Editing Bell test (section)

== Loopholes ==
Though the series of increasingly sophisticated Bell test experiments has convinced the physics community that local hidden-variable theories are indefensible; they can never be excluded entirely.<ref>{{Cite journal |title = Bell nonlocality|journal = Rev. Mod. Phys. |date=2014-04-18 |volume = 86|issue = 2|doi = 10.1103/RevModPhys.86.419|first = N.|last = Brunner|bibcode=2014RvMP...86..419B|pages=419–478 |arxiv=1303.2849|s2cid = 119194006 }}</ref> For example, the hypothesis of [[superdeterminism]] in which all experiments and outcomes (and everything else) are predetermined can never be excluded (because it is [[Falsifiability|unfalsifiable]]).<ref name="larsson14"/>

Up to 2015, the outcome of all experiments that violate a Bell inequality could still theoretically be explained by exploiting the detection loophole and/or the locality loophole. The locality (or communication) loophole means that since in actual practice the two detections are separated by a [[Spacetime#Time-like interval|time-like interval]], the first detection may influence the second by some kind of signal. To avoid this loophole, the experimenter has to ensure that particles travel far apart before being measured, and that the measurement process is rapid. More serious is the detection (or unfair sampling) loophole, because particles are not always detected in both wings of the experiment. It can be imagined that the complete set of particles would behave randomly, but instruments only detect a subsample showing [[quantum correlation]]s, by letting detection be dependent on a combination of local hidden variables and detector setting.{{citation needed|date=July 2022}}

Experimenters had repeatedly voiced that loophole-free tests could be expected in the near future.<ref name="García-Patrón-2004">{{cite journal |author1=R. García-Patrón |author2=J. Fiurácek |author3=N. J. Cerf |author4=J. Wenger |author5=R. Tualle-Brouri |author6=Ph. Grangier |year=2004 |title=Proposal for a Loophole-Free Bell Test Using Homodyne Detection |journal=Phys. Rev. Lett. |volume=93 |issue=13 |page=130409 |arxiv=quant-ph/0403191 |doi=10.1103/PhysRevLett.93.130409|bibcode = 2004PhRvL..93m0409G |pmid=15524691|s2cid=10147610 }}</ref><ref name="Gill-2003">{{cite encyclopedia |author-link=Richard D. Gill |last=Gill |first=Richard D. |year=2003 |title=Time, Finite Statistics, and Bell's Fifth Position |encyclopedia=Foundations of Probability and Physics - 2 |pages=179–206 |arxiv=quant-ph/0301059|bibcode = 2003quant.ph..1059G |publisher=[[Linnaeus University|Växjö University Press]]}}</ref> In 2015, a loophole-free Bell violation was reported using entangled diamond spins over a distance of {{convert|1.3|km|m}}<ref name="Hensen et al." /> and corroborated by two experiments using entangled photon pairs.<ref name=Zeilinger-2015 /><ref name=Kwiat-2015 />

The remaining possible theories that obey local realism can be further restricted by testing different spatial configurations, methods to determine the measurement settings, and recording devices. It has been suggested that using humans to generate the measurement settings and observe the outcomes provides a further test.<ref>{{cite journal |title= Quantum physics: Death by experiment for local realism|journal = Nature|date = 2015-10-21|volume = 526|issue = 7575|doi = 10.1038/nature15631|first = H.|last = Wiseman|pages=649–650|bibcode = 2015Natur.526..649W|pmid=26503054|doi-access = free}}</ref> David Kaiser of [[Massachusetts Institute of Technology|MIT]] told the ''New York Times'' in 2015 that a potential weakness of the "loophole-free" experiments is that the systems used to add randomness to the measurement may be predetermined in a method that was not detected in experiments.<ref>{{Cite news|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|newspaper = The New York Times|date = 2015-10-21|access-date = 2015-10-22|issn = 0362-4331|first = John|last = Markoff}}</ref>

===Detection loophole===
A common problem in optical Bell tests is that only a small fraction of the emitted photons are detected. It is then possible that the correlations of the detected photons are unrepresentative: although they show a violation of a Bell inequality, if all photons were detected the Bell inequality would actually be respected. This was first noted by Philip M. Pearle in 1970,<ref name="Pearle-1970">{{cite journal|first=Philip M. |last=Pearle |year=1970 |title=Hidden-Variable Example Based upon Data Rejection |journal=[[Physical Review D]] |volume=2 |issue=8 |pages=1418–25 |doi=10.1103/PhysRevD.2.1418 |bibcode=1970PhRvD...2.1418P}}</ref> who devised a local hidden variable model that faked a Bell violation by letting the photon be detected only if the measurement setting was favourable. The assumption that this does not happen, i.e., that the small sample is actually representative of the whole is called the ''fair sampling'' assumption.

