Symmetric Informationally Complete Measurements Pinpoint the Essential Difference between Classical and Quantum Probability Theories

I describe a general procedure for crafting a purely probabilistic representation of the Born Rule by means of a reference measurement: a minimal informationally-complete quantum measurement (MIC) and a set of linearly independent post-measurement quantum states. It follows that the Born Rule is a consistency condition between probabilities assigned to the outcomes of different, mutually exclusive experiments. The difference between quantum and classical physics is the way their physical assumptions augment bare probability theory: Classical physics corresponds to a trivial augmentation—one just applies the Law of Total Probability between the scenarios—while quantum theory makes use of the Born Rule expressed in one or another of the forms of our general procedure. To mark the essential difference between quantum and classical, one should seek the representations that minimize the disparity between the expressions. Using a symmetric informationally-complete measurement (SIC) minimizes this disparity, according to a large family of optimality criteria. This work complements recent studies in quantum computation where the deviation of the Born Rule from the LTP is measured in terms of negativity of Wigner functions.

(Joint work with J. B. DeBrota and C. A. Fuchs, arXiv:1805.08721)