Editing Bell test (section)

===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." />