Editing Principle of locality (section)

== Quantum mechanics ==
The relative positions of our few, easily distinguishable planets (for example) can be seen directly: understanding and measuring their relative location poses only  technical issues. The submicroscopic world on the other hand is known only by  measurements that average over many seemingly random ("statistical" or "probabilistic") events and measurements can show either [[wave–particle duality |particle-like or wave-like results]] depending on their design. This world is governed by [[quantum mechanics]].<ref name="FeynmanIII">{{Cite book |last1=Feynman |first1=Richard P. |title=Quantum Mechanics |url=https://www.feynmanlectures.caltech.edu/III_01.html |last2=Leighton |first2=Robert B. |last3=Sands |first3=Matthew L. |date=2007 |publisher=Addison-Wesley |isbn=978-0-201-02118-9 |series=[[The Feynman Lectures on Physics]] |volume=3 |location=Reading/Mass. |author-link1=Richard Feynman |author-link2=Robert B. Leighton |author-link3=Matthew Sands}}</ref> The concepts of locality are more complex and they are described in the language of [[probability]] and [[correlation]].

In the 1935 [[Einstein–Podolsky–Rosen paradox]] paper (EPR paper),<ref>{{Cite journal |last1=Einstein |first1=A. |last2=Podolsky |first2=B. |last3=Rosen |first3=N. |date=1935-05-15 |title=Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? |journal=Physical Review |language=en |volume=47 |issue=10 |pages=777–780 |doi=10.1103/PhysRev.47.777 |issn=0031-899X|doi-access=free |bibcode=1935PhRv...47..777E }}</ref> [[Albert Einstein]], [[Boris Podolsky]] and [[Nathan Rosen]] imagined such an experiment. They observed that quantum mechanics predicts what is now known as [[quantum entanglement]] and examined its consequences.<ref name=ReidDrummond >{{Cite journal |last1=Reid |first1=M. D. |last2=Drummond |first2=P. D. |last3=Bowen |first3=W. P. |last4=Cavalcanti |first4=E. G. |last5=Lam |first5=P. K. |last6=Bachor |first6=H. A. |last7=Andersen |first7=U. L. |last8=Leuchs |first8=G. |date=2009-12-10 |title=Colloquium : The Einstein-Podolsky-Rosen paradox: From concepts to applications |url=https://link.aps.org/doi/10.1103/RevModPhys.81.1727 |journal=Reviews of Modern Physics |language=en |volume=81 |issue=4 |pages=1727–1751 |doi=10.1103/RevModPhys.81.1727 |arxiv=0806.0270 |bibcode=2009RvMP...81.1727R |issn=0034-6861 |hdl=10072/37941 |s2cid=53407634 |hdl-access=free }}</ref> In their view, the classical principle of locality implied that "no real change can take place" at Bob's site as a result of whatever measurements Alice was doing. Since quantum mechanics does predict a [[wavefunction collapse]] that depends on Alice's choice of measurement, they concluded that this was a form of action-at-distance and that the wavefunction could not be a complete description of reality. Other physicists did not agree: they accepted the quantum wavefunction as complete and questioned the nature of locality and reality assumed in the EPR paper.<ref name=ClauserShimony1978>Clauser, John F., and Abner Shimony. "[https://duneece.wiscweb.wisc.edu/wp-content/uploads/sites/605/2019/01/J_F_Clauser_1978_Rep._Prog._Phys._41_002_bell_test.pdf Bell's theorem. Experimental tests and implications]". Reports on Progress in Physics 41.12 (1978): 1881.</ref> 

In 1964 [[John Stewart Bell]] investigated whether it might be possible to fulfill Einstein's goal—to "complete" quantum theory—with [[Local hidden-variable theory|local hidden variables]] to explain the correlations between spatially separated particles as predicted by quantum theory. Bell established a criterion to distinguish between local hidden-variables theory and quantum theory by measuring specific values of correlations between entangled particles. Subsequent experimental [[Bell test|tests]] have shown that some quantum effects do violate [[Bell's theorem|Bell's inequalities]] and cannot be reproduced by a local hidden-variables theory.<ref name=ReidDrummond/> Bell's theorem depends on careful defined models of locality.