To do away with this assumption it is necessary to detect a sufficiently large fraction of the photons. This is usually characterized in terms of the detection efficiency <math>\eta</math>, defined as the probability that a photodetector detects a photon that arrives at it. [[Anupam Garg]] and [[N. David Mermin]] showed that when using a maximally entangled state and the CHSH inequality an efficiency of <math>\eta > 2\sqrt2-2 \approx 0.83 </math> is required for a loophole-free violation.<ref name="Garg & Mermin, 1987">{{cite journal|first1=Anupam |last1=Garg |first2=N. David |last2=Mermin |year=1987 |title=Detector inefficiencies in the Einstein-Podolsky-Rosen experiment |journal=[[Physical Review D]] |volume=25 |issue=12 |pages=3831–5 |doi=10.1103/PhysRevD.35.3831 |pmid=9957644 |bibcode=1987PhRvD..35.3831G}}</ref> Later Philippe H. Eberhard showed that when using a ''partially'' entangled state a loophole-free violation is possible for <math>\eta > 2/3 \approx 0.67 </math>,<ref>{{cite journal|first=P. H. |last=Eberhard |year=1993 |title=Background level and counter efficiencies required for a loophole-free Einstein-Podolsky-Rosen experiment |journal=[[Physical Review A]] |volume=47 |issue=2 |pages=747–750|doi=10.1103/PhysRevA.47.R747 |pmid=9909100 |bibcode=1993PhRvA..47..747E }}</ref> which is the optimal bound for the CHSH inequality.<ref>{{cite journal |last1=Larsson |first1=Jan-Åke |last2=Semitecolos |first2=Jason |title=Strict detector-efficiency bounds for n-site Clauser-Horne inequalities |journal=[[Physical Review A]] |date=2001 |volume=63 |issue=2 |page=022117 |doi=10.1103/PhysRevA.63.022117 |arxiv=quant-ph/0006022 |bibcode=2001PhRvA..63b2117L |s2cid=119469607 }}</ref> Other Bell inequalities allow for even lower bounds. For example, there exists a four-setting inequality which is violated for <math>\eta > (\sqrt5-1)/2 \approx 0.62</math>.<ref>{{cite journal |last1=Vértesi |first1=Tamás |last2=Pironio |first2=Stefano |last3=Brunner |first3=Nicolas |title=Closing the detection loophole in Bell experiments using qudits |journal=[[Physical Review Letters]] |date=2010 |volume=104 |issue=6 |page=060401 |doi=10.1103/PhysRevLett.104.060401 |pmid=20366808 |arxiv=0909.3171|bibcode=2010PhRvL.104f0401V |s2cid=22053479 }}</ref>

Historically, only experiments with non-optical systems have been able to reach high enough efficiencies to close this loophole, such as trapped ions,<ref>{{cite journal|first1=M. A. |last1=Rowe |first2=D. |last2=Kielpinski |first3=V. |last3=Meyer |first4=C. A. |last4=Sackett |first5=W. M. |last5=Itano |first6=C. |last6=Monroe |first7=D. J. |last7=Wineland |display-authors=5|year=2001 |title=Experimental violation of a Bell's inequality with efficient detection |journal=[[Nature (journal)|Nature]] |volume=409 |issue=6822 |pages=791–94 |doi=10.1038/35057215 |pmid=11236986|bibcode=2001Natur.409..791R |url=https://deepblue.lib.umich.edu/bitstream/2027.42/62731/1/409791a0.pdf |hdl=2027.42/62731 |s2cid=205014115 |hdl-access=free }}</ref> superconducting qubits,<ref name=Ansmann>{{cite journal|last1=Ansmann|first1=M. |last2=Wang|first2=H.|last3=Bialczak|first3=R. C.|last4=Hofheinz|first4=M.|last5=Lucero|first5=E.|last6=Neeley|first6=M.|last7=O'Connell|first7=A. D.|last8=Sank|first8=D.|last9=Weides|first9=M.|last10=Wenner|first10=J.|last11=Cleland|first11=A. N.|last12=Martinis|first12=J. M.|display-authors=5|date=24 September 2009|title=Violation of Bell's inequality in Josephson phase qubits| journal=[[Nature (journal)|Nature]] |volume=461|issue=7263|pages=504–506|doi=10.1038/nature08363|pmid=19779447|bibcode=2009Natur.461..504A|s2cid=4401494 }}</ref> and [[nitrogen-vacancy center]]s.<ref>{{cite journal|last1=Pfaff |first1=W. |last2=Taminiau |first2=T. H. |last3=Robledo |first3=L. |last4=Bernien |first4=H. |last5=Markham |first5=M. |last6=Twitchen |first6=D. J. |last7=Hanson |first7=R. |display-authors=5|year=2013 |title=Demonstration of entanglement-by-measurement of solid-state qubits |journal=[[Nature Physics]] |volume=9 |issue=1 |pages=29–33 |doi=10.1038/nphys2444|arxiv=1206.2031 |bibcode=2013NatPh...9...29P |s2cid=2124119 }}</ref> These experiments were not able to close the locality loophole, which is easy to do with photons. More recently, however, optical setups have managed to reach sufficiently high detection efficiencies by using superconducting photodetectors,<ref name=Zeilinger-2015 /><ref name=Kwiat-2015 /> and hybrid setups have managed to combine the high detection efficiency typical of matter systems with the ease of distributing entanglement at a distance typical of photonic systems.<ref name="Hensen et al." />

