11.2 Bell's inequalities and Bell's theorem

Bell's inequalities and theorem are game-changers in quantum mechanics. They challenge our classical understanding of reality by showing that local hidden variable theories can't explain quantum phenomena. This mind-bending concept shakes up everything we thought we knew about physics.

These ideas are crucial to grasping quantum entanglement and the EPR paradox. By proving that quantum mechanics violates classical limits, Bell's work reveals the true weirdness of the quantum world and forces us to rethink causality and reality itself.

Local Hidden Variable Theories and Bell's Theorem

Challenging Classical Assumptions

• Bell's theorem disproves local hidden variable theories in quantum mechanics
• Local hidden variable theories attempt to explain quantum phenomena using classical physics principles
• Quantum non-locality describes the instantaneous influence between entangled particles regardless of distance
• Probabilistic predictions in quantum mechanics contrast with deterministic classical physics

Implications of Bell's Theorem

• Bell's theorem demonstrates the incompatibility between quantum mechanics and local realism
• Local realism assumes physical properties exist independently of measurement and cannot be influenced faster than light
• Quantum entanglement violates local realism by exhibiting correlations that exceed classical limits
• Bell's theorem led to a paradigm shift in understanding the nature of reality and causality

Bell's Inequalities and the CHSH Inequality

Mathematical Formulation of Bell's Inequalities

• Bell's inequalities mathematically express the limits of correlations in local hidden variable theories
• CHSH inequality (Clauser-Horne-Shimony-Holt) provides a specific formulation of Bell's inequalities
• CHSH inequality states that for local hidden variable theories, the absolute value of a specific combination of correlations must be less than or equal to 2
• Quantum mechanics predicts violations of the CHSH inequality for certain entangled states

Quantum Violation and Entanglement

• Quantum violation occurs when experimental results exceed the bounds set by Bell's inequalities
• Entangled states exhibit correlations that surpass the classical limits imposed by Bell's inequalities
• The maximum quantum violation of the CHSH inequality is $2\sqrt{2}$ (Tsirelson's bound)
• Quantum entanglement enables stronger correlations than those allowed by local hidden variable theories

Experimental Verification

Bell Test Experiments

• Bell test experiments aim to empirically verify the predictions of quantum mechanics against local hidden variable theories
• Entangled particle pairs (photons, electrons) are typically used in Bell test experiments
• Quantum non-locality manifests in the correlated measurements of entangled particles
• Probabilistic predictions of quantum mechanics are confirmed through statistical analysis of experimental results

Loophole-Free Tests and Technological Advancements

• Early Bell test experiments faced potential loopholes that could allow for local hidden variable explanations
• Locality loophole arises from the possibility of communication between detectors during measurement
• Detection loophole occurs due to inefficient particle detection, potentially biasing results
• Recent loophole-free Bell tests have closed these gaps using advanced technology (high-efficiency detectors, fast random number generators)
• Experimental results consistently support quantum mechanical predictions, violating Bell's inequalities