In the first part of this thesis Bell's theorem is revisited. It points at a difference between the quantum and the classical world. This difference is often behind the advantages of solutions using quantum mechanics. New and more general versions of Bell inequalities are presented. These inequalities involve multiple settings per observer. Compared with the two-setting inequalities, the new ones reveal the non-classical character of a broader class of states. Some of them are also proven to be optimal (tight). Next, we go beyond Bell's theorem. It is shown, both in theory and in experiment, that incompatibility between quantum mechanics and realistic theories can be extended into an important class of nonlocal models. We also show that the violation of Bell inequalities disqualifies local realistic models with a limited lack of the experimenter's freedom. This, at first glance quite philosophical result, has its down-to-earth implications for quantum communication. In the second part of the thesis well-known examples of quantum communication are reviewed. Next, new results concerning quantum cryptography and quantum communication complexity are given.
In the first part of this thesis Bell's theorem is revisited. It points at a difference between the quantum and the classical world. This difference is often behind the advantages of solutions using quantum mechanics. New and more general versions of Bell inequalities are presented. These inequalities involve multiple settings per observer. Compared with the two-setting inequalities, the new ones reveal the non-classical character of a broader class of states. Some of them are also proven to be optimal (tight). Next, we go beyond Bell's theorem. It is shown, both in theory and in experiment, that incompatibility between quantum mechanics and realistic theories can be extended into an important class of nonlocal models. We also show that the violation of Bell inequalities disqualifies local realistic models with a limited lack of the experimenter's freedom. This, at first glance quite philosophical result, has its down-to-earth implications for quantum communication. In the second part of the thesis well-known examples of quantum communication are reviewed. Next, new results concerning quantum cryptography and quantum communication complexity are given.
Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck's inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck's inequality. In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson's bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.
It is one of the most remarkable features of quantum physics that measurements on spatially separated systems cannot always be described by a locally causal theory. In such a theory, the outcomes of local measurements are determined in advance solely by some unknown (or hidden) variables and the choice of local measurements. Correlations that are allowed within the framework of a locally causal theory are termed classical. Typically, the fact that quantum mechanics does not always result in classical correlations is revealed by the violation of Bell inequalities, which are constraints that have to be satisfied by any classical correlations. It has been known for a long time that entanglement is necessary to demonstrate nonclassical correlations, and hence a Bell inequality violation. However, since some entangled quantum states are known to admit explicit locally causal models, the exact role of entanglement in Bell inequality violation has remained obscure. This thesis provides both a comprehensive review on these issues as well as a report on new discoveries made to clarify the relationship between entanglement and Bell inequality violation. In particular, within the framework of a standard Bell experiment, i.e., a Bell inequality test that is directly performed on a single copy of a quantum state r, we have derived two algorithms to determine, respectively, a lower bound and an upper bound on the strength of correlations that r can offer for any given Bell inequality. Both of these algorithms make use convex optimization techniques in the form of a semidefinite program. By examples, we show that these algorithms can often be used in tandem, in conjunction with convexity arguments, to determine if a quantum state can offer nonclassical correlations and hence violates a given Bell inequality. On the other hand, since a standard Bell experiment typically involves measurements over many copies of the quantum systems, we have also investigated the possibility of enhancing the strength of nonclassical correlation by, instead, performing collective measurements on multiple copies of the quantum systems. Our findings show that even without postselection, such joint measurements may also lead to stronger nonclassical correlations, and hence a better Bell inequality violation. Meanwhile, previous studies have indicated that entangled state admitting locally causal models may still lead to observable nonclassical correlations if, prior to a standard Bell experiment, the state is subjected to some appropriate local preprocessing. This phenomenon of hidden nonlocality was discovered more than a decade ago, but to date, it is still not known if all entangled states can demonstrate nonclassical correlations through these more sophisticated Bell experiments. A key result in this thesis then consists of showing that for all bipartite entangled states, observable nonclassical correlations, in the form of a Bell-CHSH inequality violation, can indeed be derived if we allow both local preprocessing and the usage of shared ancillary state which by itself does not violate the Bell-CHSH inequality. This establishes a kind of equivalence between bipartite entanglement and states that cannot be simulated by classical correlations. In summary, for a standard Bell experiment where no local preprocessing on a quantum state r is allowed, we have provided two algorithms that can be used in tandem to determine if r can be simulated by a locally causal theory, whereas in the scenario where local preprocessing is allowed, we have demonstrated that bipartite entangled states are precisely those which cannot always be simulated classically.
