The boundary between classical and quantum correlations is well characterized by linear constraints called Bell inequalities. It is much harder to characterize the boundary of the quantum set itself in the space of no-signaling correlations. For the points on the quantum boundary that violate maximally some Bell inequalities, J. Oppenheim and S. Wehner [Science 330, 1072 (2010)] pointed out a complex property: Alice's optimal measurements steer Bob's local state to the eigenstate of an effective operator corresponding to its maximal eigenvalue. This effective operator is the linear combination of Bob's local operators induced by the coefficients of the Bell inequality, and it can be interpreted as defining a fine-grained uncertainty relation. It is natural to ask whether the same property holds for other points on the quantum boundary, using the Bell expression that defines the tangent hyperplane at each point. We prove that this is indeed the case for a large set of points, including some that were believed to provide counterexamples. The price to pay is to acknowledge that the Oppenheim-Wehner criterion does not respect equivalence under the no-signaling constraint: for each point, one has to look for specific forms of writing the Bell expressions.
The boundary between classical and quantum correlations is well characterized by linear constraints called Bell inequalities. It is much harder to characterize the boundary of the quantum set itself in the space of no-signaling correlations. For the points on the quantum boundary that violate maximally some Bell inequalities, J. Oppenheim and S. Wehner [Science 330, 1072 (2010)] pointed out a complex property: Alice's optimal measurements steer Bob's local state to the eigenstate of an effective operator corresponding to its maximal eigenvalue. This effective operator is the linear combination of Bob's local operators induced by the coefficients of the Bell inequality, and it can be interpreted as defining a fine-grained uncertainty relation. It is natural to ask whether the same property holds for other points on the quantum boundary, using the Bell expression that defines the tangent hyperplane at each point. We prove that this is indeed the case for a large set of points, including some that were believed to provide counterexamples. The price to pay is to acknowledge that the Oppenheim-Wehner criterion does not respect equivalence under the no-signaling constraint: for each point, one has to look for specific forms of writing the Bell expressions.
Quantum nonlocality and the uncertainty principle are two fundamental cornerstones of quantum theory, reflecting totally different aspects that distinguish it from classical theory. However, Oppenheim and Wehner introduced fine-grained uncertainty relations (FGURs) to account for the degree of quantum nonlocality. In the simplest Bell scenario, they showed that the quantum nonlocality measured by the maximal violation of some Bell inequality was determined by the FGUR. Along this line of research, we first derive explicit FGURs with general weights about probability distributions of two general projective measurements on local systems. Then we split those joint probabilities of Bell inequalities into a single-party conditional probability and the remaining joint probabilities so that the weighted FGURs can be applied straightforwardly to obtain the maximal violation in various scenarios ranging from bipartite, tripartite, to a complete set of multipartite Bell inequalities. Furthermore, in the bipartite scenario, the exact correlation boundary can be shown to be solely determined by our weighted local FGURs.
We study the faces of the set of quantum correlations, i.e., the Bell and noncontextuality inequalities without any quantum violation. First, we investigate the question of whether every proper (facet-defining) Bell inequality for two parties, other than the trivial ones from positivity, normalization, and no-signaling, can be violated by quantum correlations, i.e., whether the classical Bell polytope or the smaller correlation polytope share any facets with their respective quantum sets. To do this, we develop a recently derived bound on the quantum value of linear games based on the norms of game matrices to give a simple sufficient condition to identify linear games with no quantum advantage. Additionally we show how this bound can be extended to the general class of unique games. We then show that the paradigmatic examples of correlation Bell inequalities with no quantum violation, namely the nonlocal computation games, do not constitute facet-defining Bell inequalities, not even for the correlation polytope. We also extend this to an arbitrary prime number of outcomes for a specific class of these games. We then study the faces in the simplest Clauser-Horne-Shimony-Holt Bell scenario of binary dichotomic measurements, and identify edges in the set of quantum correlations in this scenario. Finally, we relate the noncontextual polytope of single-party correlation inequalities with the cut polytope $\text{CUT}(\mathbf{\ensuremath{\nabla}}G)$, where G denotes the compatibility graph of observables in the contextuality scenario and $\mathbf{\ensuremath{\nabla}}G$ denotes the suspension graph of G. We observe that there exist facet-defining noncontextuality inequalities with no quantum violation, and furthermore that this set of inequalities is beyond those implied by the consistent exclusivity principle.
