Comparing the degrees of incompatibility inherent in probabilistic physical theories

TL;DR

This paper introduces a new measure for quantifying the incompatibility of observables in probabilistic physical theories, revealing quantum theory's maximal incompatibility and identifying theories with even greater incompatibility under refined measures.

Contribution

It proposes a novel quantitative framework for comparing the incompatibility of observables across probabilistic theories, including quantum mechanics.

Findings

Quantum observables can be as incompatible as any in probabilistic theories.

Refined measures show some theories surpass quantum in incompatibility.

The framework enables systematic comparison of nonclassical features.

Abstract

We introduce a new way of quantifying the degrees of incompatibility of two ob- servables in a probabilistic physical theory and, based on this, a global measure of the degree of incompatibility inherent in such theories, across all observable pairs. This opens up a novel and flexible way of comparing probabilistic theories with respect to the nonclassical feature of incompatibility, raising many interesting questions, some of which will be answered here. We show that quantum theory contains observables that are as incompatible as any probabilistic physical theory can have if arbitrary pairs of observables are considered. If one adopts a more refined measure of the degree of incompatibility, for instance, by restricting the comparison to binary observables, it turns out that there are probabilistic theories whose inherent degree of incompatibility is greater than that of quantum mechanics.