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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}}

Physics:Quantum Bell's theorem - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template