Abstract
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.
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