Physics:Quantum contextuality - Revision history

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	==Kochen and Specker==		==Kochen and Specker==
	~~{{main\|~~Kochen–Specker theorem}}		''Related topic:'' Kochen–Specker theorem
	The need for contextuality was discussed informally in 1935 by Grete Hermann,<ref>{{cite book \|last1=Crull \|first1=Elise \|last2=Bacciagaluppi \|first2=Guido \|title=Grete Hermann - Between Physics and Philosophy \|date=2016 \|publisher=Springer \|location=Netherlands \|isbn=978-94-024-0968-0 \|page=154 \|ref=46}}</ref> but it was more than 30 years later when Simon B. Kochen and Ernst Specker, and separately John Bell, constructed proofs that any realistic hidden-variable theory able to explain the phenomenology of quantum mechanics is contextual for systems of Hilbert space dimension three and greater. The Kochen–Specker theorem proves that realistic noncontextual hidden-variable theories cannot reproduce the empirical predictions of quantum mechanics.<ref>{{Cite journal \|last=Carsten \|first=Held \|date=2000-09-11 \|title=The Kochen–Specker Theorem \|url=https://plato.stanford.edu/archives/spr2018/entries/kochen-specker/ \|access-date=2018-11-17 \|website=Stanford Encyclopedia of Philosophy}}</ref> Such a theory would suppose the following.		The need for contextuality was discussed informally in 1935 by Grete Hermann,<ref>{{cite book \|last1=Crull \|first1=Elise \|last2=Bacciagaluppi \|first2=Guido \|title=Grete Hermann - Between Physics and Philosophy \|date=2016 \|publisher=Springer \|location=Netherlands \|isbn=978-94-024-0968-0 \|page=154 \|ref=46}}</ref> but it was more than 30 years later when Simon B. Kochen and Ernst Specker, and separately John Bell, constructed proofs that any realistic hidden-variable theory able to explain the phenomenology of quantum mechanics is contextual for systems of Hilbert space dimension three and greater. The Kochen–Specker theorem proves that realistic noncontextual hidden-variable theories cannot reproduce the empirical predictions of quantum mechanics.<ref>{{Cite journal \|last=Carsten \|first=Held \|date=2000-09-11 \|title=The Kochen–Specker Theorem \|url=https://plato.stanford.edu/archives/spr2018/entries/kochen-specker/ \|access-date=2018-11-17 \|website=Stanford Encyclopedia of Philosophy}}</ref> Such a theory would suppose the following.

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	Contextuality was first demonstrated to be a feature of quantum phenomenology by the Bell–Kochen–Specker theorem.<ref name=":1" /><ref name=":2" /> The study of contextuality has developed into a major topic of interest in [[Physics:Quantum foundations\|quantum foundations]] as the phenomenon crystallises certain non-classical and counter-intuitive aspects of quantum theory. A number of powerful mathematical frameworks have been developed to study and better understand contextuality, from the perspective of sheaf theory,<ref name=":3">{{Cite journal\|last1=Abramsky\|first1=Samson\|last2=Brandenburger\|first2=Adam\|date=2011-11-28\|title=The Sheaf-Theoretic Structure Of Non-Locality and Contextuality\|journal=New Journal of Physics\|volume=13\|issue=11\|article-number=113036\|arxiv=1102.0264\|bibcode=2011NJPh...13k3036A\|doi=10.1088/1367-2630/13/11/113036\|s2cid=17435105\|issn=1367-2630}}</ref> graph theory,<ref>{{Cite journal\|last1=Cabello\|first1=Adan\|last2=Severini\|first2=Simone\|last3=Winter\|first3=Andreas\|date=2014-01-27\|title=Graph-Theoretic