Bell inequalities from group theory

Asher Peres

Robert R. Tucci

This paper presents a new, computer-implementable algorithm for determining Bell inequalities. The algorithm is very general: one can use it to derive the Bell inequalities for any of the experiments that are usually considered for this purpose, such as the Bohm-Bell experiment, the Clauser-Horne experiment, or the experiments with spin 1, 3/2, 2 and 5/2 particles proposed by Mermin, Schwarz and Garg. As an example, this paper applies the algorithm to the Clauser-Horne experiment. The algorithm, which is based on simple notions from n-dimensional Euclidean geometry, gives a simple geometrical interpretation to the Bell inequalities. The algorithm allows one to demonstrate that a set of Bell inequalities is complete. The algorithm uses special hidden variables that were first used by Wigner and Belinfante. This paper also shows how to use finite group theory to express a set of Bell inequalities in a form that makes its invariance under relabellings explicit.

Denis Rosset, Jean-Daniel Bancal, Nicolas Gisin

Since John S. Bell demonstrated the interest of studying linear combinations of probabilities in relation with the EPR paradox in 1964, Bell inequalities have lead to numerous developments. Unfortunately, the description of Bell inequalities is subject to several degeneracies, which make any exchange of information about them unnecessarily hard. Here, we analyze these degeneracies and propose a decomposition for Bell-like inequalities based on a set of reference expressions which is not affected by them. These reference expressions set a common ground for comparing Bell inequalities. We provide algorithms based on finite group theory to compute this decomposition. Implementing these algorithms allows us to set up a compendium of reference Bell-like inequalities, available online at http://www.faacets.com . This website constitutes a platform where registered Bell-like inequalities can be explored, new inequalities can be compared to previously-known ones and relevant information on Bell inequalities can be added in a collaborative manner.

G. Coddens

This work is part of a reconstruction of quantum mechanics from scratch with the aim to understand what it means. In previous work we have shown that the Dirac equation can be derived by classical reasoning just using relativity and group theory. This implies that contrary to common belief quantum mechanics (QM) is not magical or radically different from classical mechanics. It is also not incompatible with the theory of relativity but completely part of it. As the experimental violations of the Bell inequalities seem to take exception with this general scheme for making sense of QM we have scrutinized the derivation of the Bell inequalities to figure out the limitations of our approach. We were able to show that the derivation contains a logical error based on wrong modelling. In the present paper we complete this investigation by showing that Malus' law can be derived by classical reasoning using group representation theory, both for electrons and photons. This is a further, be it somewhat more indirect proof that the derivation of the Bell inequalities contains an error. But more importantly it shows that the philosophy of our approach remains intact, by proving that we can also understand Malus' law with our methods and thereby validating our approach also for this physical phenomenon.

Enrico G. Beltrametti, Maciej Mączyński