Generalised Probabilistic Theories

We give an introduction to research associated with the generalised probabilistic theories framework, also known as the convex framework. States are real vectors representing lists of probabilities of measurement outcomes. Convex combinations of the vectors represent probabilistic combinations of different state preparations. Transformations are real matrices. Measurement outcomes are represented by functionals of the states, inner products of the state with a real vector, whose values are the probability of the measurement outcome in question. The framework generalises quantum theory. We describe the operational meaning of the framework, and how the concepts can be defined in terms of cones of states and measurement outcome vectors. We describe how the classical and quantum probability theories are represented in the framework. We describe Bell non-locality and the theory with super-quantum non-locality known as box world. We discuss generalised Hamiltonian mechanics in the discrete case and in continuous phase space, including the role of negativity of the phase space density in contextuality and tunnelling.