===Locality loophole===
One of the assumptions of Bell's theorem is the one of locality, namely that the choice of setting at a measurement site does not influence the result of the other. The motivation for this assumption is the [[special relativity|theory of relativity]], that prohibits communication faster than light. For this motivation to apply to an experiment, it needs to have space-like separation between its measurements events. That is, the time that passes between the choice of measurement setting and the production of an outcome must be shorter than the time it takes for a light signal to travel between the measurement sites.<ref name="Bell-1987b">{{cite journal|first=J. S. |last=Bell |year=1980 |title=Atomic-cascade photons and quantum-mechanical nonlocality|journal=Comments on Atomic and Molecular Physics |volume=9 |pages=121–126}} Reprinted as {{cite book|chapter=Chapter 13 |first=J. S. |last=Bell |title=Speakable and Unspeakable in Quantum Mechanics |publisher=Cambridge University Press |year=1987|page=109}}</ref>

The first experiment that strived to respect this condition was Aspect's 1982 experiment.<ref name="Aspect-1982b">{{cite journal |first1=Alain |last1=Aspect |first2=Jean |last2=Dalibard |first3=Gérard |last3=Roger |year=1982 |title=Experimental Test of Bell's Inequalities Using Time-Varying Analyzers |journal=[[Physical Review Letters]] |volume=49 |issue=25 |pages=1804–7 |doi=10.1103/PhysRevLett.49.1804|bibcode = 1982PhRvL..49.1804A|doi-access=free }}</ref> In it the settings were changed fast enough, but deterministically. The first experiment to change the settings randomly, with the choices made by a [[quantum random number generator]], was Weihs et al.'s 1998 experiment.<ref name="Weihs-1998">{{cite journal|first1=G. |last1=Weihs |first2=T. |last2=Jennewein |first3=C. |last3=Simon |first4=H. |last4=Weinfurter |first5=A. |last5=Zeilinger |year=1998 |title=Violation of Bell's inequality under strict Einstein locality conditions |journal=[[Physical Review Letters]] |volume=81 |issue=23 |pages=5039–5043 |arxiv=quant-ph/9810080 |doi=10.1103/PhysRevLett.81.5039 |bibcode=1998PhRvL..81.5039W|s2cid=29855302 }}</ref> Scheidl et al. improved on this further in 2010 by conducting an experiment between locations separated by a distance of {{cvt|144|km|mi}}.<ref name="Scheidl-2010">{{cite journal|first1=Thomas |last1=Scheidl |first2=Rupert |last2=Ursin |first3=Johannes |last3=Kofler |first4=Sven |last4=Ramelow |first5=Xiao-Song |last5=Ma |first6=Thomas |last6=Herbst |first7=Lothar |last7=Ratschbacher |first8=Alessandro |last8=Fedrizzi |first9=Nathan K. |last9=Langford |first10=Thomas |last10=Jennewein |first11=Anton |last11=Zeilinger |year=2010|title=Violation of local realism with freedom of choice |journal=[[Proceedings of the National Academy of Sciences of the United States of America|PNAS]] |volume=107 |issue=46 |pages=19708–19713 |doi=10.1073/pnas.1002780107|display-authors=etal|bibcode=2010PNAS..10719708S |arxiv=0811.3129 |pmid=21041665 |pmc=2993398 |doi-access=free}}</ref>

