Characterizing Bell nonlocality and EPR steering

Bell nonlocality and Einstein-Podolsky-Rosen (EPR) steering are very important quantum correlations in composite quantum systems. Bell nonlocality of a bipartite state is observed in some local quantum measurements, while EPR steering was first observed by Schrodinger in the context of famous EPR paradox. In this paper, we discuss the Bell nonlocality and EPR steering of bipartite states, including mathematical definitions and characterizations of these two quantum correlations, the convexity as well as the closedness of the sets of all Bell local states and all EPR unsteerable states, respectively. We also derive sufficient conditions for a state to be steerable; these conditions imply that Alice can steer Bob's state whenever Alice has two POV measurements such that the sets of Bob's normalized conditional states become two disjoint sets of pure states, or whenever she has one POV measurement such that Bob's normalized conditional states become a linearly independent set of pure states.