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Bell's theorem prevents local Kolmogorov-simulations of the singlet state oftwo 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 stationaryautocorrelation functions of mean-square-continuous, $2\pi $-periodic,$\pm1$-valued, stochastic processes. This bound is given by $\Vert C-\cos\Vert\geq(1-\frac{8}{\pi^{2}}) /\sqrt{2}\approx0.133\,95.$

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