===Locality and hidden variables===
{{main article|Bell's theorem}}
Bell described local causality in terms of probability needed for analysis of quantum mechanics. Using the notation that <math>P(\mathbf{r} \mid g)</math> for the probability of a result <math>\mathbf{r}</math> with given state <math>g</math>, Bell investigated the probability distribution
<math display=block>
 P(\mathbf{ab} \mid AB, \lambda),
</math>
where <math>\lambda</math> represents hidden state variables set (locally) when the two particles are initially co-located. If local causality holds, then the probabilities observed by Alice and by Bob should be only coupled by the hidden variables, and we can show that
<math display=block>
 P(\mathbf{ab} \mid AB, \lambda) = P(\mathbf{a} \mid A, \lambda) P(\mathbf{b} \mid B, \lambda).
</math> 
Bell proved that a consequence of this factorization are limits on the correlations observed by Alice and Bob known as Bell inequalities. Since quantum mechanics predicts correlations stronger than this limit, locally set hidden variables cannot be added to "complete" quantum theory as desired by the EPR paper.<ref name=Brunner>{{Cite journal |last1=Brunner |first1=Nicolas |last2=Cavalcanti |first2=Daniel |last3=Pironio |first3=Stefano |last4=Scarani |first4=Valerio |last5=Wehner |first5=Stephanie |date=2014-04-18 |title=Bell nonlocality |url=https://link.aps.org/doi/10.1103/RevModPhys.86.419 |journal=Reviews of Modern Physics |language=en |volume=86 |issue=2 |pages=419–478 |doi=10.1103/RevModPhys.86.419 |issn=0034-6861|arxiv=1303.2849 |bibcode=2014RvMP...86..419B |s2cid=119194006 }}</ref> 

Numerous experiments specifically designed to probe the issues of locality confirm the predictions of quantum mechanics; these include experiments where the two measurement locations are more than a kilometer apart.<ref name=Brunner/><ref>{{Cite journal |last1=Holmes |first1=Rebecca |year=2017 |title=Local realism is dead, long live local realism? |journal=Physics World |volume=30 |issue=6 |pages=21–25 |doi=10.1088/2058-7058/30/6/41|bibcode=2017PhyW...30f..21H }}</ref> 
The 2022 [[Nobel Prize in Physics]] was awarded to [[Alain Aspect]], [[John Clauser]] and [[Anton Zeilinger]], in part "for experiments with entangled photons, establishing the violation of Bell inequalities".<ref>{{Cite web |date=4 October 2022 |title=The Nobel Prize in Physics 2022 |url=https://www.nobelprize.org/prizes/physics/2022/press-release/ |access-date=6 October 2022 |publisher=[[Nobel Foundation]] |archive-date=4 October 2022 |archive-url=https://web.archive.org/web/20221004095754/https://www.nobelprize.org/prizes/physics/2022/press-release/ |url-status=live }}</ref> The specific aspect of quantum theory that leads to these correlations is termed [[quantum entanglement]], and versions of Bell's scenario are now used to verify entanglement experimentally.<ref name=Brunner/>

==== {{Anchor|Local realism}}Terminology ====
{{main article|Quantum nonlocality}}
{{See also|Bell's theorem#Interpretations|Reality#Realism and locality in physics}}

Bell's mathematical results, when compared to experimental data, eliminate local hidden-variable mathematical quantum theories. But the interpretation of the math with respect to the physical world remains under debate. Bell described the assumptions behind his work as "local causality", shortened to "locality"; later authors referred to the assumptions as '''local realism'''.<ref>{{Cite journal |last=Laudisa |first=Federico |date=Feb 2023 |title=How and when did locality become 'local realism'? A historical and critical analysis (1963–1978) |url=https://linkinghub.elsevier.com/retrieve/pii/S0039368122001753 |journal=Studies in History and Philosophy of Science |language=en |volume=97 |pages=44–57 |doi=10.1016/j.shpsa.2022.11.008 |pmid=36549108 |arxiv=2205.05452 |s2cid=248693366 }}</ref> These different names do not alter the mathematical assumptions. 

