A Critical Review of Works Pertinent to the Einstein-Bohr Debate and Bell’s Theorem

This review is related to the Einstein-Bohr debate and to Einstein–Podolsky–Rosen’s (EPR) and Bohm’s (EPRB) Gedanken-experiments as well as their realization in actual experiments. I examine a significant number of papers, from my minority point of view and conclude that the well-known theorems of Bell and Clauser, Horne, Shimony and Holt (CHSH) deal with mathematical abstractions that have only a tenuous relation to quantum theory and the actual EPRB experiments. It is also shown that, therefore, Bell-CHSH cannot be used to assess the nature of quantum entanglement, nor can physical features of entanglement be used to prove Bell-CHSH. Their proofs are, among other factors, based on a statistical sampling argument that is invalid for general physical entities and processes and only applicable for finite “populations”; not for elements of physical reality that are linked, for example, to a time-like continuum. Bell-CHSH have, furthermore, neglected the subtleties of the theorem of Vorob’ev that includes their theorems as special cases. Vorob’ev found that certain combinatorial-topological cyclicities of classical random variables form a necessary and sufficient condition for the constraints that are now known as Bell-CHSH inequalities. These constraints, however, must not be linked to the observables of quantum theory nor to the actual EPRB experiments for a variety of reasons, including the existence of continuum-related variables and appropriate considerations of symmetry.

The Violation of Bell-CHSH Inequalities Leads to Different Conclusions Depending on the Description Used

Since the experimental observation of the violation of the Bell-CHSH inequalities, much has been said about the non-local and contextual character of the underlying system. However, the hypothesis from which Bell’s inequalities are derived differ according to the probability space used to write them. The violation of Bell’s inequalities can, alternatively, be explained by assuming that the hidden variables do not exist at all, that they exist but their values cannot be simultaneously assigned, that the values can be assigned but joint probabilities cannot be properly defined, or that averages taken in different contexts cannot be combined. All of the above are valid options, selected by different communities to provide support to their particular research program.

Device-independent quantum key distribution from generalized CHSH inequalities

Device-independent quantum key distribution aims at providing security guarantees even when using largely uncharacterised devices. In the simplest scenario, these guarantees are derived from the CHSH score, which is a simple linear combination of four correlation functions. We here derive a security proof from a generalisation of the CHSH score, which effectively takes into account the individual values of two correlation functions. We show that this additional information, which is anyway available in practice, allows one to get higher key rates than with the CHSH score. We discuss the potential advantage of this technique for realistic photonic implementations of device-independent quantum key distribution.

A local-realistic quantum mechanical model of spin and spin entanglement

This paper aims at reproducing quantum mechanical (QM) spin and spin entanglement results using a realist, stochastic, and local approach, without the standard QM mathematical formulation. The concrete model proposed includes the description of Stern–Gerlach apparatuses and of Bell test experiments. Single particle trajectories are explicitly evaluated as a function of a few stochastic variables that they assumedly carry on. QM predictions are retrieved as probability distributions of similarly-prepared ensembles of particles. Notably, it is shown that the proposed model, despite being both local and realist, is able to violate the Bell–CHSH inequalities by exploiting the coincidence loophole and thus intrinsically renouncing to one of the Bell’s assumptions.

Violation of Bell-CHSH Inequalities through Optimal Local Filters in the Vacuum

We investigate quantum correlations appearing for two-qubit detectors which are initially uncorrelated and locally coupled to a massless scalar field in a vacuum state. Under the perturbation up to the second order in the coupling, the state of the detectors can be entangled through the interaction with the scalar field but satisfies the Bell-CHSH inequality. The violation of the Bell-CHSH inequality for such an entangled state is revealed by local filtering operations. In this paper, we construct the optimal filtering operations for the qubit detectors and derive the success probability of the filtering. The success probability characterizes the reliability of revealing the violation of the Bell-CHSH inequality by the filtering operations. Through these analyses, we demonstrate a trade-off relation between the success probability and the size of parameter region showing the violation of the Bell-CHSH inequality.

Kolmogorov proof of the Clauser, Horne, Shimony and Holt inequalities

Boolean logic is used to prove the CHSH inequalities. The proof elucidates the connection between Einstein elements of reality and quantum non-locality. The violation of the CHSH inequality by quantum theory is discussed and the two-stage view of quantum measurement relevance to incompatible observables is outlined.