Editing Bell's theorem (section)

===Bell (1964)===
Bell's 1964 paper shows that a very simple local hidden-variable model can [[Local hidden-variable theory#Models|in restricted circumstances]] reproduce the predictions of quantum mechanics, but then he demonstrates that, in general, such models give different predictions.<ref name=Bell1964>{{cite journal | last1 = Bell | first1 = J. S. | author-link = John Stewart Bell | 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 }}</ref><ref name = "ND Mermin 1993-07"/>{{rp|806}} Bell considers a refinement by [[David Bohm]] of the Einstein–Podolsky–Rosen (EPR) thought experiment. In this scenario, a pair of particles are formed together in such a way that they are described by a [[singlet state|spin singlet state]] (which is an example of an entangled state). The particles then move apart in opposite directions. Each particle is measured by a [[Stern–Gerlach experiment|Stern–Gerlach device]], a measuring instrument that can be oriented in different directions and that reports one of two possible outcomes, representable by <math>+1</math> and <math>-1</math>. The configuration of each measuring instrument is represented by a unit [[Euclidean vector|vector]], and the quantum-mechanical prediction for the [[Quantum correlation|correlation]] between two detectors with settings <math>\vec{a}</math> and <math>\vec{b}</math> is
<math display="block">P(\vec{a}, \vec{b}) = -\vec{a} \cdot \vec{b}.</math>
In particular, if the orientation of the two detectors is the same (<math>\vec{a} = \vec{b}</math>), then the outcome of one measurement is certain to be the negative of the outcome of the other, giving <math>P(\vec{a}, \vec{a}) = -1</math>. And if the orientations of the two detectors are orthogonal (<math>\vec{a} \cdot \vec{b} = 0</math>), then the outcomes are uncorrelated, and <math>P(\vec{a}, \vec{b}) = 0</math>. Bell proves by example that these special cases ''can'' be explained in terms of hidden variables, then proceeds to show that the full range of possibilities involving intermediate angles ''cannot''.

Bell posited that a local hidden-variable model for these correlations would explain them in terms of an integral over the possible values of some hidden parameter <math>\lambda</math>:<math display="block">P(\vec{a}, \vec{b}) = \int d\lambda\, \rho(\lambda) A(\vec{a}, \lambda) B(\vec{b}, \lambda),</math>
where <math>\rho(\lambda)</math> is a [[probability density function]]. The two functions <math>A(\vec{a}, \lambda)</math> and <math>B(\vec{b}, \lambda)</math> provide the responses of the two detectors given the orientation vectors and the hidden variable:<math display="block">A(\vec{a}, \lambda) = \pm 1, \, B(\vec{b}, \lambda) = \pm 1.</math>
Crucially, the outcome of detector <math>A</math> does not depend upon <math>\vec{b}</math>, and likewise the outcome of <math>B</math> does not depend upon <math>\vec{a}</math>, because the two detectors are physically separated. Now we suppose that the experimenter has a ''choice'' of settings for the second detector: it can be set either to <math>\vec{b}</math> or to <math>\vec{c}</math>. Bell proves that the difference in correlation between these two choices of detector setting must satisfy the inequality<math display="block">|P(\vec{a}, \vec{b}) - P(\vec{a}, \vec{c})| \leq 1 + P(\vec{b}, \vec{c}).</math>
However, it is easy to find situations where quantum mechanics violates the Bell inequality.<ref>{{Cite book |last=Griffiths |first=David J. |author-link=David J. Griffiths |title=Introduction to Quantum Mechanics |title-link=Introduction to Quantum Mechanics (book) |date=2005 |publisher=Pearson Prentice Hall |isbn=0-13-111892-7 |edition=2nd |location=Upper Saddle River, NJ |oclc=53926857}}</ref>{{Rp|425–426}} For example, let the vectors <math>\vec{a}</math> and <math>\vec{b}</math> be orthogonal, and let <math>\vec{c}</math> lie in their plane at a 45° angle from both of them. Then<math display="block">P(\vec{a}, \vec{b}) = 0,</math>
while
<math display="block">P(\vec{a}, \vec{c}) = P(\vec{b}, \vec{c}) = -\frac{\sqrt{2}}{2},</math>
but
<math display="block">\frac{\sqrt{2}}{2} \nleq 1 - \frac{\sqrt{2}}{2}.</math>
Therefore, there is no local hidden-variable model that can reproduce the predictions of quantum mechanics for all choices of <math>\vec{a}</math>, <math>\vec{b}</math>, and <math>\vec{c}.</math> Experimental results contradict the classical curves and match the curve predicted by quantum mechanics as long as experimental shortcomings are accounted for.<ref name="Stanford"/>

Bell's 1964 theorem requires the possibility of perfect anti-correlations: the ability to make a completely certain prediction about the result from the second detector, knowing the result from the first.<ref name="Stanford" /> The theorem builds upon the "EPR criterion of reality", a concept introduced in the 1935 paper by Einstein, Podolsky, and Rosen. This paper posits: "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity."<ref name="EPR">{{cite journal | title = Can Quantum-Mechanical Description of Physical Reality be Considered Complete? | date = 1935-05-15 | first1 = A. | last1 = Einstein |first2=B. |last2 = Podolsky |first3=N. |last3 = Rosen | author-link1 = Albert Einstein | author-link2 = Boris Podolsky | author-link3 = Nathan Rosen | journal = [[Physical Review]] | volume = 47 | issue = 10 | pages = 777–780 | bibcode = 1935PhRv...47..777E |doi = 10.1103/PhysRev.47.777 | doi-access = free }}</ref> Bell noted that this applies when the two detectors are oriented in the same direction (<math>\vec{a} = \vec{b}</math>), and so the EPR criterion would imply that some element of reality must predetermine the measurement result. Because the quantum description of a particle does not include any such element, the quantum description would have to be incomplete. In other words, Bell's 1964 paper shows that, assuming locality, the EPR criterion implies hidden variables and then he demonstrates that local hidden variables are incompatible with quantum mechanics.<ref>{{Cite journal |last1=Home |first1=D. |last2=Selleri |first2=F. |date=September 1991 |title=Bell's theorem and the EPR paradox |journal=[[La Rivista del Nuovo Cimento]] |language=en |volume=14 |issue=9 |pages=1–95 |doi=10.1007/BF02811227 |bibcode=1991NCimR..14i...1H |issn=1826-9850}}</ref><ref>{{Cite journal |last1=Clauser |first1=J. F. |author-link1=John Clauser |last2=Shimony |first2=A. |author-link2=Abner Shimony |date=1978-12-01 |title=Bell's theorem. Experimental tests and implications |journal=[[Reports on Progress in Physics]] |volume=41 |issue=12 |pages=1881–1927 |doi=10.1088/0034-4885/41/12/002 |bibcode=1978RPPh...41.1881C |issn=0034-4885}}</ref> Because experiments cannot achieve perfect correlations or anti-correlations in practice, Bell-type inequalities based on derivations that relax this assumption are tested instead.<ref name="Stanford"/>