Einstein-Podolsky-Rosen steering and quantum steering ellipsoids: Optimal two-qubit states and projective measurements

We identify the families of states that maximize some recently proposed quantifiers of Einstein-Podolsky-Rosen (EPR) steering and the volume of the quantum steering ellipsoid (QSE). The optimal measurements which maximize genuine EPR steering measures are discussed and we develop a way to find them using the QSE. We thus explore the links between genuine EPR steering and the QSE and introduce states that can be the most useful for one-sided device-independent quantum cryptography for a given amount of noise.