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Bell's inequality was originally derived under the assumption thatexperimenters are free to select detector settings independently of any local"hidden variables" that might affect the outcomes of measurements on entangledparticles. This assumption has come to be known as "measurement independence"(also referred to as "freedom of choice" or "settings independence"). For atwo-setting, two-outcome Bell test, we derive modified Bell inequalities thatrelax measurement independence, for either or both observers, while remaininglocally causal. We describe the loss of measurement independence for eachobserver using the parameters $M_1$ and $M_2$, as defined by Hall in 2010, andalso by a more complete description that adds two new parameters, which we call$\hat{M}_1$ and $\hat{M}_2$, deriving a modified Bell inequality for eachdescription. These "relaxed" inequalities subsume those considered in previouswork as special cases, and quantify how much the assumption of measurementindependence needs to be relaxed in order for a locally causal model to producea given violation of the standard Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH)inequality. We show that both relaxed Bell inequalities are tight bounds on theCHSH parameter by constructing locally causal models that saturate them. Forany given Bell inequality violation, the new two-parameter and four-parametermodels each require significantly less mutual information between the hiddenvariables and measurement settings than previous models. We conjecture that thenew models, with optimal parameters, require the minimum possible mutualinformation for a given Bell violation. We further argue that, contrary tovarious claims in the literature, relaxing freedom of choice need not implysuperdeterminism.

Relaxed Bell Inequalities with Arbitrary Measurement Dependence for Each Observer