Editing Quantum entanglement (section)

=== Failure of local hidden-variable theories ===
A possible resolution to the paradox is to assume that quantum theory is incomplete, and the result of measurements depends on predetermined "[[hidden-variables theory|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> The state of the particles being measured contains some hidden variables, whose values effectively determine, right from the moment of separation, what the outcomes of the spin measurements are going to be. This would mean that each particle carries all the required information with it, and nothing needs to be transmitted from one particle to the other at the time of measurement. Einstein and others (see the previous section) originally believed this was the only way out of the paradox, and the accepted quantum mechanical description (with a random measurement outcome) must be incomplete.

[[Local hidden-variable theory|Local hidden variable theories]] fail, however, when measurements of the spin of entangled particles along different axes are considered. If a large number of pairs of such measurements are made (on a large number of pairs of entangled particles), then statistically, if the local realist or hidden variables view were correct, the results would always satisfy [[Bell's inequality]]. A [[Bell test|number of experiments]] have shown in practice that Bell's inequality is not satisfied.<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 |pages=080403 |doi=10.1103/PhysRevLett.121.080403 |last1 = Rauch |first1 = Dominik |pmid=30192604 |display-authors=etal |arxiv=1808.05966|bibcode=2018PhRvL.121h0403R }}</ref> Moreover, when measurements of the entangled particles are made in moving [[special relativity|relativistic]] reference frames, in which each measurement (in its own relativistic time frame) occurs before the other, the measurement results remain correlated.<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 |pages=22111 |arxiv=quant-ph/0007009 |bibcode=2001PhRvA..63b2111Z |doi=10.1103/PhysRevA.63.022111 |s2cid=44611890}}</ref><ref name=Gilder2009/>{{rp|321–324}}

The fundamental issue about measuring spin along different axes is that these measurements cannot have definite values at the same time―they are [[Incompatible observables|incompatible]] in the sense that these measurements' maximum simultaneous precision is constrained by the [[uncertainty principle]]. This is contrary to what is found in classical physics, where any number of properties can be measured simultaneously with arbitrary accuracy. It has been proven mathematically that compatible measurements cannot show Bell-inequality-violating correlations,<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> and thus entanglement is a fundamentally non-classical phenomenon.