Specification of the strongest possible Bell inequalities for arbitrarily complicated physical scenarios -- any number of observers choosing between any number of observables with any number of possible outcomes -- is currently an open problem. Here I provide a new set of tools, which I refer to as "multideviations", for finding and analyzing these inequalities for the fully general case. In Part I, I introduce the multideviation framework and then use it to prove an important theorem: the Bell distributions can be generated from the set of joint distributions over all observables by deeming specific degrees of freedom unobservable. In Part II, I show how the theorem provides a new method for finding tight Bell inequalities. I then specify a set of new tight Bell inequalities for arbitrary event spaces -- the "even/odd" inequalities -- which have a straightforward interpretation when expressed in terms of multideviations. The even/odd inequalities concern degrees of freedom that are independent of those involved in parameter independence, raising the possibility of a new Bell's theorem with stronger philosophical implications. Also, contrary to expectations, the violation of the inequalities by quantum mechanics increases in size with the number of systems.
Specification of the strongest possible Bell inequalities for arbitrarily complicated physical scenarios -- any number of observers choosing between any number of observables with any number of possible outcomes -- is currently an open problem. Here I provide a new set of tools, which I refer to as "multideviations", for finding and analyzing these inequalities for the fully general case. In Part I, I introduce the multideviation framework and then use it to prove an important theorem: the Bell distributions can be generated from the set of joint distributions over all observables by deeming specific degrees of freedom unobservable. In Part II, I show how the theorem provides a new method for finding tight Bell inequalities. I then specify a set of new tight Bell inequalities for arbitrary event spaces -- the "even/odd" inequalities -- which have a straightforward interpretation when expressed in terms of multideviations. The even/odd inequalities concern degrees of freedom that are independent of those involved in parameter independence, raising the possibility of a new Bell's theorem with stronger philosophical implications. Also, contrary to expectations, the violation of the inequalities by quantum mechanics increases in size with the number of systems.
Bell's theorem for systems more complicated than two qubits faces a hidden, as yet undiscussed, problem. One of the methods to derive Bell inequalities is to assume the existence of a joint probability distribution for measurement results for all settings in the given experiment. However, for spin-1 systems, one faces the problem that the eigenvalues of observables do not allow a consistent algebra if one allows all possible settings on each side (Bell's 1966 contradiction), or some specific sets (leading to a Kochen-Specker 1967 contradiction). We show here that by choosing a special set of settings which never lead to inconsistent algebra of eigenvalues, one can still derive multisetting Bell inequalities, and that they are robustly violated. Violation factors increase with the number of subsystems. The inequalities involve only spin observables, we do not allow all possible qutrit observables, still the violations are strong.
Abstract Characterizing the set of all Bell inequalities is a notably hard task. An insightful method of solving it in case of Bell correlation inequalities for scenarios with two dichotomic measurements per site—for arbitrary number of parties—was given in references [2001 Phys. Rev. A 64 010102(R)] and [2001 Phys. Rev. A 64 032112]. Using complex-valued correlation functions, we generalize their approach to a broader class of Bell scenarios, in which the parties may choose from more than two multi-outcome measurements. Although the resulting families of Bell inequalities are not always tight, their coefficients have an intuitively understandable structure. We probe their usefulness by numerically testing their ability to detect Bell nonclassicality in simple interferometric experiments. Moreover, we identify a similar structure in the CGLMP inequality expressed in a correlation-based form, which allows us to generalise it to three parties.
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.
Bell's theorem rests on the following fundamental condition for a local system: P(a,b|alpha, beta, lambda)= P(a|alpha, lambda)P(b|beta, lambda). Here a and b are the outcomes respectively for measurements alpha on one side, and beta on the other, of an experiment involving two entangled particles traveling in opposite directions from a source. The parameter lambda (the set of "hidden variables") represents a more complete description of the joint state of the two particles. Because of lambda, the joint probability of detection is now dependent only on lambda and the local measurement setting of alpha; similarly for the other side and the setting beta. From this equation John Bell derived a simple inequality that is violated by the predictions of quantum mechanics, which is generally taken to imply that quantum mechanics is a nonlocal theory. But, by combining Richard Feynman's formulation of quantum mechanics with a model of particle interaction described by David Deutsch, we develop a system (the "space of all paths," SP) that (1) is immediately seen to replicate the predictions of quantum mechanics, (2) has a single outcome for each quantum event (unlike MWI on which it is partly based), and (3) contains the set lambda of hidden variables consisting of all possible paths from the source to the detectors on each side of the two-particle experiment. However, the set lambda is nonmeasurable, and therefore the above equation is meaningless in SP.
A partial solution to the problem of generalizing Bell’s inequalities to arbitrary numbers of physical properties is proposed. It is first assumed that the considered sets of probabilities correspond to events which satisfy a postulate ensuring that they form an orthomodular partially ordered set admitting a full set of states. In this framework a theorem generalizing Bell’s inequalities to an arbitrary finite number of events is proven. An interpretation of these results in Hilbert space is indicated. Conditions characterizing the classicality of so-called correlation probabilities are then found, and a method for verifying inequalities involving measurable correlations is discussed.
Full-correlation Bell-like inequalities represent an important subclass of Bell-like inequalities that have found applications in both a better understanding of fundamental physics and in quantum information science. Loosely speaking, these are inequalities where only measurement statistics involving all parties play a role. In this paper, we provide a framework for the study of a large family of such inequalities that are symmetrical with respect to arbitrary permutations of the parties. As an illustration of the power of our framework, we derive (i) a new family of Svetlichny inequalities for arbitrary numbers of parties, settings and outcomes, (ii) a new family of two-outcome device-independent entanglement witnesses for genuine n -partite entanglement and (iii) a new family of two-outcome Tsirelson inequalities for arbitrary numbers of parties and settings. We also discuss briefly the application of these new inequalities in the characterization of quantum correlations.