Approach to Quantum Correlations\|journal=Physical Review Letters\|volume=112\|issue=4\|article-number=040401\|arxiv=1401.7081\|bibcode=2014PhRvL.112d0401C\|doi=10.1103/PhysRevLett.112.040401\|issn=0031-9007\|pmid=24580419\|s2cid=34998358}}</ref> hypergraphs,<ref name="Acín 533–628">{{Cite journal\|last1=Acín\|first1=Antonio\|last2=Fritz\|first2=Tobias\|last3=Leverrier\|first3=Anthony\|last4=Sainz\|first4=Ana Belén\|date=2015-03-01\|title=A Combinatorial Approach to Nonlocality and Contextuality\|journal=Communications in Mathematical Physics\|volume=334\|issue=2\|pages=533–628\|doi=10.1007/s00220-014-2260-1\|issn=1432-0916\|arxiv=1212.4084\|bibcode=2015CMaPh.334..533A\|s2cid=119292509}}</ref> algebraic topology,<ref>{{Cite journal\|last1=Abramsky\|first1=Samson\|last2=Mansfield\|first2=Shane\|last3=Barbosa\|first3=Rui Soares\|date=2012-10-01\|title=The Cohomology of Non-Locality and Contextuality\|journal=Electronic Proceedings in Theoretical Computer Science\|volume=95\|pages=1–14\|doi=10.4204/EPTCS.95.1\|issn=2075-2180\|arxiv=1111.3620\|s2cid=9046880}}</ref> and probabilistic couplings.<ref name=":11">{{Cite journal\|last1=Dzhafarov\|first1=Ehtibar N.\|last2=Kujala\|first2=Janne V.\|date=2016-09-07\|title=Probabilistic foundations of contextuality\|journal=Fortschritte der Physik\|volume=65\|issue=6–8\|article-number=1600040\|doi=10.1002/prop.201600040\|issn=0015-8208\|bibcode=2017ForPh..6500040D\|arxiv=1604.08412\|s2cid=56245502}}</ref>		Contextuality was first demonstrated to be a feature of quantum phenomenology by the Bell–Kochen–Specker theorem.<ref name=":1" /><ref name=":2" /> The study of contextuality has developed into a major topic of interest in [[Physics:Quantum foundations\|quantum foundations]] as the phenomenon crystallises certain non-classical and counter-intuitive aspects of quantum theory. A number of powerful mathematical frameworks have been developed to study and better understand contextuality, from the perspective of sheaf theory,<ref name=":3">{{Cite journal\|last1=Abramsky\|first1=Samson\|last2=Brandenburger\|first2=Adam\|date=2011-11-28\|title=The Sheaf-Theoretic Structure Of Non-Locality and Contextuality\|journal=New Journal of Physics\|volume=13\|issue=11\|article-number=113036\|arxiv=1102.0264\|bibcode=2011NJPh...13k3036A\|doi=10.1088/1367-2630/13/11/113036\|s2cid=17435105\|issn=1367-2630}}</ref> graph theory,<ref>{{Cite journal\|last1=Cabello\|first1=Adan\|last2=Severini\|first2=Simone\|last3=Winter\|first3=Andreas\|date=2014-01-27\|title=Graph-Theoretic Approach to Quantum Correlations\|journal=Physical Review Letters\|volume=112\|issue=4\|article-number=040401\|arxiv=1401.7081\|bibcode=2014PhRvL.112d0401C\|doi=10.1103/PhysRevLett.112.040401\|issn=0031-9007\|pmid=24580419\|s2cid=34998358}}</ref> hypergraphs,<ref name="Acín 533–628">{{Cite journal\|last1=Acín\|first1=Antonio\|last2=Fritz\|first2=Tobias\|last3=Leverrier\|first3=Anthony\|last4=Sainz\|first4=Ana Belén\|date=2015-03-01\|title=A Combinatorial Approach to Nonlocality and Contextuality\|journal=Communications in Mathematical Physics\|volume=334\|issue=2\|pages=533–628\|doi=10.1007/s00220-014-2260-1\|issn=1432-0916\|arxiv=1212.4084\|bibcode=2015CMaPh.334..533A\|s2cid=119292509}}</ref> algebraic topology,<ref>{{Cite journal\|last1=Abramsky\|first1=Samson\|last2=Mansfield\|first2=Shane\|last3=Barbosa\|first3=Rui Soares\|date=2012-10-01\|title=The Cohomology of Non-Locality and Contextuality\|journal=Electronic Proceedings in Theoretical Computer Science\|volume=95\|pages=1–14\|doi=10.4204/EPTCS.95.1\|issn=2075-2180\|arxiv=1111.3620\|s2cid=9046880}}</ref> and probabilistic couplings.<ref name=":11">{{Cite journal\|last1=Dzhafarov\|first1=Ehtibar N.\|last2=Kujala\|first2=Janne V.\|date=2016-09-07\|title=Probabilistic foundations of contextuality\|journal=Fortschritte der Physik\|volume=65\|issue=6–8\|article-number=1600040\|doi=10.1002/prop.201600040\|issn=0015-8208\|bibcode=2017ForPh..6500040D\|arxiv=1604.08412\|s2cid=56245502}}</ref>

