Optimal Bell inequalities for qubit-qudit systems

Abstract: We evaluate the maximal Bell violation for a generic qubit-qudit system, obtaining easily computable expressions in arbitrary qudit dimension. This work generalizes the well-known Horodeckis's result for a qubit-qubit system. We also give simple lower and upper bounds on that violation. We apply our general results to address a number of issues. Namely, we obtain a bound on the degree of purity required in a system to exhibit nonlocality and study the statistics of nonlocality in random density matrices. Besides, we show the impossibility of improving the amount of Bell-violation by embedding the qudit in a Hilbert space of larger dimension. We also discuss how the results are generalized to POVM measurements. Finally, the general result is illustrated with a family of density matrices in the context of a qubit-qutrit system.