Bell nonlocality places the long-standing philosophical disputes over the nature of quantum theory on a mathematically rigorous, experimentally testable footing, positioning quantum mechanics between local realism and the most general no-signaling theories. Yet the precise quantum boundary has remained elusive even in the minimal Bell scenario, where the extreme points generating the set are known. Here we characterize this boundary through a computable necessary and sufficient criterion: a correlation is quantum if and only if it can be decomposed into an extreme quantum correlation and a classical correlation. The criterion follows from our proof that every Bell functional exhibiting quantum advantage admits a unique quantum maximizer. We further classify the finitely many maximal-face structures containing each nonlocal exposed point according to their local deterministic vertices. By sharply distinguishing physically realizable correlations from nonphysical ones, these results resolve a long-standing open problem and provide a new foundation for identifying the principles underlying quantum theory and determining the ultimate information-theoretic limits of quantum correlations.
In device-independent quantum information, correlations between local measurement outcomes observed by spatially separated parties in a Bell test play a fundamental role. Even though it is long-known that the set of correlations allowed in quantum theory lies strictly between the Bell-local set and the no-signaling set, many questions concerning the geometry of the quantum set remain unanswered. Here, we revisit the problem of when the boundary of the quantum set coincides with the no-signaling set in the simplest Bell scenario. In particular, for each Class of these common boundaries containing k zero probabilities, we provide a (5−k) -parameter family of quantum strategies realizing these (extremal) correlations. We further prove that self-testing is possible in all nontrivial Classes beyond the known examples of Hardy-type correlations, and provide numerical evidence supporting the robustness of these self-testing results. Candidates of one-parameter families of self-testing correlations from some of these Classes are identified. As a byproduct of our investigation, if the qubit strategies leading to an extremal nonlocal correlation are local-unitarily equivalent, a self-testing statement provably follows. Interestingly, all these self-testing correlations found on the no-signaling boundary are provably non-exposed. An analogous characterization for the set M of quantum correlations arising from finite-dimensional maximally entangled states is also provided. En route to establishing this last result, we show that all correlations of M in the simplest Bell scenario are attainable as convex combinations of those achievable using a Bell pair and projective measurements. In turn, we obtain the maximal Clauser-Horne-Shimony-Holt Bell inequality violation by any maximally entangled two-qudit state and a no-go theorem regarding the self-testing of such states.
A problem in quantum information theory is to find the experimental setup that maximizes the nonlocality of correlations with respect to some suitable measure such as the violation of Bell inequalities. There are however some complications with Bell inequalities. First and foremost it is unfeasible to determine the whole set of Bell inequalities already for a few measurements and thus unfeasible to find the experimental setup maximizing their violation. Second, the Bell violation suffers from an ambiguity stemming from the choice of the normalization of the Bell coefficients. An alternative measure of nonlocality with a direct information-theoretic interpretation is the minimal amount of classical communication required for simulating nonlocal correlations. In the case of many instances simulated in parallel, the minimal communication cost per instance is called nonlocal capacity, and its computation can be reduced to a convex-optimization problem. This quantity can be computed for a higher number of measurements and turns out to be useful for finding the optimal experimental setup. Focusing on the bipartite case, we present a simple method for maximizing the nonlocal capacity over a given configuration space and, in particular, over a set of possible measurements, yielding the corresponding optimal setup. Furthermore, we show that there is a functional relationship between Bell violation and nonlocal capacity. The method is illustrated with numerical tests and compared with the maximization of the violation of CGLMP-type Bell inequalities on the basis of entangled two-qubit as well as two-qutrit states. Remarkably, the anomaly of nonlocality displayed by qutrits turns out to be even stronger if the nonlocal capacity is employed as a measure of nonlocality.
The upper bound of the fine-grained uncertainty relation is different for classical physics, quantum physics, and no-signaling theories with maximal nonlocality (superquantum correlation), as was shown in the case of bipartite systems [J. Oppenheim and S. Wehner, Science 330, 1072 (2010)]. Here, we extend the fine-grained uncertainty relation to the case of tripartite systems. We show that the fine-grained uncertainty relation determines the nonlocality of tripartite systems as manifested by the Svetlichny inequality, discriminating between classical physics, quantum physics, and superquantum correlations.