Revealing the Boundary between Quantum Mechanics and Classical Model by EPR-Steering Inequality

In quantum information, the Werner state is a benchmark to test the boundary between quantum mechanics and classical models. There have been three well-known critical values for the two-qubit Werner state, i.e., V_ c^ E=1/3 characterizing the boundary between entanglement and separable model, V_ c^ B≈ 0.6595 characterizing the boundary between Bell's nonlocality and the local-hidden-variable (LHV) model, while V_ c^ S=1/2 characterizing the boundary between Einstein-Podolsky- Rosen (EPR) steering and the local-hidden-state (LHS) model. So far, the problem of V_ c^ E=1/3 has been completely solved by an inequality involving in the positive-partial-transpose criterion, while how to reveal the other two critical values by the inequality approach are still open. In this work, we focus on EPR steering, which is a form of quantum nonlocality intermediate between entanglement and Bell's nonlocality. By proposing the optimal N-setting linear EPR-steering inequalities, we have successfully obtained the desired value V_ c^ S=1/2 for the two-qubit Werner state, thus resolving the long-standing problem.