Relevant multi-setting tight Bell inequalities for qubits and qutrits

Abstract In the celebrated paper [D. Collins, N. Gisin, J. Phys. A Math. Gen. 37 (2004) 1775], Collins and Gisin presented for the first time a three-setting Bell inequality (here we call it CG inequality for simplicity) which is relevant to the Clauser–Horne–Shimony–Holt (CHSH) inequality. Inspired by their brilliant ideas, we obtained some multi-setting tight Bell inequalities, which are relevant to the CHSH inequality and the CG inequality. Moreover, we generalized the method in the paper [J.L. Chen, D.L. Deng, Phys. Rev. A 79 (2009) 012115] to construct Bell inequality for qubits to higher dimensional system. Based on the generalized method, we present, for the first time, a three-setting tight Bell inequality for two qutrits, which is maximally violated by nonmaximally entangled states and relevant to the Collins–Gisin–Linden–Massar–Popescu inequality.