Editing Faster-than-light (section)

==Superluminal travel of non-information==
{{Main|Superluminal motion}}

In the context of this article, "faster-than-light" means the transmission of information or matter faster than ''c'', a constant equal to the [[speed of light]] in vacuum, which is 299,792,458&nbsp;m/s (by definition of the metre)<ref>{{cite web|url=https://www.bipm.org/en/CGPM/db/17/1/|title=The 17th Conférence Générale des Poids et Mesures (CGPM) : Definition of the metre|website=bipm.org|access-date=July 5, 2020|archive-date=May 27, 2020|archive-url=https://web.archive.org/web/20200527104823/https://www.bipm.org/en/CGPM/db/17/1/|url-status=dead}}</ref> or about 186,282.397 miles per second. This is not quite the same as traveling faster than light, since:
*Some processes propagate faster than ''c'', but cannot carry information (see examples in the sections immediately following).
*In some materials where light travels at speed ''c/n'' (where ''n'' is the [[refractive index]]) other particles can travel faster than ''c/n'' (but still slower than ''c''), leading to [[Cherenkov radiation]] (see [[#Phase_velocities_above_c|phase velocity below]]).
Neither of these phenomena violates [[special relativity]] or creates problems with [[causality]], and thus neither qualifies as faster-than-light as described here.

In the following examples, certain influences may appear to travel faster than light, but they do not convey energy or information faster than light, so they do not violate special relativity.

===Daily sky motion===
For an earth-bound observer, objects in the sky complete one revolution around the Earth in one day. [[Proxima Centauri]], the nearest star outside the [[Solar System]], is about four and a half [[light-year]]s away.<ref name="A2 Student Book">
{{cite book
|author=University of York Science Education Group
|year=2001
|title=Salter Horners Advanced Physics A2 Student Book
|publisher=Heinemann
|pages=302–303
|isbn=978-0435628925
}}</ref> In this frame of reference, in which Proxima Centauri is perceived to be moving in a circular trajectory with a radius of four light years, it could be described as having a speed many times greater than ''c'' as the rim speed of an object moving in a circle is a product of the radius and angular speed.<ref name="A2 Student Book"/> It is also possible on a [[geostatic orbit|geostatic]] view, for objects such as comets to vary their speed from subluminal to superluminal and vice versa simply because the distance from the Earth varies. Comets may have orbits which take them out to more than 1000 [[astronomical unit|AU]].<ref>
{{cite web
|date=15 April 1996
|title=The Furthest Object in the Solar System
|url=http://www.oarval.org/furthest.htm
|work=Information Leaflet No. 55
|publisher=Royal Greenwich Observatory
}}</ref> The circumference of a circle with a radius of 1000 AU is greater than one light day. In other words, a comet at such a distance is superluminal in a geostatic, and therefore non-inertial, frame.

===Light spots and shadows===
If a laser beam is swept across a distant object, the spot of laser light can seem to move across the object at a speed greater than ''c''.<ref name="Gibbs"/> Similarly, a shadow projected onto a distant object seems to move across the object faster than ''c''.<ref name="Gibbs">
{{cite web
|last=Gibbs |first=P.
|year=1997
|url=http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/FTL.html
|title=Is Faster-Than-Light Travel or Communication Possible?
|work=The Original Usenet Physics FAQ
|access-date=20 August 2008
}}</ref> In neither case does the light travel from the source to the object faster than ''c'', nor does any information travel faster than light. No object is moving in these examples. For comparison, consider water squirting out of a garden hose as it is swung side to side: water does not instantly follow the direction of the hose.<ref name="Gibbs"/><ref>
{{cite book
|last1=Salmon |first1=W. C.
|year=2006
|title=Four Decades of Scientific Explanation
|url=https://books.google.com/books?id=FHqOXCd06e8C&pg=PA107
|publisher=[[University of Pittsburgh Press]]
|isbn=978-0-8229-5926-7
|page=107
}}</ref><ref>
{{cite book
|last1=Steane |first1=A.
|year=2012
|title=The Wonderful World of Relativity: A Precise Guide for the General Reader
|url=https://books.google.com/books?id=4m14K1PpJwMC&pg=PA180
|page=180
|publisher=[[Oxford University Press]]
|isbn=978-0-19-969461-7
}}</ref>

===Closing speeds===
The rate at which two objects in motion in a single frame of reference get closer together is called the mutual or closing speed. This may approach twice the speed of light, as in the case of two particles travelling at close to the speed of light in opposite directions with respect to the reference frame.

