View source for Quantum entanglement

{{Short description|Physics phenomenon}}
{{Use American English|date=December 2024}}
{{Use dmy dates|date=December 2024}}
[[File:SPDC figure.png|thumb|[[Spontaneous parametric down-conversion]] process can split photons into type II photon pairs with mutually perpendicular polarization.]]
{{Quantum mechanics|fundamentals}}

'''Quantum entanglement''' is the phenomenon where the [[quantum state]] of each [[Subatomic particle|particle]] in a group cannot be described independently of the state of the others, even when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between [[classical physics]] and [[quantum physics]]: entanglement is a primary feature of quantum mechanics not present in classical mechanics.<ref name="horodecki2007"/>{{rp|867|q=In this way entanglement is that feature of quantum formalism which makes it impossible to simulate quantum correlations within any classical formalism.}}

[[Measurement#Quantum mechanics|Measurements]] of [[physical properties]] such as [[position (vector)|position]], [[momentum]], [[Spin (physics)|spin]], and [[polarization (waves)|polarization]] performed on entangled particles can, in some cases, be found to be perfectly [[correlated]]. For example, if a pair of entangled particles is generated such that their total spin is known to be zero, and one particle is found to have clockwise spin on a first axis, then the spin of the other particle, measured on the same axis, is found to be anticlockwise. However, this behavior gives rise to seemingly [[paradox]]ical effects: any measurement of a particle's properties results in an apparent and irreversible [[wave function collapse]] of that particle and changes the original quantum state. With entangled particles, such measurements affect the entangled system as a whole.

Such phenomena were the subject of a 1935 paper by [[Albert Einstein]], [[Boris Podolsky]], and [[Nathan Rosen]],<ref name="Einstein1935">
{{cite journal
 | last1      = Einstein | first1     = Albert | author-link1 = Albert Einstein
 | last2      = Podolsky | first2     = Boris  | author-link2 = Boris Podolsky
 | last3      = Rosen    | first3     = Nathan | author-link3 = Nathan Rosen
 | year=1935
 | title=Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?
 | journal=Phys. Rev.
 | volume=47
 | issue=10
 | pages=777–780
 | bibcode=1935PhRv...47..777E
 | doi=10.1103/PhysRev.47.777
 | doi-access=free
}}</ref> and several papers by [[Erwin Schrödinger]] shortly thereafter,<ref name="Schrödinger1935">
{{cite journal |author=Schrödinger |first=Erwin |author-link=Erwin Schrödinger |year=1935 |title=Discussion of probability relations between separated systems |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] |volume=31 |issue=4 |pages=555–563 |bibcode=1935PCPS...31..555S |doi=10.1017/S0305004100013554 |s2cid=121278681}}</ref><ref name="Schrödinger1936">
{{cite journal |author=Schrödinger |first=Erwin |author-link=Erwin Schrödinger |year=1936 |title=Probability relations between separated systems |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] |volume=32 |issue=3 |pages=446–452 |bibcode=1936PCPS...32..446S |doi=10.1017/S0305004100019137 |s2cid=122822435}}
</ref> describing what came to be known as the [[EPR paradox]]. Einstein and others considered such behavior impossible, as it violated the [[local realism]] view of [[causality]]<ref>Physicist John Bell depicts the Einstein camp in this debate in his article entitled "Bertlmann's socks and the nature of reality", p. 143 of ''Speakable and unspeakable in quantum mechanics'': "For EPR that would be an unthinkable 'spooky action at a distance'. To avoid such action at a distance they have to attribute, to the space-time regions in question, real properties in advance of observation, correlated properties, which predetermine the outcomes of these particular observations. Since these real properties, fixed in advance of observation, are not contained in quantum formalism, that formalism for EPR is incomplete. It may be correct, as far as it goes, but the usual quantum formalism cannot be the whole story." And again on p. 144 Bell says: "Einstein had no difficulty accepting that affairs in different places could be correlated. What he could not accept was that an intervention at one place could influence, immediately, affairs at the other." Downloaded 5 July 2011 from {{cite book |year=1987 |title=Speakable and Unspeakable in Quantum Mechanics |first=J. S. |last=Bell |publisher=[[CERN]] |isbn=0-521-33495-0 |url=http://philosophyfaculty.ucsd.edu/faculty/wuthrich/GSSPP09/Files/BellJohnS1981Speakable_BertlmannsSocks.pdf |archive-url=https://web.archive.org/web/20150412044550/http://philosophyfaculty.ucsd.edu/faculty/wuthrich/GSSPP09/Files/BellJohnS1981Speakable_BertlmannsSocks.pdf |archive-date=12 April 2015  }}</ref> and argued that the accepted formulation of [[quantum mechanics]] must therefore be incomplete.

Later, however, the counterintuitive predictions of quantum mechanics were verified in tests where polarization or spin of entangled particles were measured at separate locations, statistically violating [[Bell's theorem|Bell's inequality]].<ref name="Clauser"/><ref name=":0" /><ref name=":1" /><ref name=":2" /> This established that the correlations produced from quantum entanglement cannot be explained in terms of [[local hidden variable theory|local hidden variables]], i.e., properties contained within the individual particles themselves.
However, despite the fact that entanglement can produce statistical [[correlation]]s between events in widely separated places, it cannot be used for [[faster-than-light communication]].<ref>{{cite book |last=Penrose |first=Roger |title=The road to reality: a complete guide to the laws of the universe |publisher=Jonathan Cape |year=2004 |isbn=978-0-224-04447-9 |location=London |page=603 |author-link=Roger Penrose}}</ref><ref>{{cite web |last=Siegel |first=Ethan |title=No, We Still Can't Use Quantum Entanglement To Communicate Faster Than Light |url=https://www.forbes.com/sites/startswithabang/2020/01/02/no-we-still-cant-use-quantum-entanglement-to-communicate-faster-than-light/ |access-date=6 January 2023 |website=Starts with a Bang |publisher=Forbes |language=en}}</ref><ref name="Griffiths"/>{{rp|453}}
 
Quantum entanglement has been demonstrated experimentally with [[photon]]s,<ref name="Kocher1">{{cite journal |last1=Kocher |first1=C. A. |last2=Commins |first2=E. D. |year=1967 |title=Polarization Correlation of Photons Emitted in an Atomic Cascade |url=http://www.escholarship.org/uc/item/1kb7660q |journal=Physical Review Letters |volume=18 |issue=15 |pages=575–577 |bibcode=1967PhRvL..18..575K |doi=10.1103/PhysRevLett.18.575}}</ref><ref name="Kocherphd">{{cite thesis |last=Kocher |first=Carl Alvin |date=1 May 1967 |title=Polarization Correlation of Photons Emitted in an Atomic Cascade |url=https://escholarship.org/uc/item/1kb7660q |degree=PhD|publisher=University of California |language=en}}</ref> [[electron]]s,<ref name="NTR-20151021">{{cite journal |author=Hensen, B. |title=Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres |date=21 October 2015 |journal=[[Nature (journal)|Nature]] |doi=10.1038/nature15759 |display-authors=etal |volume=526 |issue=7575 |pages=682–686 |bibcode=2015Natur.526..682H |pmid=26503041 |arxiv=1508.05949 |hdl=2117/79298 |s2cid=205246446}} See also [http://www.nature.com/articles/nature15759.epdf?referrer_access_token=1QB20mTNTZW60nEXil0D79RgN0jAjWel9jnR3ZoTv0Pfu6MWINxm4Io03p2jIRZ8qX_3I3N0Kr-AlItuikCZOJrG8QbdRRghlecFwmixlbQpWuw1dtaib4Le5DQOG3u_aXHU85x1JEhOcQTa1sHi0yvW23bblxmEQZAmHL4G0gIVusG_6JWorroY5BprgbTl4FiaE8WltEgMoUMZfZBkEfbMcFDp5iR112TFx_x3ZRj88Wa23E2moEvTfKjtlued0&tracking_referrer=www.nytimes.com free online access version].</ref><ref name="NYT-20151021">{{cite news |last=Markoff |first=Jack |title=Sorry, Einstein. Quantum Study Suggests 'Spooky Action' Is Real. |url=https://www.nytimes.com/2015/10/22/science/quantum-theory-experiment-said-to-prove-spooky-interactions.html |date=21 October 2015 |work=The New York Times |access-date=21 October 2015 }}</ref> [[top quark]]s,<ref>{{cite web | url=https://physicsworld.com/a/quantum-entanglement-observed-in-top-quarks/ | title=Quantum entanglement observed in top quarks | date=11 October 2023 |website=[[Physics World]] |first=Martijn |last=Boerkamp}}</ref> molecules<ref>{{cite journal |last1=Holland |first1=Connor M. |last2=Lu |first2=Yukai |last3=Cheuk |first3=Lawrence W. |date=8 December 2023 |title=On-demand entanglement of molecules in a reconfigurable optical tweezer array |url=https://www.science.org/doi/10.1126/science.adf4272 |journal=Science |language=en |volume=382 |issue=6675 |pages=1143–1147 |doi=10.1126/science.adf4272 |pmid=38060644 |issn=0036-8075|arxiv=2210.06309 |bibcode=2023Sci...382.1143H }}</ref> and even small diamonds.<ref>{{cite journal |journal=Science |date=2 December 2011 |volume=334 |issue=6060 |pages=1253–1256 |doi=10.1126/science.1211914 |pmid=22144620 |title=Entangling macroscopic diamonds at room temperature |bibcode = 2011Sci...334.1253L |last1=Lee |first1=K. C. |last2=Sprague |first2=M. R. |last3=Sussman |first3=B. J. |last4=Nunn |first4=J. |last5=Langford |first5=N. K. |last6=Jin |first6=X.-M. |last7=Champion |first7=T. |last8=Michelberger |first8=P. |last9=Reim |first9=K. F. |last10=England |first10=D. |last11=Jaksch |first11=D. |last12=Walmsley |first12=I. A. |s2cid=206536690 |display-authors=4 |url=http://scholarbank.nus.edu.sg/handle/10635/112433 }}</ref> The use of quantum entanglement in [[quantum communication|communication]] and [[quantum computing|computation]] is an active area of research and development.

== History ==
{{hatnote| Background: [[History of quantum mechanics]]}}
[[File:NYT May 4, 1935.jpg|right|thumb|268x268px|Article headline regarding the [[EPR paradox|Einstein–Podolsky–Rosen (EPR) paradox]] paper, in the 4 May 1935 issue of ''[[The New York Times]]'']]
Albert Einstein and Niels Bohr engaged in a long-running collegial dispute over the interpretation of quantum mechanics, now known as the [[Bohr–Einstein debates]]. During these debates, Einstein introduced a [[thought experiment]] involving a box that emits a photon. He noted that the experimenter's choice of which measurement to make on the box would change what can be predicted about the photon, even when the photon is very far away. This argument, which Einstein had formulated by 1931, was an early recognition of what would later be called entanglement.<ref>{{cite book|first=Don |last=Howard |chapter=''Nicht Sein Kann Was Nicht Sein Darf'', or The Prehistory of EPR, 1909–1935: Einstein's Early Worries About The Quantum Mechanics of Composite Systems |title=Sixty-Two Years of Uncertainty |editor-first=A. I. |editor-last=Miller |publisher=Plenum Press |location=New York |year=1990 |chapter-url=http://www.ub.edu/hcub/hfq/sites/default/files/Howard1990-1.pdf |pages=61–111}}</ref> That same year, [[Hermann Weyl]] observed in his textbook on [[group theory]] and quantum mechanics that quantum systems composed of multiple interacting parts exhibit a kind of ''[[Gestalt psychology|Gestalt]],'' in which "the whole is greater than the sum of its parts".<ref>{{cite book|first=Hermann |last=Weyl |author-link=Hermann Weyl |title=Gruppentheorie und Quantenmechanik |title-link=Gruppentheorie und Quantenmechanik |trans-title=Group Theory and Quantum Mechanics |translator-first=H. P. |translator-last=Robertson |translator-link=Howard P. Robertson |year=1931 |edition=2nd |pages=92–93}}</ref><ref>{{cite journal|first=Adrian |last=Heathcote |title=Multiplicity and indiscernability |doi=10.1007/s11229-020-02600-8 |journal=Synthese |volume=198 |pages=8779–8808 |year=2021 |issue=9 |quote=For Weyl clearly anticipated entanglement by noting that the pure state of a coupled system need not be determined by the states of the composites [...] Weyl deserves far more credit than he has received for laying out the basis for entanglement — more than six years before Schrödinger coined the term.}}</ref> In 1932, [[Erwin Schrödinger]] derived the defining equations of quantum entanglement but left them unpublished.<ref>{{cite thesis |last=Christandl |first=Matthias |date=2006 |degree=PhD |publisher=University of Cambridge |title=The Structure of Bipartite Quantum States – Insights from Group Theory and Cryptography |journal= |pages=vi, iv |arxiv=quant-ph/0604183 |bibcode=2006PhDT.......289C }}</ref> In 1935, [[Grete Hermann]] studied the mathematics of an electron interacting with a photon and noted the phenomenon that would later be called entanglement.<ref>{{cite book |first=Thomas |last=Filk |chapter=Carl Friedrich von Weizsäcker's 'Ortsbestimmung eines Elektrons' and its Influence on Grete Hermann |doi=10.1007/978-94-024-0970-3_5 |title=Grete Hermann – Between Physics and Philosophy |publisher=Springer |series=Studies in History and Philosophy of Science |volume=42 |editor-first1=Elise |editor-last1=Crull |editor-first2=Guido |editor-last2=Bacciagaluppi |year=2016 |isbn=978-94-024-0968-0 |page=76}}</ref> Later that same year, Einstein, [[Boris Podolsky]] and [[Nathan Rosen]] published a paper on what is now known as the [[Einstein–Podolsky–Rosen paradox|Einstein–Podolsky–Rosen (EPR) paradox]], a thought experiment that attempted to show that "the [[quantum-mechanical]] description of physical reality given by wave functions is not complete".<ref name="Einstein1935" /> Their thought experiment considered two systems that interact and then separate, and they argued that, afterward, quantum mechanics could not describe the two systems individually.

