Structure of Quantum Theory

QBism views quantum mechanics as subjective Bayesian probability theory supplemented with additional nonclassical coherence conditions. The fundamental nonclassical coherence relation is a gentle modification of the law of total probability: it can be derived on very general grounds, and corresponds to the Born rule in quantum theory. The key conceptual trick is to demand probabilistic coherence with a so-called reference measurement: from this the framework of generalized probabilistic theories naturally emerges. We propose a similarly gentle modification of the classical rule for the product of valuations on the outcomes of a reference measurement. A valuation on any measurement has an equivalent valuation on the reference measurement: imposing a moment-matching condition implies that the theory is in fact a Euclidean Jordan algebra. Remarkably, the entire structure of the algebra is encoded in the probabilities which characterize the reference measurement itself. Moreover, specializing to quantum theory implies that the reference measurement corresponds to a complex projective 3-design. Our result not only provides a QBist motivation for the Jordan algebraic structure of quantum theory, but also suggests a practical method for certifying quantum 3-designs in a theory agnostic way.