Dissertations, Theses, and Capstone Projects

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Mark Hillery

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Bell's theorem; Bell theorem; quantum nonlocality; quantum non-locality


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.

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