Complementary contributions of indeterminism and signalling to quantum correlations

TL;DR

This paper introduces measures of indeterminism and signalling to quantify their roles in quantum correlations, deriving bounds that relate these resources to violations of Bell inequalities and highlighting their complementary contributions.

Contribution

It provides a quantitative framework linking indeterminism and signalling to Bell inequality violations, revealing their complementary roles in quantum correlations.

Findings

Derived a generalized Bell-type inequality with bounds B(I,S).

Quantified the minimal indeterminism and signalling needed for quantum violations.

Proposed a conjecture on stronger relations for singlet state simulations.

Abstract

Simple quantitative measures of indeterminism and signalling, $I$ and $S$, are defined for models of statistical correlations. It is shown that any such model satisfies a generalised Bell-type inequality, with tight upper bound B(I,S). This upper bound explicitly quantifies the complementary contributions required from indeterminism and signalling, for modelling any given violation of the standard Bell-CHSH inequality. For example, all models of the maximum quantum violation must either assign no more than 80% probability of occurrence to some underlying event, and/or allow a nonlocal change of at least 60% in an underlying marginal probability of one observer in response to a change in measurement setting by a distant observer. The results yield a corresponding complementarity relation between the numbers of local random bits and nonlocal signalling bits required to model a given violation. A stronger relation is conjectured for simulations of singlet states. Signalling appears to be a useful resource only if a `gap' condition is satisfied, corresponding to being able to nonlocally flip some underlying marginal probability $p$ to its complementary value 1-p.