A review of papers<ref name=Lambare>{{Cite journal |last=Lambare |first=Justo Pastor |date=Oct 2022 |title=On the Meaning of Local Realism |url=https://link.springer.com/10.1007/s10701-022-00618-1 |journal=Foundations of Physics |language=en |volume=52 |issue=5 |page=98 |doi=10.1007/s10701-022-00618-1 |bibcode=2022FoPh...52...98L |s2cid=252107202 |issn=0015-9018}}</ref> using this phrase suggests that a common (classical) physics definition of '''realism'''<!--boldface per WP:R#PLA; 'Realism in physics' redirects here--> is 
{{quote | the assumption that measurement outcomes are well defined prior to and independent of the measurements.<ref>{{Cite journal |last1=Paterek |first1=Tomasz |last2=Fedrizzi |first2=Alessandro |last3=Gröblacher |first3=Simon |last4=Jennewein |first4=Thomas |last5=Żukowski |first5=Marek |last6=Aspelmeyer |first6=Markus |last7=Zeilinger |first7=Anton |date=2007-11-21 |title=Experimental Test of Nonlocal Realistic Theories Without the Rotational Symmetry Assumption |url=https://link.aps.org/doi/10.1103/PhysRevLett.99.210406 |journal=Physical Review Letters |language=en |volume=99 |issue=21 |page=210406 |doi=10.1103/PhysRevLett.99.210406 |pmid=18233201 |issn=0031-9007|arxiv=0708.0813 |bibcode=2007PhRvL..99u0406P |s2cid=21746600 }}</ref>}}

This definition includes classical concepts like "well-defined", which conflicts with [[quantum superposition]], and "prior to&nbsp;... measurements", which implies (metaphysical) preexistence of properties. Specifically, the term local realism in the context of Bell's theorem cannot be viewed as a kind of "realism" involving locality other than the kind implied by the Bell screening assumption. This conflict between common ideas of realism and quantum mechanics requires careful analysis whenever local realism is discussed.<ref name=Lambare/>{{rp|98}}
Adding a "locality" modifier, that the results of two spatially well-separated measurements cannot causally affect each other,<ref name=ReidDrummond/> does not make the combination relate to Bell's proof; the only interpretation that Bell assumed was the one he called local causality.<ref name=Lambare/>{{rp|98}} Consequently, Bell's theorem does not restrict the possibility of nonlocal variables as well as theories based on [[retrocausality]] or [[superdeterminism]].<ref name="WhartonArgaman" />

Because of the probabilistic nature of wave function collapse, this apparent violation of locality in quantum mechanics cannot be used to transmit [[information]] faster than light, in accordance to the [[no communication theorem]].<ref>{{Cite book |last1=Susskind |first1=Leonard |url=https://books.google.com/books?id=LX2-AQAAQBAJ |title=Quantum Mechanics: The Theoretical Minimum |last2=Friedman |first2=Art |date=2014-02-25 |publisher=Penguin Books Limited |isbn=978-0-14-197782-9 |language=en}}</ref> [[Asher Peres]] distinguishes between ''weak'' and ''strong nonlocality'', the latter referring to the theories that allow [[Superluminal communication|faster-than-light communication]]. Under these terms, quantum mechanics would allow weakly nonlocal correlations but not strong nonlocality.<ref>{{Cite book |last=Peres |first=A. |url=https://books.google.com/books?id=pQXSBwAAQBAJ&q=asher+peres |title=Quantum Theory: Concepts and Methods |date=2006-06-01 |publisher=Springer Science & Business Media |isbn=978-0-306-47120-9 |language=en}}</ref>