  Entanglement of high-dimensional and multipartite quantum systems offer
promising perspectives in quantum information processing. However, the
characterization and measure of such kind of entanglement is of great
challenge. Here we consider the overlaps between the maximal quantum mean
values and the classical bound of the CHSH inequalities for pairwise-qubit
states in two-dimensional subspaces. We show that the concurrence of a pure
state in any high-dimensional multipartite system can be equivalently
represented by these overlaps. Here we consider the projections of an arbitrary
high-dimensional multipartite state to two-qubit states. We investigate the
non-localities of these projected two-qubit sub-states by their violations of
CHSH inequalities. From these violations, the overlaps between the maximal
quantum mean values and the classical bound of the CHSH inequality, we show
that the concurrence of a high-dimensional multipartite pure state can be
exactly expressed by these overlaps. We further derive a lower bound of the
concurrence for any quantum states, which is tight for pure states. The lower
bound not only imposes restriction on the non-locality distributions among the
pairwise qubit states, but also supplies a sufficient condition for
distillation of bipartite entanglement. Effective criteria for detecting
genuine tripartite entanglement and the lower bound of concurrence for genuine
tripartite entanglement are also presented based on such non-localities.