Imagine two fast-moving particles approaching each other from opposite sides of a [[particle accelerator]] of the collider type. The closing speed would be the rate at which the distance between the two particles is decreasing. From the point of view of an observer standing at rest relative to the accelerator, this rate will be slightly less than twice the speed of light.

[[Special relativity]] does not prohibit this. It tells us that it is wrong to use [[Galilean relativity]] to compute the velocity of one of the particles, as would be measured by an observer traveling alongside the other particle. That is, special relativity gives the correct [[velocity-addition formula]] for computing such [[relative velocity]].

It is instructive to compute the relative velocity of particles moving at ''v'' and −''v'' in accelerator frame, which corresponds to the closing speed of 2''v''&nbsp;>&nbsp;''c''. Expressing the speeds in units of ''c'', ''β''&nbsp;=&nbsp;''v''/''c'':
:<math>\beta_\text{rel} = \frac{\beta + \beta}{1 + \beta ^2} = \frac{2\beta}{1 + \beta^2} \leq 1.</math>

===Proper speeds===
If a spaceship travels to a planet one light-year (as measured in the Earth's rest frame) away from Earth at high speed, the time taken to reach that planet could be less than one year as measured by the traveller's clock (although it will always be more than one year as measured by a clock on Earth). The value obtained by dividing the distance traveled, as determined in the Earth's frame, by the time taken, measured by the traveller's clock, is known as a proper speed or a [[proper velocity]]. There is no limit on the value of a proper speed as a proper speed does not represent a speed measured in a single inertial frame. A light signal that left the Earth at the same time as the traveller would always get to the destination before the traveller would.

===Phase velocities above ''c''===
The [[phase velocity]] of an [[electromagnetic wave]], when traveling through a medium, can routinely exceed ''c'', the vacuum velocity of light. For example, this occurs in most glasses at [[X-ray]] frequencies.<ref>
{{cite book
|last=Hecht |first=E.
|year=1987
|title=Optics
|page=62 |edition=2nd
|publisher=[[Addison Wesley]]
|isbn=978-0-201-11609-0
}}</ref> However, the phase velocity of a wave corresponds to the propagation speed of a theoretical single-frequency (purely [[monochromatic]]) component of the wave at that frequency. Such a wave component must be infinite in extent and of constant amplitude (otherwise it is not truly monochromatic), and so cannot convey any information.<ref>
{{cite journal
|last=Sommerfeld |first=A.
|year=1907
|title=An Objection Against the Theory of Relativity and its Removal
|journal=[[Physikalische Zeitschrift]]
|volume=8 |issue=23 |pages=841–842
|title-link=s:Translation:An Objection Against the Theory of Relativity and its Removal
}}</ref>
Thus a phase velocity above ''c'' does not imply the propagation of [[signal (information theory)|signals]] with a velocity above ''c''.<ref name="phase">
{{cite journal
|title=Phase, Group, and Signal Velocity
|url=https://www.mathpages.com/home/kmath210/kmath210.htm
|journal=American Journal of Physics
|bibcode=1954AmJPh..22..618W
|access-date=2007-04-30
|last1=Weber
|first1=J.
|year=1954
|volume=22
|issue=9
|page=618
|doi=10.1119/1.1933858
|url-access=subscription
}}</ref>

