Extending Bell's Theorem: Nonlocality via Measurement Dependence

Besides well-known conditions of locality or factorisability, deriving the Bell inequalities requires assuming that the distribution of hidden variables and Alice’s and Bob’s measurement settings be independent of each other. We show that (analogously to violations of locality due to action at a distance) certain violations of this Measurement Independence assumption can be associated with a notion of signalling in principle, thus making them also testable in principle, and spell out the appropriate conditions. Accordingly, we show that by imposing no-signalling one can prove a version of Bell’s theorem that does not require the assumption of Measurement Independence. We discuss the “Schulman model” as an example, as well as lessons for “experimental metaphysics”.

💡 Research Summary

The paper revisits the derivation of Bell inequalities, focusing on the three standard assumptions: Outcome Independence (OI), Parameter Independence (PI), and Measurement Independence (MI). While OI and PI are widely discussed as sources of non‑locality, MI – the requirement that the hidden‑variable distribution λ be statistically independent of the measurement settings chosen by Alice and Bob – has received less attention. The authors argue that violations of MI can be linked to a form of “signalling in principle,” a type of influence that may be difficult to exploit experimentally but is allowed by the dynamics of a theory.

First, the authors formalise what they mean by signalling in principle. They construct an explicit signalling protocol that works even when OI and PI hold, provided that the hidden‑variable distribution depends on the settings. In such a scenario Alice’s choice of measurement context can alter the statistical ensemble arriving at Bob’s side, giving Bob, in principle, a way to distinguish Alice’s setting choices. This establishes a new category of non‑locality that is distinct from the usual OI‑ or PI‑type violations.

Crucially, the paper shows that if one adds the no‑signalling condition to the usual OI and PI, the Bell inequalities follow without any need for MI. In other words, “no‑signalling + OI + PI ⇒ Bell inequalities” is proved, providing an extended version of Bell’s theorem that does not rely on the often‑criticised MI assumption. This result overturns the common belief that MI is indispensable for deriving the inequalities.

To illustrate the abstract arguments, the authors analyse the “Schulman model,” a concrete hidden‑variable construction that reproduces certain quantum statistics (e.g., spin‑½ singlet correlations) while explicitly violating MI. In this model the dependence of λ on the settings arises from global constraints or retro‑causal influences, rather than from any superluminal signalling. The model demonstrates that only a modest amount of correlation between settings and hidden variables is sufficient to reproduce the observed Bell‑inequality violations.

The paper situates these ideas within broader discussions of super‑determinism, retrocausality, and global‑constraint theories (e.g., ’t Hooft’s cellular automaton, Palmer’s invariant‑set theory). It argues that labeling all MI‑violating theories as “conspiratorial” is misleading; many such models involve systematic, law‑like mechanisms rather than ad‑hoc fine‑tuning. Moreover, the authors stress that causation can be treated as an operational notion—signalling in principle—without committing to a particular metaphysical stance on the nature of cause.

Finally, the authors propose an “experimental metaphysics” program: by designing experiments that test for the specific signalling protocols associated with MI violations, one can empirically distinguish between OI, PI, and MI‑type non‑locality. They discuss how recent “cosmic Bell” experiments, which use distant quasars or human‑generated randomness to set measurement choices, already place strong constraints on conspiratorial MI violations, but do not entirely rule out systematic, law‑like dependencies.

In summary, the paper expands Bell’s theorem by showing that MI is not a necessary assumption when no‑signalling is imposed, introduces a principled notion of signalling associated with MI violations, demonstrates these ideas with the Schulman model, and outlines a research agenda for experimentally probing the different flavors of quantum non‑locality. This work deepens our conceptual understanding of the foundations of quantum theory and opens new avenues for testing the limits of locality, realism, and free‑choice assumptions.