	[[Physics:Quantum nonlocality\|Nonlocality]], in the sense of Bell's theorem, may be viewed as a special case of the more general phenomenon of contextuality, in which measurement contexts contain measurements that are distributed over ~~[[Physics:Causal structure#Curves\|~~spacelike]] separated regions. This follows from [[Physics:Quantum contextuality#Fine's theorem\|Fine's theorem]].<ref name=":6">{{Cite journal\|last=Fine\|first=Arthur\|date=1982-02-01\|title=Hidden Variables, Joint Probability, and the Bell Inequalities\|journal=Physical Review Letters\|volume=48\|issue=5\|pages=291–295\|doi=10.1103/PhysRevLett.48.291\|bibcode=1982PhRvL..48..291F}}</ref><ref name=":3" />		[[Physics:Quantum nonlocality\|Nonlocality]], in the sense of Bell's theorem, may be viewed as a special case of the more general phenomenon of contextuality, in which measurement contexts contain measurements that are distributed over spacelike separated regions. This follows from [[Physics:Quantum contextuality#Fine's theorem\|Fine's theorem]].<ref name=":6">{{Cite journal\|last=Fine\|first=Arthur\|date=1982-02-01\|title=Hidden Variables, Joint Probability, and the Bell Inequalities\|journal=Physical Review Letters\|volume=48\|issue=5\|pages=291–295\|doi=10.1103/PhysRevLett.48.291\|bibcode=1982PhRvL..48..291F}}</ref><ref name=":3" />

	Quantum contextuality has been identified as a source of quantum computational speedups and quantum advantage in quantum computing.<ref>{{Cite journal\|last=Raussendorf\|first=Robert\|date=2013-08-19\|title=Contextuality in measurement-based quantum computation\|journal=Physical Review A\|volume=88\|issue=2\|article-number=022322\|doi=10.1103/PhysRevA.88.022322\|issn=1050-2947\|arxiv=0907.5449\|bibcode=2013PhRvA..88b2322R\|s2cid=118495073}}</ref><ref>{{Cite journal\|last1=Howard\|first1=Mark\|last2=Wallman\|first2=Joel\|last3=Veitch\|first3=Victor\|last4=Emerson\|first4=Joseph\|date=June 2014\|title=Contextuality supplies the 'magic' for quantum computation\|journal=Nature\|volume=510\|issue=7505\|pages=351–355\|doi=10.1038/nature13460\|pmid=24919152\|issn=0028-0836\|arxiv=1401.4174\|bibcode=2014Natur.510..351H\|s2cid=4463585}}</ref><ref name=":4">{{Cite journal\|last1=Abramsky\|first1=Samson\|last2=Barbosa\|first2=Rui Soares\|last3=Mansfield\|first3=Shane\|date=2017-08-04\|title=Contextual Fraction as a Measure of Contextuality\|journal=Physical Review Letters\|volume=119\|issue=5\|article-number=050504\|doi=10.1103/PhysRevLett.119.050504\|pmid=28949723\|issn=0031-9007\|arxiv=1705.07918\|bibcode=2017PhRvL.119e0504A\|s2cid=206295638}}</ref><ref>{{Cite journal\|last1=Bermejo-Vega\|first1=Juan\|last2=Delfosse\|first2=Nicolas\|last3=Browne\|first3=Dan E.\|last4=Okay\|first4=Cihan\|last5=Raussendorf\|first5=Robert\|date=2017-09-21\|title=Contextuality as a Resource for Models of Quantum Computation with Qubits\|journal=Physical Review Letters\|volume=119\|issue=12\|article-number=120505\|doi=10.1103/PhysRevLett.119.120505\|pmid=29341645\|issn=0031-9007\|bibcode=2017PhRvL.119l0505B\|arxiv=1610.08529\|s2cid=34682991}}</ref> Contemporary research has increasingly focused on exploring its utility as a computational resource.		