Editing Bell's theorem (section)

==Variations and related results==
===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"/>

===GHZ–Mermin (1990)===
{{main|GHZ experiment}}

[[Daniel Greenberger]], [[Michael Horne (physicist)|Michael A. Horne]], and [[Anton Zeilinger]] presented a four-particle thought experiment in 1990, which [[N. David Mermin|David Mermin]] then simplified to use only three particles.<ref name="GHZ1990">{{cite journal |first1=D. |last1=Greenberger |author-link1=Daniel Greenberger |first2=M. |last2=Horne |author-link2=Michael A. Horne |first3=A. |last3=Shimony |author-link3=Abner Shimony |first4=A. |last4=Zeilinger |author-link4=Anton Zeilinger |title=Bell's theorem without inequalities |journal=[[American Journal of Physics]] |volume=58 |issue=12 |pages=1131 |year=1990|bibcode = 1990AmJPh..58.1131G |doi = 10.1119/1.16243 |doi-access=free }}</ref><ref name="mermin1990">{{cite journal |first=N. David |last=Mermin |author-link=N. David Mermin |title=Quantum mysteries revisited |journal=[[American Journal of Physics]] |volume=58 |issue=8 |pages=731–734 |year=1990|bibcode = 1990AmJPh..58..731M |doi = 10.1119/1.16503}}</ref> In this thought experiment, Victor generates a set of three spin-1/2 particles described by the quantum state<math display="block">|\psi\rangle = \frac{1}{\sqrt{2}}(|000\rangle - |111\rangle) \, , </math>
where as above, <math>|0\rangle</math> and <math>|1\rangle</math> are the eigenvectors of the Pauli matrix <math>\sigma_z</math>. Victor then sends a particle each to Alice, Bob, and Charlie, who wait at widely separated locations. Alice measures either <math>\sigma_x</math> or <math>\sigma_y</math> on her particle, and so do Bob and Charlie. The result of each measurement is either <math>+1</math> or <math>-1</math>. Applying the Born rule to the three-qubit state <math>|\psi\rangle</math>, Victor predicts that whenever the three measurements include one <math>\sigma_x</math> and two <math>\sigma_y</math>'s, the product of the outcomes will always be <math>+1</math>. This follows because <math>|\psi\rangle</math> is an eigenvector of <math>\sigma_x \otimes \sigma_y \otimes \sigma_y</math> with eigenvalue <math>+1</math>, and likewise for <math>\sigma_y \otimes \sigma_x \otimes \sigma_y</math> and <math>\sigma_y \otimes \sigma_y \otimes \sigma_x</math>. Therefore, knowing Alice's result for a <math>\sigma_x</math> measurement and Bob's result for a <math>\sigma_y</math> measurement, Victor can predict with probability 1 what result Charlie will return for a <math>\sigma_y</math> measurement. According to the EPR criterion of reality, there would be an "element of reality" corresponding to the outcome of a <math>\sigma_y</math> measurement upon Charlie's qubit. Indeed, this same logic applies to both measurements and all three qubits. Per the EPR criterion of reality, then, each particle contains an "instruction set" that determines the outcome of a <math>\sigma_x</math> or <math>\sigma_y</math> measurement upon it. The set of all three particles would then be described by the instruction set<math display="block">(a_x,a_y,b_x,b_y,c_x,c_y) \, , </math>
with each entry being either <math>-1</math> or <math>+1</math>, and each <math>\sigma_x</math> or <math>\sigma_y</math> measurement simply returning the appropriate value.

If Alice, Bob, and Charlie all perform the <math>\sigma_x</math> measurement, then the product of their results would be <math>a_x b_x c_x</math>. This value can be deduced from<math display="block">(a_x b_y c_y) (a_y b_x c_y) (a_y b_y c_x) = a_x b_x c_x a_y^2 b_y^2 c_y^2 = a_x b_x c_x \, , </math>
because the square of either <math>-1</math> or <math>+1</math> is <math>1</math>. Each factor in parentheses equals <math>+1</math>, so<math display="block">a_x b_x c_x = +1 \, , </math>
and the product of Alice, Bob, and Charlie's results will be <math>+1</math> with probability unity. But this is inconsistent with quantum physics: Victor can predict using the state <math>|\psi\rangle</math> that the measurement <math>\sigma_x \otimes \sigma_x \otimes \sigma_x</math> will instead yield <math>-1</math> with probability unity.

