Tilted Hardy paradoxes for device-independent randomness extraction

The device-independent paradigm has had spectacular successes in randomness generation, key distribution and self-testing, however most of these results have been obtained under the assumption that parties hold trusted and private random seeds. In efforts to relax the assumption of measurement independence, Hardy's non-locality tests have been proposed as ideal candidates. In this paper, we introduce a family of tilted Hardy paradoxes that allow to self-test general pure two-qubit entangled states, as well as certify up to $1$ bit of local randomness. We then use these tilted Hardy tests to obtain an improvement in the generation rate in the state-of-the-art randomness amplification protocols for Santha-Vazirani (SV) sources with arbitrarily limited measurement independence. Our result shows that device-independent randomness amplification is possible for arbitrarily biased SV sources and from almost separable states. Finally, we introduce a family of Hardy tests for maximally entangled states of local dimension $4, 8$ as the potential candidates for DI randomness extraction to certify up to the maximum possible $2 \log d$ bits of global randomness.