Relaxed Bell inequalities and Kochen-Specker theorems

TL;DR

This paper quantifies the minimal relaxations of key assumptions like no signaling and measurement independence needed to reproduce quantum correlations, providing a resource-based framework for understanding Bell and Kochen-Specker violations.

Contribution

It introduces a quantitative framework for measuring the degrees of relaxation of assumptions in quantum models, enabling comparison of resources needed for different correlations.

Findings

Measurement dependence requires only 1/15 of a bit of correlation.

Relaxation thresholds for Mermin's theorem, Hardy's, and Conway-Kochen's results are identified.

Relaxation of measurement independence affects the validity of foundational quantum 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.