Quantum_A_Spooky_Action_at_a_Distance.txt
{{Author|Harold Foppele}}
{{Physics}}
{{Quiz}}
{{Learning project}}
←[[Quantum]]
This page is an advanced conceptual overview of quantum entanglement intended for readers with prior coursework in quantum mechanics. It emphasizes conceptual structure rather than step-by-step instruction.
== Quantum Entanglemen Course Overview==
[[File:Artistic impression of an atom 2b.png|thumb|Artistic impression of an atom 2b]]
Welcome to the Wikiversity course on Quantum Entanglement. <br>This course explores the fundamental concepts, history, mathematical details, and applications of quantum entanglement, a key phenomenon in quantum mechanics. Learning Objectives By the end of this course, learners will be able to:
*Understand the basic principles of quantum entanglement and its differences from classical correlations.
*Explain the historical development, including the EPR paradox and Bell's theorem.
*Describe mathematical formulations of entangled states.
*Discuss applications in quantum information and experiments demonstrating entanglement.
'''Prerequisites'''<BR>Basic knowledge of quantum mechanics, linear algebra, and probability is recommended.
====Course Structure====
The course is divided into modules based on the provided content. Each module includes readings, key concepts, and optional discussion questions.
=== Module 1: Introduction to Quantum Entanglement ===
[[File:SPDC figure.png|thumb|[[W:Spontaneous parametric down-conversion|Spontaneous parametric down-conversion]] process can split photons into type II photon pairs with mutually perpendicular polarization.]]Quantum entanglement occurs when the quantum state of a composite system cannot be factored into independent states of its individual particles, regardless of the distance separating them. This behavior has no analogue in classical physics.<ref name="horodecki2007">Horodecki, R., Horodecki, P., Horodecki, M., & Horodecki, K. (2009). Quantum entanglement. Reviews of Modern Physics, 81(2), 865–942. https://doi.org/10.1103/RevModPhys.81.865</ref>
Measurements of properties such as spin or polarization on entangled particles yield correlated outcomes. In a spin-singlet pair with total spin zero, measuring one particle to be spin up along a given axis guarantees the other will be spin down along that axis. The act of measurement projects the joint quantum state of the system into a definite outcome, meaning the entangled state applies to the system as a whole rather than to the individual particles.
This weirdness kicked off with a 1935 paper from 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 some follow-ups from Erwin Schrödinger,<ref name="Schrödinger1935">
{{cite journal|author=Schrödinger|first=Erwin|author-link=W:Erwin Schrödinger|year=1935|title=Discussion of probability relations between separated systems|journal=[[W:Mathematical Proceedings of the Cambridge Philosophical Society|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=W:Erwin Schrödinger|year=1936|title=Probability relations between separated systems|journal=[[W:Mathematical Proceedings of the Cambridge Philosophical Society|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> laying out what's now called the EPR paradox. Einstein and crew thought it was nuts because it messed with local realism's take on cause and effect<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|url=http://philosophyfaculty.ucsd.edu/faculty/wuthrich/GSSPP09/Files/BellJohnS1981Speakable_BertlmannsSocks.pdf|title=Speakable and Unspeakable in Quantum Mechanics|last=Bell|first=J. S.|publisher=[[W:CERN|CERN]]|year=1987|isbn=0-521-33495-0|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 figured quantum mechanics must be missing something.
But later, experiments proved quantum's predictions right, with polarization or spin measurements on distant entangled particles breaking Bell's inequality in stats.<ref name="Clauser" /><ref name=":0" /><ref name=":1" /><ref name=":2" /> You cannot explain these links with local hidden variables inside the particles.
Still, even though entanglement creates these correlations over huge distances, you cannot use it to send messages faster than the speed of light.<ref>{{cite book|title=The road to reality: a complete guide to the laws of the universe|last=Penrose|first=Roger|publisher=Jonathan Cape|year=2004|isbn=978-0-224-04447-9|location=London|page=603|author-link=W: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 with photons,<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> electrons,<ref name="NTR-20151021">{{cite journal|author=Hensen, B.|display-authors=etal|date=21 October 2015|title=Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres|journal=[[W:Nature (journal)|Nature]]|volume=526|issue=7575|pages=682–686|arxiv=1508.05949|bibcode=2015Natur.526..682H|doi=10.1038/nature15759|pmid=26503041|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 quarks,<ref>{{cite web|url=https://physicsworld.com/a/quantum-entanglement-observed-in-top-quarks/|title=Quantum entanglement observed in top quarks|last=Boerkamp|first=Martijn|date=11 October 2023|website=[[W:Physics World]]}}</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 [[W:quantum communication|communication]] and [[W:quantum computing|computation]] is an area of research and development.
==== Key Concepts ====
*Definition of entanglement.
*EPR paradox and "spooky action at a distance."
*No faster-than-light communication.
==== Discussion Question ====
How does quantum entanglement challenge classical intuitions about locality?
