9.2 Bell's inequality and its experimental verification

Quantum entanglement challenges our understanding of reality. Bell's inequality proves that quantum mechanics defies local realism, a concept Einstein believed essential. This mind-bending idea has far-reaching implications beyond physics.

Experiments have verified Bell's inequality, confirming quantum mechanics' predictions. These tests use entangled particles to show stronger correlations than classical physics allows. This opens doors to exciting applications in quantum computing and cryptography.

Bell's Theorem and Local Realism

Fundamental Concepts of Bell's Theorem

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• Bell's theorem states no local hidden variable theory can reproduce all quantum mechanics predictions
• Demonstrates quantum mechanics incompatibility with local realism combining locality and counterfactual definiteness
• Challenges Einstein-Podolsky-Rosen (EPR) argument for quantum mechanics incompleteness
• Implies abandonment of either locality or realism in understanding quantum phenomena
• Sets up framework for experimental tests distinguishing between quantum mechanics and local hidden variable theories

Implications Beyond Physics

• Influences discussions in philosophy of science and nature of reality
• Impacts understanding of causality and determinism in the quantum world
• Shapes debates on the interpretation of quantum mechanics (Copenhagen interpretation)
• Inspires research into quantum foundations and search for deeper understanding of quantum phenomena
• Contributes to development of quantum information theory and quantum cryptography

Bell's Inequality Derivation

Mathematical Formulation

• Derived from local realism assumptions providing testable prediction for local hidden variable theories
• CHSH (Clauser-Horne-Shimony-Holt) version commonly used in experimental tests $|S| ≤ 2$, where S combines correlation coefficients
• Involves probability theory and statistical correlations between measurement outcomes
• Considers correlated measurements on entangled particle pairs (spin-1/2 particles)
• Demonstrates constraints on local hidden variable theories predictions while quantum mechanics remains unconstrained

Quantum Mechanical Predictions

• Quantum mechanics predicts violations of Bell's inequality for certain entangled states
• Singlet state of two spin-1/2 particles serves as a prime example of violation
• Predicts stronger correlations between entangled particles than allowed by local hidden variable theories
• Quantum entanglement plays crucial role in violation of Bell's inequality
• Theoretical maximum violation in quantum mechanics $2\sqrt{2}$ (Tsirelson's bound)

Experimental Verification of Bell's Inequality

Historical Experiments

• First experimental tests performed by Freedman and Clauser in 1972
• More precise experiments conducted by Aspect et al. in the 1980s
• Early experiments used entangled photon pairs from atomic cascades
• Faced challenges including low detection efficiencies and potential loopholes
• Provided initial evidence supporting quantum mechanics predictions

Modern Experimental Techniques

• Use entangled photon pairs from parametric down-conversion
• Employ high-efficiency detectors to close potential loopholes
• Ensure space-like separation of measurements to address locality loophole
• Achieve sufficiently high detection efficiencies to close detection loophole
• Utilize random number generators for measurement basis selection (freedom of choice)
• Recent experiments (Hensen et al. 2015, Giustina et al. 2015) closed all major loopholes simultaneously

Bell's Theorem and Quantum Mechanics

Non-Classical Nature of Quantum Mechanics

• Provides clear demarcation between classical and quantum physics
• Establishes quantum entanglement as fundamentally non-classical phenomenon without classical analog
• Resolves EPR paradox showing incompatibility of "elements of reality" with quantum mechanics
• Supports Copenhagen interpretation challenging deterministic interpretations
• Demonstrates limitations of classical concepts in describing quantum world

Quantum Information Applications

• Bell's theorem foundational for quantum information technologies
• Enables secure quantum key distribution in quantum cryptography
• Crucial for quantum teleportation protocols
• Utilized in quantum computing algorithms exploiting entanglement
• Inspires development of quantum sensors with enhanced precision