Quantum contextuality has been identified as a source of quantum computational speedups and quantum advantage in quantum computing.<ref>{{Cite journal\|last=Raussendorf\|first=Robert\|date=2013-08-19\|title=Contextuality in measurement-based quantum computation\|journal=Physical Review A\|volume=88\|issue=2\|article-number=022322\|doi=10.1103/PhysRevA.88.022322\|issn=1050-2947\|arxiv=0907.5449\|bibcode=2013PhRvA..88b2322R\|s2cid=118495073}}</ref><ref>{{Cite journal\|last1=Howard\|first1=Mark\|last2=Wallman\|first2=Joel\|last3=Veitch\|first3=Victor\|last4=Emerson\|first4=Joseph\|date=June 2014\|title=Contextuality supplies the 'magic' for quantum computation\|journal=Nature\|volume=510\|issue=7505\|pages=351–355\|doi=10.1038/nature13460\|pmid=24919152\|issn=0028-0836\|arxiv=1401.4174\|bibcode=2014Natur.510..351H\|s2cid=4463585}}</ref><ref name=":4">{{Cite journal\|last1=Abramsky\|first1=Samson\|last2=Barbosa\|first2=Rui Soares\|last3=Mansfield\|first3=Shane\|date=2017-08-04\|title=Contextual Fraction as a Measure of Contextuality\|journal=Physical Review Letters\|volume=119\|issue=5\|article-number=050504\|doi=10.1103/PhysRevLett.119.050504\|pmid=28949723\|issn=0031-9007\|arxiv=1705.07918\|bibcode=2017PhRvL.119e0504A\|s2cid=206295638}}</ref><ref>{{Cite journal\|last1=Bermejo-Vega\|first1=Juan\|last2=Delfosse\|first2=Nicolas\|last3=Browne\|first3=Dan E.\|last4=Okay\|first4=Cihan\|last5=Raussendorf\|first5=Robert\|date=2017-09-21\|title=Contextuality as a Resource for Models of Quantum Computation with Qubits\|journal=Physical Review Letters\|volume=119\|issue=12\|article-number=120505\|doi=10.1103/PhysRevLett.119.120505\|pmid=29341645\|issn=0031-9007\|bibcode=2017PhRvL.119l0505B\|arxiv=1610.08529\|s2cid=34682991}}</ref> Contemporary research has increasingly focused on exploring its utility as a computational resource.

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	{{Author\|Harold Foppele}}		{{Author\|Harold Foppele}}

	{{Sourceattribution\|Quantum contextuality\|1}}		{{Sourceattribution\|Physics:Quantum contextuality\|1}}

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	Quantum contextuality has been identified as a source of quantum computational speedups and [[Quantum supremacy\|quantum advantage]] in [[Quantum computing\|quantum computing]].<ref>{{Cite journal\|last=Raussendorf\|first=Robert\|date=2013-08-19\|title=Contextuality in measurement-based quantum computation\|journal=Physical Review A\|volume=88\|issue=2\|article-number=022322\|doi=10.1103/PhysRevA.88.022322\|issn=1050-2947\|arxiv=0907.5449\|bibcode=2013PhRvA..88b2322R\|s2cid=118495073}}</ref><ref>{{Cite journal\|last1=Howard\|first1=Mark\|last2=Wallman\|first2=Joel\|last3=Veitch\|first3=Victor\|last4=Emerson\|first4=Joseph\|date=June 2014\|title=Contextuality supplies the 'magic' for quantum computation\|journal=Nature\|volume=510\|issue=7505\|pages=351–355\|doi=10.1038/nature13460\|pmid=24919152\|issn=0028-0836\|arxiv=1401.4174\|bibcode=2014Natur.510..351H\|s2cid=4463585}}</ref><ref name=":4">{{Cite journal\|last1=Abramsky\|first1=Samson\|last2=Barbosa\|first2=Rui Soares\|last3=Mansfield\|first3=Shane\|date=2017-08-04\|title=Contextual Fraction as a Measure of Contextuality\|journal=Physical Review Letters\|volume=119\|issue=5\|article-number=050504\|doi=10.1103/PhysRevLett.119.050504\|pmid=28949723\|issn=0031-9007\|arxiv=1705.07918\|bibcode=2017PhRvL.119e0504A\|s2cid=206295638}}</ref><ref>{{Cite journal\|last1=Bermejo-Vega\|first1=Juan\|last2=Delfosse\|first2=Nicolas\|last3=Browne\|first3=Dan E.