This thought experiment can also be recast as a traditional Bell inequality or, equivalently, as a nonlocal game in the same spirit as the CHSH game.<ref name="Brassard 2004">{{Cite journal|arxiv = quant-ph/0408052|last1 = Brassard|first1 = Gilles|title = Recasting Mermin's multi-player game into the framework of pseudo-telepathy|last2 = Broadbent|first2 = Anne|last3 = Tapp|first3 = Alain|year = 2005 |journal=Quantum Information and Computation |volume=5 |issue=7 |pages=538–550|doi = 10.26421/QIC5.7-2|bibcode = 2004quant.ph..8052B |author-link1 = Gilles Brassard |author-link2 = Anne Broadbent }}</ref> In it, Alice, Bob, and Charlie receive bits <math>x,y,z</math> from Victor, promised to always have an even number of ones, that is, <math>x\oplus y\oplus z = 0</math>, and send him back bits <math>a,b,c</math>. They win the game if <math>a,b,c</math> have an odd number of ones for all inputs except <math>x=y=z=0</math>, when they need to have an even number of ones. That is, they win the game [[if and only if]] <math>a \oplus b \oplus c = x \lor y \lor z</math>. With local hidden variables the highest probability of victory they can have is 3/4, whereas using the quantum strategy above they win it with certainty. This is an example of [[quantum pseudo-telepathy]].

===Kochen–Specker theorem (1967)===
{{main|Kochen–Specker theorem}}
In quantum theory, orthonormal bases for a [[Hilbert space]] represent measurements that can be performed upon a system having that Hilbert space. Each vector in a basis represents a possible outcome of that measurement.{{refn|group=note|In more detail, as developed by [[Paul Dirac]],<ref>{{cite book|first=Paul Adrien Maurice |last=Dirac |author-link=Paul Dirac |title=The Principles of Quantum Mechanics |title-link=The Principles of Quantum Mechanics |publisher=Clarendon Press |location=Oxford |year=1930}}</ref> [[David Hilbert]],<ref>{{cite book|first=David |last=Hilbert |author-link=David Hilbert |title=Lectures on the Foundations of Physics 1915–1927: Relativity, Quantum Theory and Epistemology |publisher=Springer |doi=10.1007/b12915 |editor-first1=Tilman |editor-last1=Sauer |editor-first2=Ulrich |editor-last2=Majer |year=2009 |isbn=978-3-540-20606-4 |oclc=463777694}}</ref> [[John von Neumann]],<ref>{{cite book|first=John |last=von Neumann |author-link=John von Neumann |title=Mathematische Grundlagen der Quantenmechanik |publisher=Springer |location=Berlin |year=1932}} English translation: {{cite book|title=Mathematical Foundations of Quantum Mechanics |title-link=Mathematical Foundations of Quantum Mechanics |publisher=Princeton University Press |year=1955 |translator-first=Robert T. |translator-last=Beyer |translator-link=Robert T. Beyer}}</ref> and [[Hermann Weyl]],<ref>{{cite book|first=Hermann |last=Weyl |author-link=Hermann Weyl |title=The Theory of Groups and Quantum Mechanics |title-link=Gruppentheorie und Quantenmechanik |orig-year=1931 |publisher=Dover |year=1950 |isbn=978-0-486-60269-1 |translator-first=H. P. |translator-last=Robertson |translator-link=Howard P. Robertson}} Translated from the German {{cite book |title=Gruppentheorie und Quantenmechanik |year=1931 |edition=2nd |publisher={{ill|S. Hirzel Verlag|de}}}}</ref> the state of a quantum mechanical system is a vector <math>|\psi\rangle</math> belonging to a ([[Separable space|separable]]) Hilbert space <math>\mathcal H</math>. Physical quantities of interest — position, momentum, energy, spin — are represented by "observables", which are [[self-adjoint operator|self-adjoint]] linear [[Operator (physics)|operator]]s acting on the Hilbert space. When an observable is measured, the result will be one of its eigenvalues with probability given by the [[Born rule]]: in the simplest case the eigenvalue <math>\eta</math> is non-degenerate and the probability is given by <math>|\langle \eta|\psi\rangle|^2</math>, where <math>|\eta\rangle</math> is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by <math>\langle \psi|P_\eta\psi\rangle</math>, where <math>P_\eta</math> is the projector onto its associated eigenspace. For the purposes of this discussion, we can take the eigenvalues to be non-degenerate.}} Suppose that a hidden variable <math>\lambda</math> exists, so that knowing the value of <math>\lambda</math> would imply certainty about the outcome of any measurement. Given a value of <math>\lambda</math>, each measurement outcome – that is, each vector in the Hilbert space – is either ''impossible'' or ''guaranteed.'' A Kochen–Specker configuration is a finite set of vectors made of multiple interlocking bases, with the property that a vector in it will always be ''impossible'' when considered as belonging to one basis and ''guaranteed'' when taken as belonging to another. In other words, a Kochen–Specker configuration is an "uncolorable set" that demonstrates the inconsistency of assuming a hidden variable <math>\lambda</math> can be controlling the measurement outcomes.<ref>{{cite book|first=Asher |last=Peres |author-link=Asher Peres |title=Quantum Theory: Concepts and Methods |title-link=Quantum Theory: Concepts and Methods |year=1993 |publisher=[[Kluwer]] |isbn=0-7923-2549-4 |oclc=28854083}}</ref>{{Rp|196–201}}

