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.
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.
Abstract This chapter presents a detailed discussion of Bell inequalities and their experimental tests. Starting from the Aharonov-Bohm version of the Einstein-Podolsky-Rosen (EPR) paradox, we review EPR’s criteria for realism and completeness and Bell’s locality condition, before deriving the Clauser-Horne-Shimony-Holt (CHSH) inequality. This is followed by an exposition on Bell’s original inequality and other Bell inequalities by Wigner, as well as Clauser and Horne. We then discuss the milestones in the experimental violation of Bell inequalities, from the experiment of Freedman and Clauser in the 1970s, to the efforts of Aspect et al. in the 1980s, the experiments by the teams of Zeilinger and Gisin in the 1990s, and culminating the loophole-free tests in 2015. Finally we turn to a brief discussion of the interpretations of quantum mechanics, focusing on contrasting the realist views held by, e.g., Einstein and Bell, and the informational views with proponents such as Brukner and Zeilinger
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.
Bell inequalities are derived for any number of observers, any number of alternative setups for each one of them, and any number of distinct outcomes for each experiment. It is shown that if a physical system consists of several distant subsystems, and if the results of tests performed on the latter are determined by local variables with objective values, then the joint probabilities for triggering any given set of distant detectors are convex combinations of a finite number of Boolean arrays, whose components are either 0 or 1 according to a simple rule. This convexity property is both necessary and sufficient for the existence of local objective variables. It leads to a simple graphical method which produces a large number of generalized Clauser- Horne inequalities corresponding to the faces of a convex polytope. It is plausible that quantum systems whose density matrix has a positive partial transposition satisfy all these inequalities, and therefore are compatible with local objective variables, even if their quantum properties are essentially nonlocal.
Following an approach proposed by Eberhard, a graphical representation of entangled particles is used to provide insight into the meaning and direct experimental consequences of local realism. The most famous Bell-type theorems, like the original inequality developed by Bell, as well as the ones presented by Clauser, Horne, Shimony, and Holt, Clauser and Horn, Eberhard, d'Espagnat, and Maccone, are then derived by means of the graphical method. The universality and efficiency of the presented method in achieving different Bell's inequalities follow from making local realism explicitly a starting point in our general reasoning. The pedagogical and illustrative nature of the approach makes it easy to see directly how all the miscellaneous inequalities are actually rooted in the local realist view on microscopic particles.
Bell's theorem can be proved through simple geometrical reasoning, without the need for the Psi function, probability distributions, or calculus. The proof is based on N. David Mermin's explication of the Einstein-Podolsky-Rosen-Bohm experiment, which involves Stern-Gerlach detectors which flash red or green lights when detecting spin-up or spin-down. The statistics of local hidden variable theories for this experiment can be arranged in colored strips from which simple inequalities can be deduced. These inequalities lead to a demonstration of Bell's theorem. Moreover, all local hidden variable theories can be graphed in such a way as to enclose their statistics in a pyramid, with the quantum-mechanical result lying a finite distance beneath the base of the pyramid.
In this work we look for novel classes of Bell's inequalities and methods to produce them. We also find their quantum violations including, if possible, the maximum one. The Jordan bases method that we explain in Chapter 2 is about using a pair of certain type of orthonormal bases whose spans are subspaces related to measurement outcomes of incompatible quantities on the same physical system. Jordan vectors are the briefest way of expressing the relative orientation of any two subspaces. This feature helps us to reduce the dimensionality of the parameter space on which we do searches for optimization. The work is published in [24]. In Chapter 3, we attempt to find a connection between group theory and Bell's theorem. We devise a way of generating terms of a Bell's inequality that are related to elements of an algebraic group. The same group generates both the terms of the Bell's inequality and the observables that are used to calculate the quantum value of the Bell expression. Our results are published in [25][26]. In brief, Bell's theorem is the main tool of a research program that was started by Einstein, Podolsky, Rosen [19] and Bohr [8] in the early days of quantum mechanics in their discussions about the core nature of physical systems. These debates were about a novel type of physical states called superposition states, which are introduced by quantum mechanics and manifested in the apparent inevitable randomness in measurement outcomes of identically prepared systems. Bell's huge contribution was to find a means of quantifying the problem and hence of opening the way to experimental verification by rephrasing the questions as limits on certain combinations of correlations between measurement results of spatially separate systems [7]. Thanks to Bell, the fundamental questions related to the nature of quantum mechanical systems became quantifiable [6]. According to Bell's theorem, some correlations between quantum entangled systems that involve incompatible quantities are not allowed by classical mechanics, a feature that is called as "quantum nonlocality". An experimental observation of those correlations, in other words, a violation of the limits imposed by classical physics, implies the correctness of quantum description and invalidates the classical, local realistic models. The first Bell experiments were proposed by Clauser, Horne, Shimony, and Holt, who invented the most famous Bell's inequality [13]. Later, the Aspect experiments were satisfactory enough for the physics community to be conclusive about the validation of quantum mechanics [1][3][4][2]. Ekert's work on applications of quantum nonlocality to communication resulted in the new field of quantum communication and cryptography, and turned the research program into a practical one [20]. Pitowsky showed a method to find all expressions of limitations due to local realism, all Bell's inequalities, for a given physical scenario. He also proved that the problem is, unfortunately, NP-complete and hence as the scenarios get more complex, they also become computationally intractable [33][34]. Therefore, different methods for the solution of special cases of the problem are necessary. Inequalities found for those special cases can be called classes of Bell's inequalities. For example, Werner and Wolf [41] and Collins, Gisin, Linden, Massar, and Popescu [16] found classes that cover a wide range of scenarios. Our work is a similar kind of effort to produce and study new types of Bell's inequalities.