\|last4=Okay\|first4=Cihan\|last5=Raussendorf\|first5=Robert\|date=2017-09-21\|title=Contextuality as a Resource for Models of Quantum Computation with Qubits\|journal=Physical Review Letters\|volume=119\|issue=12\|article-number=120505\|doi=10.1103/PhysRevLett.119.120505\|pmid=29341645\|issn=0031-9007\|bibcode=2017PhRvL.119l0505B\|arxiv=1610.08529\|s2cid=34682991}}</ref> Contemporary research has increasingly focused on exploring its utility as a computational resource.		Quantum contextuality has been identified as a source of quantum computational speedups and [[Quantum supremacy\|quantum advantage]] in [[Quantum computing\|quantum computing]].<ref>{{Cite journal\|last=Raussendorf\|first=Robert\|date=2013-08-19\|title=Contextuality in measurement-based quantum computation\|journal=Physical Review A\|volume=88\|issue=2\|article-number=022322\|doi=10.1103/PhysRevA.88.022322\|issn=1050-2947\|arxiv=0907.5449\|bibcode=2013PhRvA..88b2322R\|s2cid=118495073}}</ref><ref>{{Cite journal\|last1=Howard\|first1=Mark\|last2=Wallman\|first2=Joel\|last3=Veitch\|first3=Victor\|last4=Emerson\|first4=Joseph\|date=June 2014\|title=Contextuality supplies the 'magic' for quantum computation\|journal=Nature\|volume=510\|issue=7505\|pages=351–355\|doi=10.1038/nature13460\|pmid=24919152\|issn=0028-0836\|arxiv=1401.4174\|bibcode=2014Natur.510..351H\|s2cid=4463585}}</ref><ref name=":4">{{Cite journal\|last1=Abramsky\|first1=Samson\|last2=Barbosa\|first2=Rui Soares\|last3=Mansfield\|first3=Shane\|date=2017-08-04\|title=Contextual Fraction as a Measure of Contextuality\|journal=Physical Review Letters\|volume=119\|issue=5\|article-number=050504\|doi=10.1103/PhysRevLett.119.050504\|pmid=28949723\|issn=0031-9007\|arxiv=1705.07918\|bibcode=2017PhRvL.119e0504A\|s2cid=206295638}}</ref><ref>{{Cite journal\|last1=Bermejo-Vega\|first1=Juan\|last2=Delfosse\|first2=Nicolas\|last3=Browne\|first3=Dan E.\|last4=Okay\|first4=Cihan\|last5=Raussendorf\|first5=Robert\|date=2017-09-21\|title=Contextuality as a Resource for Models of Quantum Computation with Qubits\|journal=Physical Review Letters\|volume=119\|issue=12\|article-number=120505\|doi=10.1103/PhysRevLett.119.120505\|pmid=29341645\|issn=0031-9007\|bibcode=2017PhRvL.119l0505B\|arxiv=1610.08529\|s2cid=34682991}}</ref> Contemporary research has increasingly focused on exploring its utility as a computational resource.
	~~<div style="float:right; border:1px solid #e0d890; background:#fff8cc; padding:6px; margin:0 0 1em 1em; width:420px;">~~
	<div style="font-size:90%;">Quantum contextuality. In a noncontextual model, the value assigned to an observable is independent of which compatible measurements are performed alongside it. In quantum mechanics, the outcome can depend on the measurement context.</div>
	~~</div>~~
	</div>		</div>

	<div style="width:300px;">		<div style="width:300px;">
	[[File:Quantum_contextuality_yellow.png\|thumb\|280px\|Quantum contextuality.]]		[[File:Quantum_contextuality_yellow.png\|thumb\|280px\|Quantum contextuality. In a noncontextual model, the value assigned to an observable is independent of which compatible measurements are performed alongside it. In quantum mechanics, the outcome can depend on the measurement context.]]
