CHSH Inequality Test

Introduction

The CHSH inequality, named after John Clauser, Michael Horne, Abner Shimony, and Richard Holt, is a specific case of Bell’s inequalities and is used to test whether the predictions of quantum mechanics about entangled particles are borne out by experiment, or whether local hidden variable theories can be sustained.

Bell’s Theorem and Local Hidden Variables

Bell’s theorem states that no physical theory of local hidden variables can reproduce all of the predictions of quantum mechanics. Local hidden variables theories propose that the outcomes of measurements made on entangled particles are determined by local properties that preexist the measurement, which is in contrast to quantum mechanics that considers the measurement outcomes as fundamentally random and correlated in a way that cannot be explained by any local theory.

CHSH Inequality

The CHSH inequality is given by:

where is the expected value of the product of the results of measurements on two entangled particles, and are the possible measurements for the first particle, and and are the possible measurements for the second particle.

Experimental Tests

Experiments testing the CHSH inequality have consistently found violations, which is in line with the predictions of quantum mechanics and contradicts local hidden variables theories. These experiments usually involve measuring the polarization of entangled photons.

Implications

The violation of the CHSH inequality supports the quantum mechanical view of the world, which includes fundamental randomness and nonlocal correlations known as quantum entanglement. It rules out a broad class of local hidden variables theories, which sought to restore determinism and locality to physics. However, the interpretation of these results remains an active area of debate in the foundations of quantum mechanics.

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