Symmetric Informationally Complete Measurements Identify the Irreducible Difference between Classical and Quantum Systems

TL;DR

This paper demonstrates that symmetric informationally-complete measurements (SICs) provide an optimal probabilistic representation of the Born Rule, highlighting the fundamental difference between classical and quantum systems through minimal disparity in their probabilistic frameworks.

Contribution

The authors prove that SIC-based representations of the Born Rule minimize the difference between quantum and classical probability structures in two key ways, revealing the irreducible quantum-classical divide.

Findings

SIC measurements minimize the disparity between quantum and classical probability rules.

The representation based on SICs reduces the difference in unitarily invariant distance measures.

SIC-based representations optimize the volume in the probability simplex, indicating a new form of uncertainty principle.

Abstract

We describe a general procedure for associating a minimal informationally-complete quantum measurement (or MIC) and a set of linearly independent post-measurement quantum states with a purely probabilistic representation of the Born Rule. Such representations are motivated by QBism, where the Born Rule is understood 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 (or SIC) 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 new kind of uncertainty principle). Both of these arise from a significant majorization result. 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.