===Group velocities above ''c''===
The [[group velocity]] of a wave may also exceed ''c'' in some circumstances.<ref>
{{cite journal
|last1=Wang |first1=L. J.
|last2=Kuzmich |first2=A.
|last3=Dogariu |first3=A.
|year=2000
|title=Gain-assisted superluminal light propagation
|journal=[[Nature (journal)|Nature]]
|volume=406 |issue=6793 |pages=277–279
|doi=10.1038/35018520
|pmid=10917523
|doi-access= |bibcode=2000Natur.406..277W|s2cid=4358601
}}</ref><ref>
{{cite journal
|last1=Bowlan |first1=P.
|last2=Valtna-Lukner |first2=H.
|last3=Lõhmus |first3=M.
|last4=Piksarv |first4=P.
|last5=Saari |first5=P.
|last6=Trebino |first6=R.
|s2cid=122056218
|year=2009
|title=Measurement of the spatiotemporal electric field of ultrashort superluminal Bessel-X pulses
|journal=[[Optics and Photonics News]]
|volume=20 |issue=12 |page=42
|bibcode=2009OptPN..20...42M
|doi=10.1364/OPN.20.12.000042
}}</ref> In such cases, which typically at the same time involve rapid attenuation of the intensity, the maximum of the envelope of a pulse may travel with a velocity above ''c''. However, even this situation does not imply the propagation of [[signal (information theory)|signals]] with a velocity above ''c'',<ref>
{{cite book |last=Brillouin |first=L. |url=https://archive.org/details/wavepropagationg00bril_0 |title=Wave Propagation and Group Velocity |publisher=[[Academic Press]] |year=1960 |url-access=registration}}</ref> even though one may be tempted to associate pulse maxima with signals. The latter association has been shown to be misleading, because the information on the arrival of a pulse can be obtained before the pulse maximum arrives. For example, if some mechanism allows the full transmission of the leading part of a pulse while strongly attenuating the pulse maximum and everything behind (distortion), the pulse maximum is effectively shifted forward in time, while the information on the pulse does not come faster than ''c'' without this effect.<ref>
{{cite journal
|last1=Withayachumnankul |first1=W.
|last2=Fischer |first2=B. M.
|last3=Ferguson |first3=B.
|last4=Davis |first4=B. R.
|last5=Abbott |first5=D.
|year=2010
|title=A Systemized View of Superluminal Wave Propagation
|url=http://www.eleceng.adelaide.edu.au/personal/dabbott/publications/PIE_withayachumnankul2010.pdf
|journal=[[Proceedings of the IEEE]]
|volume=98 |issue=10 |pages=1775–1786
|doi=10.1109/JPROC.2010.2052910
|s2cid=15100571
}}</ref> However, group velocity [[Gouy phase shift|can exceed]] ''c'' in some parts of a [[Gaussian beam]] in vacuum (without attenuation). The [[diffraction]] causes the peak of the pulse to propagate faster, while overall power does not.<ref>
{{cite journal
|last1=Horváth |first1=Z. L.
|last2=Vinkó |first2=J.
|last3=Bor |first3=Zs.
|last4=von der Linde |first4=D.
|year=1996
|title=Acceleration of femtosecond pulses to superluminal velocities by Gouy phase shift
|url=http://www.ilp.physik.uni-essen.de/vonderLinde/Publikationen/APB96_gouy.pdf |archive-url=https://web.archive.org/web/20030403070745/http://www.ilp.physik.uni-essen.de/vonderLinde/Publikationen/APB96_gouy.pdf |archive-date=2003-04-03 |url-status=live
|journal=[[Applied Physics B]]
|volume=63 |issue=5 |pages=481–484
|bibcode=1996ApPhB..63..481H
|doi=10.1007/BF01828944
|s2cid=54757568
}}</ref>

