Abstract Bell’s theorem is often said to imply that quantum mechanics violates local causality, and that local causality cannot be restored with a hidden-variables theory. This however is only correct if the hidden-variables theory fulfils an assumption called Statistical Independence. Violations of Statistical Independence are commonly interpreted as correlations between the measurement settings and the hidden variables (which determine the measurement outcomes). Such correlations have been discarded as “fine-tuning” or a “conspiracy”. We here point out that the common interpretation is at best physically ambiguous and at worst incorrect. The problem with the common interpretation is that Statistical Independence might be violated because of a non-trivial measure in state space, a possibility we propose to call “supermeasured”. We use Invariant Set Theory as an example of a supermeasured theory that violates the Statistical Independence assumption in Bell’s theorem without requiring correlations between hidden variables and measurement settings (physical statistical independence).
The precision with which we can measure operators that do not commute with conserved quantities is limited by the need to preserve the associated global symmetries. We show how to construct a local hidden-variable model that violates Bell inequalities by interpreting this measurement error as altering the measure space of the hidden variables. This provides a physically-motivated example of a supermeasured model, where the statistical independence assumption used to form Bell inequalities is violated without there being a causal dependence of the measurement settings on the hidden variables (superdeterminism) or a causal dependence of the hidden variables on the measurement settings (retrocausality). The model also gives subtly different predictions to quantum mechanics, that could be tested experimentally.
Bell's theorem is often said to imply that quantum mechanics violates local causality, and that local causality cannot be restored with a hidden-variables theory. This however is only correct if the hidden-variables theory fulfils an assumption called Statistical Independence. Violations of Statistical Independence are commonly interpreted as correlations between the measurement settings and the hidden variables (which determine the measurement outcomes). Such correlations have been discarded as finetuning or a conspiracy. We here point out that the common interpretation is at best physically ambiguous and at worst incorrect. The problem with the common interpretation is that Statistical Independence might be violated because of a non-trivial measure in state space, a possibility we propose to call supermeasured. We use Invariant Set Theory as an example of a supermeasured theory that violates the Statistical Independence assumption in Bell's theorem without requiring correlations between hidden variables and measurement settings.
In a recent note, Hance and Hossenfelder (arXiv:2211.01331) recall that "locally causal completions of quantum mechanics are possible, if they violate the assumption [called statistical independence or measurement independence] that the hidden variables do not in any way depend on measurement settings" and that, consequently, the experimental violations of Bell inequalities "show that maintaining local causality requires violating statistical independence". However, Hance and Hossenfelder also argue that "we should (...) look for independent experimental evidence that can distinguish the two different options: non-locality and statistical independence, or locality and violations of statistical independence" and that "the unwillingness to consider theories without statistical independence may be the reason we do not yet have a locally causal theory for the foundations of physics that is consistent with general relativity". Here, we recall that there is a third option, namely, rejecting that measurement outcomes are governed in any way by hidden variables. Moreover, we argue that some recent results in the search for principles singling out the sets of quantum correlations for Bell and Kochen-Specker contextuality scenarios point out that this third option is scientifically more plausible and answers the question of why and how nature produces quantum nonlocality.
Bell’s theorem is purported to demonstrate the impossibility of a local “hidden variable” theory underpinning quantum mechanics. It relies on the well-known assumption of ‘locality’, and also on a little-examined assumption called ‘statistical independence ’ (SI). Violations of this assumption have variously been thought to suggest “backward causation”, a “conspiracy ” on the part of nature, or the denial of “free will”. It will be shown here that these are spurious worries, and that denial of SI simply implies nonlocal correlation between spacelike degrees of freedom. Lorentz-invariant theories in which SI does not hold are easily constructed: two are exhibited here. It is conjectured, on this basis, that quantum-mechanical phenomena may be modeled by a local theory after all. 1
Abstract Bell’s theorem demonstrates that any physical theory that is consistent with the predictions of quantum mechanics, and which satisfies some apparently innocuous assumptions, must violate the principle of local causality . It may therefore be possible to maintain local causality by rejecting one of these other assumptions instead. One possibility that has recently received significant attention involves rejecting the principle of statistical independence (SI). In this paper, we consider the frequency interpretation of SI, which states that
ρ(λ)≈ρ(λ∣Z) , where
ρ(λ) is the relative frequency of an element of an ensemble being in the state
λ , and Z is a label that separates the ensemble into apparently randomly selected sub-ensembles. SI is violated when the sub-ensemble frequency
ρ(λ∣Z) fails to be representative of the ensemble frequency
ρ(λ) . We argue that physical theories that systematically violate SI should all be understood as superdeterministic . This perspective on SI sheds light on a number of issues that are being debated in the superdeterminism literature, especially concerning its scope and philosophical consequences. Regarding scope, we argue that superdeterministic theories fall into three categories, deterministic theories with fine-tuned initial conditions, fluke theories, and nomic exclusion theories. We also argue that retrocausal and invariant set theories need not violate SI, which is contrary to how they are normally presented. Regarding philosophical implications, we argue that superdeterminism is incompatible with free will according to some prominent compatibilist accounts. We also argue that although superdeterminism is conspiratorial, it is not unscientific, but pre-scientific.