Shortly after this paper appeared, [[Erwin Schrödinger]] wrote a letter to Einstein in [[German language|German]] in which he used the word ''Verschränkung'' (translated by himself as ''entanglement'') to describe situations like that of the EPR scenario.<ref name=MK>{{cite book |last=Kumar |first=Manjit |title=Quantum: Einstein, Bohr, and the Great Debate about the Nature of Reality |publisher=W. W. Norton & Company |year=2010 |page=313 |isbn=978-0-393-07829-9}}</ref> Schrödinger followed up with a full paper defining and discussing the notion of ''entanglement'',<ref name="Schroeder-2017">{{cite journal |last=Schroeder |first=Daniel V. |date=1 November 2017 |title=Entanglement isn't just for spin |url=https://pubs.aip.org/ajp/article/85/11/812/1057936/Entanglement-isn-t-just-for-spin |journal=American Journal of Physics |volume=85 |issue=11 |pages=812–820 |arxiv=1703.10620 |doi=10.1119/1.5003808 |bibcode=2017AmJPh..85..812S |issn=0002-9505}}</ref> saying "I would not call [entanglement] ''one'' but rather ''the'' characteristic trait of quantum mechanics, the one that enforces its entire departure from [[Classical mechanics|classical]] lines of thought."<ref name="Schrödinger1935"/> 
Like Einstein, Schrödinger was dissatisfied with the concept of entanglement, because it seemed to violate the speed limit on the transmission of information implicit in the [[theory of relativity]].<ref>{{cite book|editor-first1=Alisa |editor-last1=Bokulich |editor-first2=Gregg |editor-last2=Jaeger |title=Philosophy of Quantum Information and Entanglement |publisher=Cambridge University Press |year=2010 |isbn=978-0-511-67655-0 |chapter=Introduction |page=xv}}</ref> Einstein later disparaged quantum mechanics for seemingly exhibiting "''spukhafte Fernwirkung''"<ref name="spukhafte">Letter from Einstein to Max Born, 3 March 1947; ''The Born-Einstein Letters; Correspondence between Albert Einstein and Max and Hedwig Born from 1916 to 1955'', Walker, New York, 1971. Cited in {{cite journal |author=Hobson |first=M. P. |display-authors=etal |year=1998 |title=Quantum Entanglement and Communication Complexity |journal=SIAM J. Comput. |volume=30 |issue=6 |pages=1829–1841 |citeseerx=10.1.1.20.8324}})</ref> or "[[spooky action at a distance]]", meaning the acquisition of a value of a property at one location resulting from a measurement at a distant location.<ref name="MerminMoon-1985">{{Cite journal |last=Mermin |first=N. David |author-link=N. David Mermin |date=1985 |title=Is the Moon There When Nobody Looks? Reality and the Quantum Theory |url=https://archive.org/details/mermin_moon |journal=Physics Today |volume=38 |number=4 |pages=38–47 |doi=10.1063/1.880968|bibcode=1985PhT....38d..38M }}</ref>

In 1946, [[John Archibald Wheeler]] suggested studying the [[polarization (physics)|polarization]] of pairs of [[gamma-ray]] photons produced by electron–[[positron]] annihilation.<ref>{{cite journal|first=J. A. |last=Wheeler |author-link=John Archibald Wheeler |title=Polyelectrons |journal=Annals of the New York Academy of Sciences |volume=48 |number=3 |pages=219–238 |year=1946 |doi=10.1111/j.1749-6632.1946.tb31764.x}}</ref> [[Chien-Shiung Wu]] and I. Shaknov carried out this experiment in 1949,<ref name=":3">
{{cite journal
 |last1=Wu |first1=C. S.
 |last2=Shaknov |first2=I.
 |year=1950
 |title=The Angular Correlation of Scattered Annihilation Radiation
 |journal=[[Physical Review]]
 |volume=77 |issue= 1|page=136
 |bibcode=1950PhRv...77..136W
 |doi=10.1103/PhysRev.77.136
}}</ref> thereby demonstrating that the entangled particle pairs considered by EPR could be created in the laboratory.<ref>
{{cite journal
 |last1=Duarte |first1=F. J. |author1-link=F. J. Duarte
 |year=2012
 |title=The origin of quantum entanglement experiments based on polarization measurements
 |journal=[[European Physical Journal H]]
 |volume=37 |issue=2 |pages=311–318
 |bibcode=2012EPJH...37..311D
 |doi=10.1140/epjh/e2012-20047-y
}}</ref>

Despite Schrödinger's claim of its importance, little work on entanglement was published for decades after his paper was published.<ref name="Schroeder-2017"/> In 1964 [[John S. Bell]] demonstrated an upper limit, seen in [[Bell's inequality]], regarding the strength of correlations that can be produced in any theory obeying [[local realism]], and showed that quantum theory predicts violations of this limit for certain entangled systems.<ref name=":4">{{cite journal |author=Bell |first=J. S. |author-link=John Stewart Bell |year=1964 |title=On the Einstein Poldolsky Rosen paradox |journal=[[Physics Physique Физика]] |volume=1 |issue=3 |pages=195–200 |doi=10.1103/PhysicsPhysiqueFizika.1.195 |doi-access=free}}</ref><ref>{{cite journal |last=Mermin |first=N. David |date=1981 |title=Quantum Mysteries for Anyone |journal=The Journal of Philosophy |volume=78 |issue=7 |pages=397–408 |doi=10.2307/2026482 |jstor=2026482 |issn=0022-362X}}</ref>{{rp|405}} His inequality is experimentally testable, and there have been numerous [[Bell test experiments|relevant experiments]], starting with the pioneering work of [[Stuart Freedman]] and [[John Clauser]] in 1972<ref name="Clauser">{{cite journal|doi=10.1103/PhysRevLett.28.938|last1=Freedman|first1=Stuart J.|last2=Clauser|first2=John F.|title=Experimental Test of Local Hidden-Variable Theories|journal=Physical Review Letters |volume=28 |issue=14 |pages=938–941|year=1972 |bibcode=1972PhRvL..28..938F|url=https://escholarship.org/uc/item/2f18n5nk|doi-access=free}}</ref> and [[Alain Aspect]]'s experiments in 1982.<ref name="Aspect1982">
{{cite journal
 | last1 = Aspect | first1 = Alain
 | last2 = Grangier | first2 = Philippe
 | last3 = Roger | first3 = Gérard
 | title = Experimental Realization of Einstein–Podolsky–Rosen–Bohm Gedankenexperiment: A New Violation of Bell's Inequalities
 | journal = Physical Review Letters
 | volume = 49
 | issue = 2
 | pages = 91–94
 | year = 1982
 | doi = 10.1103/PhysRevLett.49.91 | doi-access = free
 | bibcode=1982PhRvL..49...91A
}}</ref><ref name="hanson">{{cite journal|last1=Hanson|first1=Ronald|title=Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres|journal=Nature|volume=526|issue=7575|pages=682–686|doi=10.1038/nature15759|arxiv=1508.05949|bibcode = 2015Natur.526..682H|pmid=26503041|year=2015|s2cid=205246446}}</ref><ref>{{cite journal |last=Aspect |first=Alain |date=16 December 2015 |title=Closing the Door on Einstein and Bohr's Quantum Debate |journal=[[Physics (magazine)|Physics]] |volume=8 |article-number=123 |bibcode=2015PhyOJ...8..123A |doi=10.1103/Physics.8.123 |doi-access=free}}</ref>

While Bell actively discouraged students from pursuing work like his as too esoteric, after a talk at Oxford a student named [[Artur Ekert]] suggested that the violation of a Bell inequality could be used as a resource for communication.<ref name=Gilder2009/>{{rp|315}} Ekert followed up by  publishing a [[quantum key distribution]]  protocol called [[E91 protocol|E91]] based on it.<ref name=ekert91/><ref name="horodecki2007"/>{{rp|874|q=The first discovery within quantum information theory, which involves entanglement, is due to Ekert 1991.}}

In 1992, the entanglement concept was leveraged to propose [[quantum teleportation]],<ref name="BBCJPW93">{{cite journal |last1=Bennett |first1=Charles H. |author-link1=Charles H. Bennett (computer scientist) |last2=Brassard |first2=Gilles |author-link2=Gilles Brassard |last3=Crépeau |first3=Claude |author-link3=Claude Crépeau |last4=Jozsa |first4=Richard |author-link4=Richard Jozsa |last5=Peres |first5=Asher |author-link5=Asher Peres |last6=Wootters |first6=William K. |author-link6=William Wootters |date=29 March 1993 |title=Teleporting an Unknown Quantum State via Dual Classical and Einstein–Podolsky–Rosen Channels |journal=[[Physical Review Letters]] |volume=70 |issue=13 |pages=1895–1899 |doi=10.1103/PhysRevLett.70.1895 |pmid=10053414 |bibcode=1993PhRvL..70.1895B |doi-access=free |citeseerx=10.1.1.46.9405}}</ref> an effect that was realized experimentally in 1997.<ref>{{cite journal |last=Lindley |first=David |date=8 January 2010 |title=Landmarks: Teleportation is not Science Fiction |url=https://physics.aps.org/story/v25/st1 |volume=25 |journal=[[Physical Review Focus|Physics (Physical Review Focus)]]}}</ref><ref name="Bouwmeester-1997">{{cite journal |last1=Bouwmeester |first1=Dik |author-link1=Dirk Bouwmeester |last2=Pan |first2=Jian-Wei |last3=Mattle |first3=Klaus |last4=Eibl |first4=Manfred |last5=Weinfurter |first5=Harald |last6=Zeilinger |first6=Anton |date=1 December 1997 |title=Experimental quantum teleportation |journal=Nature |volume=390 |issue=6660 |pages=575–579 |doi=10.1038/37539 |arxiv=1901.11004 |bibcode=1997Natur.390..575B |s2cid=4422887}}</ref><ref name="Rome1998">
{{cite journal
 |last1=Boschi |first1=D. |last2=Branca |first2=S. |last3=De Martini |first3=F. |last4=Hardy |first4=L. |last5=Popescu |first5=S.
 |journal=[[Physical Review Letters]]
 |volume=80
 |issue=6
 |pages=1121–1125
 |doi= 10.1103/PhysRevLett.80.1121
 |title=Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein–Podolsky–Rosen Channels
 |date=9 February 1998
 |arxiv = quant-ph/9710013 |bibcode = 1998PhRvL..80.1121B |s2cid=15020942
}}</ref>

Beginning in the mid-1990s, [[Anton Zeilinger]] used the generation of entanglement via [[Spontaneous parametric down-conversion|parametric down-conversion]] to develop [[entanglement swapping]]<ref name=Gilder2009>{{cite book |last=Gilder |first=Louisa |title=The age of entanglement: when quantum physics was reborn |date=2009 |publisher=Vintage Books |isbn=978-1-4000-9526-1 |edition=1. Vintage Book |location=New York, NY}}</ref>{{rp|317}} and demonstrate [[quantum cryptography]] with entangled photons.<ref>{{cite journal|first1=T. |last1=Jennewein |first2=C. |last2=Simon |first3=G. |last3=Weihs |first4=H. |last4=Weinfurter |first5=A. |last5=Zeilinger |author-link5=Anton Zeilinger |title=Quantum Cryptography with Entangled Photons |journal=Physical Review Letters |volume=84 |pages=4729–4732 |year=2000 |issue=20 |doi=10.1103/PhysRevLett.84.4729|pmid=10990782 |arxiv=quant-ph/9912117 |bibcode=2000PhRvL..84.4729J }}</ref><ref>{{cite journal|last1=Del Santo |first1=F |last2=Schwarzhans |first2=E. |year=2022 |title="Philosophysics" at the University of Vienna: The (Pre-) History of Foundations of Quantum Physics in the Viennese Cultural Context |journal=Physics in Perspective |volume=24 |number=2–3 |pages=125–153 |doi=10.1007/s00016-022-00290-y |pmid=36437910 |pmc=9678993 |arxiv=2011.11969|bibcode=2022PhP....24..125D }}</ref>

In 2022, the [[Nobel Prize in Physics]] was awarded to Aspect, Clauser, and Zeilinger "for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science".<ref name="NobelPrize">{{cite press release |url=https://www.nobelprize.org/prizes/physics/2022/press-release/ |title=The Nobel Prize in Physics 2022 |date=4 October 2022 |publisher=[[The Royal Swedish Academy of Sciences]] |access-date=5 October 2022}}</ref>

== Concept ==

=== Meaning of entanglement ===
Just as [[energy]] is a resource that facilitates mechanical operations, entanglement is a resource that facilitates performing tasks that involve communication and computation.<ref name="Nielsen-2010"/>{{rp|106}}<ref name="Rieffel2011"/>{{rp|218}}<ref name="Bengtsson2017">{{cite book|first1=Ingemar |last1=Bengtsson |first2=Karol |last2=Życzkowski |author-link2=Karol Życzkowski |title=Geometry of Quantum States: An Introduction to Quantum Entanglement |title-link=Geometry of Quantum States |year=2017 |publisher=Cambridge University Press |edition=2nd |isbn=978-1-107-02625-4}}</ref>{{rp|435}}<ref name="Bub2023">{{cite SEP|url-id=qt-entangle |author-first=Jeffrey |author-last=Bub |author-link=Jeffrey Bub |title=Quantum Entanglement and Information |date=2023-05-02}}</ref> The mathematical definition of entanglement can be paraphrased as saying that maximal knowledge about the whole of a system does not imply maximal knowledge about the individual parts of that system.<ref name="Rau2021">{{cite book|first=Jochen |last=Rau |title=Quantum Theory: An Information Processing Approach |publisher=Oxford University Press |year=2021 |isbn=978-0-19-289630-8}}</ref> If the quantum state that describes a pair of particles is entangled, then the results of measurements upon one half of the pair can be strongly correlated with the results of measurements upon the other. However, entanglement is not the same as "correlation" as understood in classical probability theory and in daily life. Instead, entanglement can be thought of as ''potential'' correlation that can be used to generate actual correlation in an appropriate experiment.<ref name="Fuchs2011">{{cite book |first=Christopher A. |last=Fuchs |title=Coming of Age with Quantum Information |date=6 January 2011 |publisher=Cambridge University Press |isbn=978-0-521-19926-1 }}</ref>{{rp|130}} The correlations generated from an entangled quantum state cannot in general be replicated by classical probability.<ref name="Holevo2001">{{cite book|first=Alexander S. |last=Holevo |author-link=Alexander Holevo |title=Statistical Structure of Quantum Theory |publisher=Springer |series=[[Lecture Notes in Physics|Lecture Notes in Physics. Monographs]] |year=2001 |isbn=3-540-42082-7}}</ref>{{rp|33}}

An example of entanglement is a [[subatomic particle]] that [[Particle decay|decays]] into an entangled pair of other particles. The decay events obey the various [[conservation laws]], and as a result, the measurement outcomes of one daughter particle must be highly correlated with the measurement outcomes of the other daughter particle (so that the total momenta, angular momenta, energy, and so forth remains roughly the same before and after this process). For instance, a [[Spin (physics)|spin]]-zero particle could decay into a pair of spin-1/2 particles. If there is no orbital angular momentum, the total spin angular momentum after this decay must be zero (by the [[conservation of angular momentum]]). Whenever the first particle is measured to be [[Spin (physics)#Direction|spin up]] on some axis, the other, when measured on the same axis, is always found to be [[Spin (physics)#Direction|spin down]]. This is called the spin anti-correlated case and the pair is said to be in the [[singlet state]]. Perfect anti-correlations like this could be explained by "hidden variables" within the particles. For example, we could hypothesize that the particles are made in pairs such that one carries a value of "up" while the other carries a value of "down". Then, knowing the result of the spin measurement upon one particle, we could predict that the other will have the opposite value. Bell illustrated this with a story about a colleague, Bertlmann, who always wore socks with mismatching colors. "Which colour he will have on a given foot on a given day is quite unpredictable," Bell wrote, but upon observing "that the first sock is pink you can be already sure that the second sock will not be pink."<ref>{{cite journal|first=J. |last=Bell |title=Bertlmann's Socks and the Nature of Reality |journal=Journal de Physique Colloques |year=1981 |volume=42 (C2) |pages=41–62 |doi=10.1051/jphyscol:1981202 |url=https://hal.science/jpa-00220688v1}}</ref> Revealing the remarkable features of quantum entanglement requires considering multiple distinct experiments, such as spin measurements along different axes, and comparing the correlations obtained in these different configurations.<ref name="Zwiebach2022">{{cite book|first=Barton |last=Zwiebach |title=Mastering Quantum Mechanics: Essentials, Theory, and Applications |author-link=Barton Zwiebach |publisher=MIT Press |year=2022 |isbn=978-0-262-04613-8}}</ref>{{rp|§18.8}}