===Cosmic expansion===

According to [[Hubble's law]], the [[expansion of the universe]] causes distant galaxies to appear to recede from us faster than the speed of light. However, the recession speed associated with [[Hubble's law]], defined as the rate of increase in [[comoving and proper distances|proper distance]] per interval of [[cosmological time]], is not a velocity in a relativistic sense. Moreover, in [[general relativity]], velocity is a local notion, and there is not even a unique definition for the relative velocity of a cosmologically distant object.<ref>
{{cite web |last=Wright |first=E. L. |date=12 June 2009 |title=Cosmology Tutorial – Part 2 |url=http://www.astro.ucla.edu/~wright/cosmo_02.htm#MD |access-date=2011-09-26 |work=Ned Wright's Cosmology Tutorial |publisher=[[UCLA]]}}</ref> Faster-than-light cosmological recession speeds are entirely a [[coordinate conditions|coordinate]] effect.

There are many galaxies visible in telescopes with [[redshift]] numbers of 1.4 or higher. All of these have cosmological recession speeds greater than the speed of light. Because the [[Hubble's law#Interpretation|Hubble parameter]] is decreasing with time, there can actually be cases where a galaxy that is receding from us faster than light does manage to emit a signal which reaches us eventually.<ref>See the last two paragraphs in {{cite web
|last1=Rothstein |first1=D.
|date=10 September 2003
|title=Is the universe expanding faster than the speed of light?
|url=http://curious.astro.cornell.edu/the-universe/cosmology-and-the-big-bang/104-the-universe/cosmology-and-the-big-bang/expansion-of-the-universe/616-is-the-universe-expanding-faster-than-the-speed-of-light-intermediate
|website=Ask an Astronomer
}}</ref><ref name="ly93">
{{cite web
|last1=Lineweaver |first1=C.
|last2=Davis |first2=T. M.
|date=March 2005
|title=Misconceptions about the Big Bang
|url=http://www.mso.anu.edu.au/~charley/papers/LineweaverDavisSciAm.pdf |archive-url=https://web.archive.org/web/20060527202701/http://www.mso.anu.edu.au/%7Echarley/papers/LineweaverDavisSciAm.pdf |archive-date=2006-05-27 |url-status=live
|work=[[Scientific American]]
|pages=36–45
|access-date=2008-11-06
}}</ref><ref>
{{cite journal |last1=Davis |first1=T. M. |last2=Lineweaver |first2=C. H. |year=2004 |title=Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the universe |journal=[[Publications of the Astronomical Society of Australia]] |volume=21 |issue=1 |pages=97–109 |arxiv=astro-ph/0310808 |bibcode=2004PASA...21...97D |doi=10.1071/AS03040 |s2cid=13068122}}</ref>

However, because [[accelerating expansion of the universe|the expansion of the universe is accelerating]], it is projected that most galaxies will eventually cross a type of cosmological [[event horizon]] where any light they emit past that point will never be able to reach us at any time in the infinite future,<ref>
{{cite journal
|last=Loeb |first=A.
|year=2002
|title=The Long-Term Future of Extragalactic Astronomy
|journal=[[Physical Review D]]
|volume=65 |issue=4 |pages=047301
|arxiv=astro-ph/0107568
|bibcode=2002PhRvD..65d7301L
|doi=10.1103/PhysRevD.65.047301
|s2cid=1791226
}}</ref> because the light never reaches a point where its "peculiar velocity" towards us exceeds the expansion velocity away from us (these two notions of velocity are also discussed in {{Section link|Comoving and proper distances|Uses of the proper distance}}). The current distance to this cosmological event horizon is about 16 billion light-years, meaning that a signal from an event happening at present would eventually be able to reach us in the future if the event was less than 16 billion light-years away, but the signal would never reach us if the event was more than 16 billion light-years away.<ref name="ly93"/>

