Relaxed Bell inequalities and Kochen-Specker theorems

Abstract

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 the existence of an outcome independent model is equivalent to the existence of a deterministic model.

19 pages (including 3 appendices); v3: minor clarifications, to appear in PRA

References in corpus (7)

• The Free Will Theorem
• The Communication Cost of Simulating Bell Correlations
• Local deterministic model of singlet state correlations based on relaxing measurement independence
• Testing quantum correlations versus single-particle properties within Leggett's model and beyond
• Bell inequalities for continuous-variable correlations
• Local Causality and Completeness: Bell vs. Jarrett
• Complementary contributions of indeterminism and signalling to quantum correlations

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• Bell scenarios with communication
• Randomness Requirement on CHSH Bell Test in the Multiple Run Scenario
• Demonstration of Quantum Nonlocality in presence of Measurement Dependence
• Are quantum correlations genuinely quantum?
• Is nonlocality responsible for the violation of Bell's inequalities?
• Bell's Conspiracy, Schrödinger's Black Cat and Global Invariant Sets
• Computational Perspectives on Bell Inequalities and Many-body Quantum Correlations