	</div>		</div>

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	Contextuality was first demonstrated to be a feature of quantum phenomenology by the Bell–Kochen–Specker theorem.<ref name=":1" /><ref name=":2" /> The study of contextuality has developed into a major topic of interest in [[Physics:Quantum foundations\|quantum foundations]] as the phenomenon crystallises certain non-classical and counter-intuitive aspects of quantum theory. A number of powerful mathematical frameworks have been developed to study and better understand contextuality, from the perspective of sheaf theory,<ref name=":3">{{Cite journal\|last1=Abramsky\|first1=Samson\|last2=Brandenburger\|first2=Adam\|date=2011-11-28\|title=The Sheaf-Theoretic Structure Of Non-Locality and Contextuality\|journal=New Journal of Physics\|volume=13\|issue=11\|article-number=113036\|arxiv=1102.0264\|bibcode=2011NJPh...13k3036A\|doi=10.1088/1367-2630/13/11/113036\|s2cid=17435105\|issn=1367-2630}}</ref> graph theory,<ref>{{Cite journal\|last1=Cabello\|first1=Adan\|last2=Severini\|first2=Simone\|last3=Winter\|first3=Andreas\|date=2014-01-27\|title=Graph-Theoretic Approach to Quantum Correlations\|journal=Physical Review Letters\|volume=112\|issue=4\|article-number=040401\|arxiv=1401.7081\|bibcode=2014PhRvL.112d0401C\|doi=10.1103/PhysRevLett.112.040401\|issn=0031-9007\|pmid=24580419\|s2cid=34998358}}</ref> hypergraphs,<ref name="Acín 533–628">{{Cite journal\|last1=Acín\|first1=Antonio\|last2=Fritz\|first2=Tobias\|last3=Leverrier\|first3=Anthony\|last4=Sainz\|first4=Ana Belén\|date=2015-03-01\|title=A Combinatorial Approach to Nonlocality and Contextuality\|journal=Communications in Mathematical Physics\|volume=334\|issue=2\|pages=533–628\|doi=10.1007/s00220-014-2260-1\|issn=1432-0916\|arxiv=1212.4084\|bibcode=2015CMaPh.334..533A\|s2cid=119292509}}</ref> algebraic topology,<ref>{{Cite journal\|last1=Abramsky\|first1=Samson\|last2=Mansfield\|first2=Shane\|last3=Barbosa\|first3=Rui Soares\|date=2012-10-01\|title=The Cohomology of Non-Locality and Contextuality\|journal=Electronic Proceedings in Theoretical Computer Science\|volume=95\|pages=1–14\|doi=10.4204/EPTCS.95.1\|issn=2075-2180\|arxiv=1111.3620\|s2cid=9046880}}</ref> and probabilistic couplings.<ref name=":11">{{Cite journal\|last1=Dzhafarov\|first1=Ehtibar N.\|last2=Kujala\|first2=Janne V.\|date=2016-09-07\|title=Probabilistic foundations of contextuality\|journal=Fortschritte der Physik\|volume=65\|issue=6–8\|article-number=1600040\|doi=10.1002/prop.201600040\|issn=0015-8208\|bibcode=2017ForPh..6500040D\|arxiv=1604.08412\|s2cid=56245502}}</ref>		Contextuality was first demonstrated to be a feature of quantum phenomenology by the Bell–Kochen–Specker theorem.<ref name=":1" /><ref name=":2" /> The study of contextuality has developed into a major topic of interest in [[Physics:Quantum foundations\|quantum foundations]] as the phenomenon crystallises certain non-classical and counter-intuitive aspects of quantum theory. A number of powerful mathematical frameworks have been developed to study and better understand contextuality, from the perspective of sheaf theory,<ref name=":3">{{Cite journal\|last1=Abramsky\|first1=Samson\|last2=Brandenburger\|first2=Adam\|date=2011-11-28\|title=The Sheaf-Theoretic Structure Of Non-Locality and Contextuality\|journal=New Journal of