=== Module 2: History of Quantum Entanglement ===
[[File:Portrait_of_Albert_Einstein_and_Others_%281879-1955%29%2C_Physicist_-_Restoration1.jpg|thumb|Portrait of Albert Einstein and Others]]
{{hatnote| Background: [[W:History of quantum mechanics|History of quantum mechanics]]}}
Albert Einstein and Niels Bohr got into a long, argument about how to interpret quantum mechanics, the Bohr-Einstein debates. Einstein came up with this thought experiment about a box with a photon, pointing out that what you measure at the box changes what you can tell about the photon over there. He worked this out by 1931, basically looking at what we later call 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 wrote in his book on group theory and quantum mechanics that when parts of a system interact, the whole thing has this Gestalt quality, the whole's more than just the parts added up.<ref>{{cite book|title=Gruppentheorie und Quantenmechanik|title-link=Gruppentheorie und Quantenmechanik|last=Weyl|first=Hermann|year=1931|edition=2nd|pages=92–93|translator-last=Robertson|translator-first=H. P.|trans-title=Group Theory and Quantum Mechanics|author-link=W:Hermann Weyl|translator-link=Howard P. Robertson}}</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 figured out the key equations for entanglement but didn't publish them.<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 looked at the math of an electron and photon interacting and spotted what we'd call 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 year, Einstein, Podolsky, and Rosen put out their paper on the EPR paradox, arguing that quantum mechanics' wave function doesn't give a complete picture of reality.<ref name="Einstein1935" /> They talked about two systems that interact and then split apart, and after that, quantum mechanics can't describe them on their own.
Right after that paper dropped, Erwin Schrödinger wrote Einstein a letter in German, using the word "Verschränkung" (which he translated as "entanglement") for those EPR situations.<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 then wrote a full paper explaining 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> calling it not just one but ''the'' key feature of quantum mechanics that sets it apart from classical thinking.<ref name="Schrödinger1935" />
Like Einstein, Schrödinger wasn't thrilled with entanglement, it seemed to break the relativity rule on how fast info can travel.<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 mocked it as "spukhafte Fernwirkung" or "spooky action at a distance," where measuring something here instantly sets a property over there.<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><ref name="MerminMoon-1985">{{Cite journal|last=Mermin|first=N. David|author-link=W: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|pages=38–47|bibcode=1985PhT....38d..38M|doi=10.1063/1.880968|number=4}}</ref>
In 1946, John Archibald Wheeler suggested checking the polarization of gamma-ray photon pairs from electron-positron annihilation.<ref>{{cite journal|last=Wheeler|first=J. A.|author-link=W:John Archibald Wheeler|year=1946|title=Polyelectrons|journal=Annals of the New York Academy of Sciences|volume=48|pages=219–238|doi=10.1111/j.1749-6632.1946.tb31764.x|number=3}}</ref> Chien-Shiung Wu and I. Shaknov did the 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=[[W:Physical Review]]|volume=77|issue=1|page=136|bibcode=1950PhRv...77..136W|doi=10.1103/PhysRev.77.136}}</ref> showing you could make EPR-type entangled pairs in a lab.<ref>
{{cite journal|last1=Duarte|first1=F. J.|year=2012|title=The origin of quantum entanglement experiments based on polarization measurements|journal=[[W:European Physical Journal H]]|volume=37|issue=2|pages=311–318|bibcode=2012EPJH...37..311D|doi=10.1140/epjh/e2012-20047-y|author1-link=W:F. J. Duarte}}</ref>
Even though Schrödinger called it crucial, not much got written about entanglement for years after his paper.<ref name="Schroeder-2017" /> Then in 1964, John S. Bell showed there's a limit, Bell's inequality, on how strong correlations can be in any local realism theory, and quantum predicts breaking that for some entangled systems.<ref name=":4">{{cite journal|author=Bell|first=J. S.|author-link=W:John Stewart Bell|year=1964|title=On the Einstein Poldolsky Rosen paradox|journal=[[W: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}} You can test this, and experiments started with 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 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>
Bell was not keen on students chasing this stuff, he thought it was too fringe, but after a lecture at Oxford, a student named Artur Ekert suggested using Bell inequality violations for communication.<ref name="ekert91">{{cite journal
|last1=Ekert
|first1=Artur K.
