  Robustness in the violation of Collins-Linden-Gisin-Masser-Popescu (CGLMP)
inequality is investigated from the dual perspective of noise in measurements
as well as in states. To quantify it, we introduce a quantity called area of
nonlocal region which reveals a dimensional advantage. Specifically, we report
that with the increase of dimension, the maximally violating states (MVS) show
a greater enhancement in the area of nonlocal region in comparison to the
maximally entangled states (MES) and the scaling of the increment, in this
case, grows faster than visibility. Moreover, we examine the robustness in the
sequential violation of CGLMP inequality using weak measurements, and find that
even for higher dimensions, two observers showing a simultaneous violation of
the CGLMP inequality as obtained for two-qubit states persists. We notice that
the complementarity between information gain and disturbance by measurements is
manifested by the decrease of the visibility in the first round and the
increase of the same in the second round with dimensions. Furthermore, the
amount of white noise that can be added to an MES so that it gives two rounds
of the violation, decreases with the dimension, while the same does not
appreciably change for the MVS.