Quantum [[physical system|systems]] can become entangled through various types of interactions. For some ways in which entanglement may be achieved for experimental purposes, see the section below on [[#Methods of creating entanglement|methods]]. Entanglement is broken when the entangled particles [[quantum decoherence|decohere]] through interaction with the environment; for example, when a measurement is made. In more detail, this process involves the particles becoming entangled with the environment, as a consequence of which, the quantum state describing the particles themselves is no longer entangled.<ref name="Peres1993">{{cite book|first=Asher |last=Peres |author-link=Asher Peres |title=Quantum Theory: Concepts and Methods |title-link=Quantum Theory: Concepts and Methods |publisher=Kluwer |year=1993 |isbn=0-7923-2549-4 }}</ref>{{rp|369}}<ref>{{cite journal|doi=10.1016/j.physrep.2019.10.001 |first=Max |last=Schlosshauer |title=Quantum decoherence |journal=Physics Reports |volume=831 |date=25 October 2019 |pages=1–57 |arxiv=1911.06282|bibcode=2019PhR...831....1S }}</ref>

Mathematically, an entangled system can be defined to be one whose quantum state cannot be factored as a product of states of its local constituents; that is to say, they are not individual particles but are an inseparable whole. When entanglement is present, one constituent cannot be fully described without considering the other(s).<ref name="Mermin2007">{{cite book|first=N. David |last=Mermin |author-link=N. David Mermin |title=Quantum Computer Science: An Introduction |publisher=Cambridge University Press |year=2007 |isbn=978-0-521-87658-2}}</ref>{{rp|18–19}}<ref name="Zwiebach2022"/>{{rp|§1.5}} The state of a composite system is always expressible as a sum, or [[quantum superposition|superposition]], of products of states of local constituents; it is entangled if this sum cannot be written as a single product term.<ref name="Rieffel2011">{{Cite book |last1=Rieffel |first1=Eleanor |author-link1=Eleanor Rieffel |title=Quantum Computing: A Gentle Introduction |title-link=Quantum Computing: A Gentle Introduction |last2=Polak |first2=Wolfgang |date=2011 |publisher=MIT Press |isbn=978-0-262-01506-6 |series=Scientific and engineering computation |location=Cambridge, Mass}}</ref>{{Rp|page=39}}

=== Paradox ===
{{main|EPR paradox}}
The singlet state described above is the basis for one version of the EPR paradox. In this variant, introduced by [[David Bohm]], a source emits particles and sends them in opposite directions. The state describing each pair is entangled.<ref>{{cite book|first=David |last=Bohm |author-link=David Bohm |title=Quantum Theory |orig-date=1951 |year=1989 |publisher=Dover |edition=reprint |isbn=0-486-65969-0 |pages=611–622}}</ref> In the standard textbook presentation of quantum mechanics, performing a spin measurement on one of the particles causes the wave function for the whole pair to [[wave function collapse|collapse]] into a state in which each particle has a definite spin (either up or down) along the axis of measurement. The outcome is random, with each possibility having a probability of 50%. However, if both spins are measured along the same axis, they are found to be anti-correlated. This means that the random outcome of the measurement made on one particle seems to have been transmitted to the other, so that it can make the "right choice" when it too is measured.<ref name="Zwiebach2022"/>{{rp|§18.8}}<ref name="Griffiths">{{cite book|first1=David J. |last1=Griffiths |author-link1=David J. Griffiths |first2=Darrell F. |last2=Schroeter |title=Introduction to Quantum Mechanics |title-link=Introduction to Quantum Mechanics (book) |edition=3rd |year=2018 |publisher=Cambridge University Press |isbn=978-1-107-18963-8 }}</ref>{{rp|447–448}}

The distance and timing of the measurements can be chosen so as to make the interval between the two measurements [[spacelike]], hence, any causal effect connecting the events would have to travel faster than light. According to the principles of [[special relativity]], it is not possible for any information to travel between two such measuring events. It is not even possible to say which of the measurements came first. For two spacelike separated events {{math|''x''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}} there are [[inertial frame]]s in which {{math|''x''<sub>1</sub>}} is first and others in which {{math|''x''<sub>2</sub>}} is first. Therefore, the correlation between the two measurements cannot be explained as one measurement determining the other: different observers would disagree about the role of cause and effect.<ref>{{cite journal|first=Asher |last=Peres |author-link=Asher Peres |doi=10.1103/PhysRevA.61.022117 |title=Classical interventions in quantum systems. II. Relativistic invariance |journal=Physical Review A |volume=61 |article-number=022117 |date=2000-01-18|issue=2 |arxiv=quant-ph/9906034 |bibcode=2000PhRvA..61b2117P }}</ref>

=== Failure of local hidden-variable theories ===
A possible resolution to the paradox is to assume that quantum theory is incomplete, and the result of measurements depends on predetermined "[[hidden-variables theory|hidden variables]]".<ref name="Gibney2017">
{{cite journal
 | last = Gibney
 | first = Elizabeth
 | title = Cosmic Test Bolsters Einstein's "Spooky Action at a Distance"
 | journal = Scientific American
 | url = https://www.scientificamerican.com/article/cosmic-test-bolsters-einsteins-ldquo-spooky-action-at-a-distance-rdquo/
 | year = 2017
}}</ref> The state of the particles being measured contains some hidden variables, whose values effectively determine, right from the moment of separation, what the outcomes of the spin measurements are going to be. This would mean that each particle carries all the required information with it, and nothing needs to be transmitted from one particle to the other at the time of measurement. Einstein and others (see the previous section) originally believed this was the only way out of the paradox, and the accepted quantum mechanical description (with a random measurement outcome) must be incomplete.

[[Local hidden-variable theory|Local hidden variable theories]] fail, however, when measurements of the spin of entangled particles along different axes are considered. If a large number of pairs of such measurements are made (on a large number of pairs of entangled particles), then statistically, if the local realist or hidden variables view were correct, the results would always satisfy [[Bell's inequality]]. A [[Bell test|number of experiments]] have shown in practice that Bell's inequality is not satisfied.<ref name = "Clauser"/><ref>{{cite journal|last1=Dehlinger |first1=Dietrich |first2=M. W. |last2=Mitchell |title=Entangled photons, nonlocality, and Bell inequalities in the undergraduate laboratory |journal=American Journal of Physics |volume=70 |number=9 |year=2002 |pages=903–910 |arxiv=quant-ph/0205171 |doi=10.1119/1.1498860|bibcode=2002AmJPh..70..903D }}</ref><ref>{{cite journal|date=May 2018|title=Challenging local realism with human choices |journal=Nature |volume=557 |issue=7704 |pages=212–216 |doi=10.1038/s41586-018-0085-3 |bibcode=2018Natur.557..212B |author1=BIG Bell Test Collaboration |pmid=29743691 |arxiv=1805.04431 }}</ref><ref>{{cite journal|title=Cosmic Bell Test Using Random Measurement Settings from High-Redshift Quasars|date=20 August 2018 |journal=Physical Review Letters |volume=121 |number=8 |article-number=080403 |doi=10.1103/PhysRevLett.121.080403 |last1 = Rauch |first1 = Dominik |pmid=30192604 |display-authors=etal |arxiv=1808.05966|bibcode=2018PhRvL.121h0403R }}</ref> Moreover, when measurements of the entangled particles are made in moving [[special relativity|relativistic]] reference frames, in which each measurement (in its own relativistic time frame) occurs before the other, the measurement results remain correlated.<ref>{{cite journal |author=Zbinden |first=H. |author2=Gisin |author3=Tittel |display-authors=1 |year=2001 |title=Experimental test of nonlocal quantum correlations in relativistic configurations |url=http://archive-ouverte.unige.ch/unige:37034 |journal=Physical Review A |volume=63 |issue=2 |article-number=22111 |arxiv=quant-ph/0007009 |bibcode=2001PhRvA..63b2111Z |doi=10.1103/PhysRevA.63.022111 |s2cid=44611890}}</ref><ref name=Gilder2009/>{{rp|321–324}}

The fundamental issue about measuring spin along different axes is that these measurements cannot have definite values at the same time―they are [[Incompatible observables|incompatible]] in the sense that these measurements' maximum simultaneous precision is constrained by the [[uncertainty principle]]. This is contrary to what is found in classical physics, where any number of properties can be measured simultaneously with arbitrary accuracy. It has been proven mathematically that compatible measurements cannot show Bell-inequality-violating correlations,<ref>{{cite journal|last1=Cirel'son|first1=B. S.|title=Quantum generalizations of Bell's inequality |journal=Letters in Mathematical Physics |volume=4|issue=2|pages=93–100| year=1980|doi=10.1007/BF00417500|bibcode=1980LMaPh...4...93C |s2cid=120680226}}</ref> and thus entanglement is a fundamentally non-classical phenomenon.

== Nonlocality and entanglement ==
As discussed above, entanglement is necessary to produce a violation of a [[Bell's theorem|Bell inequality]]. However, the mere presence of entanglement alone is insufficient,<ref name="Brunner-RMP2014">
{{cite journal
 | title=Bell nonlocality
 | last1 = Brunner | first1 = Nicolas
 | last2 = Cavalcanti | first2 = Daniel
 | last3 = Pironio | first3 = Stefano
 | last4 = Scarani | first4 = Valerio
 | last5 = Wehner | first5 = Stephanie
 | journal= Reviews of Modern Physics
 | volume=86
 | issue=2
 | pages=419–478
 | date=2014
 | doi=10.1103/RevModPhys.86.419
 | arxiv=1303.2849
 | bibcode=2014RvMP...86..419B
 | s2cid=119194006
}}</ref> as Bell himself noted in his 1964 paper.<ref name=":4" /> This is demonstrated, for example, by [[Werner state]]s, which are a family of states describing pairs of particles. For appropriate choices of the key parameter that identifies a given Werner state within the full set thereof, the Werner states exhibit entanglement. Yet pairs of particles described by Werner states always admit a local hidden variable model. In other words, these states cannot power the violation of a Bell inequality, despite possessing entanglement.<ref name="werner1989">{{cite journal |last=Werner |first=R. F. |author-link=Reinhard F. Werner |year=1989 |title=Quantum States with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model |journal=[[Physical Review A]] |volume=40 |issue=8 |pages=4277–4281 |bibcode=1989PhRvA..40.4277W |doi=10.1103/PhysRevA.40.4277 |pmid=9902666}}</ref> This can be generalized from pairs of particles to larger collections as well.<ref name="Augusiak2015">
{{cite journal
 | last1 = Augusiak | first1 = R.
 | last2 = Demianowicz | first2 = M.
 | last3 = Tura | first3 = J.
 | last4 = Acín | first4 = A.
 | title = Entanglement and nonlocality are inequivalent for any number of parties
 | journal = Physical Review Letters
 | volume = 115
 | issue = 3
 | article-number = 030404
 | year = 2015
 | arxiv = 1407.3114
 | doi = 10.1103/PhysRevLett.115.030404
 | pmid = 26230773
 | hdl = 2117/78836
 | bibcode = 2015PhRvL.115c0404A
 | s2cid = 29758483
}}</ref>

The violation of Bell inequalities is often called ''[[quantum nonlocality]].'' This term is not without controversy.<ref>{{cite SEP|title=Action at a Distance in Quantum Mechanics |url-id=qm-action-distance |author-first=Joseph |author-last=Berkovitz |date=2007-01-26}}</ref> It is sometimes argued that using the term ''nonlocality'' carries the unwarranted implication that the violation of Bell inequalities must be explained by physical, faster-than-light signals.<ref name="Scarani">{{cite book|first=Valerio |last=Scarani |title=Bell Nonlocality |publisher=Oxford University Press |year=2019 |isbn=978-0-19-878841-6 |page=8}}</ref> In other words, the failure of local hidden-variable models to reproduce quantum mechanics is not necessarily a sign of true nonlocality in quantum mechanics itself.<ref>{{cite book|first=Roland |last=Omnès |author-link=Roland Omnès |title=The Interpretation of Quantum Mechanics |publisher=Princeton University Press |year=1994 |isbn=978-0-691-03669-4 |pages=399–400}}</ref><ref>{{cite journal|last=Mermin |first=N. D. |author-link=N. David Mermin |title=What Do These Correlations Know About Reality? Nonlocality and the Absurd |journal=[[Foundations of Physics]] |volume=29 |year=1999 |issue=4 |pages=571–587 |arxiv=quant-ph/9807055 |bibcode=1998quant.ph..7055M |doi=10.1023/A:1018864225930}}</ref><ref>{{cite book|last=Żukowski |first=Marek |title=Quantum &#91;Un&#93;Speakables II |chapter=Bell's Theorem Tells Us Not What Quantum Mechanics is, but What Quantum Mechanics is Not |date=2017 |series=The Frontiers Collection |pages=175–185 |editor-last=Bertlmann |editor-first=Reinhold |place=Cham |publisher=Springer International Publishing |doi=10.1007/978-3-319-38987-5_10 |isbn=978-3-319-38985-1 |editor2-last=Zeilinger |editor2-first=Anton |editor-link2=Anton Zeilinger |arxiv=1501.05640}}</ref> Despite these reservations, the term ''nonlocality'' has become a widespread convention.<ref name="Scarani"/>

The term ''nonlocality'' is also sometimes applied to other concepts besides the nonexistence of a local hidden-variable model, such as [[unextendible product basis|whether states can be distinguished by local measurements]].<ref>{{cite journal |last1=Bennett |first1=Charles H. |last2=DiVincenzo |first2=David P. |last3=Fuchs |first3=Christopher A. |last4=Mor |first4=Tal |last5=Rains |first5=Eric |last6=Shor |first6=Peter W. |last7=Smolin |first7=John A. |last8=Wootters |first8=William K. |year=1999 |title=Quantum nonlocality without entanglement |journal=Physical Review A |volume=59 |issue=2 |pages=1070–1091 |arxiv=quant-ph/9804053 |bibcode=1999PhRvA..59.1070B |doi=10.1103/PhysRevA.59.1070 |s2cid=15282650}}</ref> Moreover, [[quantum field theory]] is often said to be ''local'' because [[observable]]s defined within spacetime regions that are [[spacelike]] separated must commute.<ref name="Brunner-RMP2014"/><ref>{{cite book|last=Haag |first=Rudolf |author-link=Rudolf Haag |title=Local Quantum Physics: Fields, Particles, Algebras |edition=2nd |publisher=Springer |pages=107–108 |isbn=3-540-61451-6 |year=1996}}</ref> These other uses of ''local'' and ''nonlocal'' are not discussed further here.

== Mathematical details ==
The following subsections use the formalism and theoretical framework developed in the articles [[bra–ket notation]] and [[mathematical formulation of quantum mechanics]].

=== Pure states ===
Consider two arbitrary quantum systems {{mvar|A}} and {{mvar|B}}, with respective [[Hilbert space]]s {{mvar|H<sub>A</sub>}} and {{mvar|H<sub>B</sub>}}. The Hilbert space of the composite system is the [[tensor product]]
: <math> H_A \otimes H_B.</math>

If the first system is in state <math>| \psi \rangle_A</math> and the second in state <math>| \phi \rangle_B</math>, the state of the composite system is
: <math>|\psi\rangle_A \otimes |\phi\rangle_B.</math>

States of the composite system that can be represented in this form are called separable states, or [[product state]]s. However, not all states of the composite system are separable. Fix a [[basis (linear algebra)|basis]] <math> \{|i \rangle_A\}</math> for {{mvar|H<sub>A</sub>}} and a basis <math> \{|j \rangle_B\}</math> for {{mvar|H<sub>B</sub>}}. The most general state in {{math|''H<sub>A</sub>'' ⊗ ''H<sub>B</sub>''}} is of the form
: <math>|\psi\rangle_{AB} = \sum_{i,j} c_{ij} |i\rangle_A \otimes |j\rangle_B</math>.

