One of the main goals in the study of quantum nonlocality is to determine the maximum violation achieved by quantum correlations in a Bell scenario. However, given a Bell inequality, there is no general algorithm to perform this task. As an intermediate step, the development of efficiently computable bounds has played an important role for the advance of the field. In this thesis we phrase the problem of determining the quantum value of a Bell expression in the framework of computational complexity. Then we present our contributions exploring efficiently computable bounds (based on the spectral norm of some matrices) to the quantum value of a particular class of Bell inequalities: the linear games. We derive several results for the bipartite and the multipartite scenarios.
One of the main goals in the study of quantum nonlocality is to determine the maximum violation achieved by quantum correlations in a Bell scenario. However, given a Bell inequality, there is no general algorithm to perform this task. As an intermediate step, the development of efficiently computable bounds has played an important role for the advance of the field. In this thesis we phrase the problem of determining the quantum value of a Bell expression in the framework of computational complexity. Then we present our contributions exploring efficiently computable bounds (based on the spectral norm of some matrices) to the quantum value of a particular class of Bell inequalities: the linear games. We derive several results for the bipartite and the multipartite scenarios.
Determination of classical and quantum values of bipartite Bell inequalities plays a central role in quantum nonlocality. In this work, we characterize in a simple way bipartite Bell inequalities, free of marginal terms, for which the quantum value can be achieved by considering a classical strategy, for any number of measurement settings and outcomes. These findings naturally generalize known results about nonlocal computation and quantum XOR games. Additionally, our technique allows us to determine the classical value for a wide class of Bell inequalities, having quantum advantage or not, in any bipartite scenario.
The violations of Bell inequalities by measurements on quantum states give rise to the phenomenon of quantum non-locality and express the advantage of using quantum resources over classical ones for certain information-theoretic tasks. The relative degree of quantum violations has been well studied in the bipartite scenario and in the multipartite scenario with respect to fully local behaviours. However, the multipartite setting entails a more complex classification in which different notions on non-locality can be established. In particular, genuine multipartite non-local distributions apprehend truly multipartite effects, given that these behaviours cannot be reproduced by bilocal models that allow correlations among strict subsets of the parties beyond a local common cause. We show here that, while in the so-called correlation scenario the relative violation of bilocal Bell inequalities by quantum resources is bounded, i.e. it does not grow arbitrarily with the number of inputs, it turns out to be unbounded in the general case. We identify Bell functionals that take the form of non-local games for which the ratio of the quantum and bilocal values grows unboundedly as a function of the number of inputs and outputs.
Multipartite Bell nonlocality provides a device-independent probe of many-body quantum correlations, but its characterization is limited by the rapid growth of the underlying classical and quantum optimization problems. We develop a scalable method for constructing and certifying permutationally invariant Bell inequalities using only one- and two-body correlators. The construction gives families of inequalities with robust quantum violations for general m measurements as the number of parties N becomes large. To improve robustness against noise, we optimize the ratio of the quantum value to the classical bound for these families in the large-N limit. We then certify the resulting quantum violation using semidefinite programming. For the broad class of Bell inequalities studied here, the infinite-N ratios take simple rational values for finite m and converge to coth(1) as m→∞. The optimized inequalities efficiently detect many-body Bell nonlocality with collective measurements, with more measurement settings leading to stronger violations.
Abstract The violations of Bell inequalities by measurements on quantum states give rise to the phenomenon of quantum non-locality and express the advantage of using quantum resources over classical ones for certain information-theoretic tasks. The relative degree of quantum violations has been well studied in the bipartite scenario and in the multipartite scenario with respect to fully local behaviours. However, the multipartite setting entails a more complex classification in which different notions on non-locality can be established. In particular, genuine multipartite non-local distributions apprehend truly multipartite effects, given that these behaviours cannot be reproduced by bilocal models that allow correlations among strict subsets of the parties beyond a local common cause. We show here that, while in the so-called correlation scenario the relative violation of bilocal Bell inequalities by quantum resources is bounded, i.e. it does not grow arbitrarily with the number of inputs, it turns out to be unbounded in the general case. We identify Bell functionals that take the form of non-local games for which the ratio of the quantum and bilocal values grows unboundedly as a function of the number of inputs and outputs.
It is a well-known fact that both quantum entanglement and nonlocality (implied by the violation of Bell inequalities) constitute quantum correlations that cannot be arbitrarily shared among subsystems. They are both monogamous, albeit in a different fashion. In the present contribution we focus on nonlocality monogamy relations such as the Toner-Verstraete, the Seevinck, and a derived monogamy inequality for three parties and compare them with multipartite nonlocality measures for the whole set of pure states distributed according to the Haar measure. In this numerical endeavor, we also see that, although monogamy relations for nonlocality cannot exist for more than three parties, in practice the exploration of the whole set of states for different numbers of qubits will return effective bounds on the maximum value of all bipartite Bell violations among subsystems. Hence, we shed light on the effective nonlocality monogamy bounds in the multiqubit case.
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.