A 2-Categorical Analysis of Complementary Families, Quantum Key Distribution and the Mean King Problem

The 2-categorical approach to quantum information is now well-developed in its basic aspects [4, 12, 14]. The central aim is to use the structure of a symmetric monoidal 2-category to describe quantum procedures in an abstract way, such that the ordinary versions of these procedures are recovered when we apply the formalism in 2Hilb, the symmetric monoidal 2-category of 2-Hilbert spaces [3]. This builds on the highly successful categorical quantum mechanics research programme of Abramsky, Coecke and collaborators [1, 2], in which quantum procedures are axiomatized in terms of monoidal 1-categories. The key advantage of the 2-categorical setup is that many important structures, such as teleportation, dense coding and complementary observables, can be defined by single 2-categorical equations. These typically have direct physical interpretations, with the defining equation for a structure following immediately from a careful physical description of its required properties. A computer algebra system TwoVect [10, 11] allows these equations to be directly evaluated computationally. In this paper, we show that the formalism can be applied successfully to more sophisticated quantum procedures: measurements in a complementary family of bases, quantum key distribution (QKD), and the Mean King problem. For each scenario, we write down a 2-categorical equation that defines the entire procedure in a precise way. For example, here is the defining diagram for BB84 QKD: