Non-locality in Categorical Quantum Mechanics

Interest has grown in recent years in the construction of ‘quantum-like’ theories, toy theories which exhibit some but not all features of quantum mechanics. Such theories are expressed in diverse mathematical terms which may impede comparison of their properties. In this thesis we present a unifying mathematical framework in which we can compare a variety of ‘quantum-like’ theories, based on Abramsky and Coecke’s work on applying category theory to quantum mechanics. By doing so we hope to gain a clearer insight into the precise ways in which these theories differ mathematically, and whether this relates to the differences in phenomena which they predict. As an example of this kind of approach, we express Spekkens’s toy bit theory within the categorical framework, in the process proving its consistency. The toy bit theory reproduces many features of quantum mechanics, and this is reflected in the fact that within the categorical framework it shares many structural features with quantum mechanics. It differs however, in that it is, by construction, a local hidden variable theory. We develop a categorical treatment of hidden variables, and then demonstrate that the categorical structures which differ between quantummechanics and the toy theory are exactly those which relate to the question of hidden variables. We extend this to a general result applying to a wider range of theories.