Bell Inequalities Tailored to Maximally Entangled States

Bell inequalities have traditionally been used to demonstrate that quantum theory is nonlocal, in the sense that there exist correlations generated from composite quantum states that cannot be explained by means of local hidden variables. With the advent of device-independent quantum information protocols, Bell inequalities have gained an additional role as certificates of relevant quantum properties. In this work, we consider the problem of designing Bell inequalities that are tailored to detect maximally entangled states. We introduce a class of Bell inequalities valid for an arbitrary number of measurements and results, derive analytically their tight classical, nonsignaling, and quantum bounds and prove that the latter is attained by maximally entangled states. Our inequalities can therefore find an application in device-independent protocols requiring maximally entangled states.

Figure 1

Minimum fidelity of the state in the black box to the maximally entangled state of two qutrits, as a function of the violation of ${\tilde{I}}_{3,2}$. At the maximal violation 4, the fidelity is equal to 1, meaning that the quantum state used in the Bell experiment must be maximally entangled. The numerical method that we used does not yield a positive lower bound on the fidelity below ${\tilde{I}}_{3,2}\approx 3.79$ (for comparison, the classical bound is ${\tilde{I}}_{3,2}\le (1+3\sqrt{3})/2\approx 3.01$).