Physics\|volume=13\|issue=11\|article-number=113036\|arxiv=1102.0264\|bibcode=2011NJPh...13k3036A\|doi=10.1088/1367-2630/13/11/113036\|s2cid=17435105\|issn=1367-2630}}</ref> graph theory,<ref>{{Cite journal\|last1=Cabello\|first1=Adan\|last2=Severini\|first2=Simone\|last3=Winter\|first3=Andreas\|date=2014-01-27\|title=Graph-Theoretic Approach to Quantum Correlations\|journal=Physical Review Letters\|volume=112\|issue=4\|article-number=040401\|arxiv=1401.7081\|bibcode=2014PhRvL.112d0401C\|doi=10.1103/PhysRevLett.112.040401\|issn=0031-9007\|pmid=24580419\|s2cid=34998358}}</ref> hypergraphs,<ref name="Acín 533–628">{{Cite journal\|last1=Acín\|first1=Antonio\|last2=Fritz\|first2=Tobias\|last3=Leverrier\|first3=Anthony\|last4=Sainz\|first4=Ana Belén\|date=2015-03-01\|title=A Combinatorial Approach to Nonlocality and Contextuality\|journal=Communications in Mathematical Physics\|volume=334\|issue=2\|pages=533–628\|doi=10.1007/s00220-014-2260-1\|issn=1432-0916\|arxiv=1212.4084\|bibcode=2015CMaPh.334..533A\|s2cid=119292509}}</ref> algebraic topology,<ref>{{Cite journal\|last1=Abramsky\|first1=Samson\|last2=Mansfield\|first2=Shane\|last3=Barbosa\|first3=Rui Soares\|date=2012-10-01\|title=The Cohomology of Non-Locality and Contextuality\|journal=Electronic Proceedings in Theoretical Computer Science\|volume=95\|pages=1–14\|doi=10.4204/EPTCS.95.1\|issn=2075-2180\|arxiv=1111.3620\|s2cid=9046880}}</ref> and probabilistic couplings.<ref name=":11">{{Cite journal\|last1=Dzhafarov\|first1=Ehtibar N.\|last2=Kujala\|first2=Janne V.\|date=2016-09-07\|title=Probabilistic foundations of contextuality\|journal=Fortschritte der Physik\|volume=65\|issue=6–8\|article-number=1600040\|doi=10.1002/prop.201600040\|issn=0015-8208\|bibcode=2017ForPh..6500040D\|arxiv=1604.08412\|s2cid=56245502}}</ref>

	[[Physics:Quantum nonlocality\|Nonlocality]], in the sense of Bell's theorem, may be viewed as a special case of the more general phenomenon of contextuality, in which measurement contexts contain measurements that are distributed over ~~[[Physics:Causal structure#Curves\|~~spacelike]] separated regions. This follows from [[Physics:Quantum contextuality#Fine's theorem\|Fine's theorem]].<ref name=":6">{{Cite journal\|last=Fine\|first=Arthur\|date=1982-02-01\|title=Hidden Variables, Joint Probability, and the Bell Inequalities\|journal=Physical Review Letters\|volume=48\|issue=5\|pages=291–295\|doi=10.1103/PhysRevLett.48.291\|bibcode=1982PhRvL..48..291F}}</ref><ref name=":3" />		[[Physics:Quantum nonlocality\|Nonlocality]], in the sense of Bell's theorem, may be viewed as a special case of the more general phenomenon of contextuality, in which measurement contexts contain measurements that are distributed over spacelike separated regions. This follows from [[Physics:Quantum contextuality#Fine's theorem\|Fine's theorem]].<ref name=":6">{{Cite journal\|last=Fine\|first=Arthur\|date=1982-02-01\|title=Hidden Variables, Joint Probability, and the Bell Inequalities\|journal=Physical Review Letters\|volume=48\|issue=5\|pages=291–295\|doi=10.1103/PhysRevLett.48.291\|bibcode=1982PhRvL..48..291F}}</ref><ref name=":3" />

