Editing Einstein–Podolsky–Rosen paradox (section)

=== Bell's theorem ===
{{main|Bell's theorem}}
In 1964, [[John Stewart Bell]] published a paper<ref name=Bell1964>{{cite journal | last1 = Bell | first1 = J. S. | year = 1964 | title = On the Einstein Podolsky Rosen Paradox | url = https://cds.cern.ch/record/111654/files/vol1p195-200_001.pdf | journal = [[Physics Physique Физика]] | volume = 1 | issue = 3| pages = 195–200 | doi = 10.1103/PhysicsPhysiqueFizika.1.195 | doi-access = free }}</ref> investigating the puzzling situation at that time: on one hand, the EPR paradox purportedly showed that quantum mechanics was nonlocal, and suggested that a hidden-variable theory could heal this nonlocality. On the other hand, David Bohm had recently developed the first successful hidden-variable theory, but it had a grossly nonlocal character.<ref>{{cite journal |last1=Bohm |first1=D. |title=A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I |journal=Physical Review |date=1952 |volume=85 |issue=2 |page=166 |doi=10.1103/PhysRev.85.166|bibcode=1952PhRv...85..166B }}</ref><ref>{{cite journal |last1=Bohm |first1=D. |title=A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. II |journal=Physical Review |date=1952 |volume=85 |issue=2 |page=180 |doi=10.1103/PhysRev.85.180|bibcode=1952PhRv...85..180B }}</ref> Bell set out to investigate whether it was indeed possible to solve the nonlocality problem with hidden variables, and found out that first, the correlations shown in both EPR's and Bohm's versions of the paradox could indeed be explained in a local way with hidden variables, and second, that the correlations shown in his own variant of the paradox couldn't be explained by ''any'' local hidden-variable theory. This second result became known as the Bell theorem.

To understand the first result, consider the following toy hidden-variable theory introduced later by J.J. Sakurai:<ref name=Sakurai>{{cite book  | last1 = Sakurai  | first1 = J. J.  |last2 = Napolitano  | first2 = Jim  | title = [[Modern Quantum Mechanics]] | edition = 2nd  | publisher = Addison-Wesley  | year = 2010|isbn =978-0805382914}}</ref>{{rp|239–240}} in it, quantum spin-singlet states emitted by the source are actually approximate descriptions for "true" physical states possessing definite values for the ''z''-spin and ''x''-spin. In these "true" states, the positron going to Bob always has spin values opposite to the electron going to Alice, but the values are otherwise completely random. For example, the first pair emitted by the source might be "(+''z'', −''x'') to Alice and (−''z'', +''x'') to Bob", the next pair "(−''z'', −''x'') to Alice and (+''z'', +''x'') to Bob", and so forth. Therefore, if Bob's measurement axis is aligned with Alice's, he will necessarily get the opposite of whatever Alice gets; otherwise, he will get "+" and "−" with equal probability.

Bell showed, however, that such models can only reproduce the singlet correlations when Alice and Bob make measurements on the same axis or on perpendicular axes. As soon as other angles between their axes are allowed, local hidden-variable theories become unable to reproduce the quantum mechanical correlations. This difference, expressed using inequalities known as "[[Bell's inequalities]]", is in principle experimentally testable. After the publication of Bell's paper, a variety of [[Bell test|experiments to test Bell's inequalities]] were carried out, notably by the group of [[Alain Aspect]] in the 1980s;<ref name=Aspect1999>{{cite journal | title = Bell's inequality test: more ideal than ever | journal = Nature | date = 1999-03-18 | author = Aspect A | volume = 398 | pages = 189–90| doi = 10.1038/18296 | url = http://www.rpi.edu/dept/phys/Courses/PHYS4510/AspectNature.pdf |bibcode = 1999Natur.398..189A | issue=6724| s2cid = 44925917 | doi-access = free }}</ref> all experiments conducted to date have found behavior in line with the predictions of quantum mechanics. The present view of the situation is that quantum mechanics flatly contradicts Einstein's philosophical postulate that any acceptable physical theory must fulfill "local realism". The fact that quantum mechanics violates Bell inequalities indicates that any hidden-variable theory underlying quantum mechanics must be non-local; whether this should be taken to imply that quantum mechanics ''itself'' is non-local is a matter of continuing debate.<ref>{{cite journal|last=Werner |first=R. F. |title=Comment on 'What Bell did' |journal=Journal of Physics A |year=2014 |volume=47 |issue=42 |pages=424011 |doi=10.1088/1751-8113/47/42/424011|bibcode=2014JPhA...47P4011W |s2cid=122180759 }}</ref><ref>{{cite journal|first1=M. |last1=Żukowski |first2=Č. |last2=Brukner |author-link2=Časlav Brukner |title=Quantum non-locality—it ain't necessarily so... |journal=[[Journal of Physics A]] |volume=47 |year=2014 |issue=42 |pages=424009 |doi=10.1088/1751-8113/47/42/424009|arxiv=1501.04618|bibcode=2014JPhA...47P4009Z |s2cid=119220867 }}</ref>