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

TL;DR

This paper establishes a quantitative lower bound on how closely classical stochastic processes can approximate quantum spin correlations, highlighting fundamental differences between classical and quantum models.

Contribution

It derives a specific positive lower bound for the L2-distance between quantum spin singlet correlations and any classical autocorrelation approximation.

Findings

Lower bound of approximately 0.134 for classical approximation of quantum correlations.

Demonstrates fundamental limitations of classical models in replicating quantum spin correlations.

Quantifies the gap between classical and quantum correlation functions.

Abstract

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 $\Vert C-\cos\Vert \geq(1-\frac{8}{\pi^{2}}) /\sqrt{2}\approx0.133\,95.$