A 3-design is all you need

Can all of d-dimensional quantum theory be derived from studying the behavior of a single measuring device? The answer is yes, if the measuring device corresponds to a complex-projective 3-design. Through its Jordan algebraic structure, quantum state-space is fully determined by its third moment, allowing one to characterize the pure states of the theory by their probabilities with respect to a 3-design measurement. In fact, pure state probability distributions live in the non-negative orthant, in the intersection of three spheres: a 1-norm sphere, a 2-norm sphere, and a 3-norm sphere, of specified radii. QBism proposes that quantum theory is normative guidance on an agent's gambles on a world without hidden variables, and re-interprets the quantum formalism as a set of consistency conditions on an agent's probability assignments across different experiments. Thus consistency with a single 3-design "reference device," properly interpreted, implies consistency with all of quantum theory.