Optimal quantum violation of Clauser–Horne–Shimony–Holt like steering inequality

We study a recently proposed Einstein–Podolsky–Rosen steering inequality (Cavalcanti et al 2015 J. Opt. Soc. Am. B 32 A74–A81). Analogous to Clauser–Horne–Shimony–Holt (CHSH) inequality for Bell nonlocality, in the simplest scenario, i.e., two parties, two measurements per party and two outcomes per measurement, this newly proposed inequality has been proved to be necessary and sufficient for steering. In this article we find the optimal violation amount of this inequality in quantum theory. Interestingly, the optimal violation amount matches with optimal quantum violation of CHSH inequality, i.e., Cirel'son quantity. We further study the optimal violation of this inequality for different classes of 2-qubit quantum states. (paper)