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We introduce Bell inequalities based on covariance, one of the most common measures of correlation. Explicit examples are discussed, and violations in quantum theory are demonstrated. A crucial feature of these covariance Bell inequalities is their nonlinearity; this has nontrivial consequences for the derivation of their local bound, which is not reached by deterministic local correlations. For our simplest inequality, we derive analytically tight bounds for both local and quantum correlations. An interesting application of covariance Bell inequalities is that they can act as shared randomness witnesses: specifically, the value of the Bell expression gives device-independent lower bounds on both the dimension and the entropy of the shared random variable in a local model.
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Bell inequalities are important tools in contrasting classical and quantum behaviors. To date, most Bell inequalities are linear combinations of statistical correlations between remote parties. Nevertheless, finding the classical and quantum mechanical (Tsirelson) bounds for a given Bell inequality in a general scenario is a difficult task which rarely leads to closed-form solutions. Here we introduce a new class of Bell inequalities based on products of correlators that alleviate these issues. Each such Bell inequality is associated with a unique coordination game. In the simplest case, Alice and Bob, each having two random variables, attempt to maximize the area of a rectangle and the rectangles area is represented by a certain parameter. This parameter, which is a function of the correlations between their random variables, is shown to be a Bell parameter, i.e. the achievable bound using only classical correlations is strictly smaller than the achievable bound using non-local quantum correlations We continue by generalizing to the case in which Alice and Bob, each having now n random variables, wish to maximize a certain volume in n-dimensional space. We term this parameter a multiplicative Bell parameter and prove its Tsirelson bound. Finally, we investigate the case of local hidden variables and show that for any deterministic strategy of one of the players the Bell parameter is a harmonic function whose maximum approaches the Tsirelson bound as the number of measurement devices increases. Some theoretical and experimental implications of these results are discussed.
Understanding the limits of quantum theory in terms of uncertainty and correlation has always been a topic of foundational interest. Surprisingly this pursuit can also bear interesting applications such as device-independent quantum cryptography and tomography or self-testing. Building upon a series of recent works on the geometry of quantum correlations, we are interested in the problem of computing quantum Bell inequalities or the boundary between quantum and post-quantum world. Better knowledge of this boundary will lead to more efficient device-independent quantum processing protocols. We show that computing quantum Bell inequalities is an instance of a quantifier elimination problem, and apply these techniques to the bipartite scenario in which each party can have three measurement settings. Due to heavy computational complexity, we are able to obtain the characterization of certain linear relaxation of the quantum set for this scenario. The resulting quantum Bell inequalities are shown to be equivalent to the Tsirelson-Landau-Masanes arcsin inequality, which is the only type of quantum Bell inequality found since 1987.
A technique, which we call homogenization, is applied to transform CH-type Bell inequalities, which contain lower order correlations, into CHSH-type Bell inequalities, which are defined for highest order correlation functions. A homogenization leads to inequalities involving more settings, that is a choice of one more observable is possible for each party. We show that this technique preserves the tightness of Bell inequalities: a homogenization of a tight CH-type Bell inequality is still a tight CHSH-type Bell inequality. As an example we obtain $3times3times3$ CHSH-type Bell inequalities by homogenization of $2times 2times 2$ CH-type Bell inequalities derived by Sliwa in [Phys. Lett. A {bf 317}, 165 (2003)].
Finding all Bell inequalities for a given number of parties, measurement settings, and measurement outcomes is in general a computationally hard task. We show that all Bell inequalities which are symmetric under the exchange of parties can be found by examining a symmetrized polytope which is simpler than the full Bell polytope. As an illustration of our method, we generate 238885 new Bell inequalities and 1085 new Svetlichny inequalities. We find, in particular, facet inequalities for Bell experiments involving two parties and two measurement settings that are not of the Collins-Gisin-Linden-Massar-Popescu type.
We propose a method to generate analytical quantum Bell inequalities based on the principle of Macroscopic Locality. By imposing locality over binary processings of virtual macroscopic intensities, we establish a correspondence between Bell inequalities and quantum Bell inequalities in bipartite scenarios with dichotomic observables. We discuss how to improve the latter approximation and how to extend our ideas to scenarios with more than two outcomes per setting.
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