Hardy's paradox for multi-settings and high-dimensional systems

Recently, Chen et al introduced an alternative form of Hardy's paradox for $2$-settings and high-dimensional systems [Phy. Rev. A 88, 062116 (2013)], in which there is a great progress in improving the maximum probability of the nonlocal event. Here, we construct a general Hardy's paradox for multi-settings and high-dimensional systems, which (i) includes the paradox in [Phy. Rev. A 88, 062116 (2013)] as a special case, (ii) for spin-1/2 systems, is equivalent to the ladder proof of nonlocality without inequalities in [Phy. Rev. Lett. 13, 2755 (1997)], (iii) for spin-1 systems, increases the maximum probability of the nonlocal event by adding the number of settings, specially, with only 5-settings it can be improved to 0.40184, which is more than two times higher than 0.171, the maximal success probability to prove nonlocality in Adan's paradox [Phy. Rev. A 58, 1687 (1998)].