===Free will theorem===
{{main|Free will theorem}}
The Kochen–Specker type of argument, using configurations of interlocking bases, can be combined with the idea of measuring entangled pairs that underlies Bell-type inequalities. This was noted beginning in the 1970s by Kochen,<ref>{{Cite journal |last1=Redhead |first1=Michael |author-link1=Michael Redhead |last2=Brown |first2=Harvey |author-link2=Harvey R. Brown |date=1991-07-01 |title=Nonlocality in Quantum Mechanics |journal=[[Aristotelian Society|Proceedings of the Aristotelian Society, Supplementary Volumes]] |language=en |volume=65 |issue=1 |pages=119–160 |doi=10.1093/aristoteliansupp/65.1.119 |issn=0309-7013 |jstor=4106773 |quote=A similar approach was arrived at independently by Simon Kochen, although never published (private communication).}}</ref> Heywood and Redhead,<ref>{{Cite journal|last1=Heywood|first1=Peter|last2=Redhead|first2=Michael L. G. |author-link2=Michael Redhead |date=May 1983|title=Nonlocality and the Kochen–Specker paradox |journal=[[Foundations of Physics]] |language=en|volume=13|issue=5|pages=481–499|doi=10.1007/BF00729511|bibcode=1983FoPh...13..481H |s2cid=120340929|issn=0015-9018}}</ref> Stairs,<ref>{{Cite journal|last=Stairs|first=Allen|date=December 1983|title=Quantum Logic, Realism, and Value Definiteness|journal=[[Philosophy of Science (journal)|Philosophy of Science]] |language=en|volume=50|issue=4|pages=578–602|doi=10.1086/289140|s2cid=122885859|issn=0031-8248}}</ref> and Brown and Svetlichny.<ref>{{Cite journal|last1=Brown |first1=H. R. |author-link1=Harvey Brown (philosopher) |last2=Svetlichny|first2=G.|date=November 1990|title=Nonlocality and Gleason's lemma. Part I. Deterministic theories|journal=[[Foundations of Physics]] |language=en|volume=20|issue=11|pages=1379–1387|doi=10.1007/BF01883492|bibcode=1990FoPh...20.1379B |s2cid=122868901 |issn=0015-9018}}</ref> As EPR pointed out, obtaining a measurement outcome on one half of an entangled pair implies certainty about the outcome of a corresponding measurement on the other half. The "EPR criterion of reality" posits that because the second half of the pair was not disturbed, that certainty must be due to a physical property belonging to it.<ref>{{Cite journal|last1=Glick|first1=David|last2=Boge|first2=Florian J.|date=2019-10-22|title=Is the Reality Criterion Analytic?|journal=[[Erkenntnis]]|language=en|volume=86|issue=6|pages=1445–1451|arxiv=1909.11893|bibcode=2019arXiv190911893G|doi=10.1007/s10670-019-00163-w|s2cid=202889160|issn=0165-0106}}</ref> In other words, by this criterion, a hidden variable <math>\lambda</math> must exist within the second, as-yet unmeasured half of the pair. No contradiction arises if only one measurement on the first half is considered. However, if the observer has a choice of multiple possible measurements, and the vectors defining those measurements form a Kochen–Specker configuration, then some outcome on the second half will be simultaneously impossible and guaranteed.