This thesis is the synthesis of theoretical and experimental works performed in the area of quantum foundations, particularly on quantum correlations and experimental tests of multiplicative Bell inequalities. First we begin with a comprehensive theoretical work performed on the foundations of quantum mechanics, focusing on the puzzling concepts of quantum entanglement, and hidden variable theories. Specifically, we present a broad overview of different classes of hidden variable theories such as local, crypto-nonlocal, contextual and non-local theories, along with several Bell like inequalities for these theories, providing theoretical proofs based on quantum mechanics for the falsification of some of these theories. Second we present a body of experimental, and theoretical works performed on a new class of Bell inequalities, i.e., the multiplicative Bell inequalities. We experimentally report the observation of the Bell parameters close to the Tsirelson (quantum) limit, upto a large number of measurement devices (n), and compare the results with a particular deterministic strategy. We also obtain classical bounds for some n, and report the experimental violation of these classical limits. We theoretically derive new richer bounds on the CHSH inequality (named after John Clauser, Michael Horne, Abnor Shimony and Richard Holt) and the multiplicative Bell parameter for n=2, based on the principle of ``relativistic independence'', and experimentally observe the distribution of Bell parameters as predicted by these bounds.
Quantum mechanics has been a paradigm shift and a revolutionary framework in our understanding of the physical world. The theory has been able to describe a wide variety of phenomena and has been experimentally verified with great precision. Despite that, to this day, the seemingly ad hoc formulation of quantum theory and several of its features still elude an intuitive understanding. At the same time, the applications of quantum mechanics in information theory have emerged as a promising avenue. Namely, its potential advantages in computation, in the so-called nascent field of quantum computation and in communication and information processing tasks such as quantum teleportation and super-dense coding. In all these endeavours, a significant effort has been made in understanding the maximum extent to which quantum mechanics can provide an advantage. This thesis explores an intersection of these two research lines, i.e. to have a physical explanation of some of the features of quantum mechanics and at the same time, study the outer limits of quantum mechanics. The combining insight rests on a newer foundational idea: that information itself is physical rather than an abstract entity and thus, is subject to physical laws. In this thesis, we explore the principle of information causality, which states that in any communication scenario, the amount of information carried is bounded by the degrees of freedom of the carrier. Information causality was initially proposed as a physical principle aimed at deriving the predictions of quantum mechanics on the type of correlations observed in a Bell scenario, which is a communication scenario involving multiple parties with access to a shared randomness resource. This principle has been proposed to recover some of the well-known features of quantum mechanics, specifically Tsirelson’s bound of the Clauser-Horne-Shimony-Holt inequality. Earlier results found limited generalisations due to the difficulty of deducing implications of the information causality principle. In this thesis, we greatly expand the scope of applicability of the principle. We develop a general-purpose algorithm to derive constraints on the set of quantum correlations. We extend the principle to be applicable to a wider variety of Bell scenarios, derive new families of quantum Bell inequalities and prove new bounds on the success probabilities of Entanglement Assisted Random Access Codes. We introduce an extension of the principle which applies to any distributed computing scenario. Using this extension, we improve on several previously known results in the field, and in certain cases, we even recover part of the quantum set as well. This extension provides rich insights into the communication complexity of distributed functions, using which we prove theory-independent randomised communication complexity bounds for several well-known functions and provide another general-purpose algorithm to bound the randomised quantum communication complexity of any function. As a consequence of the extension, we also establish that the extension supersedes the principle of non-trivial communication complexity.
New types of Bell inequalities and new versions of the Bell theorem will be presented here. They lead to stronger constraints for local realistic theories than the standard inequalities. It will be shown that the fact that such inequalities can be violated indicates not only that we have to abandon classical concepts, but is also a direct indication that quantum states which lead to such violations can, without any further processing, like distillation, etc., be applied to perform tasks with efficiency unattainable for classical systems. Misinterpretations of Bell theorem will be discussed. On one hand, the root of some erroneous claims that there exists a local realistic model for EPR-Bell correlations will be exposed. On the other hand, over-interpretations using vague terms like quantum non-locality or just non-locality will be criticized.