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Over the past few decades, experimental tests of Bell-type inequalities have been at the forefront of understanding quantum mechanics and its implications. These strong bounds on specific measurements on a physical system originate from…

This text is an introduction to an operational outlook on Bell inequalities, which has been very fruitful in the past few years. It has lead to the recognition that Bell tests have their own place in applied quantum technologies, because…

Bell's theorem cannot be proved if complementary measurements have to be represented by random variables which cannot be added or multiplied. One such case occurs if their domains are not identical. The case more directly related to the…

Counterfactual definiteness must be used as at least one of the postulates or axioms that are necessary to derive Bell-type inequalities. It is considered by many to be a postulate that is not only commensurate with classical physics (as…

Bell's inequality is derived from three assumptions: measurement independence, outcome independence, and parameter independence. Among these, measurement independence, often taken for granted, holds that hidden variables are statistically…

We derive two classes of multi-mode Bell inequalities under local realistic assumptions, which are violated only by the entangled states negative under partial transposition in accordance with the Peres conjecture. Remarkably, the failure…

Relevant aspects for testing Bell inequalities with entangled meson-antimeson systems are analyzed. In particular, we argue that the result of A. Go, J. Mod. Optics 51, 991 (2004), which nicely illustrate the quantum entanglement of B-meson…

We explore the challenges posed by the violation of Bell-like inequalities by $d$-dimensional systems exposed to imperfect state-preparation and measurement settings. We address, in particular, the limit of high-dimensional systems,…

John Bell showed that a big class of local hidden-variable models stands in conflict with quantum mechanics and experiment. Recently, there were suggestions that empirical adequate hidden-variable models might exist, which presuppose a…

A disagreement of the empirical results with quantum mechanical predictions is pointed out in the experiment by B. G. Christensen et al. The anomaly is quite similar to one present in the analogous experiment by M. Giustina et al.…

The interpretation of the violation of Bell-Clauser-Horne inequalities is revisited, in relation with the notion of extension of QM predictions to unmeasurable correlations. Such extensions are compatible with QM predictions in many cases,…

Bell's Theorem was developed on the basis of considerations involving a linear combination of spin correlation functions, each of which has a distinct pair of arguments. The simultaneous presence of these different pairs of arguments in the…

We derive a new class of correlation Bell-type inequalities. The inequalities are valid for any number of outcomes of two observables per each of n parties, including continuous and unbounded observables. We show that there are no…

In a local realist world view, physical properties are defined prior to and independent of measurement, and no physical influence can propagate faster than the speed of light. Proper experimental violation of a Bell inequality would show…

The algebraic derivation of the numerical limits of Bell inequalities in either three or four random variables is independent of the assumption of randomness.The limits of the inequalities follow as mathematical consequences of their…

The violation of Bell inequalities seems to establish an important fact about the world: that it is non-local. However, this result relies on the assumption of the statistical independence of the measurement settings with respect to…

It is argued that the concept of ``physical quantities possessed by the system'' is both redundant and inappropriate. We have examined two versions of the concept of ``possessed values'': one identical with the observed values and the other…

We discuss the role of counter-factual meaningfulness (a weaker cousin of "counter-factual definiteness") as a premise in the derivation of the Bell and CHSH inequalities. The basic question motivating the discussion is this: can the CHSH…

This is a review devoted to the complementarity-contextuality interplay with connection to the Bell inequalities. Starting discussion with complementarity, we point out to contextuality as its seed. {\it Bohr-contextuality} is dependence of…

We discuss the possibility of devising cosmological observables which violate Bell's inequalities. Such observables could be used to argue that cosmic scale features were produced by quantum mechanical effects in the very early universe. As…