Physics:Quantum Bell's theorem - Revision history

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	'''Bell's theorem''' ~~is a Book I topic in the Quantum Collection. 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''' 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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	{{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}}

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	'''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>		'''Bell's theorem''' is a Book I topic in the Quantum Collection. 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.

	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
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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.

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	'''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>		'''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>

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	→ If a theory is local, it cannot agree with quantum mechanics		→ If a theory is local, it cannot agree with quantum mechanics
	→ If it agrees with quantum mechanics, it must be nonlocal		→ If it agrees with quantum mechanics, it must be nonlocal
	[[File:Bell_theorem_entanglement.jpg\|thumb\|~~400px~~\|~~Illustration of Bell’s~~ theorem~~: measurements on entangled particles exhibit correlations that violate classical (local realistic) expectations~~.]]		</div>

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			[[File:Bell_theorem_entanglement.jpg\|thumb\|280px\|Quantum Bell's theorem.]]
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	== Conceptual background ==		== Conceptual background ==

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

{{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>

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
[[File:Bell_theorem_entanglement.jpg|thumb|400px|Illustration of Bell’s theorem: measurements on entangled particles exhibit correlations that violate classical (local realistic) expectations.]]

== 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>

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:

<math>
|\langle A_0 B_0 \rangle + \langle A_0 B_1 \rangle + \langle A_1 B_0 \rangle - \langle A_1 B_1 \rangle| \leq 2
</math>

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:

<math>
|\psi\rangle = \frac{|0\rangle|1\rangle - |1\rangle|0\rangle}{\sqrt{2}}
</math>

the CHSH expression reaches:

<math>
2\sqrt{2}
</math>

This exceeds the classical limit of 2 and is known as the '''Tsirelson bound'''.<ref>{{cite book |last=Rau |first=Jochen |title=Quantum Theory: An Information Processing Approach |publisher=Oxford University Press |year=2021}}</ref>

Thus:

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

== Experimental tests ==

[[Physics:Bell test|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.<ref name="Hensen2015">{{cite journal |last=Hensen |first=B. |title=Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres |journal=Nature |volume=526 |pages=682–686 |year=2015 |doi=10.1038/nature15759}}</ref>

== 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
* [[Physics:Quantum contextuality|Quantum contextuality]] → measurement outcomes depend on context
* [[Physics:Free will theorem|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:

* [[Physics:Quantum information theory|Quantum information]]
* [[Physics:Quantum cryptography|Quantum cryptography]]
* [[Physics:Quantum computing|Quantum computing]]

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

== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum Bell's theorem|1}}