Depolarizing Reference Devices in Generalized Probabilistic Theories

Abstract: QBism is an interpretation of quantum theory which views quantum mechanics as standard probability theory supplemented with a few extra normative constraints. The fundamental gambit is to represent states and measurements, as well as time evolution, with respect to an informationally complete reference device. From this point of view, the Born rule appears as a coherence condition on probability assignments across several different experiments which manifests as a deformation of the law of total probability (LTP). In this work, we fully characterize those reference devices for which this deformation takes a "simplest possible" (term-wise affine) form. Working in the framework of generalized probability theories (GPTs), we show that, given any reference measurement, a set of post-measurement reference states can always be chosen to give its probability rule this very form. The essential condition is that the corresponding measure-and-prepare channel be depolarizing. We also relate our construction to Szymusiak and S{\l}omczy\'nski's recently introduced notion of morphophoricity and re-examine critically a matrix-norm-based measure of LTP deformation in light of our results. What stands out for the QBist project from this analysis is that it is not only the pure form of the Born rule that must be understood normatively, but the constants within it as well. It is they that carry the details of quantum theory.