This state is separable if there exist vectors <math> [c^A_i], [c^B_j]</math> so that <math> c_{ij}= c^A_i c^B_j,</math> yielding <math display="inline"> |\psi\rangle_A = \sum_{i} c^A_{i} |i\rangle_A</math> and <math display="inline"> |\phi\rangle_B = \sum_{j} c^B_{j} |j\rangle_B.</math> It is inseparable if for any vectors <math> [c^A_i],[c^B_j]</math> at least for one pair of coordinates <math> c^A_i,c^B_j</math> we have <math> c_{ij} \neq c^A_ic^B_j.</math> If a state is inseparable, it is called an 'entangled state'.<ref name="Rieffel2011"/>{{rp|218}}<ref name="Zwiebach2022"/>{{rp|§1.5}}

For example, given two basis vectors <math> \{|0\rangle_A, |1\rangle_A\}</math> of {{mvar|H<sub>A</sub>}} and two basis vectors <math> \{|0\rangle_B, |1\rangle_B\}</math> of {{mvar|H<sub>B</sub>}}, the following is an entangled state:
: <math>\tfrac{1}{\sqrt{2}} \left ( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B \right ).</math>

If the composite system is in this state, it is impossible to attribute to either system {{mvar|A}} or system {{mvar|B}} a definite [[pure state]]. Another way to say this is that while the [[von Neumann entropy]] of the whole state is zero (as it is for any pure state), the entropy of the subsystems is greater than zero. In this sense, the systems are "entangled". The above example is one of four [[Bell states]], which are (maximally) entangled pure states (pure states of the {{math|''H<sub>A</sub>'' ⊗ ''H<sub>B</sub>''}} space, but which cannot be separated into pure states of each {{mvar|H<sub>A</sub>}} and {{mvar|H<sub>B</sub>}}).<ref name="Zwiebach2022"/>{{rp|§18.6}}

Now suppose Alice is an observer for system {{mvar|A}}, and Bob is an observer for system {{mvar|B}}. If in the entangled state given above Alice makes a measurement in the <math> \{|0\rangle, |1\rangle\}</math> eigenbasis of {{mvar|A}}, there are two possible outcomes, occurring with equal probability: Alice can obtain the outcome 0, or she can obtain the outcome 1. If she obtains the outcome 0, then she can predict with certainty that Bob's result will be 1. Likewise, if she obtains the outcome 1, then she can predict with certainty that Bob's result will be 0. In other words, the results of measurements on the two qubits will be perfectly anti-correlated. This remains true even if the systems {{mvar|A}} and {{mvar|B}} are spatially separated. This is the foundation of the EPR paradox.<ref name="Nielsen-2010" />{{Rp|pages=113–114}}

The outcome of Alice's measurement is random. Alice cannot decide which state to collapse the composite system into, and therefore cannot transmit information to Bob by acting on her system. Causality is thus preserved, in this particular scheme. For the general argument, see [[no-communication theorem]].

=== Ensembles ===
As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space. More generally, if one has less information about the system, then one calls it an 'ensemble' and describes it by a [[density matrix]], which is a [[positive-semidefinite matrix]], or a [[trace class]] when the state space is infinite-dimensional, and which has trace 1. By the [[spectral theorem]], such a matrix takes the general form:
: <math>\rho = \sum_i w_i |\alpha_i\rangle \langle\alpha_i|,</math>
where the ''w''<sub>''i''</sub> are positive-valued probabilities (they sum up to 1), the vectors {{math|''α''<sub>''i''</sub>}} are unit vectors, and in the infinite-dimensional case, we would take the closure of such states in the trace norm. We can interpret {{mvar|ρ}} as representing an ensemble where <math> w_i </math> is the proportion of the ensemble whose states are <math>|\alpha_i\rangle</math>. When a mixed state has rank 1, it therefore describes a 'pure ensemble'. When there is less than total information about the state of a quantum system we need [[#Reduced density matrices|density matrices]] to represent the state.<ref name="Peres1993"/>{{rp|73–74}}<ref name="Holevo2001"/>{{rp|13–15}}<ref name="Zwiebach2022"/>{{rp|§22.2}}

Experimentally, a mixed ensemble might be realized as follows. Consider a "black box" apparatus that spits [[electron]]s towards an observer. The electrons' Hilbert spaces are [[identical particles|identical]]. The apparatus might produce electrons that are all in the same state; in this case, the electrons received by the observer are then a pure ensemble. However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state <math>|\mathbf{z}+\rangle</math> with spins aligned in the positive {{math|'''z'''}} direction, and the other with state <math>|\mathbf{y}-\rangle</math> with spins aligned in the negative {{math|'''y'''}} direction. Generally, this is a mixed ensemble, as there can be any number of populations, each corresponding to a different state.

Following the definition above, for a bipartite composite system, mixed states are just density matrices on {{math|''H<sub>A</sub>'' ⊗ ''H<sub>B</sub>''}}. That is, it has the general form
: <math>\rho =\sum_{i} w_i\left[\sum_{j} \bar{c}_{ij} (|\alpha_{ij}\rangle\otimes|\beta_{ij}\rangle)\right]\left[\sum_k c_{ik} (\langle\alpha_{ik}|\otimes\langle\beta_{ik}|)\right]
</math>
where the ''w''<sub>''i''</sub> are positively valued probabilities, <math display="inline">\sum_j |c_{ij}|^2=1</math>, and the vectors are unit vectors. This is self-adjoint and positive and has trace 1.

Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as<ref name=Laloe>{{cite journal|last=Laloe|first=Franck|year=2001|title=Do We Really Understand Quantum Mechanics|journal=American Journal of Physics |volume=69 |issue=6|pages=655–701 |arxiv=quant-ph/0209123 |bibcode=2001AmJPh..69..655L |doi=10.1119/1.1356698|s2cid=123349369 }}</ref>{{rp|131–132}}
: <math>\rho = \sum_i w_i \rho_i^A \otimes \rho_i^B, </math>
where the {{math|''w''<sub>''i''</sub>}} are positively valued probabilities and the <math>\rho_i^A</math>s and <math>\rho_i^B</math>s are themselves mixed states (density operators) on the subsystems {{mvar|A}} and {{mvar|B}} respectively. In other words, a state is separable if it is a probability distribution over uncorrelated states, or product states. By writing the density matrices as sums of pure ensembles and expanding, we may assume without loss of generality that <math>\rho_i^A</math> and <math>\rho_i^B</math> are themselves pure ensembles. A state is then said to be entangled if it is not separable.

In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be [[NP-hard]].<ref>{{cite book |last=Gurvits |first=L. |title=Proceedings of the thirty-fifth annual ACM symposium on Theory of computing |year=2003 |isbn=978-1-58113-674-6 |page=10 |language=en |chapter=Classical deterministic complexity of Edmonds' Problem and quantum entanglement |doi=10.1145/780542.780545 |arxiv=quant-ph/0303055 |s2cid=5745067}}</ref> For the {{math|2 × 2}} and {{math|2 × 3}} cases, a necessary and sufficient criterion for separability is given by the famous [[Peres-Horodecki criterion|Positive Partial Transpose (PPT)]] condition.<ref>{{cite journal |vauthors=Horodecki M, Horodecki P, Horodecki R |title=Separability of mixed states: necessary and sufficient conditions |journal=Physics Letters A |volume=223 |issue=1 |page=210 |year=1996 |doi=10.1016/S0375-9601(96)00706-2 |bibcode=1996PhLA..223....1H|arxiv = quant-ph/9605038 |citeseerx=10.1.1.252.496 |s2cid=10580997 }}</ref>

=== Reduced density matrices ===
The idea of a reduced density matrix was introduced by [[Paul Dirac]] in 1930.<ref name="Dirac1930">
{{cite journal
 | last = Dirac | first = Paul Adrien Maurice | author-link = Paul Dirac
 | title = Note on exchange phenomena in the Thomas atom
 | journal = [[Mathematical Proceedings of the Cambridge Philosophical Society]]
 | volume = 26
 | number = 3
 | pages = 376–385
 | doi = 10.1017/S0305004100016108
 | bibcode=1930PCPS...26..376D
 | year = 1930
 | url = https://www.cambridge.org/core/services/aop-cambridge-core/content/view/6C5FF7297CD96F49A8B8E9E3EA50E412/S0305004100016108a.pdf/note-on-exchange-phenomena-in-the-thomas-atom.pdf
 | doi-access=free
}}</ref> Consider as above systems {{mvar|A}} and {{mvar|B}} each with a Hilbert space {{mvar|H<sub>A</sub>, H<sub>B</sub>}}. Let the state of the composite system be
: <math> |\Psi \rangle \in H_A \otimes H_B. </math>

As indicated above, in general there is no way to associate a pure state to the component system {{mvar|A}}. However, it still is possible to associate a density matrix. Let
: <math>\rho_T = |\Psi\rangle \; \langle\Psi|</math>.
which is the [[projection operator]] onto this state. The state of {{mvar|A}} is the [[partial trace]] of {{mvar|ρ<sub>T</sub>}} over the basis of system {{mvar|B}}:
: <math>\rho_A \ \stackrel{\mathrm{def}}{=}\ \sum_j^{N_B} \left( I_A \otimes \langle j|_B \right) \left( |\Psi\rangle \langle\Psi| \right)\left( I_A \otimes |j\rangle_B \right) = \hbox{Tr}_B \; \rho_T.</math>

The sum occurs over <math>N_B := \dim(H_B)</math> and <math>I_A</math> the identity operator in <math>H_A</math>. {{mvar|ρ<sub>A</sub>}} is sometimes called the reduced density matrix of {{mvar|ρ}} on subsystem {{mvar|A}}. Colloquially, we "trace out" or "trace over" system {{mvar|B}} to obtain the reduced density matrix on {{mvar|A}}.<ref name="Rieffel2011"/>{{rp|207–212}}<ref name="Rau2021"/>{{rp|133}}<ref name="Zwiebach2022"/>{{rp|§22.4}}

For example, the reduced density matrix of {{mvar|A}} for the entangled state
: <math>\tfrac{1}{\sqrt{2}} \left ( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B \right),</math>
discussed above is<ref name="Zwiebach2022"/>{{rp|§22.4}}
: <math>\rho_A = \tfrac{1}{2} \left ( |0\rangle_A \langle 0|_A + |1\rangle_A \langle 1|_A \right ).</math>

This demonstrates that the reduced density matrix for an entangled pure ensemble is a mixed ensemble. In contrast, the density matrix of {{mvar|A}} for the pure product state <math>|\psi\rangle_A \otimes |\phi\rangle_B</math> discussed above is<ref name="Nielsen-2010"/>{{rp|106}}
: <math>\rho_A = |\psi\rangle_A \langle\psi|_A,</math>
the projection operator onto <math>|\psi\rangle_A</math>.

In general, a bipartite pure state ''ρ'' is entangled if and only if its reduced states are mixed rather than pure.<ref name="Rau2021"/>{{rp|131}}

=== Entanglement as a resource ===
In quantum information theory, entangled states are considered a 'resource', i.e., something costly to produce and that allows implementing valuable transformations.<ref name="Chitambar2019">
{{cite journal
 | last1 = Chitambar | first1 = Eric
 | last2 = Gour | first2 = Gilad
 | title = Quantum resource theories
 | journal = Reviews of Modern Physics
 | volume = 91
 | number = 2
 | article-number = 025001
 | doi = 10.1103/RevModPhys.91.025001
 | arxiv = 1806.06107
 | year = 2019
 | bibcode = 2019RvMP...91b5001C
 | s2cid = 119194947
}}</ref><ref name="GG-2022">
{{cite journal
 | last1 = Georgiev
 | first1 = Danko D.
 | last2 = Gudder
 | first2 = Stanley P.
 | title = Sensitivity of entanglement measures in bipartite pure quantum states
 | journal = Modern Physics Letters B
 | volume = 36
 | number = 22
 | pages = 2250101–2250255
 | doi = 10.1142/S0217984922501019
 | arxiv = 2206.13180
 | year = 2022
 | bibcode = 2022MPLB...3650101G
 | s2cid = 250072286
}}</ref> The setting in which this perspective is most evident is that of "distant labs", i.e., two quantum systems labelled "A" and "B" on each of which arbitrary [[quantum operation]]s can be performed, but which do not interact with each other quantum mechanically. The only interaction allowed is the exchange of classical information, which combined with the most general local quantum operations gives rise to the class of operations called [[LOCC]] (local operations and classical communication). These operations do not allow the production of entangled states between systems A and B. But if A and B are provided with a supply of entangled states, then these, together with LOCC operations can enable a larger class of transformations.

If Alice and Bob share an entangled state, Alice can tell Bob over a telephone call how to reproduce a quantum state <math>|\Psi\rangle</math> she has in her lab. Alice performs a joint measurement on <math>|\Psi\rangle</math> together with her half of the entangled state  and tells Bob the results. Using Alice's results Bob operates on his half of the entangled state to make it equal to <math>|\Psi\rangle</math>. Since Alice's measurement necessarily erases the quantum state of the system in her lab, the state <math>|\Psi\rangle</math> is not copied, but transferred: it is said to be "[[quantum teleportation|teleported]]" to Bob's laboratory through this protocol.<ref name="Nielsen-2010">{{cite book |last1=Nielsen |first1=Michael A. |title=Quantum Computation and Quantum Information |title-link=Quantum Computation and Quantum Information |last2=Chuang |first2=Isaac L. |publisher=Cambridge Univ. Press |year=2010 |isbn=978-0-521-63503-5 |edition=10th anniversary|location=Cambridge}}</ref>{{rp|27}}<ref name="horodecki2007"/>{{rp|875}}<ref>{{cite journal|arxiv=1505.07831 |title=Advances in Quantum Teleportation |first1=S. |last1=Pirandola |first2=J. |last2=Eisert |first3=C. |last3=Weedbrook |first4=A. |last4=Furusawa |first5=S. L. |last5=Braunstein |journal=Nature Photonics |volume=9 |pages=641–652 |year=2015 |issue=10 |doi=10.1038/nphoton.2015.154|bibcode=2015NaPho...9..641P }}</ref>

[[File:Entanglement swapping.svg|thumb|Entanglement of states from independent sources can be swapped through Bell state measurement.<ref name="GuoReview2023">{{cite journal |last1=Hu |first1=Xiao-Min |last2=Guo |first2=Yu |last3=Liu |first3=Bi-Heng |last4=Li |first4=Chuan-Feng |last5=Guo |first5=Guang-Can |date=June 2023 |title=Progress in quantum teleportation |url=https://www.nature.com/articles/s42254-023-00588-x |journal=Nature Reviews Physics |language=en |volume=5 |issue=6 |pages=339–353 |doi=10.1038/s42254-023-00588-x |bibcode=2023NatRP...5..339H |issn=2522-5820|url-access=subscription }}</ref>{{rp|341}}]]
[[Entanglement swapping]] is variant of teleportation that allows two parties that have never interacted to share an entangled state. The swapping protocol begins with two EPR sources. One source emits an entangled pair of particles A and B, while the other emits a second entangled pair of particles C and D. Particles B and C are subjected to a measurement in the basis of Bell states. The state of the remaining particles, A and D, collapses to a Bell state, leaving them entangled despite never having interacted with each other.<ref name="horodecki2007"/><ref name="Pan1998">{{Cite journal |last1=Pan |first1=J.-W. |last2=Bouwmeester |first2=D. |last3=Weinfurter |first3=H. |last4=Zeilinger |first4=A. |author-link4=Anton Zeilinger |year=1998 |title=Experimental entanglement swapping: Entangling photons that never interacted |journal=[[Physical Review Letters]] |volume=80 |number=18 |pages=3891–3894 |doi=10.1103/PhysRevLett.80.3891 |bibcode=1998PhRvL..80.3891P }}</ref>

