Bell's Inequality

Bell's Inequality is a mathematical constraint derived by physicist John Bell in 1964 that demonstrates the incompatibility of local hidden-variable theories with the predictions of quantum mechanics.

The key aspects of Bell's Inequality are:

1. Local Hidden-Variable Theories:

   • Local hidden-variable theories propose that quantum particles have underlying "hidden" properties that determine the outcomes of measurements, in accordance with the principle of locality.

   • These theories assume that the measurement of one particle in an entangled pair does not instantaneously affect the measurement of the other particle, even when they are spatially separated.

2. Derivation of the Inequality:

   • Bell showed that if quantum systems obey local hidden-variable theories, then the correlations between measurements performed on the systems must satisfy a specific mathematical inequality.

3. Quantum Mechanical Predictions:

   • Bell then demonstrated that the predictions of quantum mechanics for certain experimental setups involving entangled particles could violate this inequality.

4. Experimental Verification:

   • Experiments, such as the one conducted by Freedman and Clauser in 1972, have repeatedly shown that the experimental data violates Bell's inequality, in agreement with the predictions of quantum mechanics.

The significance of Bell's Inequality is that it provides a testable, mathematical criterion for distinguishing between local hidden-variable theories and quantum mechanics. The experimental violations of Bell's inequality have firmly established that the quantum world cannot be explained by classical, local models, and that entanglement is a fundamental, non-local property of quantum systems.

This has had profound implications for our understanding of the nature of reality, as it demonstrates the need to abandon the classical, intuitive notions of causality and locality that had previously underpinned much of physics. The exploration of Bell's Inequality and its experimental verification have been instrumental in the development of quantum information science and the ongoing investigation of the foundational principles of quantum mechanics.