|year=1991
|title=Quantum cryptography based on Bell's theorem
|journal=Physical Review Letters
|volume=67
|issue=6
|pages=661–663
|doi=10.1103/PhysRevLett.67.661
|pmid=10044956
|bibcode=1991PhRvL..67..661E
}}</ref><ref name="horodecki2007" />{{rp|874|q=The first discovery within quantum information theory, which involves entanglement, is due to Ekert 1991.}}
In 1992, academics started using entanglement to suggest quantum teleportation,<ref name="BBCJPW93">{{cite journal|last1=Bennett|first1=Charles H.|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=[[W:Physical Review Letters]]|volume=70|issue=13|pages=1895–1899|bibcode=1993PhRvL..70.1895B|citeseerx=10.1.1.46.9405|doi=10.1103/PhysRevLett.70.1895|pmid=10053414|doi-access=free|author-link1=Charles H. Bennett (computer scientist)}}</ref> and have experimented this by 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|journal=[[W:Physical Review Focus|Physics (Physical Review Focus)]]|volume=25}}</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.|date=9 February 1998|title=Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein–Podolsky–Rosen Channels|journal=[[W:Physical Review Letters|Physical Review Letters]]|volume=80|issue=6|pages=1121–1125|arxiv=quant-ph/9710013|bibcode=1998PhRvL..80.1121B|doi=10.1103/PhysRevLett.80.1121|s2cid=15020942}}</ref> In 1990, Anton Zeilinger used parametric down-conversion to create entanglement, leading to 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 showing 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=2011.05669|bibcode=2022PhP....24..125D }}</ref>
In 2022, the Nobel Prize in Physics went to Aspect, Clauser, and Zeilinger for their entangled photon experiments, proving Bell inequality violations, and starting quantum info 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=[[W:The Royal Swedish Academy of Sciences|The Royal Swedish Academy of Sciences]]|access-date=5 October 2022}}</ref>
==== Key Concepts ====
*Bohr-Einstein debates.
*Contributions from Schrödinger, Bell, and others.
*First experiments and Nobel recognition.
==== Discussion Question ====
Why did Einstein describe entanglement as "spooky action at a distance"?
=== Module 3: Core Concepts and Paradox ===[[File:Eprheaders.gif|thumb|Eprheaders]][[File:Quantum Entanglement Experiment via Spontaneous Parametric Down-Conversion (SPDC).jpg|thumb|Quantum Entanglement Experiment via Spontaneous Parametric Down-Conversion (SPDC)]]
* Concept: Meaning of entanglementLike energy drives machines, entanglement powers tasks in communication and computing.<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}}Basically, knowing everything about the whole system doesn't mean you know everything about its parts.<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> For a pair of entangled particles, measuring one can tie strongly to what you get from the other. But it is not the everyday correlation, the potential for correlation that turns reality in the right setup.<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}} These links from entangled states can't be mimicked by classical odds.<ref name="Holevo2001">{{cite book|title=Statistical Structure of Quantum Theory|last=Holevo|first=Alexander S.|publisher=Springer|year=2001|isbn=3-540-42082-7|series=[[W:Lecture Notes in Physics|Lecture Notes in Physics. Monographs]]|author-link=W:Alexander Holevo}}</ref>{{rp|33}}
A subatomic particle splitting into an entangled duo. The split follows standard rules, so measuring one predicts the other (keeping totals like momentum or energy steady). A spin-zero particle breaks into two spin-1/2 ones. No orbital spin means total spin post-split is zero. Measure the first as spin up on an axis, the second's down on that axis. That's the anti-correlated singlet state. You might think hidden variables inside explain it, like one has "up," the other "down." Bell used the story of his pal Bertlmann, who always wore odd-colored socks: see one pink, know the other's not.<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> But to see quantum entanglement's true weirdness, you need various experiments, like spins on different axes, and compare those correlations.<ref name="Zwiebach2022">{{cite book|title=Mastering Quantum Mechanics: Essentials, Theory, and Applications|last=Zwiebach|first=Barton|publisher=MIT Press|year=2022|isbn=978-0-262-04613-8|author-link=W:Barton Zwiebach}}</ref>{{rp|§18.8}}
Systems get entangled through different interactions. Check the methods section below for lab ways to make it happen. It breaks when particles decohere from environmental pokes, like measurements—the particles entangle with the surroundings, losing their own entanglement.<ref name="Peres1993">{{cite book|title=Quantum Theory: Concepts and Methods|title-link=Quantum Theory: Concepts and Methods|last=Peres|first=Asher|publisher=Kluwer|year=1993|isbn=0-7923-2549-4|author-link=W:Asher Peres}}</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>
Math-wise, an entangled system's state can't break down into products of its parts' states, they are one unit. One part need the other.<ref name="Mermin2007">{{cite book|title=Quantum Computer Science: An Introduction|last=Mermin|first=N. David|publisher=Cambridge University Press|year=2007|isbn=978-0-521-87658-2|author-link=W:N. David Mermin}}</ref>{{rp|18–19}}<ref name="Zwiebach2022" />{{rp|§1.5}} A combined system's state is a sum or superposition of local products; entangled if not reducible to one 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 ===
[[File:EPR illustration.svg|thumb|EPR illustration]]
{{main|W:EPR paradox|l1 = EPR paradox]}}
The singlet state is key to one take on the EPR paradox. In David Bohm's version, a source shoots particles opposite ways. Each pair's state is entangled.<ref>{{cite book|title=Quantum Theory|last=Bohm|first=David|publisher=Dover|year=1989|isbn=0-486-65969-0|edition=reprint|pages=611–622|author-link=W:David Bohm|orig-date=1951}}</ref> Textbooks say measuring spin on one collapses the pair's wave function, giving each a definite spin (up or down) on that axis. It's random, 50-50. But same-axis measures are always opposite. So one measurement's random result seems sent to the other to match.<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}}
You can set distances and timings so the measurements are spacelike, any cause linking them would beat light speed. Relativity says no info travels that fast. You can't even say which happened first; frames differ on order for spacelike events x1 and x2. So correlations aren't one determining the other, observers would argue over cause and effect.<ref>{{cite journal|last=Peres|first=Asher|author-link=W:Asher Peres|date=2000-01-18|title=Classical interventions in quantum systems. II. Relativistic invariance|journal=Physical Review A|volume=61|issue=2|arxiv=quant-ph/9906034|bibcode=2000PhRvA..61b2117P|doi=10.1103/PhysRevA.61.022117|article-number=022117}}</ref>
One fix: quantum theory's incomplete, outcomes from preset "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> Particles carry split-time info setting spins, no need for transmission. Einstein thought this solved it, making quantum's randomness incomplete.
