The derivation of Bell inequalities requires an assumption of measurement independence, related to the amount of free will experimenters have in choosing measurement settings. Violation of these inequalities by singlet state correlations brings this assumption into question. A simple measure of the degree of measurement independence is defined for correlation models, and it is shown that all spin correlations of a singlet state can be modeled via giving up just 14% of measurement independence. The underlying model is deterministic and no signaling. It may thus be favorably compared with other underlying models of the singlet state, which require maximum indeterminism or maximum signaling. A local deterministic model is also given that achieves the maximum possible violation of the well-known Bell-Clauser-Horne-Shimony-Holt inequality, at a cost of only 1/3 of measurement independence.
The derivation of Bell inequalities requires an assumption of measurement independence, related to the amount of free will experimenters have in choosing measurement settings. Violation of these inequalities by singlet state correlations brings this assumption into question. A simple measure of the degree of measurement independence is defined for correlation models, and it is shown that all spin correlations of a singlet state can be modeled via giving up just 14% of measurement independence. The underlying model is deterministic and no signaling. It may thus be favorably compared with other underlying models of the singlet state, which require maximum indeterminism or maximum signaling. A local deterministic model is also given that achieves the maximum possible violation of the well-known Bell-Clauser-Horne-Shimony-Holt inequality, at a cost of only 1/3 of measurement independence.
The combination of various physically plausible properties, such as no signaling, determinism, and experimental free will, is known to be incompatible with quantum correlations. Hence, these properties must be individually or jointly relaxed in any model of such correlations. The necessary degrees of relaxation are quantified here via natural distance and information-theoretic measures. This allows quantitative comparisons between different models in terms of the resources, such as the number of bits of randomness, communication, and/or correlation, that they require. For example, measurement dependence is a relatively strong resource for modeling singlet-state correlations, with only 1/15 of one bit of correlation required between measurement settings and the underlying variable. It is shown how various ``relaxed'' Bell inequalities may be obtained, which precisely specify the complementary degrees of relaxation required to model any given violation of a standard Bell inequality. The robustness of a class of Kochen-Specker theorems, to relaxation of measurement independence, is also investigated. It is shown that a theorem of Mermin remains valid unless measurement independence is relaxed by 1/3. The Conway-Kochen ``free will'' theorem and a result of Hardy are less robust, failing if measurement independence is relaxed by only 6.5% and 4.5%, respectively. An appendix shows that existence of an outcome-independent model is equivalent to existence of a deterministic model.
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 M1 and M2, 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.
Simple quantitative measures of indeterminism and signaling, I and S, are defined for models of statistical correlations. It is shown that any such model satisfies a generalized Bell-type inequality, with tight upper bound B(I,S). This upper bound explicitly quantifies the complementary contributions required from indeterminism and signaling, for modeling any given violation of the standard Bell--Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. For example, all models of the maximum quantum violation must either assign no more than 80 probability of occurrence to some underlying event, and/or allow a nonlocal change of at least 60 in an underlying marginal probability of one observer in response to a change in measurement setting by a distant observer. The results yield a corresponding complementarity relation between the numbers of local random bits and nonlocal signaling bits required to model a given violation. A stronger relation is conjectured for simulations of singlet states. Signaling appears to be a useful resource only if a ``gap'' condition is satisfied, corresponding to being able to nonlocally flip some underlying marginal probability p to its complementary value $1\ensuremath{-}p$.
Quantum correlations that violate the Bell inequality cannot be explained by any local-hidden-variable theory that is measurement independent. However, this violation merely signifies the incompatibility of the underlying assumptions of reality, locality, and measurement independence, without providing a quantitative measure of the extent to which each assumption is violated. In contrast, Hall introduced a measure for each of the following assumptions: indeterminism, signaling, and measurement dependence, and Hall also generalized the Bell-Clauser-Horne-Shimony-Holt inequality, which provides a quantitative trade-off relationship between these assumptions. In this paper, we consider the introduction of hidden variables to be an essential assumption of Bell's theorem and introduce a quantification of hidden variables, which we term ``hiddenness.'' We derive a trade-off relation between hiddenness and measurement dependence that applies to any local-hidden-variable theory.
The assumption of measurement independence is required for a local deterministic model to conduct a Bell test. The violation of a Bell inequality by such a model implies that this assumption must be relaxed. The degree to which the assumption needs to be relaxed to achieve violation of some bipartite Bell inequalities, has been investigated recently in [Phys. Rev. Lett. 105, 250404(2010), Phys. Rev. A 99, 012121(2019)]. In this work, we study the minimum degree of relaxation required to simulate violations of various well-known tripartite Bell inequalities and opens the possibility of relaxation in multipartite scenario. Local deterministic models are also provided to achieve the violations of these Bell inequalities.
Abstract The assumption of measurement independence is required for a local deterministic model to conduct a Bell test. So, the violation of a Bell inequality by such a model implies that this assumption must be relaxed. The degree to which this assumption needs to be relaxed to achieve violation of some bipartite Bell inequalities, has been investigated in [Phys. Rev. Lett. 105, 250404(2010), Phys. Rev. A 99, 012121(2019)]. In this work, we study the minimum degree of relaxation required to simulate violations of various well-known tripartite Bell inequalities. Local deterministic models are also provided to achieve the violations of these Bell inequalities. PACS numbers: 03.67.Mn, 03.65.Ud.;