On the inequivalence of the CH and CHSH inequalities due to finite statistics

Abstract

Different variants of a Bell inequality, such as CHSH and CH, are known to be equivalent when evaluated on nonsignaling outcome probability distributions. However, in experimental setups, the outcome probability distributions are estimated using a finite number of samples. Therefore the nonsignaling conditions are only approximately satisfied and the robustness of the violation depends on the chosen inequality variant. We explain that phenomenon using the decomposition of the space of outcome probability distributions under the action of the symmetry group of the scenario, and propose a method to optimize the statistical robustness of a Bell inequality. In the process, we describe the finite group composed of relabeling of parties, measurement settings and outcomes, and identify correspondences between the irreducible representations of this group and properties of outcome probability distributions such as normalization, signaling or having uniform marginals.

arXiv comment27 pages, 12 figures

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• Algorithmic construction of local models for entangled quantum states: optimization for two-qubit states
• Experimental comparison of tomography and self-testing in certifying entanglement
• RepLAB: a computational/numerical approach to representation theory
• Closing the detection loophole in multipartite Bell experiments with a limited number of efficient detectors
• Correcting for finite statistics effects in a quantum steering experiment