But local hidden variable ideas flop with different-axis spins. Stats-wise, many pairs would meet Bell's inequality if local realism held. Experiments say no.<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> And in moving frames where one measure comes before the other, correlations hold.<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 big problem with different-axis spins: they can't have set values simultaneously, they're incompatible, limited by uncertainty. Unlike classical, where you measure anything together precisely. Math shows compatible measures can't violate Bell,<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> so entanglement's purely quantum.
'''Key Concepts'''
==== Singlet state and correlations. ====
*EPR paradox resolution via Bell inequalities.
*Incompatibility with local hidden variables.
==== Discussion Question ====
How do Bell inequalities demonstrate the non-classical nature of entanglement?
===Module 4: Nonlocality, Resources, and Mathematical Details===
[[File:PHYSIQUE QUANTIQUE LES TRAVAUX D'ALAIN ASPECT, PROFESSEUR À L’X, COURONNÉS DU PRIX NOBEL 2022 (52489082838).jpg|thumb|300x300px|Alain Aspect explaining his experiment]]
==== Nonlocality and entanglement ====
Alain Aspect's pioneering experiments in the early 1980s marked a turning point, providing strong evidence for quantum nonlocality. Using entangled photon pairs from calcium atomic cascades and rapidly switching polarizers (via acousto-optic modulators), his setup ensured measurement settings were chosen in a space-like separated manner, closing the locality loophole and demonstrating clear violations of Bell inequalities.
You need entanglement to break a Bell inequality. But just having it is not enough,<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> like Bell pointed out in '64.<ref name=":4" /> Look at Werner states for pairs: some show entanglement but fit local hidden models, no Bell break.<ref name="werner1989">{{cite journal|last=Werner|first=R. F.|author-link=W:Reinhard F. Werner|year=1989|title=Quantum States with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model|journal=[[W:Physical Review A]]|volume=40|issue=8|pages=4277–4281|bibcode=1989PhRvA..40.4277W|doi=10.1103/PhysRevA.40.4277|pmid=9902666}}</ref> Same for bigger groups.<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>
Breaking Bell inequalities gets called quantum nonlocality. The term stirs debate. Some say it hints wrongly at superluminal physical 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> Failing local hidden models does not mean quantum's truly nonlocal.<ref>{{cite book|title=The Interpretation of Quantum Mechanics|last=Omnès|first=Roland|publisher=Princeton University Press|year=1994|isbn=978-0-691-03669-4|pages=399–400|author-link=W:Roland Omnès}}</ref><ref>{{cite journal|last=Mermin|first=N. D.|author-link=W:N. David Mermin|year=1999|title=What Do These Correlations Know About Reality? Nonlocality and the Absurd|journal=[[W:Foundations of Physics]]|volume=29|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 [Un]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> But "nonlocality" stuck around anyway.<ref name="Scarani" />
Sometimes "nonlocality" means other things, like if states can be told apart locally.<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> Quantum field theory's called local because observables in spacelike spots commute.<ref name="Brunner-RMP2014" /><ref>{{cite book|title=Local Quantum Physics: Fields, Particles, Algebras|last=Haag|first=Rudolf|publisher=Springer|year=1996|isbn=3-540-61451-6|edition=2nd|pages=107–108|author-link=W:Rudolf Haag}}</ref> We won't dig into those other meanings here.
== Mathematical details ==
The following subsections use the formalism and theoretical framework developed in the articles [[W:bra–ket notation|bra–ket notation]] and [[W:mathematical formulation of quantum mechanics|mathematical formulation of quantum mechanics]].
=== Pure states ===
[[File:Bell local hidden variables geometry with screening (green).png|thumb|Spacetime diagram for Bell's local hidden variable proof. Dash lines show relativistically valid region.]]
