Bell Inequalities, Grothendieck’s Constant, and Root Two

B. S. Tsirelson showed that comparisons between probabilities in “classical” physics and probabilities in quantum mechanics yield discrepancy measures Kₙ for finite n × n real matrices that approach Grothendieck’s constant KG as n gets large. It is known that K₂ = K₃ = √2 and that KG ≥ π /2 = 1.57 ⋯, but examples of n × n matrices for specified n that demonstrate Kₙ > √2 have eluded researchers. A series of elementary examples are provided, which yield lower bounds on Kk ( k - 1 ) that approach 3/2 as k gets large. A uniform change along the main diagonal of our basic example shows that K₂₀ = 10/7 = 1.42 ⋯.