In this talk I will discuss recent results on how much device-independent key can be extracted from small non-locality, and how much device-independent randomness can be extracted from locally d-dimensional quantum systems.

While it is known that Bell non-locality is necessary for device-independent quantum key distribution (DIQKD), we recently learnt that it is not sufficient in most protocols. This raises the question whether some minimal amount of non-locality is necessary for DIQKD. I will show that this is not the case---in fact, correlations arbitrarily close to the set of local correlations allow for unbounded device-independent key rates.

Then, I will move on to randomness generation, and show that the maximal theoretically possible 2*log(d) bits of device-independent randomness can be extracted from a locally d-dimensional maximally entangled state in every dimension d in which a symmetric informationally complete (SIC) POVM exists.

This talk is mostly based on arxiv:2310.08635 and a work in preparation with Ranyiliu Chen, Sigurd Anker Laursen Storgaard, Laura Mančinska and Jurij Volčič.