Consider two arbitrary quantum systems {{mvar|A}} and {{mvar|B}}, with respective [[W:Hilbert space|Hilbert space]]s {{mvar|HA}} and {{mvar|HB}}. The Hilbert space of the composite system is the [[W:tensor product|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 [[W:product state|product state]]s. However, not all states of the composite system are separable. Fix a [[W:basis (linear algebra)|basis]] <math>{|i\rangle_A}</math> for {{mvar|HA}} and a basis <math>{|j\rangle_B}</math> for {{mvar|HB}}. The most general state in {{math|''HA'' ⊗ ''HB''}} 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]</math>, <math>[c^B_j]</math> so that <math>c_{ij}= c^A_i c^B_j</math>, yielding <math>|\psi\rangle_A = \sum_{i} c^A_{i} |i\rangle_A</math> and <math>|\phi\rangle_B = \sum_{j} c^B_{j} |j\rangle_B.</math> It is inseparable if for any vectors <math>[c^A_i]</math>, <math>[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_i c^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|HA}} and two basis vectors <math>|0\rangle_B, |1\rangle_B</math> of {{mvar|HB}}, 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 [[W:pure state|pure state]]. Another way to say this is that while the [[W:von Neumann entropy|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 [[W:Bell states|Bell states]], which are (maximally) entangled pure states (pure states of the <math>H_A \otimes H_B</math> space, but which cannot be separated into pure states of each {{mvar|HA}} and {{mvar|HB}}).<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.
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 [[W:no-communication theorem|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 it's called an 'ensemble', described by a [[W:density matrix|density matrix]], which is a [[W:positive-semidefinite matrix|positive-semidefinite matrix]], or a [[W:trace class|trace class]] when the state space is infinite-dimensional, and which has trace 1. By the [[W:spectral theorem|spectral theorem]], such a matrix takes the general form:
: <math>\rho = \sum_i w_i |\alpha_i\rangle \langle\alpha_i|,</math>
where the <math>w_i</math> are positive-valued probabilities (they sum up to 1), the vectors <math>|\alpha_i\rangle</math> are unit vectors, and in the infinite-dimensional case, we would take the closure of such states in the trace norm. Interpret <math>\rho</math> 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 a [[W:#Reduced density matrices|density matrices]] is used 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 [[W:Electron|electron]]s towards an observer. The electrons' Hilbert spaces are [[W: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>\mathbf{z}</math> direction, and the other with state <math>|\mathbf{y}-\rangle</math> with spins aligned in the negative <math>\mathbf{y}</math> 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_A \otimes H_B.</math> 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 <math>w_i</math> are positively valued probabilities, <math>\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
: <math>\rho = \sum_i w_i \rho_i^A \otimes \rho_i^B,</math>
where the <math>w_i</math> 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 [[W:NP-hard|NP-hard]].<ref>{{cite book |last=Gurvits |first=L. |title=Proceedings of the thirty-five annual ACM symposium on Theory of computing |year=2003 |page=10 |chapter=Classical deterministic complexity of Edmonds' Problem and quantum entanglement |doi=10.1145/780542.780545}}</ref> For the <math>2\times2</math> and <math>2\times3</math> cases, a necessary and sufficient criterion for separability is given by the famous [[W: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 }}</ref>
=== Reduced density matrices ===
[[File:Bloch sphere representation of optimal POVM and states for unambiguous quantum state discrimination (yellow).png|thumb|Bloch sphere representation of optimal POVM and states for unambiguous quantum state discrimination (yellow)]]
The idea of a reduced density matrix was introduced by [[W:Paul Dirac|Paul Dirac]] in 1930.
Consider as above systems {{mvar|A}} and {{mvar|B}} each with a Hilbert space {{mvar|HA, HB}}. 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 [[W:projection operator|projection operator]] onto this state. The state of {{mvar|A}} is the [[W:partial trace|partial trace]] of {{mvar|ρT}} 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) = \mathrm{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|ρA}} is sometimes called the reduced density matrix of {{mvar|ρ}} on subsystem {{mvar|A}}.
For example, the reduced density matrix of {{mvar|A}} for the entangled state
: <math>\frac{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 = \frac{1}{2}\left(|0\rangle_A\langle0|_A + |1\rangle_A\langle1|_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
: <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 <math>\rho</math> is entangled if and only if its reduced states are mixed rather than pure.<ref name="Rau2021"/>{{rp|131}}
=== Entanglement as a resource ===
[[File:Quantum teleportatie en verstrengeling hulpbron.png|thumb|250px|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.