An interaction between a qubit of A and a qubit of B can be realized by first teleporting A's qubit to B, then letting it interact with B's qubit (which is now a LOCC operation, since both qubits are in B's lab) and then teleporting the qubit back to A. Two maximally entangled states of two qubits are used up in this process. Thus entangled states are a resource that enables the realization of quantum interactions (or of quantum channels) in a setting where only LOCC are available, but they are consumed in the process. There are other applications where entanglement can be seen as a resource, e.g., private communication or distinguishing quantum states.<ref name="horodecki2007" />

===Multipartite entanglement===
{{main|Multipartite entanglement}}
Quantum states describing systems made of more than two pieces can also be entangled. An example for a three-qubit system is the [[Greenberger–Horne–Zeilinger state|Greenberger–Horne–Zeilinger (GHZ) state]],
<math display="block">|\mathrm{GHZ}\rangle = \frac{|000\rangle + |111\rangle}{\sqrt{2}}.</math>
Another three-qubit example is the [[W state]]:
<math display="block">|\mathrm{W}\rangle = \frac{|001\rangle + |010\rangle + |100\rangle}{\sqrt{3}}.</math>
Tracing out any one of the three qubits turns the GHZ state into a separable state, whereas the result of tracing over any of the three qubits in the W state is still entangled. This illustrates how multipartite entanglement is a more complicated topic than bipartite entanglement: systems composed of three or more parts can exhibit multiple qualitatively different types of entanglement.<ref name="Bengtsson2017"/>{{rp|493–497}} A single particle cannot be maximally entangled with more than a particle at a time, a property called [[Monogamy of entanglement|monogamy]].<ref>{{Cite book |last1=Bertlmann |first1=Reinhold |url=https://books.google.com/books?id=uzHaEAAAQBAJ&dq=monogamy+of+entanglement&pg=PA511 |title=Modern Quantum Theory: From Quantum Mechanics to Entanglement and Quantum Information |last2=Friis |first2=Nicolai |date=2023-10-05 |publisher=Oxford University Press |isbn=978-0-19-150634-5 |language=en |page=511}}</ref>

=== Classification of entanglement ===
Not all quantum states are equally valuable as a resource. One method to quantify this value is to use an [[#Entanglement measures|entanglement measure]] that assigns a numerical value to each quantum state. However, it is often interesting to settle for a coarser way to compare quantum states. This gives rise to different classification schemes. Most entanglement classes are defined based on whether states can be converted to other states using LOCC or a subclass of these operations. The smaller the set of allowed operations, the finer the classification. Important examples are:
* If two states can be transformed into each other by a local unitary operation, they are said to be in the same ''LU class''. This is the finest of the usually considered classes. Two states in the same LU class have the same value for entanglement measures and the same value as a resource in the distant-labs setting. There is an infinite number of different LU classes (even in the simplest case of two qubits in a pure state).<ref name="GRB1998">{{cite journal |author1=Grassl, M. |author2=Rötteler, M. |author3=Beth, T. |title=Computing local invariants of quantum-bit systems |journal=Phys. Rev. A |volume=58 |issue=3 |pages=1833–1839 |year=1998 |doi=10.1103/PhysRevA.58.1833 |arxiv=quant-ph/9712040|bibcode=1998PhRvA..58.1833G |s2cid=15892529 }}</ref><ref name="Kraus2010">{{cite journal |author=Kraus |first=Barbara |author-link=Barbara Kraus |year=2010 |title=Local unitary equivalence of multipartite pure states |journal=Physical Review Letters |volume=104 |issue=2 |article-number=020504 |arxiv=0909.5152 |bibcode=2010PhRvL.104b0504K |doi=10.1103/PhysRevLett.104.020504 |pmid=20366579 |s2cid=29984499}}</ref>
* If two states can be transformed into each other by local operations including measurements with probability larger than 0, they are said to be in the same 'SLOCC class' ("stochastic LOCC"). Qualitatively, two states <math>\rho_1</math> and <math>\rho_2</math> in the same SLOCC class are equally powerful, since one can transform each into the other, but since the transformations <math>\rho_1\to\rho_2</math> and <math>\rho_2\to\rho_1</math> may succeed with different probability, they are no longer equally valuable. E.g., for two pure qubits there are only two SLOCC classes: the entangled states (which contains both the (maximally entangled) Bell states and weakly entangled states like <math>|00\rangle+0.01|11\rangle</math>) and the separable ones (i.e., product states like <math>|00\rangle</math>).<ref>{{cite journal |author=Nielsen |first=M. A. |year=1999 |title=Conditions for a Class of Entanglement Transformations |journal=Physical Review Letters |volume=83 |issue=2 |page=436 |arxiv=quant-ph/9811053 |bibcode=1999PhRvL..83..436N |doi=10.1103/PhysRevLett.83.436 |s2cid=17928003}}</ref><ref name="GoWa2010">{{cite journal |author1=Gour, G. |author2=Wallach, N. R. |title=Classification of Multipartite Entanglement of All Finite Dimensionality |journal=Phys. Rev. Lett. |volume=111 |issue=6 |article-number=060502 |year=2013 |doi=10.1103/PhysRevLett.111.060502 |pmid=23971544 |arxiv=1304.7259 |bibcode=2013PhRvL.111f0502G |s2cid=1570745}}</ref>
* Instead of considering transformations of single copies of a state (like <math>\rho_1\to\rho_2</math>) one can define classes based on the possibility of multi-copy transformations. E.g., there are examples when <math>\rho_1\to\rho_2</math> is impossible by LOCC, but <math>\rho_1\otimes\rho_1\to\rho_2</math> is possible. A very important (and very coarse) classification is based on the property whether it is possible to transform an arbitrarily large number of copies of a state <math>\rho</math> into at least one pure entangled state. States that have this property are called [[Entanglement distillation|distillable]]. These states are the most useful quantum states since, given enough of them, they can be transformed (with local operations) into any entangled state and hence allow for all possible uses. It came initially as a surprise that not all entangled states are distillable; those that are not are called "[[Bound entanglement|bound entangled]]".<ref name="HHH97">{{cite journal |author1=Horodecki, M. |author2=Horodecki, P. |author3=Horodecki, R. |title=Mixed-state entanglement and distillation: Is there a ''bound'' entanglement in nature? |journal=Phys. Rev. Lett. |volume=80 |issue=1998 |pages=5239–5242 |year=1998 |arxiv=quant-ph/9801069|doi=10.1103/PhysRevLett.80.5239 |bibcode=1998PhRvL..80.5239H |s2cid=111379972 }}</ref><ref name="horodecki2007" />

A different entanglement classification is based on what the quantum correlations present in a state allow A and B to do: one distinguishes three subsets of entangled states: (1) the ''[[Quantum nonlocality|non-local]] states'', which produce correlations that cannot be explained by a local hidden variable model and thus violate a Bell inequality, (2) the ''[[Quantum steering|steerable]] states'' that contain sufficient correlations for A to modify ("steer") by local measurements the conditional reduced state of B in such a way, that A can prove to B that the state they possess is indeed entangled, and finally (3) those entangled states that are neither non-local nor steerable. All three sets are non-empty.<ref name="WJD2007">{{cite journal |last1=Wiseman |first1=H. M. |last2=Jones |first2=S. J. |last3=Doherty |first3=A. C. |year=2007 |title=Steering, Entanglement, Nonlocality, and the Einstein–Podolsky–Rosen Paradox |journal=Physical Review Letters |volume=98 |issue=14 |article-number=140402 |arxiv=quant-ph/0612147 |bibcode=2007PhRvL..98n0402W |doi=10.1103/PhysRevLett.98.140402 |pmid=17501251 |s2cid=30078867}}</ref>

=== Entropy ===
In this section, the entropy of a mixed state is discussed as well as how it can be viewed as a measure of quantum entanglement.

==== Definition ====
In classical [[information theory]] {{mvar|H}}, the [[Shannon entropy]], is associated to a probability distribution, <math>p_1, \cdots, p_n</math>, in the following way:<ref name="SE">{{cite journal |url=http://authors.library.caltech.edu/5516/1/CERpra97b.pdf#page=10 |title=Information-theoretic interpretation of quantum error-correcting codes |journal=Physical Review A |date=September 1997 |volume=56 |number=3 |pages=1721–1732 |arxiv=quant-ph/9702031 |doi=10.1103/PhysRevA.56.1721 |first1=Nicolas J. |last1=Cerf |first2=Richard |last2=Cleve |bibcode=1997PhRvA..56.1721C }}</ref>
: <math>H(p_1, \cdots, p_n ) = - \sum_i p_i \log_2 p_i.</math>

Since a mixed state {{mvar|ρ}} is a probability distribution over an ensemble, this leads naturally to the definition of the [[von Neumann entropy]]:<ref name="Peres1993"/>{{rp|264}}
: <math>S(\rho) = - \hbox{Tr} \left( \rho \log_2 {\rho} \right),</math>
which can be expressed in terms of the [[eigenvalue]]s of {{mvar|ρ}}: 
: <math>S(\rho) = - \hbox{Tr} \left( \rho \log_2 {\rho} \right) = - \sum_i \lambda_i \log_2 \lambda_i</math>.

Since an event of probability 0 should not contribute to the entropy, and given that
: <math> \lim_{p \to 0} p \log p = 0,</math>
the convention {{math|0 log(0) {{=}} 0}} is adopted. When a pair of particles is described by the spin singlet state discussed above, the von Neumann entropy of either particle is {{math|log(2)}}, which can be shown to be the maximum entropy for {{math|2 × 2}} mixed states.<ref name="Holevo2001"/>{{rp|15}}

==== As a measure of entanglement ====
Entropy provides one tool that can be used to quantify entanglement, although other entanglement measures exist.<ref name="Plenio">{{cite journal|last1=Plenio |first1=Martin B. |first2=Shashank |last2=Virmani|title=An introduction to entanglement measures|year=2007|pages=1–51|volume=1|journal=Quant. Inf. Comp. |arxiv=quant-ph/0504163|bibcode=2005quant.ph..4163P}}</ref><ref name="Vedral2002">{{cite journal | last = Vedral | first = Vlatko |author-link = Vlatko Vedral | doi = 10.1103/RevModPhys.74.197 | arxiv = quant-ph/0102094 | bibcode=2002RvMP...74..197V | volume=74 | issue = 1 | title=The role of relative entropy in quantum information theory | year=2002 | journal=Reviews of Modern Physics | pages=197–234 | s2cid = 6370982 }}</ref> If the overall system is pure, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems. For bipartite pure states, the von Neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure.<ref>{{cite journal |last1=Hill |first1=S |last2=Wootters |first2=W. K. |title=Entanglement of a Pair of Quantum Bits |journal=Phys. Rev. Lett. |arxiv=quant-ph/9703041 |doi =10.1103/PhysRevLett.78.5022 |year=1997 |volume=78 |issue=26 |pages=5022–5025 |bibcode=1997PhRvL..78.5022H |s2cid=9173232 }}</ref>

It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {{mset|1/''n'', ..., 1/''n''}}.<ref name="Nielsen-2010"/>{{rp|505}} Therefore, a bipartite pure state {{math|''ρ'' ∈ ''H''<sub>A</sub> ⊗ ''H''<sub>B</sub>}} is said to be a ''maximally entangled state'' if the reduced state of each subsystem of {{mvar|ρ}} is the diagonal matrix<ref>{{Cite journal |last1=Enríquez |first1=M. |last2=Wintrowicz |first2=I. |last3=Życzkowski |first3=K. |author-link3=Karol Życzkowski |date=March 2016 |title=Maximally Entangled Multipartite States: A Brief Survey |journal=Journal of Physics: Conference Series |volume=698 |issue=1 |article-number=012003 |doi=10.1088/1742-6596/698/1/012003 |bibcode=2016JPhCS.698a2003E |issn=1742-6588|doi-access=free }}</ref>
: <math>\begin{bmatrix} \frac{1}{n}& & \\ & \ddots & \\ & & \frac{1}{n}\end{bmatrix}.</math>

For mixed states, the reduced von Neumann entropy is not the only reasonable entanglement measure.<ref name="Bengtsson2017"/>{{rp|471}}

[[Rényi entropy]] also can be used as a measure of entanglement.<ref name="Bengtsson2017"/>{{rp|447,480}}<ref>{{cite journal |last1=Wang |first1=Yu-Xin |last2=Mu |first2=Liang-Zhu |last3=Vedral |first3=Vlatko |last4=Fan |first4=Heng |date=17 February 2016 |title=Entanglement Rényi α entropy |url=https://link.aps.org/doi/10.1103/PhysRevA.93.022324 |journal=Physical Review A |language=en |volume=93 |issue=2 |article-number=022324 |arxiv=1504.03909 |doi=10.1103/PhysRevA.93.022324 |bibcode=2016PhRvA..93b2324W |issn=2469-9926}}</ref>

=== Entanglement measures ===
Entanglement measures quantify the amount of entanglement in a (often viewed as a bipartite) quantum state. As aforementioned, [[entropy of entanglement|entanglement entropy]] is the standard measure of entanglement for pure states (but no longer a measure of entanglement for mixed states). For mixed states, there are some entanglement measures in the literature<ref name="Plenio" /> and no single one is standard.
* Entanglement cost
* [[entanglement distillation|Distillable entanglement]]
* [[Entanglement of formation]]
* [[Concurrence (quantum computing)|Concurrence]]
* [[quantum relative entropy|Relative entropy of entanglement]]
* [[Squashed entanglement]]
* [[Negativity (quantum mechanics)#Logarithmic negativity|Logarithmic negativity]]
Most (but not all) of these entanglement measures reduce for pure states to entanglement entropy, and are difficult ([[NP-hard]]) to compute for mixed states as the dimension of the entangled system grows.<ref>{{cite journal|last1=Huang|first1=Yichen|title=Computing quantum discord is NP-complete|journal=New Journal of Physics|date=21 March 2014|volume=16|issue=3|article-number=033027|doi=10.1088/1367-2630/16/3/033027|bibcode=2014NJPh...16c3027H|arxiv = 1305.5941 |s2cid=118556793}}</ref>

=== Quantum field theory ===
The [[Reeh–Schlieder theorem]] of [[quantum field theory]] is sometimes interpreted as saying that entanglement is omnipresent in the [[quantum vacuum]].<ref>{{cite book|first=Stephen J. |last=Summers |chapter=Yet More Ado About Nothing: The Remarkable Relativistic Vacuum State |arxiv=0802.1854 |title=Deep Beauty: Understanding the Quantum World through Mathematical Innovation |pages=317–341 |editor-first=Hans |editor-last=Halvorson |publisher=Cambridge University Press |year=2011 |isbn=978-1-139-49922-4}}</ref>

== Applications ==

Entanglement has many applications in [[quantum information theory]]. With the aid of entanglement, otherwise impossible tasks may be achieved.

