Maximally steerable mixed state based on the linear steering inequality and the Clauser-Horne-Shimony-Holt–like steering inequality

The two-qubit maximally steerable mixed state (MSMS), defined as one that violates to the most degree a steering inequality for any fixed linear entropy, is derived mathematically here based on the two-setting (three-setting) linear steering inequality and the two-setting Clauser-Horne-Shimony-Holt--like steering inequality. Interestingly, the form of the MSMS based on such different steering inequalities obtained here is identical to the two-qubit maximally nonlocal mixed state (MNMS). It is clearly shown that finding any two-qubit state, of which the state mixedness exceeds $2/3$, thus violating the three-setting steering inequalities, is impossible. The violation of the inequalities with the Werner state and with the maximally entangled mixed state, respectively, as well as the relations between their optimal violation and the linear entropy, is also discussed comprehensively. In particular, within the range $\ensuremath{\varepsilon}(\ensuremath{\rho})\ensuremath{\in}[0,2/3]$ of the fixed linear entropy, the Werner state reaches the same violation as the MSMS does of the three-setting linear steering inequality, but not of the two-setting steering inequalities (for the latter inequalities the violation with the Werner state is generally less than that with the MSMS). For the MEMS, the optimal violation is always lower than that of the MSMS for any fixed linear entropy.