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 [[W:quantum operation|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 [[W:LOCC|LOCC]] (local operations and classical communication). These operations do n|LOCCot 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 "[[W:quantum teleportation|teleported]]" to Bob's laboratory through this protocol.<ref name="horodecki2007" />{{rp|875}}<ref>{{cite journal|arxiv=1505.07831 |title=Advances in Quantum Teleportation |first1=S. |last1=Pirandola |journal=Nature Photonics |volume=9 |pages=641–652 |year=2015 |issue=10 |doi=10.1038/nphoton.2015.154}}</ref>
[[W:Entanglement swapping|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=W:Anton Zeilinger|year=1998|title=Experimental entanglement swapping: Entangling photons that never interacted|journal=[[W:Physical Review Letters]]|volume=80|pages=3891–3894|bibcode=1998PhRvL..80.3891P|doi=10.1103/PhysRevLett.80.3891|number=18}}</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. So 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" />
==== Key Concepts ====
*Quantum nonlocality and Bell inequalities.
*Pure and mixed states, density matrices.
*Entanglement as a resource for teleportation and swapping.
===== Discussion Question =====
How does entanglement enable quantum teleportation without violating relativity?
=== Module 5: Multipartite Entanglement, Measures, and Applications ===
====== Multipartite entanglement ======
[[File:Multipartite Entanglement GHZ vs W states.png|thumb|Multipartite Entanglement GHZ vs W states]]
{{main|W:Multipartite entanglement|l1 = 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 [[W:Greenberger–Horne–Zeilinger state|Greenberger–Horne–Zeilinger (GHZ) state]],
: <math>|\mathrm{GHZ}\rangle = \frac{|000\rangle + |111\rangle}{\sqrt{2}}.</math>
Another three-qubit example is the [[W:W state|W state]]:
: <math>|\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 [[W: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=W:Barbara Kraus|year=2010|title=Local unitary equivalence of multipartite pure states|journal=Physical Review Letters|volume=104|issue=2|arxiv=0909.5152|bibcode=2010PhRvL.104b0504K|doi=10.1103/PhysRevLett.104.020504|pmid=20366579|article-number=020504|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 [[W: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 "[[W: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 ''[[W: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 ''[[W: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 ===
[[File:Entropie as Entanglement Measure.png|thumb|300px|Von Neumann Entropie as Entanglement Measure]]
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 [[W:information theory|H]], the [[W:Shannon entropy|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 <math>\rho</math> is a probability distribution over an ensemble, this leads naturally to the definition of the [[W:von Neumann entropy|von Neumann entropy]]:<ref name="Peres1993"/>{{rp|264}}
: <math>S(\rho) = - \mathrm{Tr} \left( \rho \log_2 \rho \right),</math>
which can be expressed in terms of the [[W:eigenvalue|eigenvalue]]s of <math>\rho</math>:
: <math>S(\rho) = - \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</math> 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)</math>, which can be shown to be the maximum entropy for <math>2 \times 2</math> 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=W:Vlatko Vedral|year=2002|title=The role of relative entropy in quantum information theory|journal=Reviews of Modern Physics|volume=74|issue=1|pages=197–234|arxiv=quant-ph/0102094|bibcode=2002RvMP...74..197V|doi=10.1103/RevModPhys.74.197|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 <math>{1/n, \dots, 1/n}</math>.
Therefore, a bipartite pure state <math>\rho \in \mathcal{H}_A \otimes \mathcal{H}_B</math> is said to be a ''maximally entangled state'' if the reduced state of each subsystem of <math>\rho</math> 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}}
[[W:Rényi entropy|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, [[W: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====
*[[W:entanglement distillation|Distillable entanglement]]
*[[W:Entanglement of formation|Entanglement of formation]]
*[[W:Concurrence (quantum computing)|Concurrence]]
*[[W:quantum relative entropy|Relative entropy of entanglement]]
*[[W:Squashed entanglement|Squashed entanglement]]
*[[W: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 ([[W:NP-hard|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 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 ===
[[File:Reeh-Schlieder Theorem.jpg|thumb|225x225px|Reeh-Schlieder Theorem]]
The [[W:Reeh–Schlieder theorem|Reeh–Schlieder theorem]] of [[W:quantum field theory|quantum field theory]] is sometimes interpreted as saying that entanglement is omnipresent in the [[W:quantum vacuum|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 [[W:quantum information theory|quantum information theory]]. With the aid of entanglement, otherwise impossible tasks may be achieved. Among the best-known applications of entanglement are [[W:superdense coding|superdense coding]] and [[W:quantum teleportation|quantum teleportation]].<ref name="Bouwmeester-1997" /> Most researchers believe that entanglement is necessary to realize [[W: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 [[W:quantum cryptography|quantum cryptography]],<ref name="ekert91" /> but to prove the security of [[W:quantum key distribution|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 ''[[W: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 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> This idea has been adapted to make infrared images using standard cameras 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 ===
[[File:Canonical Entangled States in Quantum Theory.jpg|thumb|350px|Canonical Entangled States in Quantum Theory]]
There are several canonical entangled states that appear often in theory and experiments.