	Quantum contextuality has been identified as a source of quantum computational speedups and quantum advantage in quantum computing.<ref>{{Cite journal\|last=Raussendorf\|first=Robert\|date=2013-08-19\|title=Contextuality in measurement-based quantum computation\|journal=Physical Review A\|volume=88\|issue=2\|article-number=022322\|doi=10.1103/PhysRevA.88.022322\|issn=1050-2947\|arxiv=0907.5449\|bibcode=2013PhRvA..88b2322R\|s2cid=118495073}}</ref><ref>{{Cite journal\|last1=Howard\|first1=Mark\|last2=Wallman\|first2=Joel\|last3=Veitch\|first3=Victor\|last4=Emerson\|first4=Joseph\|date=June 2014\|title=Contextuality supplies the 'magic' for quantum computation\|journal=Nature\|volume=510\|issue=7505\|pages=351–355\|doi=10.1038/nature13460\|pmid=24919152\|issn=0028-0836\|arxiv=1401.4174\|bibcode=2014Natur.510..351H\|s2cid=4463585}}</ref><ref name=":4">{{Cite journal\|last1=Abramsky\|first1=Samson\|last2=Barbosa\|first2=Rui Soares\|last3=Mansfield\|first3=Shane\|date=2017-08-04\|title=Contextual Fraction as a Measure of Contextuality\|journal=Physical Review Letters\|volume=119\|issue=5\|article-number=050504\|doi=10.1103/PhysRevLett.119.050504\|pmid=28949723\|issn=0031-9007\|arxiv=1705.07918\|bibcode=2017PhRvL.119e0504A\|s2cid=206295638}}</ref><ref>{{Cite journal\|last1=Bermejo-Vega\|first1=Juan\|last2=Delfosse\|first2=Nicolas\|last3=Browne\|first3=Dan E.\|last4=Okay\|first4=Cihan\|last5=Raussendorf\|first5=Robert\|date=2017-09-21\|title=Contextuality as a Resource for Models of Quantum Computation with Qubits\|journal=Physical Review Letters\|volume=119\|issue=12\|article-number=120505\|doi=10.1103/PhysRevLett.119.120505\|pmid=29341645\|issn=0031-9007\|bibcode=2017PhRvL.119l0505B\|arxiv=1610.08529\|s2cid=34682991}}</ref> Contemporary research has increasingly focused on exploring its utility as a computational resource.		Quantum contextuality has been identified as a source of quantum computational speedups and quantum advantage in quantum computing.<ref>{{Cite journal\|last=Raussendorf\|first=Robert\|date=2013-08-19\|title=Contextuality in measurement-based quantum computation\|journal=Physical Review A\|volume=88\|issue=2\|article-number=022322\|doi=10.1103/PhysRevA.88.022322\|issn=1050-2947\|arxiv=0907.5449\|bibcode=2013PhRvA..88b2322R\|s2cid=118495073}}</ref><ref>{{Cite journal\|last1=Howard\|first1=Mark\|last2=Wallman\|first2=Joel\|last3=Veitch\|first3=Victor\|last4=Emerson\|first4=Joseph\|date=June 2014\|title=Contextuality supplies the 'magic' for quantum computation\|journal=Nature\|volume=510\|issue=7505\|pages=351–355\|doi=10.1038/nature13460\|pmid=24919152\|issn=0028-0836\|arxiv=1401.4174\|bibcode=2014Natur.510..351H\|s2cid=4463585}}</ref><ref name=":4">{{Cite journal\|last1=Abramsky\|first1=Samson\|last2=Barbosa\|first2=Rui Soares\|last3=Mansfield\|first3=Shane\|date=2017-08-04\|title=Contextual Fraction as a Measure of Contextuality\|journal=Physical Review Letters\|volume=119\|issue=5\|article-number=050504\|doi=10.1103/PhysRevLett.119.050504\|pmid=28949723\|issn=0031-9007\|arxiv=1705.07918\|bibcode=2017PhRvL.119e0504A\|s2cid=206295638}}</ref><ref>{{Cite journal\|last1=Bermejo-Vega\|first1=Juan\|last2=Delfosse\|first2=Nicolas\|last3=Browne\|first3=Dan E.\|last4=Okay\|first4=Cihan\|last5=Raussendorf\|first5=Robert\|date=2017-09-21\|title=Contextuality as a Resource for Models of Quantum Computation with Qubits\|journal=Physical Review Letters\|volume=119\|issue=12\|article-number=120505\|doi=10.1103/PhysRevLett.119.120505\|pmid=29341645\|issn=0031-9007\|bibcode=2017PhRvL.119l0505B\|arxiv=1610.08529\|s2cid=34682991}}</ref> Contemporary research has increasingly focused on exploring its utility as a computational resource.