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

Abstract

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., $V_{\mathrm{c}}^{\mathrm{E}}=1/3$ characterizing the boundary between entanglement and separable model, $V_{\mathrm{c}}^{\mathrm{B}}=1/K_G(3)$ characterizing the boundary between Bell's nonlocality and the local-hidden-variable model, while $V_{\mathrm{c}}^{\mathrm{S}}=1/2$ characterizing the boundary between Einstein-Podolsky-Rosen (EPR) steering and the local-hidden-state model. So far, the problem of $V_{\mathrm{c}}^{\mathrm{E}}=1/3$ 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 $V_{\mathrm{c}}^{\mathrm{S}}=1/2$ for the two-qubit Werner state, thus resolving the long-standing problem.

Main text: 6 pages, 2 figures; SM: 8 pages, 5 figures. Adding some references

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