  We analyze the correlation structure of bipartite arbitrary-dimensional Bell
inequalities via novel conditions of correlations in terms of differences of
joint probabilities called correlators. The conditions of correlations are
shown to be necessary for the multi-level Bell state. In particular, we find
that the bipartite arbitrary-dimensional Bell-type inequalities introduced by
Collins-Gisin-Linden-Massar-Popescu [Phys. Rev. Lett. 88, 040404 (2002)] and
Son-Lee-Kim [Phys. Rev. Lett. 96, 060406 (2006)] are composed of correlators,
and we reveal that the maximal violations by the Bell state just fit the
conditions of quantum correlations. Correlators can be considered as essential
elements of Bell inequalities.