Among the best-known applications of entanglement are [[superdense coding]] and quantum teleportation.<ref name="Bouwmeester-1997"/>

Most researchers believe that entanglement is necessary to realize [[quantum computer|quantum computing]] (although this is disputed by some).<ref name="jozsa02">{{cite journal |last1=Jozsa |first1=Richard |last2=Linden |first2=Noah |year=2002 |title=On the role of entanglement in quantum computational speed-up |journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |volume=459 |issue=2036 |pages=2011–2032 |arxiv=quant-ph/0201143 |bibcode=2003RSPSA.459.2011J |citeseerx=10.1.1.251.7637 |doi=10.1098/rspa.2002.1097 |s2cid=15470259}}</ref>

Entanglement is used in some protocols of [[quantum cryptography]],<ref name="ekert91">{{cite journal |title=Quantum cryptography based on Bell's theorem |year=1991 |last1=Ekert |first1=Artur K. |journal=Physical Review Letters |volume=67 |issue=6 |pages=661–663 |doi=10.1103/PhysRevLett.67.661 |pmid=10044956 |bibcode=1991PhRvL..67..661E |s2cid=27683254}}</ref><ref name="horodecki10">{{cite journal |author1=Yin |first=Juan |author2=Yu-Huai Li |author3=Sheng-Kai Liao |author4=Meng Yang |author5=Yuan Cao |author6=Liang Zhang |author7=Ji-Gang Ren |author8=Wen-Qi Cai |author9=Wei-Yue Liu |author10=Shuang-Lin Li |author11=Rong Shu |author12=Yong-Mei Huang |author13=Lei Deng |author14=Li Li |author15=Qiang Zhang |year=2020 |title=Entanglement-based secure quantum cryptography over 1,120 kilometres |journal=Nature |volume=582 |issue=7813 |pages=501–505 |bibcode=2020Natur.582..501Y |doi=10.1038/s41586-020-2401-y |pmid=32541968 |s2cid=219692094 |author16=Nai-Le Liu |author17=Yu-Ao Chen |author18=Chao-Yang Lu |author19=Xiang-Bin Wang |author20=Feihu Xu |author21=Jian-Yu Wang |author22=Cheng-Zhi Peng |author23=Artur K. Ekert |author24=Jian-Wei Pan}}</ref> but to prove the security of [[quantum key distribution]] (QKD) under standard assumptions does not require entanglement.<ref>{{cite journal |last1=Renner |first1=R. |last2=Gisin |first2=N. |last3=Kraus |first3=B. |year=2005 |title=An information-theoretic security proof for QKD protocols |journal=Physical Review A |volume=72 |article-number=012332 |arxiv=quant-ph/0502064 |doi=10.1103/PhysRevA.72.012332 |s2cid=119052621}}</ref> However, the ''[[device-independent quantum cryptography|device independent]]'' security of QKD is shown exploiting entanglement between the communication partners.<ref>{{cite journal |author1=Pirandola |first=S. |author2=U. L. Andersen |author3=L. Banchi |author4=M. Berta |author5=D. Bunandar |author6=R. Colbeck |author7=D. Englund |author8=T. Gehring |author9=C. Lupo |author10=C. Ottaviani |author11=J. L. Pereira |author12=M. Razavi |author13=J. Shamsul Shaari |author14=M. Tomamichel |author15=V. C. Usenko |year=2020 |title=Advances in quantum cryptography |journal=Adv. Opt. Photon. |volume=12 |issue=4 |pages=1012–1236 |arxiv=1906.01645 |bibcode=2020AdOP...12.1012P |doi=10.1364/AOP.361502 |s2cid=174799187 |author16=G. Vallone |author17=P. Villoresi |author18=P. Wallden}}</ref>

In August 2014, Brazilian researcher Gabriela Barreto Lemos, from the University of Vienna, and team were able to "take pictures" of objects using photons that had not interacted with the subjects, but were entangled with photons that did interact with such objects.<ref>{{cite journal |url=http://www.nature.com/news/entangled-photons-make-a-picture-from-a-paradox-1.15781 |title=Entangled photons make a picture from a paradox |journal=Nature |access-date=13 October 2014 |doi=10.1038/nature.2014.15781 |year=2014 |last1=Gibney |first1=Elizabeth |s2cid=124976589|doi-access=free |url-access=subscription }}</ref> The idea has been adapted to make infrared images using only standard cameras that are insensitive to infrared.<ref>{{cite journal |last1=Pearce |first1=Emma |last2=Gemmell |first2=Nathan R. |last3=Flórez |first3=Jefferson |last4=Ding |first4=Jiaye |last5=Oulton |first5=Rupert F. |last6=Clark |first6=Alex S. |last7=Phillips |first7=Chris C. |date=15 November 2023 |title=Practical quantum imaging with undetected photons |url=https://opg.optica.org/abstract.cfm?URI=optcon-2-11-2386 |journal=Optics Continuum |language=en |volume=2 |issue=11 |page=2386 |doi=10.1364/OPTCON.507154 |issn=2770-0208|arxiv=2307.06225 }}</ref>

=== Entangled states ===
There are several canonical entangled states that appear often in theory and experiments.

For two [[qubits]], the [[Bell state]]s are
: <math>|\Phi^\pm\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B \pm |1\rangle_A \otimes |1\rangle_B)</math>
: <math>|\Psi^\pm\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B \pm |1\rangle_A \otimes |0\rangle_B).</math>

These four pure states are all maximally entangled and form an [[orthonormal]] [[basis (linear algebra)|basis]] of the Hilbert space of the two qubits.<ref name="Rieffel2011"/>{{rp|38–39}}<ref name="Nielsen-2010"/>{{rp|98}} They provide examples of how quantum mechanics can violate [[Bell's theorem|Bell-type inequalities]].<ref name="Rieffel2011"/>{{rp|62}}<ref name="Nielsen-2010"/>{{rp|116}}

For {{nowrap|''M'' > 2}} qubits, the [[Greenberger–Horne–Zeilinger state|GHZ state]] is
: <math>|\mathrm{GHZ}\rangle = \frac{|0\rangle^{\otimes M} + |1\rangle^{\otimes M}}{\sqrt{2}},</math>
which reduces to the Bell state <math>|\Phi^+\rangle</math> for {{nowrap|1=''M'' = 2}}. The traditional GHZ state was defined for {{nowrap|1=''M'' = 3}}. GHZ states are occasionally extended to [[qudit]]s, i.e., systems of ''d'' rather than 2 dimensions.<ref>{{Cite journal|last1=Caves |first1=Carlton M. |author-link=Carlton M. Caves |last2=Fuchs |first2=Christopher A. |last3=Schack |first3=Rüdiger |date=2002-08-20 |title=Unknown quantum states: The quantum de Finetti representation |journal=[[Journal of Mathematical Physics]] |volume=43 |number=9 |pages=4537–4559 |arxiv=quant-ph/0104088 |doi=10.1063/1.1494475 |quote=Mermin was the first to point out the interesting properties of this three-system state, following the lead of D. M. Greenberger, M. Horne, and A. Zeilinger, "Going beyond Bell's Theorem," in Bell's Theorem, Quantum Theory and Conceptions of the Universe, edited by M. Kafatos (Kluwer, Dordrecht, 1989), p. 69, where a similar four-system state was proposed. |bibcode=2002JMP....43.4537C}}</ref><ref>{{cite journal|first1=Yulin |last1=Chi |display-authors=etal |title=A programmable qudit-based quantum processor |journal=Nature Communications |year=2022 |volume=13 |issue=1 |page=1136 |article-number=1166 |doi=10.1038/s41467-022-28767-x |pmid=35246519 |bibcode=2022NatCo..13.1166C |pmc=8897515 }}</ref>

Also for {{nowrap|''M'' > 2}} qubits, there are [[Spin squeezing|spin squeezed states]], a class of [[squeezed coherent states]] satisfying certain restrictions on the uncertainty of spin measurements, which are necessarily entangled.<ref>{{cite journal |last1=Kitagawa |first1=Masahiro |last2=Ueda |first2=Masahito |year=1993 |title=Squeezed Spin States |url=https://ir.library.osaka-u.ac.jp/repo/ouka/all/77656/PhysRevA_47_06_005138.pdf |journal=Physical Review A |volume=47 |issue=6 |pages=5138–5143 |bibcode=1993PhRvA..47.5138K |doi=10.1103/physreva.47.5138 |pmid=9909547 |hdl-access=free |hdl=11094/77656}}</ref> Spin squeezed states are good candidates for enhancing precision measurements using quantum entanglement.<ref>{{cite journal |last1=Wineland |first1=D. J. |last2=Bollinger |first2=J. J. |last3=Itano |first3=W. M. |last4=Moore |first4=F. L. |last5=Heinzen |first5=D. J. |year=1992 |title=Spin squeezing and reduced quantum noise in spectroscopy |journal=Physical Review A |volume=46 |issue=11 |pages=R6797–R6800 |bibcode=1992PhRvA..46.6797W |doi=10.1103/PhysRevA.46.R6797 |pmid=9908086}}</ref>

For two [[boson]]ic modes, a [[NOON state]] is
: <math>|\psi_\text{NOON} \rangle = \frac{|N \rangle_a |0\rangle_b + |{0}\rangle_a |{N}\rangle_b}{\sqrt{2}}. </math>

This is like the Bell state <math>|\Psi^+\rangle</math> except the basis states <math>|0\rangle</math> and <math>|1\rangle</math> have been replaced with "the ''N'' photons are in one mode" and "the ''N'' photons are in the other mode".<ref name="Kishore2007">{{cite journal|first1=Kishore T. |last1=Kapale |first2=Jonathan P. |last2=Dowling |author-link2=Jonathan Dowling |title=A Bootstrapping Approach for Generating Maximally Path-Entangled Photon States |arxiv=quant-ph/0612196 |journal=Physical Review Letters |volume=99 |article-number=053602 |year=2007 |issue=5 |doi=10.1103/PhysRevLett.99.053602|pmid=17930751 |bibcode=2007PhRvL..99e3602K }}</ref>

Finally, there also exist [[twin Fock states]] for bosonic modes, which can be created by feeding a [[Fock state]] into two arms leading to a beam splitter. They are the sum of multiple NOON states, and can be used to achieve the [[Heisenberg limit]].<ref>{{cite journal |doi = 10.1103/PhysRevLett.71.1355|pmid = 10055519|title = Interferometric detection of optical phase shifts at the Heisenberg limit|journal = Physical Review Letters|volume = 71|issue = 9|pages = 1355–1358|year = 1993|last1 = Holland|first1 = M. J|last2 = Burnett|first2 = K|bibcode = 1993PhRvL..71.1355H}}</ref>

For the appropriately chosen measures of entanglement, Bell, GHZ, and NOON states are maximally entangled while spin squeezed and twin Fock states are only partially entangled.<ref>{{cite journal|doi=10.1126/science.1097522 |year=2004 |volume=304 |journal=Science |first1=Christian F. |last1=Roos |display-authors=etal |title=Control and Measurement of Three-Qubit Entangled States|issue=5676 |pages=1478–1480 |pmid=15178795 |bibcode=2004Sci...304.1478R }}</ref><ref name="Kishore2007"/><ref>{{cite journal|last1=Pezzè |first1=L. |last2=Smerzi |first2=A. |last3=Oberthaler |first3=M. K. |last4=Schmied |first4=R. |last5=Treutlein |first5=P. |year=2018 |title=Quantum metrology with nonclassical states of atomic ensembles |journal=Reviews of Modern Physics |volume=90 |number=3 |article-number=035005 |doi=10.1103/revmodphys.90.035005 |arxiv=1609.01609|bibcode=2018RvMP...90c5005P }}</ref>

=== Methods of creating entanglement ===
Entanglement is usually created by direct interactions between subatomic particles. These interactions can take numerous forms. One of the most commonly used methods is [[spontaneous parametric down-conversion]] to generate a pair of photons entangled in polarization.<ref name="horodecki2007">
{{cite journal
 |last1 = Horodecki |first1 = Ryszard
 |last2 = Horodecki |first2 = Pawel
 |last3 = Horodecki |first3 = Michal
 |last4 = Horodecki |first4 = Karol
 |title = Quantum entanglement
 |journal = Reviews of Modern Physics
 |arxiv=quant-ph/0702225
 |doi =10.1103/RevModPhys.81.865
 |year=2009
 |pages=865–942
 |bibcode=2009RvMP...81..865H
 |volume=81
 |issue=2
 |s2cid=59577352
}}</ref><ref name="Shadbolt2012">
{{cite journal
 |last1=Shadbolt |first1=P. J.
 |last2=Verde |first2=M. R.
 |last3=Peruzzo |first3=A.
 |last4=Politi |first4=A.
 |last5=Laing |first5=A.
 |last6=Lobino |first6=M.
 |last7=Matthews |first7=J. C. F.
 |last8=Thompson |first8=M. G.
 |last9=O'Brien |first9=J. L.
 |title=Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit
 |journal=Nature Photonics
 |year=2012 |volume=6 |issue=1 |pages=45–59
 |arxiv=1108.3309 |doi=10.1038/nphoton.2011.283
 |bibcode = 2012NaPho...6...45S |s2cid=56206588
 }}</ref> Other methods include the use of a [[Fiber-optic adapter|fiber coupler]] to confine and mix photons, photons emitted from decay cascade of the bi-exciton in a [[quantum dot]],<ref>{{cite journal |last=Akopian |first=N. |date=2006 |title=Entangled Photon Pairs from Semiconductor Quantum Dots |journal=Physical Review Letters |volume=96 |issue=2 |page=130501 |arxiv=quant-ph/0509060 |bibcode=2006PhRvL..96b0501D |doi=10.1103/PhysRevLett.96.020501 |pmid=16486553 |s2cid=22040546 }}</ref> or the use of the [[Hong–Ou–Mandel effect]].<ref>{{cite journal|last1=Lee |first1=Hwang |last2=Kok |first2=Pieter |last3=Dowling |first3=Jonathan P. |author-link3=Jonathan Dowling |title=A quantum Rosetta stone for interferometry |journal=Journal of Modern Optics |volume=49 |number=14–15 |year=2002 |pages=2325–2338 |doi=10.1080/0950034021000011536 |arxiv=quant-ph/0202133|bibcode=2002JMOp...49.2325L}}</ref> Quantum entanglement of a [[elementary particle|particle]] and its [[antiparticle]], such as an electron and a [[positron]], can be created by partial overlap of the corresponding [[quantum wave function]]s in [[Hardy's paradox|Hardy's interferometer]].<ref name="Hardy1992">
{{cite journal
 | last = Hardy | first = Lucien
 | title = Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories
 | journal = Physical Review Letters
 | volume = 68
 | number = 20
 | pages = 2981–2984
 | doi = 10.1103/PhysRevLett.68.2981
 | year = 1992
 | pmid = 10045577
 | bibcode = 1992PhRvL..68.2981H
}}</ref><ref name="Georgiev2022">
{{cite journal
 | last1 = Georgiev | first1 = Danko
 | last2 = Cohen | first2 = Eliahu
 | title = Entanglement measures for two-particle quantum histories
 | journal = Physical Review A
 | volume = 106
 | number = 6
 | article-number = 062437
 | doi = 10.1103/PhysRevA.106.062437
 | arxiv = 2212.07502
 | year = 2022
 | bibcode = 2022PhRvA.106f2437G
 | s2cid = 254685902
}}</ref> In the earliest tests of Bell's theorem, the entangled particles were generated using [[atomic cascade]]s.<ref name="Clauser"/>

It is also possible to create entanglement between quantum systems that never directly interacted, through the use of [[Quantum teleportation#Entanglement swapping|entanglement swapping]]. Two independently prepared, identical particles may also be entangled if their wave functions merely spatially overlap, at least partially.<ref>{{cite journal |last1=Lo Franco |first1=Rosario |last2=Compagno |first2=Giuseppe |date=14 June 2018 |title=Indistinguishability of Elementary Systems as a Resource for Quantum Information Processing |journal=Physical Review Letters |volume=120 |issue=24 |article-number=240403 |arxiv=1712.00706 |bibcode=2018PhRvL.120x0403L |doi=10.1103/PhysRevLett.120.240403 |pmid=29957003 |s2cid=49562954}}</ref>

=== Testing a system for entanglement ===

A density matrix ''ρ'' is called separable if it can be written as a convex sum of product states, namely
<math display="block">{\rho=\sum_j p_j \rho_j^{(A)}\otimes\rho_j^{(B)}}</math>
with <math>0\le p_j\le 1</math> probabilities. By definition, a state is entangled if it is not separable.

