Extremal points of the quantum set in the Clauser-Horne-Shimony-Holt scenario: Conjectured analytical solution

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.

Automated summary

Quantum mechanics has the potential to revolutionize modern information processing in various fields including cryptography, computing, and metrology. Quantum cryptography enables device-independent protocols with proven security. Analytical families of quantum extremal points are introduced in this study, along with an analysis of extremality criteria in the CHSH scenario. A set of conditions is proposed based on the technical details and physical interpretation, and its predictions on quantum extremal points are verified.