For two qubits, the Bell states are:
<math>
|\Phi^\pm\rangle = \frac{1}{\sqrt{2}} \left( |0\rangle_A \otimes |0\rangle_B \pm |1\rangle_A \otimes |1\rangle_B \right)
</math>
<math>
|\Psi^\pm\rangle = \frac{1}{\sqrt{2}} \left( |0\rangle_A \otimes |1\rangle_B \pm |1\rangle_A \otimes |0\rangle_B \right)
</math>
These four pure states are all maximally entangled and form an [[W:orthonormal|orthonormal]] [[W:basis (linear algebra)|basis]] of the Hilbert space of the two qubits.
For <math>M > 2</math> qubits, the [[W: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 <math>M = 2</math>. GHZ states are occasionally extended to [[W:qudit|qudit]]s, i.e., systems of <math>d</math> dimensions.<ref>{{Cite journal|last1=Caves|first1=Carlton M.|last2=Fuchs|first2=Christopher A.|last3=Schack|first3=Rüdiger|date=2002-08-20|title=Unknown quantum states: The quantum de Finetti representation|journal=[[W:Journal of Mathematical Physics]]|volume=43|pages=4537–4559|arxiv=quant-ph/0104088|bibcode=2002JMP....43.4537C|doi=10.1063/1.1494471|number=9}}</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>
Spin squeezed states for <math>M > 2</math> qubits are necessarily entangled and useful for precision measurements.<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><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 [[W:boson|boson]]ic modes, a [[W:NOON state|NOON state]] is:
<math> |\psi_\text{NOON}\rangle = \frac{|N\rangle_a |0\rangle_b + |0\rangle_a |N\rangle_b}{\sqrt{2}}. </math>
Finally, [[W:twin Fock states|twin Fock states]] for bosonic modes can be used to reach the [[W:Heisenberg limit|Heisenberg limit]]|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>
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>{{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 usually comes from direct particle interactions. Common methods include:
*Spontaneous parametric down-conversion (photon pairs entangled in polarization).<ref name="horodecki2007" />
*Fiber couplers to mix photons or bi-exciton decay in quantum dots.<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=22040551 }}</ref>
*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>
*Particle-antiparticle partial wavefunction overlap (Hardy's interferometer).
*Systems that never interact directly can be entangled via swapping or wavefunction overlap.<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 <math>\rho</math> is called separable if it can be written as a convex sum of product states:
<math>\rho = \sum_j p_j \rho_j^{(A)} \otimes \rho_j^{(B)}, \quad 0 \le p_j \le 1.</math>
By definition, a state is entangled if it is not separable. For 2-qubit and qubit-qutrit systems, the [[W:Peres–Horodecki criterion|Peres–Horodecki criterion]] is necessary and sufficient; for general cases, it is merely necessary. Other criteria include the [[W:range criterion|range criterion]], [[W:reduction criterion|reduction criterion]], and uncertainty relation-based tests.
Continuous variable systems use Simon's condition for 1⊕1-mode Gaussian states, generalized for higher modes. Entropic measures also provide entanglement criteria.
=== In quantum gravity ===
Entanglement may explain the "problem of time": in quantum mechanics, time is a fixed backdrop, while in general relativity, it is dynamic. The [[W:Wheeler–DeWitt equation|Wheeler–DeWitt equation]] suggests a static universe.
Page and Wootters proposed that the universe appears to evolve internally due to entanglement between an evolving subsystem and a clock. AdS/CFT models suggest spacetime emerges from entangled quantum bits on its boundary.
== Experiments demonstrating and using entanglement ==
===Bell tests===
[[File:Two channel bell test.svg|thumb|Two channel bell test]]
{{main|W:Bell test|l1 = Bell test}}
A Bell test, or Bell inequality test or experiment, is a lab setup to pit quantum mechanics against local hidden variables. They check Bell's theorem predictions. So far, every one shows local hidden variables don't match reality. Labs run many to fix design flaws that might skew earlier results. closing loopholes. Early ones couldn't rule out sneaky signals from one site to the other.<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> But "loophole-free" tests space sites so light-speed comms take longer, one case, 10,000 times longer, than measurement time.<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
|title =Bounding the speed of 'spooky action at a distance
|journal =Physical Review Letters |volume=110 |issue =26 |article-number=260407 |year =2013
|arxiv =1303.0614
|bibcode =2013PhRvL.110z0407Y
|doi = 10.1103/PhysRevLett.110.260407
|pmid =23848853
|last1 =Yin |first1 =Juan |last2 =Cao |first2 =Yuan |last3 =Yong |first3 =Hai-Lin |last4 =Ren |first4 =Ji-Gang
|last5 =Liang |first5 =Hao |last6 =Liao |first6 =Sheng-Kai |last7 =Zhou |first7 =Fei |last8 =Liu |first8 =Chang
|last9 =Wu |first9 =Yu-Ping |last10 =Pan |first10 =Ge-Sheng |last11 =Li |first11 =Li |last12 =Liu |first12 =Nai-Le
|last13 =Zhang |first13 =Qiang |last14 =Peng |first14 =Cheng-Zhi |last15 =Pan |first15 =Jian-Wei
|display-authors=4
|s2cid = 119293698