===Coincidence loophole===
In many experiments, especially those based on photon polarization, pairs of events in the two wings of the experiment are only identified as belonging to a single pair after the experiment is performed, by judging whether or not their detection times are close enough to one another. This generates a new possibility for a local hidden variables theory to "fake" quantum correlations: delay the detection time of each of the two particles by a larger or smaller amount depending on some relationship between hidden variables carried by the particles and the detector settings encountered at the measurement station.<ref name="Larsson2004">{{cite journal |last1=Larsson |first1=Jan-Åke |last2=Gill |first2=Richard |title=Bell's inequality and the coincidence-time loophole |journal=[[Europhysics Letters]] |date=2004 |volume=67 |issue=5 |page=707 |doi=10.1209/epl/i2004-10124-7 |arxiv=quant-ph/0312035|bibcode=2004EL.....67..707L |s2cid=17135877 }}</ref>

The coincidence loophole can be ruled out entirely simply by working with a pre-fixed lattice of detection windows which are short enough that most pairs of events occurring in the same window do originate with the same emission and long enough that a true pair is not separated by a window boundary.<ref name="Larsson2004"/>

===Memory loophole===
In most experiments, measurements are repeatedly made at the same two locations. A local hidden variable theory could exploit the memory of past measurement settings and outcomes in order to increase the violation of a Bell inequality. Moreover, physical parameters might be varying in time. It has been shown that, provided each new pair of measurements is done with a new random pair of measurement settings, that neither memory nor time inhomogeneity have a serious effect on the experiment.<ref>{{cite journal|title=Quantum nonlocality, Bell inequalities and the memory loophole| first1=Jonathan |last1=Barrett |first2=Daniel |last2=Collins|first3=Lucien|last3=Hardy |first4=Adrian|last4=Kent| first5=Sandu|last5=Popescu|journal=[[Physical Review A]] |volume=66 |issue=4|at=042111| year=2002|doi=10.1103/PhysRevA.66.042111| arxiv=quant-ph/0205016|bibcode=2002PhRvA..66d2111B|s2cid=6524446}}</ref><ref>{{cite book|chapter=Accardi contra Bell (cum mundi): The Impossible Coupling|first=Richard D.|last=Gill|pages=133–154|title=Mathematical Statistics and Applications: Festschrift for Constance van Eeden|editor=M. Moore|editor2=S. Froda|editor3=C. Léger|series=IMS Lecture Notes — Monograph Series|volume=42|date=2003|publisher= Institute of Mathematical Statistics|location=Beachwood, Ohio|arxiv=quant-ph/0110137}}</ref><ref name="Gill2002">{{cite book|chapter=Time, Finite Statistics, and Bell's Fifth Position|first=Richard D.|last=Gill|pages=179–206|title=Proceedings of the Conference Foundations of Probability and Physics - 2 : Växjö (Soland), Sweden, June 2-7, 2002 |volume=5|publisher=Växjö University Press|date=2002|arxiv=quant-ph/0301059 |bibcode = 2003quant.ph..1059G }}</ref>

===Superdeterminism===
{{Main|Superdeterminism}}
A necessary assumption to derive Bell's theorem is that the hidden variables are not correlated with the measurement settings. This assumption has been justified on the grounds that the experimenter has "[[free will]]" to choose the settings, and that such is necessary to do science in the first place. A (hypothetical) theory where the choice of measurement is determined by the system being measured is known as ''superdeterministic''.<ref name=larsson14>{{cite journal |last1=Larsson |first1=Jan-Åke |title=Loopholes in Bell inequality tests of local realism |journal=Journal of Physics A: Mathematical and Theoretical |date=2014 |volume=47 |issue=42 |page=424003 |doi=10.1088/1751-8113/47/42/424003 |arxiv=1407.0363 |bibcode=2014JPhA...47P4003L |s2cid=40332044 }}</ref>

=== Many-worlds loophole ===
The [[many-worlds interpretation]], also known as the [[Hugh Everett III|Hugh Everett]] interpretation, is deterministic and has local dynamics, consisting of the unitary part of quantum mechanics without collapse. Bell's theorem does not apply because of an implicit assumption that measurements have a single outcome.<ref>{{cite journal |first1=David |last1=Deutsch |author-link1=David Deutsch |first2=Patrick |last2=Hayden |author-link2=Patrick Hayden (scientist) |title=Information flow in entangled quantum systems |journal=[[Proceedings of the Royal Society A]] |date=2000 |volume=456 |issue=1999 |pages=1759–1774 |doi=10.1098/rspa.2000.0585|arxiv=quant-ph/9906007|bibcode=2000RSPSA.456.1759D |s2cid=13998168 }}</ref>