Symmetric informationally complete measurements identify the irreducible difference between classical and quantum systems

Abstract

We describe a general procedure for associating a minimal informationally complete quantum measurement with a purely probabilistic representation of the Born rule. Such representations provide a way to understand the Born rule as a consistency condition between probabilities assigned to the outcomes of one experiment in terms of the probabilities assigned to the outcomes of other experiments. In this setting, 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 (LTP) 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 irreducible difference between quantum and classical, one should seek the representations that minimize the disparity between the expressions. We prove that the representation of the Born rule obtained from a symmetric informationally complete measurement minimizes this distinction in at least two senses, the first to do with unitarily invariant distance measures between the rules and the second to do with available volume in a reference probability simplex (roughly speaking, a different kind of uncertainty principle). Both of these arise from a useful result in majorization theory. 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.