Fine-grained Einstein-Podolsky-Rosen–steering inequalities

We derive a steering inequality based on a fine-grained uncertainty relation to capture Einstein-Podolsky-Rosen steering for bipartite systems. Our steering inequality improves over previous ones since it can experimentally detect all steerable two-qubit Werner state with only two measurement settings on each side. According to our inequality, pure entangled states are maximally steerable. Moreover, by slightly changing the setting, we can express the amount of violation of our inequality as a function of their violation of the Clauser-Horne-Shimony-Holt inequality. Finally, after deriving a monogamy relation we prove that the amount of violation of our steering inequality is, up to a constant factor, a lower bound on the key rate of a one-sided device-independent quantum key distribution protocol secure against individual attacks.