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{{Short description|Fundamental theorem demonstrating incompatibility of quantum mechanics with local hidden-variable theories}}
{{Short description|Fundamental theorem demonstrating incompatibility of quantum mechanics with local hidden-variable theories}}


{{Quantum book backlink|Conceptual and interpretations}}
{{Quantum book backlink|Conceptual and interpretations}}
'''Bell's theorem''' is a foundational result in [[Physics:Quantum mechanics|quantum mechanics]] demonstrating that no theory based on [[Local hidden-variable theory|local hidden variables]] can reproduce all predictions of quantum physics. It establishes that quantum correlations arising from [[Physics:Quantum entanglement|entanglement]] are fundamentally incompatible with the classical assumptions of locality and realism.<ref name="Bell1964">{{cite journal |last=Bell |first=J. S. |title=On the Einstein Podolsky Rosen paradox |journal=Physics Physique Физика |volume=1 |issue=3 |pages=195–200 |year=1964 |doi=10.1103/PhysicsPhysiqueFizika.1.195}}</ref><ref name="Mermin1993">{{cite journal |last=Mermin |first=N. David |title=Hidden variables and the two theorems of John Bell |journal=Reviews of Modern Physics |volume=65 |issue=3 |pages=803–815 |year=1993 |doi=10.1103/RevModPhys.65.803}}</ref>
{{Quantum article nav|previous=Physics:Quantum Measurement problem|previous label=Measurement problem|next=Physics:Quantum Hidden variable theory|next label=Hidden variable theory}}


In essence, Bell showed that:
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→ If a theory is local, it cannot agree with quantum mechanics
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→ If it agrees with quantum mechanics, it must be nonlocal
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[[File:Bell_theorem_entanglement.jpg|thumb|400px|Illustration of Bell’s theorem: measurements on entangled particles exhibit correlations that violate classical (local realistic) expectations.]]
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'''Bell's theorem''' bell's theorem is a foundational result in quantum mechanics demonstrating that no theory based on local hidden variables can reproduce all predictions of quantum physics. It establishes that quantum correlations arising from entanglement are fundamentally incompatible with the classical assumptions of locality and realism. In essence, Bell showed that: → If a theory is local, it cannot agree with quantum mechanics → If it agrees with quantum mechanics, it must be nonlocal Bell's theorem is a foundational result in quantum mechanics demonstrating that no theory based on local hidden variables can reproduce all predictions of quantum physics. It establishes that quantum correlations arising from entanglement are fundamentally incompatible with the classical assumptions of locality and realism. Bell’s theorem builds on the Einstein–Podolsky–Rosen (EPR) paradox, which questioned whether quantum mechanics provides a complete description of reality.
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[[File:Bell_theorem_entanglement.jpg|thumb|280px|Quantum Bell's theorem.]]
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== Conceptual background ==
== Conceptual background ==


Bell’s theorem builds on the [[Physics:EPR paradox|Einstein–Podolsky–Rosen (EPR) paradox]], which questioned whether quantum mechanics provides a complete description of reality.<ref name="EPR1935">{{cite journal |last1=Einstein |first1=A. |last2=Podolsky |first2=B. |last3=Rosen |first3=N. |title=Can quantum-mechanical description of physical reality be considered complete? |journal=Physical Review |volume=47 |issue=10 |pages=777–780 |year=1935 |doi=10.1103/PhysRev.47.777}}</ref>
Bell’s theorem builds on the Einstein–Podolsky–Rosen (EPR) paradox, which questioned whether quantum mechanics provides a complete description of reality.<ref name="EPR1935">{{cite journal |last1=Einstein |first1=A. |last2=Podolsky |first2=B. |last3=Rosen |first3=N. |title=Can quantum-mechanical description of physical reality be considered complete? |journal=Physical Review |volume=47 |issue=10 |pages=777–780 |year=1935 |doi=10.1103/PhysRev.47.777}}</ref>


EPR considered pairs of particles in an entangled state:
EPR considered pairs of particles in an entangled state:
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== Experimental tests ==
== Experimental tests ==


[[Physics:Bell test|Bell tests]] experimentally measure correlations between entangled particles.
Bell tests experimentally measure correlations between entangled particles.


Key milestones include:
Key milestones include:
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Bell’s theorem is part of a broader class of results limiting classical interpretations:
Bell’s theorem is part of a broader class of results limiting classical interpretations:


* [[Kochen–Specker theorem]] → rules out non-contextual hidden variables   
* Kochen–Specker theorem → rules out non-contextual hidden variables   
* [[Physics:Quantum contextuality|Quantum contextuality]] → measurement outcomes depend on context   
* [[Physics:Quantum contextuality|Quantum contextuality]] → measurement outcomes depend on context   
* [[Physics:Free will theorem|Free will theorem]] → constraints on determinism and locality   
* Free will theorem → constraints on determinism and locality   


== Interpretational perspectives ==
== Interpretational perspectives ==
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* [[Physics:Quantum information theory|Quantum information]]   
* [[Physics:Quantum information theory|Quantum information]]   
* [[Physics:Quantum cryptography|Quantum cryptography]]  
* Quantum cryptography   
* [[Physics:Quantum computing|Quantum computing]]  
* Quantum computing   


and is one of the most experimentally tested principles in physics.
and is one of the most experimentally tested principles in physics.

