CGLMP and Bell–CHSH formulations of non-locality: a comparative study

The CGLMP prescription of non-locality is known to identify certain states that satisfy the Bell–CHSH inequality to be non-local. The demonstration is, however, restricted to a specific family of states. In this paper, we address the converse question: can there be states that satisfy the CGLMP inequality but violate Bell–CHSH? We find the answer to be in the affirmative. Examining coupled \(4 \times 4\) level systems, we find that there exist a large number of such states. As a direct consequence, states that violate the CGLMP inequality do not form a superset over the ones that violate Bell–CHSH.