===Astronomical observations===
Apparent [[superluminal motion]] is observed in many [[active galaxy|radio galaxies]], [[blazar]]s, [[quasar]]s, and recently also in [[microquasar]]s. The effect was predicted before it was observed by [[Martin Rees]]{{clarify|Was it predicted by Rees or observed by Rees?|date=March 2012}} and can be explained as an [[optical illusion]] caused by the object partly moving in the direction of the observer,<ref>
{{cite journal
|last1=Rees |first1=M. J.
|year=1966
|title=Appearance of relativistically expanding radio sources
|journal=[[Nature (journal)|Nature]]
|volume=211 |issue=5048 |pages=468–470
|bibcode=1966Natur.211..468R
|doi=10.1038/211468a0
|s2cid=41065207
}}</ref> when the speed calculations assume it does not. The phenomenon does not contradict the theory of [[special relativity]]. Corrected calculations show these objects have velocities close to the speed of light (relative to our reference frame). They are the first examples of large amounts of mass moving at close to the speed of light.<ref>
{{cite journal
|last1=Blandford |first1=R. D. |author-link=Roger Blandford
|last2=McKee |first2=C. F.
|last3=Rees |first3=M. J.
|year=1977
|title=Super-luminal expansion in extragalactic radio sources
|journal=[[Nature (journal)|Nature]]
|volume=267 |pages=211–216 |issue=5608
|bibcode=1977Natur.267..211B
|doi=10.1038/267211a0
|s2cid=4260167 }}</ref> Earth-bound laboratories have only been able to accelerate small numbers of elementary particles to such speeds.

===Quantum mechanics===
Certain phenomena in [[quantum mechanics]], such as [[quantum entanglement]], might give the superficial impression of allowing communication of information faster than light. According to the [[no-communication theorem]] these phenomena do not allow true communication; they only let two observers in different locations see the same system simultaneously, without any way of controlling what either sees. [[Wavefunction collapse]] can be viewed as an [[epiphenomenon]] of quantum decoherence, which in turn is nothing more than an effect of the underlying local time evolution of the wavefunction of a system and ''all'' of its environment. Since the underlying behavior does not violate local causality or allow FTL communication, it follows that neither does the additional effect of wavefunction collapse, whether real ''or'' apparent.

The [[uncertainty principle]] implies that individual photons may travel for short distances at speeds somewhat faster (or slower) than ''c'', even in vacuum; this possibility must be taken into account when enumerating [[Feynman diagram]]s for a particle interaction.<ref>
{{cite book
|last1=Grozin |first1=A.
|year=2007
|title=Lectures on QED and QCD
|url=https://archive.org/details/lecturesonqedqcd00groz |url-access=limited |page=[https://archive.org/details/lecturesonqedqcd00groz/page/n101 89]
|publisher=[[World Scientific]]
|isbn=978-981-256-914-1
}}</ref> However, it was shown in 2011 that a single photon may not travel faster than ''c''.<ref>
{{cite journal
|last1=Zhang |first1=S.
|last2=Chen |first2=J. F.
|last3=Liu |first3=C.
|last4=Loy |first4=M. M. T.
|last5=Wong |first5=G. K. L.
|last6=Du |first6=S.
|year=2011
|title=Optical Precursor of a Single Photon
|journal=[[Physical Review Letters]]
|volume=106 |issue=24 |pages=243602
|bibcode=2011PhRvL.106x3602Z
|doi=10.1103/PhysRevLett.106.243602
|pmid=21770570
|url=http://repository.ust.hk/ir/bitstream/1783.1-7246/1/PhysRevLett.106.243602.pdf |archive-url=https://web.archive.org/web/20191205034808/http://repository.ust.hk/ir/bitstream/1783.1-7246/1/PhysRevLett.106.243602.pdf |archive-date=2019-12-05 |url-status=live
}}</ref> 