For 2-qubit and qubit-qutrit systems (2&nbsp;×&nbsp;2 and 2&nbsp;×&nbsp;3 respectively) the simple [[Peres–Horodecki criterion]] provides both a necessary and a sufficient criterion for separability, and thus—inadvertently—for detecting entanglement. However, for the general case, the criterion is merely a necessary one for separability, as the problem becomes NP-hard when generalized.<ref name="NP-hard1">{{cite book|last=Gurvits |first=L. |chapter=Classical deterministic complexity of Edmonds' problem and quantum entanglement |title=Proceedings of the 35th ACM Symposium on Theory of Computing |publisher=ACM Press |location=New York |year=2003 |pages=10–19 |doi=10.1145/780542.780545|isbn=1-58113-674-9 }}</ref><ref name="NP-hard2">{{cite journal |author=Gharibian |first=Sevag |year=2010 |title=Strong NP-Hardness of the Quantum Separability Problem |journal=Quantum Information and Computation |volume=10 |pages=343–360 |arxiv=0810.4507 |doi=10.26421/QIC10.3-4-11 |s2cid=621887 |number=3&4}}</ref> Other separability criteria include (but not limited to) the [[range criterion]], [[reduction criterion]], and those based on uncertainty relations.<ref>{{cite journal |last1=Hofmann |first1=Holger F. |last2=Takeuchi |first2=Shigeki |title=Violation of local uncertainty relations as a signature of entanglement |journal=Physical Review A |date=22 September 2003 |volume=68 |issue=3 |article-number=032103 |doi=10.1103/PhysRevA.68.032103 |arxiv=quant-ph/0212090 |bibcode=2003PhRvA..68c2103H |s2cid=54893300 }}</ref><ref>{{cite journal |last1=Gühne |first1=Otfried |title=Characterizing Entanglement via Uncertainty Relations |journal=Physical Review Letters |date=18 March 2004 |volume=92 |issue=11 |article-number=117903 |doi=10.1103/PhysRevLett.92.117903|pmid=15089173 |arxiv=quant-ph/0306194 |bibcode=2004PhRvL..92k7903G |s2cid=5696147 }}</ref><ref>{{cite journal |last1=Gühne |first1=Otfried |last2=Lewenstein |first2=Maciej |title=Entropic uncertainty relations and entanglement |journal=Physical Review A |date=24 August 2004 |volume=70 |issue=2 |article-number=022316 |arxiv=quant-ph/0403219 |doi=10.1103/PhysRevA.70.022316 |bibcode=2004PhRvA..70b2316G |s2cid=118952931}}</ref><ref>{{cite journal |last1=Huang |first1=Yichen |title=Entanglement criteria via concave-function uncertainty relations |journal=Physical Review A |date=29 July 2010 |volume=82 |issue=1 |article-number=012335 |doi=10.1103/PhysRevA.82.012335 |bibcode=2010PhRvA..82a2335H }}</ref> See Ref.<ref>{{cite journal|last1=Gühne|first1=Otfried|last2=Tóth|first2=Géza| title=Entanglement detection |journal=Physics Reports|volume=474|issue=1–6|pages=1–75|doi=10.1016/j.physrep.2009.02.004|arxiv=0811.2803 |bibcode=2009PhR...474....1G |year=2009|s2cid=119288569}}</ref> for a review of separability criteria in discrete-variable systems and Ref.<ref name=FriisEtAl2019entanglement>{{cite journal|last1= Friis |first1=Nicolai |last2= Vitagliano |first2=Giuseppe |last3=Malik |first3=Mehul |last4=Huber |first4=Marcus |date=2019 |title=Entanglement certification from theory to experiment |journal=Nature Reviews Physics |language=en|volume=1|issue=|pages=72–87|doi=10.1038/s42254-018-0003-5 |issn=2522-5820 |arxiv=1906.10929 |s2cid=125658647}}</ref> for a review on techniques and challenges in experimental entanglement certification in discrete-variable systems.

A numerical approach to the problem is suggested by [[Jon Magne Leinaas]], [[Jan Myrheim]] and [[Eirik Ovrum]] in their paper "Geometrical aspects of entanglement".<ref>{{cite journal |last1=Leinaas| first1=Jon Magne| last2=Myrheim| first2=Jan| last3=Ovrum| first3=Eirik| year=2006| title=Geometrical aspects of entanglement| journal=Physical Review A| volume=74| issue=1| article-number=012313| s2cid=119443360| doi=10.1103/PhysRevA.74.012313| arxiv=quant-ph/0605079| bibcode=2006PhRvA..74a2313L}}</ref> Leinaas et al. offer a numerical approach, iteratively refining an estimated separable state towards the target state to be tested, and checking if the target state can indeed be reached.

In continuous variable systems, the Peres–Horodecki criterion also applies. Specifically, Simon<ref>{{cite journal|last1=Simon|first1=R.|title=Peres–Horodecki Separability Criterion for Continuous Variable Systems |journal=Physical Review Letters|volume=84|issue=12|pages=2726–2729|pmid=11017310 |doi=10.1103/PhysRevLett.84.2726|arxiv=quant-ph/9909044|bibcode=2000PhRvL..84.2726S|s2cid=11664720 |year=2000}}</ref> formulated a particular version of the Peres–Horodecki criterion in terms of the second-order moments of canonical operators and showed that it is necessary and sufficient for <math> 1\oplus1 </math>-mode Gaussian states (see Ref.<ref>{{cite journal|last1=Duan|first1=Lu-Ming |last2=Giedke|first2=G.|last3=Cirac|first3=J. I.|last4=Zoller|first4=P.|title=Inseparability Criterion for Continuous Variable Systems|journal=Physical Review Letters|volume=84|issue=12 |pages=2722–2725 |doi=10.1103/PhysRevLett.84.2722|pmid=11017309|arxiv=quant-ph/9908056|bibcode=2000PhRvL..84.2722D |year=2000|s2cid=9948874}}</ref> for a seemingly different but essentially equivalent approach). It was later found<ref>{{cite journal|last1=Werner|first1=R. F.|last2=Wolf|first2=M. M.|title=Bound Entangled Gaussian States|journal=Physical Review Letters|volume=86|issue=16|pages=3658–3661|pmid=11328047 |arxiv=quant-ph/0009118 |doi=10.1103/PhysRevLett.86.3658|bibcode=2001PhRvL..86.3658W|year=2001 |s2cid=20897950}}</ref> that Simon's condition is also necessary and sufficient for <math> 1\oplus n </math>-mode Gaussian states, but no longer sufficient for <math> 2\oplus2 </math>-mode Gaussian states. Simon's condition can be generalized by taking into account the higher order moments of canonical operators<ref>{{cite journal|last1=Shchukin |first1=E.|last2=Vogel |first2=W. |title=Inseparability Criteria for Continuous Bipartite Quantum States |journal=Physical Review Letters|volume=95|issue=23 |article-number=230502|doi=10.1103/PhysRevLett.95.230502|pmid=16384285|bibcode=2005PhRvL..95w0502S |arxiv=quant-ph/0508132|year=2005|s2cid=28595936}}</ref><ref>{{cite journal| last1=Hillery|first1=Mark|last2=Zubairy |first2=M.Suhail|title=Entanglement Conditions for Two-Mode States|journal=Physical Review Letters |volume=96|issue=5|article-number=050503|year=2006|doi=10.1103/PhysRevLett.96.050503|arxiv=quant-ph/0507168 |bibcode=2006PhRvL..96e0503H|pmid=16486912|s2cid=43756465}}</ref> or by using entropic measures.<ref>{{cite journal| last1=Walborn|first1=S.|last2=Taketani|first2=B.|last3=Salles|first3=A.|last4=Toscano |first4=F.|last5=de Matos Filho|first5=R.|title=Entropic Entanglement Criteria for Continuous Variables |journal=Physical Review Letters |volume=103|issue=16|doi=10.1103/PhysRevLett.103.160505|arxiv=0909.0147 |bibcode=2009PhRvL.103p0505W|pmid=19905682|article-number=160505 |year=2009 |s2cid=10523704}}</ref><ref>{{cite journal |last1=Huang |first1=Yichen |date=October 2013 |title=Entanglement Detection: Complexity and Shannon Entropic Criteria |journal=IEEE Transactions on Information Theory |volume=59 |issue=10 |pages=6774–6778 |doi=10.1109/TIT.2013.2257936 |bibcode=2013ITIT...59.6774H |s2cid=7149863}}</ref>

=== In quantum gravity ===
There is a fundamental conflict, referred to as the [[problem of time]], between the way the concept of ''time'' is used in quantum mechanics, and the role it plays in [[general relativity]]. In standard quantum theories time acts as an independent background through which states evolve, while  general relativity treats time as a dynamical variable which relates directly with matter. Part of the effort to reconcile these approaches to time results in the [[Wheeler–DeWitt equation]], which predicts the state of the universe is timeless or static, contrary to ordinary experience.<ref name=Moreva2014>{{cite journal|title= Time from quantum entanglement: an experimental illustration|arxiv=1310.4691|bibcode = 2014PhRvA..89e2122M |doi = 10.1103/PhysRevA.89.052122|volume=89|issue= 5|article-number=052122|journal=Physical Review A|year=2014 | last1 = Moreva | first1 = Ekaterina|s2cid=118638346}}</ref> 
Work started by [[Don Page (physicist)|Don Page]] and [[William Wootters]]<ref>{{cite journal |last1=Page |first1=Don N. |last2=Wootters |first2=William K. |date=15 June 1983 |title=Evolution without evolution: Dynamics described by stationary observables |url=https://link.aps.org/doi/10.1103/PhysRevD.27.2885 |journal=Physical Review D |volume=27 |issue=12 |pages=2885–2892 |doi=10.1103/PhysRevD.27.2885|bibcode=1983PhRvD..27.2885P |url-access=subscription }}</ref><ref>{{cite journal |last=Rovelli |first=Carlo |date=15 October 1990 |title=Quantum mechanics without time: A model |url=https://link.aps.org/doi/10.1103/PhysRevD.42.2638 |journal=Physical Review D |volume=42 |issue=8 |pages=2638–2646 |doi=10.1103/PhysRevD.42.2638|pmid=10013133 |bibcode=1990PhRvD..42.2638R |url-access=subscription }}</ref><ref>{{cite journal |last1=Giovannetti |first1=Vittorio |last2=Lloyd |first2=Seth |last3=Maccone |first3=Lorenzo |date=26 August 2015 |title=Quantum time |url=https://link.aps.org/doi/10.1103/PhysRevD.92.045033 |journal=Physical Review D |volume=92 |issue=4 |article-number=045033 |doi=10.1103/PhysRevD.92.045033|arxiv=1504.04215 |bibcode=2015PhRvD..92d5033G |hdl=1721.1/98287 |s2cid=85537706 }}</ref> suggests that the universe appears to evolve for observers on the inside because of energy entanglement between an evolving system and a clock system, both within the universe.<ref name=Moreva2014/> In this way the overall system can remain timeless while parts experience time via entanglement. The issue remains an open question closely related to attempts at theories of [[quantum gravity]].<ref>{{cite journal |last1=Altaie |first1=M. Basil |last2=Hodgson |first2=Daniel |last3=Beige |first3=Almut |date=3 June 2022 |title=Time and Quantum Clocks: A Review of Recent Developments |journal=Frontiers in Physics |language=English |volume=10 |article-number=897305 |doi=10.3389/fphy.2022.897305 |doi-access=free |arxiv=2203.12564 |bibcode=2022FrP....10.7305A |issn=2296-424X}}</ref><ref>{{cite book |last=Isham |first=C. J. |url=https://link.springer.com/chapter/10.1007/978-94-011-1980-1_6 |title=Integrable Systems, Quantum Groups, and Quantum Field Theories |date=1993 |publisher=Springer Netherlands |isbn=978-94-011-1980-1 |editor-last=Ibort |editor-first=L. A. |location=Dordrecht |pages=157–287 |language=en |doi=10.1007/978-94-011-1980-1_6 |arxiv=gr-qc/9210011 |editor-last2=Rodríguez |editor-first2=M. A.}}</ref>

In general relativity, gravity arises from the curvature of spacetime and that curvature derives from the distribution of matter. However, matter is governed by quantum mechanics. Integration of these two theories faces many problems. In an (unrealistic) model space called the [[anti-de Sitter space]], the [[AdS/CFT correspondence]] allows a quantum gravitational system to be related to a quantum field theory without gravity.<ref name=Swingle2018>{{cite journal |last=Swingle |first=Brian |date=10 March 2018 |title=Spacetime from Entanglement |url=https://www.annualreviews.org/doi/10.1146/annurev-conmatphys-033117-054219 |journal=Annual Review of Condensed Matter Physics |language=en |volume=9 |issue=1 |pages=345–358 |doi=10.1146/annurev-conmatphys-033117-054219 |bibcode=2018ARCMP...9..345S |issn=1947-5454|url-access=subscription }}</ref> Using this correspondence, [[Mark Van Raamsdonk]] suggested that [[spacetime]] arises as an emergent phenomenon of the quantum degrees of freedom that are entangled and live in the boundary of the spacetime.<ref>{{cite journal |last=Van Raamsdonk |first=Mark |date=2010 |title=Building up spacetime with quantum entanglement |url=https://www.worldscientific.com/doi/abs/10.1142/S0218271810018529 |journal=International Journal of Modern Physics D |language=en |volume=19 |issue=14 |pages=2429–2435 |doi=10.1142/S0218271810018529 |bibcode=2010IJMPD..19.2429V |issn=0218-2718|arxiv=1005.3035 }}</ref>

== Experiments demonstrating and using entanglement ==
===Bell tests===
{{main|Bell test}}
A [[Bell test]], also known as ''Bell inequality test'' or ''Bell experiment'', is a real-world physics experiment designed to test the theory of quantum mechanics against the hypothesis of local hidden variables. These tests empirically evaluate the implications of [[Bell's theorem]]. To date, all Bell tests have found that the hypothesis of local hidden variables is inconsistent with the way that physical systems behave. Many types of Bell tests have been performed in physics laboratories, often with the goal of ameliorating problems of experimental design or set-up that could in principle affect the validity of the findings of earlier Bell tests. This is known as "closing loopholes in Bell tests".  In earlier tests, it could not be ruled out that the result at one point could have been subtly transmitted to the remote point, affecting the outcome at the second location.<ref name=":2">{{cite web |last=Francis |first=Matthew |date=30 October 2012 |title=Quantum entanglement shows that reality can't be local |url=https://arstechnica.com/science/2012/10/quantum-entanglement-shows-that-reality-cant-be-local/ |access-date=22 August 2023 |website=Ars Technica |language=en-us}}</ref> However, so-called "loophole-free" Bell tests have since been performed where the locations were sufficiently separated that communications at the speed of light would have taken longer—in one case, 10,000 times longer—than the interval between the measurements.<ref name=":1">{{cite journal |last1=Matson |first1=John |title=Quantum teleportation achieved over record distances |journal=Nature News |date=13 August 2012 |doi=10.1038/nature.2012.11163 |s2cid=124852641}}</ref><ref name=":0">
{{cite journal 
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 |last9 =Wu |first9 =Yu-Ping |last10 =Pan |first10 =Ge-Sheng |last11 =Li |first11 =Li |last12 =Liu |first12 =Nai-Le
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 |s2cid =119293698
}}</ref><ref name="NTR-20151021"/><ref name="hanson"/>