This type of argument gained attention when an instance of it was advanced by [[John Horton Conway|John Conway]] and [[Simon B. Kochen|Simon Kochen]] under the name of the [[free will theorem]].<ref>{{cite journal | last1 = Conway | first1 = John |first2=Simon |last2=Kochen | author-link1=John Horton Conway | author-link2=Simon B. Kochen |year = 2006 | title = The Free Will Theorem | journal = [[Foundations of Physics]] | volume = 36 | issue = 10 | pages = 1441 | doi = 10.1007/s10701-006-9068-6 |arxiv = quant-ph/0604079 |bibcode = 2006FoPh...36.1441C | s2cid = 12999337 }}</ref><ref>{{Cite web |last=Rehmeyer |first=Julie |date=2008-08-15 |title=Do subatomic particles have free will? |url=https://www.sciencenews.org/article/do-subatomic-particles-have-free-will |access-date=2022-04-23 |website=[[Science News]] |language=en-US}}</ref><ref>{{Cite web |last=Thomas |first=Rachel |date=2011-12-27 |title=John Conway – discovering free will (part I) |url=https://plus.maths.org/content/john-conway-discovering-free-will-part-i |access-date=2022-04-23 |website=[[Plus Magazine]] |language=en}}</ref> The Conway–Kochen theorem uses a pair of entangled [[qutrit]]s and a Kochen–Specker configuration discovered by [[Asher Peres]].<ref>{{cite journal |last1=Conway |first1=John H. |first2=Simon |last2=Kochen | author-link1=John Horton Conway | author-link2=Simon B. Kochen |title=The strong free will theorem |journal= [[Notices of the AMS]] |volume=56 |issue=2 |year=2009 |pages=226–232 |url=http://www.ams.org/notices/200902/rtx090200226p.pdf}}</ref>

===Quasiclassical entanglement===
{{main|Spekkens toy model|Werner state}}
As Bell pointed out, some predictions of quantum mechanics can be replicated in local hidden-variable models, including special cases of correlations produced from entanglement. This topic has been studied systematically in the years since Bell's theorem. In 1989, [[Reinhard F. Werner|Reinhard Werner]] introduced what are now called [[Werner state]]s, joint quantum states for a pair of systems that yield EPR-type correlations but also admit a hidden-variable model.<ref>{{Cite journal |last=Werner |first=Reinhard F. |author-link=Reinhard F. Werner |date=1989-10-01 |title=Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model |journal=[[Physical Review A]] |language=en |volume=40 |issue=8 |pages=4277–4281 |bibcode=1989PhRvA..40.4277W |doi=10.1103/PhysRevA.40.4277 |pmid=9902666 |issn=0556-2791}}</ref> Werner states are bipartite quantum states that are invariant under [[Unitarity (physics)|unitaries]] of symmetric [[Kronecker product|tensor-product]] form: <math display="block">\rho_{AB} = (U \otimes U) \rho_{AB} (U^\dagger \otimes U^\dagger).</math>
In 2004, [[Robert Spekkens]] introduced a [[Spekkens toy model|toy model]] that starts with the premise of local, discretized degrees of freedom and then imposes a "knowledge balance principle" that restricts how much an observer can know about those degrees of freedom, thereby making them into hidden variables. The allowed states of knowledge ("epistemic states") about the underlying variables ("ontic states") mimic some features of quantum states. Correlations in the toy model can emulate some aspects of entanglement, like [[monogamy of entanglement|monogamy]], but by construction, the toy model can never violate a Bell inequality.<ref>{{Cite journal |author1-link=Robert Spekkens |last=Spekkens |first=Robert W. |date=2007-03-19 |title=Evidence for the epistemic view of quantum states: A toy theory |journal=[[Physical Review A]] |language=en |volume=75 |issue=3 |pages=032110 |arxiv=quant-ph/0401052 |bibcode=2007PhRvA..75c2110S |doi=10.1103/PhysRevA.75.032110 |s2cid=117284016 |issn=1050-2947}}</ref><ref>{{Cite journal |last1=Catani |first1=Lorenzo |last2=Browne |first2=Dan E. |date=2017-07-27 |title=Spekkens' toy model in all dimensions and its relationship with stabiliser quantum mechanics |journal=[[New Journal of Physics]] |volume=19 |issue=7 |pages=073035 |doi=10.1088/1367-2630/aa781c |bibcode=2017NJPh...19g3035C |s2cid=119428107 |issn=1367-2630 |doi-access=free |arxiv=1701.07801 }}</ref>