There have been various reports in the popular press of experiments on faster-than-light transmission in optics — most often in the context of a kind of [[quantum tunnelling]] phenomenon. Usually, such reports deal with a [[phase velocity]] or [[group velocity]] faster than the vacuum velocity of light.<ref>
{{cite book
|last1=Kåhre |first1=J.
|year=2012
|title=The Mathematical Theory of Information
|url=https://books.google.com/books?id=1ozlBwAAQBAJ&pg=PA425
|page=425 |edition=Illustrated
|publisher=[[Springer Science & Business Media]]
|isbn=978-1-4615-0975-2
}}</ref><ref>
{{cite thesis
|last1=Steinberg |first1=A. M.
|year=1994
|title=When Can Light Go Faster Than Light?
|url=https://books.google.com/books?id=E25MAQAAMAAJ
|page=100
|publisher=[[University of California, Berkeley]]
|bibcode=1994PhDT.......314S
}}</ref> However, as stated above, a superluminal phase velocity cannot be used for faster-than-light transmission of information<ref>
{{cite book
|last1=Chubb |first1=J.
|last2=Eskandarian |first2=A.
|last3=Harizanov |first3=V.
|year=2016
|title=Logic and Algebraic Structures in Quantum Computing
|url=https://books.google.com/books?id=mWVbCwAAQBAJ&pg=PA61
|page=61 |edition=Illustrated
|publisher=[[Cambridge University Press]]
|isbn=978-1-107-03339-9
}}</ref><ref>
{{cite book
|last1=Ehlers |first1=J.
|last2=Lämmerzahl |first2=C.
|year=2006
|title=Special Relativity: Will it Survive the Next 101 Years?
|url=https://books.google.com/books?id=avy6BQAAQBAJ&pg=PA506
|page=506 |edition=Illustrated
|publisher=Springer
|isbn=978-3-540-34523-7
}}</ref> 

====Hartman effect====
{{Main|Hartman effect}}

The Hartman effect is the tunneling effect through a barrier where the tunneling time tends to a constant for large barriers.<ref>
{{cite journal
|last1=Martinez |first1=J. C.
|last2=Polatdemir |first2=E.
|year=2006
|title=Origin of the Hartman effect
|journal=[[Physics Letters A]]
|volume=351 |issue=1–2 |pages=31–36
|bibcode=2006PhLA..351...31M
|doi=10.1016/j.physleta.2005.10.076
}}</ref><ref>
{{cite journal
|last1=Hartman |first1=T. E.
|year=1962
|title=Tunneling of a Wave Packet
|journal=[[Journal of Applied Physics]]
|volume=33 |issue=12 |pages=3427–3433
|bibcode=1962JAP....33.3427H
|doi=10.1063/1.1702424
}}</ref> This could, for instance, be the gap between two prisms. When the prisms are in contact, the light passes straight through, but when there is a gap, the light is refracted. There is a non-zero probability that the photon will tunnel across the gap rather than follow the refracted path. 

However, it has been claimed that the Hartman effect cannot actually be used to violate relativity by transmitting signals faster than ''c'', also because the tunnelling time "should not be linked to a velocity since evanescent waves do not propagate".<ref>
{{cite journal
|last1=Winful |first1=H. G.
|year=2006
|title=Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox
|journal=[[Physics Reports]]
|volume=436 |issue=1–2 |pages=1–69
|bibcode=2006PhR...436....1W
|doi=10.1016/j.physrep.2006.09.002
}}</ref> The evanescent waves in the Hartman effect are due to virtual particles and a non-propagating static field, as mentioned in the sections above for gravity and electromagnetism.

====Casimir effect====
{{Main|Casimir effect}}

In physics, the [[Casimir–Polder force]] is a physical force exerted between separate objects due to resonance of [[vacuum energy]] in the intervening space between the objects. This is sometimes described in terms of virtual particles interacting with the objects, owing to the mathematical form of one possible way of calculating the strength of the effect. Because the strength of the force falls off rapidly with distance, it is only measurable when the distance between the objects is extremely small. Because the effect is due to virtual particles mediating a static field effect, it is subject to the comments about static fields discussed above.

