Editing Quantum entanglement (section)

== 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 |pages=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 }}</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 |pages=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 |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 |page=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 }}</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 |page=035005 |doi=10.1103/revmodphys.90.035005 |arxiv=1609.01609}}</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
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{{cite journal
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 |arxiv=1108.3309 |doi=10.1038/nphoton.2011.283
 |bibcode = 2012NaPho...6...45S |s2cid=56206588
 }}</ref> Other methods include the use of a [[fibre 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 |pages=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
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}}</ref><ref name="Georgiev2022">
{{cite journal
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 | volume = 106
 | number = 6
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 | arxiv = 2212.07502
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}}</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 |pages=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 |page=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 |page=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 |page=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 |page=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| page=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 |pages=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|page=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|page=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 |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|pages=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 }}</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 }}</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 |pages=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 |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 |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}}</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>