Subadditity of logarithmic violation of geometrical Bell inequalities for qudits.*

Geometric inequalities for qubits [1] possess the highest robustness against the white noise among know correlation-based formulations. We present a construction of Bell inequalities for a collection of arbitrarily large subsystems (each with dimension d), with (d-1)-parameter family of local observables. The sum of outcomes of local measurments is represented as one of (d-1)-dimensional non-orthogonal vectors. We find the precise extimates of violation of the inequalities and observe that it grows exponentially with the number of (most probably) sublinearly with d. The intesting aspect is that we can compare the amount of nonclassiclity manifested in a single Bell experiment with subsystem that are composed of smaller parts with experiments conducted on these parts. The results of this comparison are counterintuitive.
[1] K. Nagata, W. Laskowski, M. Wiesniak, and M. Zukowski, Phys. Rev. Lett. 93, 23043 (2004)