Causal Networks and Freedom of Choice in Bell's Theorem

TL;DR

This paper explores how classical causal models with measurement dependence can explain quantum correlations in Bell tests, especially within networks, and derives bounds and inequalities for such models.

Contribution

It introduces a method to upper bound measurement dependence in networked Bell tests and extends Bell inequalities to complex causal networks.

Findings

Measurement dependence can be quantitatively bounded in network arrangements.

Nonlinear Bell inequalities can be derived for causal networks.

Quantum correlations can violate these network-based Bell inequalities.

Abstract

Bell's theorem is typically understood as the proof that quantum theory is incompatible with local-hidden-variable models. More generally, we can see the violation of a Bell inequality as witnessing the impossibility of explaining quantum correlations with classical causal models. The violation of a Bell inequality, however, does not exclude classical models where some level of measurement dependence is allowed, that is, the choice made by observers can be correlated with the source generating the systems to be measured. Here, we show that the level of measurement dependence can be quantitatively upper bounded if we arrange the Bell test within a network. Furthermore, we also prove that these results can be adapted in order to derive nonlinear Bell inequalities for a large class of causal networks and to identify quantumly realizable correlations that violate them.