In 2017, Yin et al. reported setting a new quantum entanglement distance record of 1,203&nbsp;km, demonstrating the survival of a two-photon pair and a violation of a Bell inequality, reaching a [[CHSH inequality|CHSH valuation]] of {{val|2.37|0.09}}, under strict Einstein locality conditions, from the [[Quantum Experiments at Space Scale|Micius satellite]] to bases in Lijian, Yunnan and Delingha, Qinghai, increasing the efficiency of transmission over prior fiberoptic experiments by an order of magnitude.<ref>{{cite journal | doi = 10.1126/science.aan3211 | volume=356 | title=Satellite-based entanglement distribution over 1200 kilometers | year=2017 | journal=Science | pages=1140–1144 | last1 = Yin | first1 = Juan | last2 = Cao | first2 = Yuan | last3 = Li | first3 = Yu-Huai | last4 = Liao | first4 = Sheng-Kai | last5 = Zhang | first5 = Liang | last6 = Ren | first6 = Ji-Gang | last7 = Cai | first7 = Wen-Qi | last8 = Liu | first8 = Wei-Yue | last9 = Li | first9 = Bo | last10 = Dai | first10 = Hui | last11 = Li | first11 = Guang-Bing | last12 = Lu | first12 = Qi-Ming | last13 = Gong | first13 = Yun-Hong | last14 = Xu | first14 = Yu | last15 = Li | first15 = Shuang-Lin | last16 = Li | first16 = Feng-Zhi | last17 = Yin | first17 = Ya-Yun | last18 = Jiang | first18 = Zi-Qing | last19 = Li | first19 = Ming | last20 = Jia | first20 = Jian-Jun | last21 = Ren | first21 = Ge | last22 = He | first22 = Dong | last23 = Zhou | first23 = Yi-Lin | last24 = Zhang | first24 = Xiao-Xiang | last25 = Wang | first25 = Na | last26 = Chang | first26 = Xiang | last27 = Zhu | first27 = Zhen-Cai | last28 = Liu | first28 = Nai-Le | last29 = Chen | first29 = Yu-Ao | last30 = Lu | first30 = Chao-Yang | last31 = Shu | first31 = Rong | last32 = Peng | first32 = Cheng-Zhi | last33 = Wang | first33 = Jian-Yu | last34 = Pan | first34 = Jian-Wei | issue=6343 | pmid = 28619937| arxiv=1707.01339 | doi-access = free |display-authors=4 }}</ref><ref>{{cite news | work=[[Science (journal)|Science]] |first=Gabriel |last=Popkin |url=https://www.science.org/content/article/china-s-quantum-satellite-achieves-spooky-action-record-distance | title=China's quantum satellite achieves 'spooky action' at record distance| date=14 June 2017}}</ref>

=== Entanglement of top quarks ===
In 2023 the [[Large Hadron Collider|LHC]] using techniques from [[quantum tomography]] measured entanglement at the highest energy so far,<ref>{{cite journal |last1=Aad |first1=G. |last2=Abbott |first2=B. |last3=Abeling |first3=K. |last4=Abicht |first4=N. J. |last5=Abidi |first5=S. H. |last6=Aboulhorma |first6=A. |last7=Abramowicz |first7=H. |last8=Abreu |first8=H. |last9=Abulaiti |first9=Y. |last10=Acharya |first10=B. S. |last11=Bourdarios |first11=C. Adam |last12=Adamczyk |first12=L. |last13=Addepalli |first13=S. V. |last14=Addison |first14=M. J. |last15=Adelman |first15=J. |date=September 2024 |title=Observation of quantum entanglement with top quarks at the ATLAS detector |journal=Nature |language=en |volume=633 |issue=8030 |pages=542–547 |doi=10.1038/s41586-024-07824-z |pmid=39294352 |pmc=11410654 |arxiv=2311.07288 |bibcode=2024Natur.633..542A |issn=1476-4687}}</ref><ref>{{cite web |date=28 September 2023 |title=ATLAS achieves highest-energy detection of quantum entanglement |url=https://atlas.cern/Updates/Briefing/Top-Entanglement |access-date=21 September 2024 |website=ATLAS |language=en}}</ref><ref>{{cite web |date=18 September 2024 |title=LHC experiments at CERN observe quantum entanglement at the highest energy yet |url=https://home.cern/news/press-release/physics/lhc-experiments-cern-observe-quantum-entanglement-highest-energy-yet |access-date=21 September 2024 |website=CERN |language=en}}</ref> a rare intersection between quantum information and high energy physics based on theoretical work first proposed in 2021.<ref>{{cite journal |last1=Afik |first1=Yoav |last2=de Nova |first2=Juan Ramón Muñoz |date=3 September 2021 |title=Entanglement and quantum tomography with top quarks at the LHC |url=https://link.springer.com/10.1140/epjp/s13360-021-01902-1 |journal=The European Physical Journal Plus |language=en |volume=136 |issue=9 |page=907 |doi=10.1140/epjp/s13360-021-01902-1 |arxiv=2003.02280 |bibcode=2021EPJP..136..907A |issn=2190-5444}}</ref> The experiment was carried by the [[ATLAS experiment|ATLAS]] detector measuring the spin of top-quark pair production and the effect was observed with a more than 5[[Standard deviation|''σ'']] level of significance, the top quark is the heaviest known particle and therefore has a very short lifetime ({{nowrap|<math>\tau</math> ≈ {{val|e=-25|u=s}}}}) being the only quark that decays before undergoing [[hadronization]] (~&nbsp;{{val|e=-23|u=s}}) and spin decorrelation (~&nbsp;{{val|e=-21|u=s}}), so the spin information is transferred without much loss to the leptonic decays products that will be caught by the detector.<ref>{{cite AV media |url=https://www.youtube.com/watch?v=nvXkn6872yk&t=847s |title=Juan Ramón Muñoz de Nova (U. Complutense) on Entanglement & quantum tomography with top quarks |date=13 January 2022 |last=IFT Webinars |access-date=28 September 2024 |via=YouTube}}</ref> The [[spin polarization]] and correlation of the particles was measured and tested for entanglement with [[Concurrence (quantum computing)|concurrence]] as well as the [[Peres–Horodecki criterion]] and subsequently the effect has been confirmed too in the [[CMS experiment|CMS]] detector.<ref>{{cite journal |last=CMS Collaboration |title=Observation of quantum entanglement in top quark pair production in proton–proton collisions at {{math|{{sqrt|''s''}}}} = 13&nbsp;TeV | journal=Reports on Progress in Physics |date=6 June 2024 | volume=87 | issue=11 | doi=10.1088/1361-6633/ad7e4d | pmid=39315475 |arxiv=2406.03976}}</ref><ref>{{cite journal |last=CMS Collaboration |title=<nowiki>Measurements of polarization and spin correlation and observation of entanglement in top quark pairs using <math><mrow>lepton+\text{jets}</mrow></math> events from proton-proton collisions at <math>\sqrt{s}=13\text{ }\text{ }TeV</math></nowiki> | journal=Physical Review D |date=17 September 2024 | volume=110 | issue=11 | article-number=112016 | doi=10.1103/PhysRevD.110.112016 |arxiv=2409.11067}}</ref>

=== Entanglement of macroscopic objects ===
In 2020, researchers reported the quantum entanglement between the [[Vibrations of a circular membrane|motion of a millimetre-sized mechanical oscillator]] and a disparate distant spin system of a cloud of atoms.<ref>{{cite journal |last1=Thomas |first1=Rodrigo A. |last2=Parniak |first2=Michał |last3=Østfeldt |first3=Christoffer |last4=Møller |first4=Christoffer B. |last5=Bærentsen |first5=Christian |last6=Tsaturyan |first6=Yeghishe |last7=Schliesser |first7=Albert |last8=Appel |first8=Jürgen |last9=Zeuthen |first9=Emil |last10=Polzik |first10=Eugene S. |title=Entanglement between distant macroscopic mechanical and spin systems |journal=Nature Physics |date=21 September 2020 |volume=17 |issue=2 |pages=228–233 |doi=10.1038/s41567-020-1031-5 |arxiv=2003.11310 |s2cid=214641162 |url=https://www.nature.com/articles/s41567-020-1031-5 |access-date=9 October 2020 |language=en |issn=1745-2481 |display-authors=4}}</ref> Later work complemented this work by quantum-entangling two mechanical oscillators.<ref>{{cite news |first=Tim |last=Wogan |title=Vibrating drumheads are entangled quantum mechanically |url=https://physicsworld.com/a/vibrating-drumheads-are-entangled-quantum-mechanically/ |access-date=14 June 2021 |work=Physics World |date=17 May 2021}}</ref><ref>{{cite journal |last1=Lépinay |first1=Laure Mercier de |last2=Ockeloen-Korppi |first2=Caspar F. |last3=Woolley |first3=Matthew J. |last4=Sillanpää |first4=Mika A. |title=Quantum mechanics–free subsystem with mechanical oscillators |journal=Science |date=7 May 2021 |volume=372 |issue=6542 |pages=625–629 |doi=10.1126/science.abf5389 |pmid=33958476 |arxiv=2009.12902 |bibcode=2021Sci...372..625M |hdl=1959.4/unsworks_79394 |s2cid=221971015 |url=https://www.science.org/doi/10.1126/science.abf5389 |access-date=14 June 2021 |language=en |issn=0036-8075}}</ref><ref>{{cite journal |last1=Kotler |first1=Shlomi |last2=Peterson |first2=Gabriel A. |last3=Shojaee |first3=Ezad |last4=Lecocq |first4=Florent |last5=Cicak |first5=Katarina |last6=Kwiatkowski |first6=Alex |last7=Geller |first7=Shawn |last8=Glancy |first8=Scott |last9=Knill |first9=Emanuel |last10=Simmonds |first10=Raymond W. |last11=Aumentado |first11=José |last12=Teufel |first12=John D. |title=Direct observation of deterministic macroscopic entanglement |journal=Science |date=7 May 2021 |volume=372 |issue=6542 |pages=622–625 |doi=10.1126/science.abf2998 |pmid=33958475 |arxiv=2004.05515 |bibcode=2021Sci...372..622K |s2cid=233872863 |url=https://www.science.org/doi/10.1126/science.abf2998 |access-date=14 June 2021 |language=en |issn=0036-8075 |display-authors=4}}</ref>

=== Entanglement of quarks and gluons in protons ===
Physicists at [[Brookhaven National Laboratory]] demonstrated quantum entanglement within [[protons]], showing [[quarks]] and [[gluons]] are interdependent rather than isolated particles.<ref>{{cite web |url=https://physicsworld.com/a/entanglement-entropy-in-protons-affects-high-energy-collisions-calculations-reveal/ |title=Entanglement entropy in protons affects high-energy collisions, calculations reveal |publisher=Physics World |date=7 January 2025 }}</ref> Using high-energy electron-proton collisions, they revealed maximal entanglement, reshaping our understanding of proton structure.<ref>{{cite journal |last=Hentschinski |first=Martin |display-authors=et al. |year=2024 |title=QCD evolution of entanglement entropy |journal=IOP Publishing |volume=87 |issue=12 |doi=10.1088/1361-6633/ad910b |pmid=39527914 |arxiv=2408.01259 |bibcode=2024RPPh...87l0501H }}</ref>

== See also ==
{{cols|colwidth=16em}}
* [[Concurrence (quantum computing)|Concurrence]]
* [[Controlled NOT gate|CNOT gate]]
* [[Einstein's thought experiments]]
* [[Entanglement witness]]
* [[ER = EPR]]
* [[Multipartite entanglement]]
* [[Normally distributed and uncorrelated does not imply independent]]
* [[Pauli exclusion principle]]
* [[Quantum coherence]]
* [[Quantum discord]]
* [[Quantum network]]
* [[Quantum phase transition]]
* [[Quantum pseudo-telepathy]]
* [[Retrocausality]]
* [[Squashed entanglement]]
* [[Stern–Gerlach experiment]]
* [[John Clive Ward|Ward's probability amplitude]]
{{colend}}
{{Portal|Physics}}

== References ==
{{reflist|30em}}

== Further reading ==
{{refbegin}}
* {{cite journal |last1=Albert |first1=David Z. |last2=Galchen |first2=Rivka |title=Was Einstein Wrong?: A Quantum Threat to Special Relativity |journal=[[Scientific American]] |volume=300 |number=3 |pages=32–39 |doi=10.1038/scientificamerican0309-32 |url=https://www.scientificamerican.com/article/was-einstein-wrong-about-relativity/ |pmid=19253771 |year=2009|url-access=subscription }}
* {{cite book |last=Cramer |first=J. G. |title=The Quantum Handshake: Entanglement, Nonlocality and Transactions |publisher=Springer Verlag |year=2015 |isbn=978-3-319-24642-0}}
* {{cite book |last=Duarte |first=F. J. |author-link=F. J. Duarte |title=Fundamentals of Quantum Entanglement |publisher=Institute of Physics |location=Bristol, United Kingdom |year=2019 |isbn=978-0-7503-2226-3}}
* {{cite journal |vauthors=Bhaskara VS, Panigrahi PK |title=Generalized concurrence measure for faithful quantification of multiparticle pure state entanglement using Lagrange's identity and wedge product |journal=Quantum Information Processing |arxiv=1607.00164 |doi=10.1007/s11128-017-1568-0 |year=2017 |volume=16 |issue=5 |article-number=118 |bibcode=2017QuIP...16..118B |s2cid=43754114}}
* {{cite journal |vauthors=Swain SN, Bhaskara VS, Panigrahi PK |title=Generalized entanglement measure for continuous-variable systems |journal=Physical Review A |arxiv=1706.01448 |doi=10.1103/PhysRevA.105.052441 |year=2022 |volume=105 |issue=5 |article-number=052441 |bibcode=2022PhRvA.105e2441S |s2cid=239885759}}
* {{cite book |year=2009 |last=Jaeger |first=G. |title=Entanglement, Information, and the Interpretation of Quantum Mechanics |location=Heildelberg, Germany |publisher=Springer |isbn=978-3-540-92127-1}}
* {{cite book |last=Steward |first=E. G. |title=Quantum Mechanics: Its Early Development and the Road to Entanglement |publisher=Imperial College Press |year=2008 |isbn=978-1-86094-978-4}}
* {{cite book|last=Wilde |first=Mark M. |author-link=Mark Wilde |title=Quantum Information Theory |edition=2nd |publisher=Cambridge University Press |year=2017 |doi=10.1017/9781316809976 <!-- whole book, not .001 like arxiv says --> |isbn=978-1-316-80997-6 |arxiv=1106.1445}}
{{refend}}

== External links ==
{{Wikiquote}}
* [https://www.youtube.com/watch?v=xM3GOXaci7w Explanatory video by ''Scientific American'' magazine]
* [https://web.archive.org/web/20121025073450/http://www.didaktik.physik.uni-erlangen.de/quantumlab/english/index.html Entanglement experiment with photon pairs – interactive]
* Audio – Cain/Gay (2009) [http://www.astronomycast.com/physics/ep-140-entanglement/ Astronomy Cast] Entanglement
* [https://www.youtube.com/watch?v=ta09WXiUqcQ "Spooky Actions at a Distance?": Oppenheimer Lecture, Prof. David Mermin (Cornell University) Univ. California, Berkeley, 2008.] Non-mathematical popular lecture on YouTube, posted Mar 2008
* [https://demonstrations.wolfram.com/QuantumEntanglementVersusClassicalCorrelation/ "Quantum Entanglement versus Classical Correlation" (Interactive demonstration)]

{{Quantum mechanics topics}}
{{authority control}}

{{DEFAULTSORT:Quantum Entanglement}}
[[Category:Quantum information science]]
[[Category:Quantum measurement]]