Latest revision as of 12:22, 20 May 2026



← Back to Conceptual and interpretations
← Previous : Measurement problem
Next : Hidden variable theory →

Bell's theorem bell's theorem is a foundational result in quantum mechanics demonstrating that no theory based on local hidden variables can reproduce all predictions of quantum physics. It establishes that quantum correlations arising from entanglement are fundamentally incompatible with the classical assumptions of locality and realism. In essence, Bell showed that: → If a theory is local, it cannot agree with quantum mechanics → If it agrees with quantum mechanics, it must be nonlocal Bell's theorem is a foundational result in quantum mechanics demonstrating that no theory based on local hidden variables can reproduce all predictions of quantum physics. It establishes that quantum correlations arising from entanglement are fundamentally incompatible with the classical assumptions of locality and realism. Bell’s theorem builds on the Einstein–Podolsky–Rosen (EPR) paradox, which questioned whether quantum mechanics provides a complete description of reality.

Quantum Bell's theorem.

Conceptual background

Bell’s theorem builds on the Einstein–Podolsky–Rosen (EPR) paradox, which questioned whether quantum mechanics provides a complete description of reality.[1]

EPR considered pairs of particles in an entangled state:

  • Measuring one particle instantaneously determines the state of the other
  • Even when separated by large distances

This suggests either:

  • Faster-than-light influence (violating locality), or
  • Pre-existing hidden variables determining outcomes

Bell formalized this dilemma mathematically.

Bell inequalities

Bell derived inequalities that any local hidden-variable theory must satisfy. The most widely used version is the **CHSH inequality**, which constrains correlations between measurements:

|⟨A0B0⟩+⟨A0B1⟩+⟨A1B0⟩−⟨A1B1⟩|≤2

This inequality relies on two key assumptions:

  • Locality: no influence propagates faster than light
  • Realism: physical properties exist prior to measurement

Quantum mechanics predicts violations of this bound.

Quantum violation

For entangled states, quantum mechanics predicts stronger correlations. For example, using a maximally entangled Bell state:

|ψ⟩=|0⟩|1⟩−|1⟩|0⟩2

the CHSH expression reaches:

22

This exceeds the classical limit of 2 and is known as the Tsirelson bound.[2]

Thus:

→ Quantum correlations violate Bell inequalities → Local hidden-variable theories cannot reproduce these results

Experimental tests

Bell tests experimentally measure correlations between entangled particles.

Key milestones include:

  • 1972 – First experimental test (Clauser & Freedman)
  • 1982 – Aspect experiments improving locality conditions
  • 2015 – Loophole-free Bell tests

All experiments consistently confirm:

  • Violation of Bell inequalities
  • Agreement with quantum mechanics

These results rule out local hidden-variable theories.[3]

Conceptual implications

Bell’s theorem has profound implications:

  • Nature is not both local and realistic
  • Quantum entanglement implies non-classical correlations
  • Classical intuitions about separability fail

It does not specify which assumption must be abandoned, leading to multiple interpretations.

Relation to other no-go theorems

Bell’s theorem is part of a broader class of results limiting classical interpretations:

  • Kochen–Specker theorem → rules out non-contextual hidden variables
  • Quantum contextuality → measurement outcomes depend on context
  • Free will theorem → constraints on determinism and locality

Interpretational perspectives

Different interpretations resolve Bell violations differently:

  • Copenhagen: abandons realism or counterfactual definiteness
  • Many-worlds: retains locality but allows multiple outcomes
  • Bohmian mechanics: retains realism but introduces nonlocality
  • Objective collapse: modifies quantum dynamics

No consensus exists on the “correct” interpretation.

Physical significance

Bell’s theorem demonstrates that:

→ Quantum mechanics is fundamentally incompatible with classical worldviews

It underpins modern developments such as:

and is one of the most experimentally tested principles in physics.

See also

Table of contents (198 articles)

Index

Full contents

9. Quantum optics and experiments (5) Back to index
Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.
Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.
14. Plasma and fusion physics (8) Back to index
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

References

  1. ↑ Einstein, A.; Podolsky, B.; Rosen, N. (1935). "Can quantum-mechanical description of physical reality be considered complete?". Physical Review 47 (10): 777–780. doi:10.1103/PhysRev.47.777. 
  2. ↑ Rau, Jochen (2021). Quantum Theory: An Information Processing Approach. Oxford University Press. 
  3. ↑ Hensen, B. (2015). "Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres". Nature 526: 682–686. doi:10.1038/nature15759. 


Author: Harold Foppele


Source attribution: Physics:Quantum Bell's theorem

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