A Volume-Maximizing Map from Quantum States to the Probability Simplex, with Applications to QBism

In this talk, I make use of a recent formulation of quantum theory in terms of minimal informationally complete quantum measurements (MICs) to promote the idea that probability is more central to quantum theory than either state vectors or operators. One advantage of this formulation is that it permits a direct comparison of classical statistical physics with quantum theory via a standard reference measurement. Classically, the possibility of perfect knowledge of a system's phase space point means any vector in the reference probability simplex is an allowable state of knowledge. By contrast, the image of quantum state space under a MIC is necessarily a proper subset of the simplex. This suggests we could take the size of the allowable subset under a given MIC to be a measure of the representation's deviation from classicality. I report work from arxiv:1805.08721, where we prove that this deviation is minimized if and only if the MIC is chosen to be a symmetric informationally complete quantum measurement (SIC)---that is, the SIC representation minimizes the deviation from classicality. Finally, I speculate that this grants the SICs a unique foundational significance. (Joint work with C. A. Fuchs and B. C. Stacey)