Born's rule as a quantum extension of Bayesian coherence

The subjective Bayesian interpretation of probability asserts that the rules of the probability calculus follow from the normative principle of Dutch-book coherence: A decision-making agent should not assign probabilities such that a series of monetary transactions based on those probabilities would lead them to expect a sure loss. Similarly, the subjective Bayesian interpretation of quantum mechanics (QBism) asserts that the Born rule is a normative rule in analogy to Dutch-book coherence, but with the addition of one or more empirical assumptions characterizing the particularities of the physical world. Here we make this link explicit for a conjectured representation of the Born rule which holds true if symmetric informationally complete quantum measurements (or SIC-POVMs) exist for every finite-dimensional Hilbert space. We prove that an agent who thinks they are gambling on the outcomes of measurements on a sufficiently quantum-like system but refuses to use this form of the Born rule when placing their bets is vulnerable to a Dutch book. The key property for being sufficiently quantum-like is that the system admits a symmetric reference measurement but that this measurement is not sampling any hidden variables.