Quantum mechanics may revolutionise many aspects of modern information processing as it promises significant advantages in several fields such as cryptography, computing and metrology. Quantum cryptography for instance allows us to implement protocols which are device-independent, i.e. they can be proven security under fewer assumptions. These protocols rely on using devices producing non-local statistics and ideally these statistics would correspond to extremal points of the quantum set in the probability space. However, even in the CHSH scenario (the simplest non-trivial Bell scenario) we do not have a full understanding of the extremal quantum points. In fact, there are only a couple of analytic families of such points. Our first contribution is to introduce two new families of analytical quantum extremal points by providing solutions to two new families of Bell functionals. In the second part we focus on developing an analytical criteria for extremality in the CHSH scenario. A well-known Tsirelson-Landau-Masanes criterion only applies to points with uniform marginals, but a generalisation has been suggested in a sequence of works by Satoshi Ishizaka. We combine these conditions into a standalone conjecture, explore their technical details and discuss their suitability. Based on the understanding acquired, we propose a new set of conditions with an elegant mathematical form and an intuitive physical interpretation. Finally, we verify that both sets of conditions give correct predictions on the new families of quantum extremal points.
Quantum mechanics may revolutionise many aspects of modern information processing as it promises significant advantages in several fields such as cryptography, computing and metrology. Quantum cryptography for instance allows us to implement protocols which are device-independent, i.e. they can be proven security under fewer assumptions. These protocols rely on using devices producing non-local statistics and ideally these statistics would correspond to extremal points of the quantum set in the probability space. However, even in the CHSH scenario (the simplest non-trivial Bell scenario) we do not have a full understanding of the extremal quantum points. In fact, there are only a couple of analytic families of such points. Our first contribution is to introduce two new families of analytical quantum extremal points by providing solutions to two new families of Bell functionals. In the second part we focus on developing an analytical criteria for extremality in the CHSH scenario. A well-known Tsirelson-Landau-Masanes criterion only applies to points with uniform marginals, but a generalisation has been suggested in a sequence of works by Satoshi Ishizaka. We combine these conditions into a standalone conjecture, explore their technical details and discuss their suitability. Based on the understanding acquired, we propose a new set of conditions with an elegant mathematical form and an intuitive physical interpretation. Finally, we verify that both sets of conditions give correct predictions on the new families of quantum extremal points.
Quantum mechanics may revolutionize many aspects of modern information processing as it promises significant advantages in several fields such as cryptography, computing, and metrology. Quantum cryptography, for instance, allows us to implement protocols which are device independent, i.e., they can be proven secure under fewer assumptions. These protocols rely on using devices producing nonlocal statistics and ideally these statistics would correspond to extremal points of the quantum set in the probability space. However, even in the Clauser-Horne-Shimony-Holt (CHSH) scenario (the simplest nontrivial Bell scenario) we do not have a full understanding of the extremal quantum points. In fact, there are only a couple of analytic families of such points. Our first contribution is to introduce two families of analytical quantum extremal points by providing solutions to two families of Bell functionals. In the second part we focus on developing an analytical criteria for extremality in the CHSH scenario. A well-known Tsirelson-Landau-Masanes criterion only applies to points with uniform marginals, but a generalization has been suggested in a sequence of works by Ishizaka. We combine these conditions into a standalone conjecture, explore their technical details, and discuss their suitability. Based on the understanding acquired, we propose a set of conditions with an elegant mathematical form and an intuitive physical interpretation. Finally, we verify that both sets of conditions give correct predictions on these families of quantum extremal points.
Investigations of the boundary of the quantum correlation set have gained increased attention in recent years. This is done through the derivation of quantum Bell inequalities, which are related to Tsirelson's problem and have significant applications in device-independent (DI) information processing. However, determining quantum Bell inequalities is a notoriously difficult task and only isolated examples are known. In this paper, we present families of (almost-)quantum Bell inequalities and highlight four foundational and DI applications. Firstly, it is known that quantum correlations on the non-signaling boundary are of crucial importance in the task of DI randomness extraction from weak sources. In the practical Bell scenario of two players with two k -outcome measurements, we derive quantum Bell inequalities that demonstrate a separation between the quantum boundary and certain portions of the boundaries of the no-signaling polytope of dimension up to 4k−8 , extending previous results from nonlocality distillation and the collapse of communication complexity. Secondly as an immediate by-product, we give a general proof of Aumann’s Agreement theorem for quantum systems as well as the almost-quantum correlations, which implies Aumann’s agreement theorem is a reasonable physical principle in the context of epistemics to pick out both quantum theory and almost-quantum correlations from general no-signaling theories. Thirdly, we present a family of quantum Bell inequalities in the two players with m binary measurements scenarios, that we prove serve to self-test the two-qubit singlet and the corresponding 2m measurements. Interestingly, this claim generalizes the result for m=2 discovered by Tsirelson-Landau-Masanes and shows an improvement over the state-of-the-art Device-Independent Randomness-Amplification (DIRA). Lastly, we use our quantum Bell inequalities to derive the general form of the principle of no advantage in nonlocal computation, which is an information-theoretic principle that serves to characterize the quantum correlation set.
Device-independent (DI) quantum protocols use Bell inequality violations to ensure security or certify quantum properties without assumptions on the devices’ internal workings. In this work, we study the role of rank-one qubit positive operator-valued measures (POVMs) in DI scenarios. This class includes all qubit extremal POVMs, i.e., those measurements that cannot be realized as mixtures of others, as well as part of non-extremal POVMs, recently shown to be useful in sequential quantum protocols. We demonstrate that any rank-one POVM can generate correlations in bipartite scenarios that saturate a Tsirelson inequality when two parties share an arbitrary entangled two-qubit state and perform specific self-tested measurements. For extremal POVMs, such saturation enables explicit computation of guessing probability and worst-case conditional von Neumann entropy. From the Tsirelson inequality, we establish a randomness certification method that facilitates numerical simulations and we validate it through a proof-of-concept experiment with three-outcome POVMs and tilted entangled states.
It is difficult to establish an analytical criterion to identify the boundaries of quantum correlations, even for the simplest Bell scenario. Here, we briefly reviewed the plausible analytical criterion, and we found a way to confirm the extremal conditions from another direction. For that purpose, we analyzed the Navascués-Pironio-Acín (NPA) hierarchy to study the algebraic structure and found that the problem could not be simplified using the 1+AB level. However, considering the plausible criterion, the 1+AB and second levels for correlations were equal, and the extremal condition in the simplest Bell scenario was replaced by that in the 1+AB level. Thus, the correctness of the plausible criterion was verified, and the results demonstrated that the plausible criterion held, thereby explaining its simplicity. It seemed plausible, but now it becomes more certain.
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.
The Bell inequalities stand at the cornerstone of the developments of quantum theory on both the foundational and applied side. The discussion started as a way to test whether the quantum description of reality is complete or not, but it developed in such a way that a new research area stemmed from it, namely quantum information. Far from being and exhausted topic, in this paper, we present a constructive and geometrically intuitive description of the local polytope and its facets in a bipartite Bell scenario with two dichotomic measurements per party.