Bounds on Petz-R\'enyi Divergences and their Applications for Device-Independent Cryptography

Variational techniques have been recently developed to find tighter bounds on the von Neumann entropy in a completely device-independent (DI) setting. This, in turn, has led to significantly improved key rates of DI protocols, in both the asymptotic limit as well as in the finite-size regime. In this paper, we discuss two approaches towards applying these variational methods for Petz-R\'enyi divergences instead. We then show how this can be used to further improve the finite-size key rate of DI protocols, utilizing a fully-R\'enyi entropy accumulation theorem developed in a partner work. Petz-R\'enyi divergences can also be applied to study DI advantage distillation, in which two-way communication is used to improve the noise tolerance of quantum key distribution (QKD) protocols. We implement these techniques to derive increased noise tolerances for DIQKD protocols, which surpass all previous known bounds.