Lower bound for the mean square distance between classical and quantum spin correlations

Bell's theorem prevents local Kolmogorov-simulations of the singlet state of two spin-1/2 particles. We derive a positive lower bound for the $L^{2}$-distance between the quantum mechanical spin singlet anticorrelation function $\cos$ and any of its classical approximants $C$ formed by the stationary autocorrelation functions of mean-square-continuous, $2\pi$-periodic, $\pm1$-valued, stochastic processes. This bound is given by $\Vert C-\cos\Vert \geq(1-\frac{8}{\pi^{2}}) /\sqrt{2}\approx0.133\,95.$