}}</ref><ref name="NTR-20151021"/><ref name="hanson"/>
In 2017, Yin and team set a 1,203 km record for entanglement, showing two-photon survival and Bell violation (CHSH 2.37±0.09) under strict locality, from Micius satellite to bases in Yunnan and Qinghai, upping efficiency ten times over fiber.<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|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|last=Popkin|first=Gabriel|date=14 June 2017|work=[[W:Science (journal)|Science]]}}</ref>
== Entanglement of top quarks ==
[[File:Quark weak interactions.svg|thumb|Quark weak interactions]]
In 2023, the [[W:Large Hadron Collider|LHC]] used techniques from [[W:quantum tomography|quantum tomography]] to measure 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> This work is based on theoretical proposals from 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 out by the [[W:ATLAS experiment|ATLAS]] detector, which measured the spin of top-quark pair production. The effect was observed with a significance of more than 5 σ. The top quark is the heaviest known particle and therefore has a very short lifetime, approximately 10⁻²⁵ s, making it the only quark that decays before undergoing hadronization (∼10⁻²³ s) and spin decorrelation (∼10⁻²¹ s). As a result, the spin information is transferred without significant loss to the leptonic decay products captured 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 [[W:spin polarization|spin polarization]] and correlation of the particles were measured and tested for entanglement using [[W:Concurrence (quantum computing)|concurrence]] as well as the [[W:Peres–Horodecki criterion|Peres–Horodecki criterion]]. The effect has also been confirmed independently by the [[W: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 \sqrt{s} = 13~\mathrm{TeV} |journal=Reports on Progress in Physics |date=6 June 2024 |volume=87 |issue=11 |doi=10.1088/1361-6633/ad7e4d |pmid=39527914 |arxiv=2406.03976}}</ref><ref>{{cite journal |last=CMS Collaboration |title=Measurements of polarization and spin correlation and observation of entanglement in top quark pairs using lepton+jets events from proton-proton collisions at \sqrt{s}=13~\mathrm{TeV} |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>
== See also ==
<div style="column-count:3; column-gap:1.2em;">
{{:Quantum/See also}}
[[Category:Quantum mechanics]]
[[Category:Physics]]
* [[W:Concurrence (quantum computing)|Concurrence]]
* [[W:Controlled NOT gate|CNOT gate]]
* [[W:Einstein's thought experiments|Einstein's thought experiments]]
* [[W:Entanglement witness|Entanglement witness]]
* [[W:ER = EPR|ER = EPR]]
* [[W:Multipartite entanglement|Multipartite entanglement]]
* [[W:Normally distributed and uncorrelated does not imply independent|Normally distributed and uncorrelated does not imply independent]]
* [[W:Pauli exclusion principle|Pauli exclusion principle]]
* [[W:Quantum coherence|Quantum coherence]]
* [[W:Quantum discord|Quantum discord]]
* [[W:Quantum network|Quantum network]]
* [[W:Quantum phase transition|Quantum phase transition]]
* [[W:Quantum pseudo-telepathy|Quantum pseudo-telepathy]]
* [[W:Retrocausality|Retrocausality]]
* [[W:Squashed entanglement|Squashed entanglement]]
* [[W:Stern–Gerlach experiment|Stern–Gerlach experiment]]
* [[W:Ward's probability amplitude|Ward's probability amplitude]]
</div>
{{Portal|Physics}}
== Key Concepts ==
{{Quantum mechanics}}
Multipartite states like GHZ and W states.
Entanglement measures and classification (LU, SLOCC).
Applications in computing, cryptography, and experiments.
==== Discussion Question ====
What are the practical implications of using entanglement in quantum computing?<br>
====Course Quiz====
Test your knowledge with this short quiz. <br>Answers are provided below.
What term did Einstein use to describe quantum entanglement?<br>a) Local realism<br>b) Spooky action at a distance<br>c) Hidden variables<br>d) Wave function collapse<br><br>Which experiment first demonstrated loophole-free Bell inequality violation using electron spins?<br>a) Aspect's experiment<br>b) Hensen et al. (2015)<br>c) Clauser's experiment<br>d) Wheeler's experiment<br>
True or False: Quantum entanglement can be used to send information faster than light.<br><br>What is the von Neumann entropy used for in the context of entanglement?<br>a) Measuring classical correlations<br>b) Quantifying entanglement in pure states<br>c) Calculating particle spin<br>d) Determining wave function probability<br>Name one application of quantum entanglement in quantum information theory.
=== 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=[[W:Scientific American|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=W: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=W: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)]
====Quiz Answers====
b) Spooky action at a distance<br>b) Hensen et al. (2015)<br>False<br>b) Quantifying entanglement in pure states<br>Quantum teleportation, superdense coding, or quantum
cryptography (any one is acceptable)
{{Quantum mechanics topics}}
=References=
<div style="column-count:3; break-inside:avoid; column-gap:2em;">
{{Reflist}}
</div>
