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arXiv:2105.09037 · Proceedings of the National Academy of Sciences

Violations of locality and free choice are equivalent resources in Bell experiments

Paweł Błasiak, Emmanuel M. Pothos, James M. Yearsley, Christoph Gallus, Ewa Borsuk

Published Apr 22, 2021• 31 citations

Abstract

Bell inequalities rest on three fundamental assumptions: realism, locality, and free choice, which lead to nontrivial constraints on correlations in very simple experiments. If we retain realism, then violation of the inequalities implies that at least one of the remaining two assumptions must fail, which can have profound consequences for the causal explanation of the experiment. We investigate the extent to which a given assumption needs to be relaxed for the other to hold at all costs, based on the observation that a violation need not occur on every experimental trial, even when describing correlations violating Bell inequalities. How often this needs to be the case determines the degree of, respectively, locality or free choice in the observed experimental behavior. Despite their disparate character, we show that both assumptions are equally costly. Namely, the resources required to explain the experimental statistics (measured by the frequency of causal interventions of either sort) are exactly the same. Furthermore, we compute such defined measures of locality and free choice for any nonsignaling statistics in a Bell experiment with binary settings, showing that it is directly related to the amount of violation of the so-called Clauser-Horne-Shimony-Holt inequalities. This result is theory independent as it refers directly to the experimental statistics. Additionally, we show how the local fraction results for quantum-mechanical frameworks with infinite number of settings translate into analogous statements for the measure of free choice we introduce. Thus, concerning statistics, causal explanations resorting to either locality or free choice violations are fully interchangeable.

DOI: 10.1073/pnas.2020569118

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A comment on: the violations of locality and free choice are equivalent resources in Bell experiments

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In a recent paper published in PNAS authors prove that locality and free choice are equivalent resources which need to be relaxed in order to fully reproduce some statistics in Bell experiments (while always maintaining realism). We explain that their assumption of free choice is simply counterfactual definiteness or noncontextuality. Therefore the resource in Bell experiments is contextuality and not the violations of locality and/or of free choice. It is definitely less mind boggling conclusion because experimenters` freedom of choice is a prerequisite of science,

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Bell inequalities rest on three fundamental assumptions: realism, locality, and free choice, which lead to nontrivial constraints on correlations in very simple experiments. If we retain realism, then violation of the inequalities implies that at least one of the remaining two assumptions must fail, which can have profound consequences for the causal explanation of the experiment. We investigate the extent to which a given assumption needs to be relaxed for the other to hold at all costs, based on the observation that a violation need not occur on every experimental trial, even when describing correlations violating Bell inequalities. How often this needs to be the case determines the degree of, respectively, locality or free choice in the observed experimental behavior. Despite their disparate character, we show that both assumptions are equally costly. Namely, the resources required to explain the experimental statistics (measured by the frequency of causal interventions of either sort) are exactly the same. Furthermore, we compute such defined measures of locality and free choice for any nonsignaling statistics in a Bell experiment with binary settings, showing that it is directly related to the amount of violation of the so-called Clauser-Horne-Shimony-Holt inequalities. This result is theory independent as it refers directly to the experimental statistics. Additionally, we show how the local fraction results for quantum-mechanical frameworks with infinite number of settings translate into analogous statements for the measure of free choice we introduce. Thus, concerning statistics, causal explanations resorting to either locality or free choice violations are fully interchangeable.

The causal modelling of Bell experiments relies on three fundamental assumptions: locality, freedom of choice and arrow-of-time. It turns out that nature violates Bell inequalities, which implies the failure of at least one of those assumptions. Since rejecting any of them, even partially, is sufficient to explain the observed correlations, it is natural to inquire about the cost in each case. This paper builds upon the findings in Blasiak et al. 2021 Proc. Natl Acad. Sci. USA 118 , e2020569118 ( doi:10.1073/pnas.2020569118 ) showing the equivalence between the locality and free choice assumptions. Here, we include retrocausal models to complete the picture of causal explanations of the observed correlations. Furthermore, we refine the discussion by considering more challenging causal scenarios which allow only single-arrow type violations of a given assumption. The figure of merit chosen for the comparison of the causal cost is defined as the minimal frequency of violation of the respective assumption required for a simulation of the observed experimental statistics. This article is part of the theme issue ‘Quantum contextuality, causality and freedom of choice’.

In a recent paper published in PNAS authors prove that locality and free choice are equivalent resources which need to be relaxed in order to fully reproduce some statistics in Bell experiments (while always maintaining realism). We explain that their assumption of free choice is simply counterfactual definiteness or noncontextuality. Therefore the resource in Bell experiments is contextuality and not the violations of locality and/or of free choice. It is definitely less mind boggling conclusion because experimenters` freedom of choice is a prerequisite of science,

Bell's inequality was originally derived under the assumption that experimenters are free to select detector settings independently of any local ``hidden variables'' that might affect the outcomes of measurements on entangled particles. This assumption has come to be known as ``measurement independence'' (also referred to as ``freedom of choice'' or ``settings independence''). For a two-setting, two-outcome Bell test, we derive modified Bell inequalities that relax measurement independence, for either or both observers, while remaining locally causal. We describe the loss of measurement independence for each observer using the parameters

{M}_{1}

and

{M}_{2}

, as defined by Hall in 2010, and also by a more complete description that adds two new parameters, which we call $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{{M}_{1}}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{{M}_{2}}$, deriving a modified Bell inequality for each description. These ``relaxed'' inequalities subsume those considered in previous work as special cases, and quantify how much the assumption of measurement independence needs to be relaxed in order for a locally causal model to produce a given violation of the standard Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. We show that both relaxed Bell inequalities are tight bounds on the CHSH parameter by constructing locally causal models that saturate them. For any given Bell inequality violation, the new two-parameter and four-parameter models each require significantly less mutual information between the hidden variables and measurement settings than previous models. We conjecture that the new models, with optimal parameters, require the minimum possible mutual information for a given Bell violation. We further argue that, contrary to various claims in the literature, relaxing freedom of choice need not imply superdeterminism.

The causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and post-quantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalities, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario require fewer input choices than their Bell scenario counterpart, which may have potential implications for device-independent protocols.

The causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and post-quantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalities, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario require fewer input choices than their Bell scenario counterpart, which may have potential implications for device-independent protocols.

Loophole-free violations of Bell inequalities imply that at least one of the assumptions behind local hidden-variable theories must fail. Here, we show that, if only one fails, then it has to fail completely, therefore excluding models that partially constrain freedom of choice or allow for partial retrocausal influences, or allow partial instantaneous actions at a distance. Specifically, we show that (i) any hidden-variable theory with outcome independence (OI) and arbitrary joint relaxation of measurement independence (MI) and parameter independence (PI) can be experimentally excluded in a Bell-like experiment with many settings on high-dimensional entangled states, and (ii) any hidden-variable theory with MI, PI and arbitrary relaxation of OI can be excluded in a Bell-like experiment with many settings on qubit-qubit entangled states. Loophole-free violations of Bell inequalities imply that at least one of the assumptions formalising local hidden-variable theories must fail. Here, the authors show that, if only one is false, then it has to fail completely, therefore excluding models that allow partial instantaneous actions at a distance, partially constrain freedom of choice or allow for partial retrocausal influences.