without daggers.