Bell Inequalities, Grothendieck's Constant and Root Two.

B. S. Cirel'son has shown that comparisons between probabilities in "classical" physics and probabilities in quantum mechanics yield discrepancy measures K sub n for finite n x n real matrices that approach Grothendieck's constant K sub G as n gets large. It is known that K sub 2 = K sub 3 = sqrt (2) and that K sub G >- pi/2 = 1.57 sup (...), but examples of n x n matrices for specified n which demonstrate K sub n > sqrt (2) have eluded researchers. We provide a series of elementary examples which yield lower bounds on K sub (k(k-1)) that approach 3/2 as k gets large. A uniform change along the main diagonal of our basic example shows that K sub (20) >- 10/7 = 1.42.