Takeaway

No local hidden-variable theory can reproduce all quantum predictions: certain correlations violate Bell inequalities, revealing nonlocality or the failure of local realism.

The problem (before → after)

  • Before: EPR argued quantum mechanics is incomplete; perhaps hidden variables restore locality and determinism.
  • After: Bell derived testable inequalities that any local hidden-variable theory must satisfy. Experiments violate them, supporting quantum mechanics.

Mental model first

Two separated boxes output ±1 when you press buttons A or B. Any pre-agreed local rule limits the pattern of joint outputs. Quantum entanglement produces patterns that exceed that limit, even without communication during the test.

Just-in-time concepts

  • Local realism: Outcomes determined by local properties, independent of distant settings.
  • CHSH inequality: |E(A B) + E(A B’) + E(A’ B) − E(A’ B’)| ≤ 2.
  • Quantum bound (Tsirelson): ≤ 2√2 with appropriate states and settings.

First-pass solution

Assume hidden variables λ with local response functions. Derive CHSH ≤ 2. Compute quantum correlations for a singlet state and show violation.

Iterative refinement

  1. Loopholes: Detection and locality loopholes were closed in modern “loophole-free” experiments.
  2. No-signaling remains intact; violations don’t enable faster-than-light communication.
  3. Device-independent cryptography leverages Bell violations for security guarantees.

Principles, not prescriptions

  • Correlations constrain theories; some classical assumptions are empirically untenable.
  • Entanglement is a resource for information tasks.

Common pitfalls

  • Equating nonlocality with signaling; Bell violations do not imply communication.
  • Thinking determinism alone is ruled out; the conjunction with locality is tested.

Connections and contrasts

  • See also: [/blog/quantum-error-correction], [/blog/quantum-decoherence], [/blog/many-body-localization] (entanglement dynamics).

Quick checks

  1. What does CHSH ≤ 2 mean? — A constraint on local hidden-variable correlations.
  2. How can QM exceed 2? — Measurement settings and entangled states align to reach up to 2√2.
  3. Does this allow FTL signaling? — No, only correlations differ.

Further reading

  • Bell, 1964 (source above)
  • Aspect, Zeilinger experiments and reviews
  • Nielsen & Chuang, Ch. 12–13