====EPR paradox====
{{Main|EPR paradox}}

The EPR paradox refers to a famous [[thought experiment]] of [[Albert Einstein]], [[Boris Podolsky]] and [[Nathan Rosen]] that was realized experimentally for the first time by [[Alain Aspect]] in 1981 and 1982 in the [[Aspect experiment]]. In this experiment, the two measurements of an [[quantum entanglement|entangled]] state are correlated even when the measurements are distant from the source and each other. However, no information can be transmitted this way; the answer to whether or not the measurement actually affects the other quantum system comes down to which [[interpretations of quantum mechanics|interpretation of quantum mechanics]] one subscribes to.

An experiment performed in 1997 by [[Nicolas Gisin]] has demonstrated quantum correlations between particles separated by over 10 kilometers.<ref>
{{cite web
|last=Suarez |first=A.
|date=26 February 2015
|title=History
|url=http://www.quantumphil.org/history.htm
|publisher=Center for Quantum Philosophy
|access-date=2017-06-07
}}</ref> But as noted earlier, the non-local correlations seen in entanglement cannot actually be used to transmit classical information faster than light, so that relativistic causality is preserved. The situation is akin to sharing a synchronized coin flip, where the second person to flip their coin will always see the opposite of what the first person sees, but neither has any way of knowing whether they were the first or second flipper, without communicating classically. See [[No-communication theorem]] for further information. A 2008 quantum physics experiment also performed by Nicolas Gisin and his colleagues has determined that in any hypothetical [[hidden-variable theory#Non-local hidden-variable theory|non-local hidden-variable theory]], the speed of the [[quantum non-local connection]] (what Einstein called "spooky action at a distance") is at least 10,000 times the speed of light.<ref>
{{cite journal
|last1=Salart |first1=D.
|last2=Baas |first2=A.
|last3=Branciard |first3=C.
|last4=Gisin |first4=N.
|last5=Zbinden |first5=H.
|year=2008
|title=Testing spooky action at a distance
|journal=[[Nature (journal)|Nature]]
|volume=454 |issue=7206 |pages=861–864
|arxiv=0808.3316
|bibcode=2008Natur.454..861S
|doi=10.1038/nature07121
|pmid=18704081
|s2cid=4401216
}}</ref>

====Delayed choice quantum eraser====
{{Main|Delayed-choice quantum eraser}}

The [[delayed-choice quantum eraser]] is a version of the EPR paradox in which the observation (or not) of interference after the passage of a photon through a [[double slit experiment]] depends on the conditions of observation of a second photon entangled with the first. The characteristic of this experiment is that the observation of the second photon can take place at a later time than the observation of the first photon,<ref>
{{cite journal
|last1=Kim |first1=Yoon-Ho
|last2=Yu |first2=Rong
|last3=Kulik |first3=Sergei P.
|last4=Shih |first4=Yanhua
|last5=Scully |first5=Marlan O.
|year=2000
|title=Delayed "Choice" Quantum Eraser
|journal=[[Physical Review Letters]]
|volume=84 |issue=1 |pages=1–5
|arxiv=quant-ph/9903047
|bibcode=2000PhRvL..84....1K
|doi=10.1103/PhysRevLett.84.1
|pmid=11015820
|s2cid=5099293
}}</ref> which may give the impression that the measurement of the later photons "retroactively" determines whether the earlier photons show interference or not, although the interference pattern can only be seen by correlating the measurements of both members of every pair and so it cannot be observed until both photons have been measured, ensuring that an experimenter watching only the photons going through the slit does not obtain information about the other photons in an faster-than-light or backwards-in-time manner.<ref>
{{cite web
|last1=Hillmer |first1=R.
|last2=Kwiat |first2=P.|author2-link=Paul Kwiat
|date=16 April 2017
|title=Delayed-Choice Experiments
|url=https://www.scientificamerican.com/article/quantum-eraser-delayed-choice-experiments/
|website=[[Scientific American]]
}}</ref><ref>
{{cite web
|last=Motl |first=L.
|date=November 2010
|title=Delayed choice quantum eraser
|url=https://motls.blogspot.com/2010/11/delayed-choice-quantum-eraser.html
|website=[[The Reference Frame]]
}}</ref>