15.1 Quantum entanglement and Bell's theorem

Quantum entanglement links particles across vast distances, defying classical physics. It challenges our understanding of reality and causality, sparking debates about the completeness of quantum mechanics and the nature of information transfer.

The EPR paradox and Bell's theorem address this mystery. Bell's inequality provides a way to test local hidden variable theories against quantum mechanics, with experiments consistently supporting quantum predictions and non-locality.

Quantum Entanglement and EPR Paradox

Quantum entanglement concept

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• Non-classical correlation between two or more quantum systems links particles across vast distances
• Systems remain correlated regardless of spatial separation defies classical intuition
• Measurement on one particle instantly affects the other creates "spooky action at a distance" (Einstein's phrase)
• Cannot be explained by classical physics challenges conventional understanding of causality
• Challenges local realism suggests reality is not entirely local or deterministic
• Suggests non-local interactions contradicts classical notions of information transfer
• Questions the completeness of quantum mechanics sparks ongoing debates in foundations of physics

EPR paradox and Bell's theorem

• Einstein-Podolsky-Rosen (EPR) thought experiment proposed in 1935 challenged quantum mechanics' completeness
• Two particles in an entangled state exhibit perfect correlations in certain measurements
• Measurement of one particle's property instantaneously determines the other particle's property
• Assumption of local realism conflicts with quantum mechanical predictions
• Incompatibility with quantum mechanical predictions led to the "EPR paradox"
• Motivated John Bell to mathematically address the paradox in 1964
• Led to formulation of Bell's inequality provided testable criterion for local hidden variable theories

Bell's Theorem and Experimental Tests

Bell's inequality derivation

• Mathematical expression: $|E(a,b) - E(a,c)| \leq 1 + E(b,c)$ quantifies correlations between measurements
• $E(x,y)$ represents correlation between measurements at detector settings x and y
• Locality assumption no faster-than-light communication between particles
• Realism assumption physical properties exist prior to measurement
• Provides testable prediction for local hidden variable theories sets upper bound on correlations
• Quantum mechanics predicts violation of the inequality in certain entangled states
• Rules out local hidden variable theories if experimentally violated
• Supports non-local nature of quantum mechanics challenges classical notions of reality

Experimental tests of Bell's theorem

• Aspect experiment (1982) first convincing demonstration of Bell inequality violation
• Innsbruck experiment (1998) improved precision and closed some experimental loopholes
• Delft experiment (2015) achieved first loophole-free Bell test
• Generation of entangled particle pairs typically uses parametric down-conversion in nonlinear crystals
• Spatially separated detectors ensure no classical communication between measurement events
• Random measurement settings prevent any pre-existing agreement between particles
• Consistent violation of Bell's inequality observed in numerous experiments
• Agreement with quantum mechanical predictions supports entanglement and non-locality
• Loophole-free experiments address potential weaknesses in earlier tests (detection loophole, locality loophole)
• Reconciling quantum non-locality with relativity remains an open challenge in theoretical physics