Editing Bell's theorem

{{Short description|Theorem in physics}}

{{Redirect|Bell inequality|the related experiments|Bell test}}

'''Bell's theorem''' is a term encompassing a number of closely related results in [[physics]], all of which determine that [[quantum mechanics]] is incompatible with [[Local hidden-variable theory|local hidden-variable theories]], given some basic assumptions about the nature of measurement. The first such result was introduced by [[John Stewart Bell]] in 1964, building upon the [[Einstein–Podolsky–Rosen paradox]], which had called attention to the phenomenon of [[quantum entanglement]].

In the context of Bell's theorem, "local" refers to the [[principle of locality]], the idea that a [[particle]] can only be influenced by its immediate surroundings, and that interactions mediated by [[Field (physics)|physical fields]] cannot propagate faster than the [[speed of light]]. "[[Hidden-variable theory|Hidden variables]]" are supposed properties of quantum particles that are not included in quantum theory but nevertheless affect the outcome of experiments. In the words of Bell, "If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."<ref>{{cite book | first = John S. | last = Bell | author-link = John Stewart Bell | title = Speakable and Unspeakable in Quantum Mechanics | publisher = Cambridge University Press | date = 1987 | page = 65 | isbn = 9780521368698 | oclc = 15053677}}</ref>

In his original paper,<ref name="Bell1964" /> Bell deduced that if measurements are performed independently on the two separated particles of an entangled pair, then the assumption that the outcomes depend upon hidden variables within each half implies a mathematical constraint on how the outcomes on the two measurements are correlated. Such a constraint would later be named a '''Bell inequality'''. Bell then showed that quantum physics predicts correlations that violate this [[Inequality (mathematics)|inequality]]. Multiple variations on Bell's theorem were put forward in the years following his original paper, using different assumptions and obtaining different Bell (or "Bell-type") inequalities.

The first rudimentary experiment designed to test Bell's theorem was performed in 1972 by [[John Clauser]] and [[Stuart Freedman]].<ref>{{cite press release |url=https://www.nobelprize.org/prizes/physics/2022/press-release/ |title=The Nobel Prize in Physics 2022 |date=October 4, 2022 |work=[[Nobel Prize]] |publisher=[[The Royal Swedish Academy of Sciences]] |access-date=6 October 2022}}</ref> More advanced experiments, known collectively as ''[[Bell test]]s'', have been performed many times since. Often, these experiments have had the goal of "closing loopholes", that is, ameliorating problems of experimental design or set-up that could in principle affect the validity of the findings of earlier Bell tests. Bell tests have consistently found that physical systems obey quantum mechanics and violate Bell inequalities; which is to say that the results of these experiments are incompatible with local hidden-variable theories.<ref name="NAT-20180509">{{cite journal |author=The BIG Bell Test Collaboration |title=Challenging local realism with human choices |date=9 May 2018 |journal=[[Nature (journal)|Nature]] |volume=557 |issue=7704 |pages=212–216 |doi=10.1038/s41586-018-0085-3 |pmid=29743691 |bibcode=2018Natur.557..212B |arxiv=1805.04431 |s2cid=13665914 }}</ref><ref>{{Cite web |url=https://www.quantamagazine.org/20170207-bell-test-quantum-loophole/ |title=Experiment Reaffirms Quantum Weirdness |last=Wolchover |first=Natalie |author-link=Natalie Wolchover |date=2017-02-07 |work=[[Quanta Magazine]] |language=en-US |access-date=2020-02-08}}</ref> 

The exact nature of the assumptions required to prove a Bell-type constraint on correlations has been debated by physicists and by [[philosophy of physics|philosophers]]. While the significance of Bell's theorem is not in doubt, different [[interpretations of quantum mechanics]] disagree about what exactly it implies.

==Theorem==
There are many variations on the basic idea, some employing stronger mathematical assumptions than others.<ref name="Stanford">{{Cite SEP|bell-theorem|title=Bell's Theorem|first = Abner | last = Shimony|author-link=Abner Shimony}}</ref> Significantly, Bell-type theorems do not refer to any particular theory of local hidden variables, but instead show that quantum physics violates general assumptions behind classical pictures of nature. The original theorem proved by Bell in 1964 is not the most amenable to experiment, and it is convenient to introduce the genre of Bell-type inequalities with a later example.<ref name="mike-and-ike"/>

Hypothetical characters [[Alice and Bob]] stand in widely separated locations. Their colleague Victor prepares a pair of particles and sends one to Alice and the other to Bob. When Alice receives her particle, she chooses to perform one of two possible measurements (perhaps by flipping a coin to decide which). Denote these measurements by <math>A_0</math> and <math>A_1</math>. Both <math>A_0</math> and <math>A_1</math> are ''binary'' measurements: the result of <math>A_0</math> is either <math>+1</math> or <math>-1</math>, and likewise for <math>A_1</math>. When Bob receives his particle, he chooses one of two measurements, <math>B_0</math> and <math>B_1</math>, which are also both binary.

Suppose that each measurement reveals a property that the particle already possessed. For instance, if Alice chooses to measure <math>A_0</math> and obtains the result <math>+1</math>, then the particle she received carried a value of <math>+1</math> for a property <math>a_0</math>.{{refn|group=note|We are for convenience assuming that the response of the detector to the underlying property is deterministic. This assumption can be replaced; it is equivalent to postulating a joint probability distribution over all the observables of the experiment.<ref>{{Cite journal |last=Fine |first=Arthur |date=1982-02-01 |title=Hidden Variables, Joint Probability, and the Bell Inequalities |url=https://link.aps.org/doi/10.1103/PhysRevLett.48.291 |journal=[[Physical Review Letters]] |language=en |volume=48 |issue=5 |pages=291–295 |doi=10.1103/PhysRevLett.48.291 |bibcode=1982PhRvL..48..291F |issn=0031-9007}}</ref><ref>{{Cite journal |last1=Braunstein |first1=Samuel L. |last2=Caves |first2=Carlton M. |author-link2=Carlton M. Caves |date=August 1990 |title=Wringing out better Bell inequalities |journal=[[Annals of Physics]] |language=en |volume=202 |issue=1 |pages=22–56 |doi=10.1016/0003-4916(90)90339-P|bibcode=1990AnPhy.202...22B }}</ref>}} Consider the combination<math display="block">a_0b_0 + a_0b_1 + a_1b_0-a_1b_1 = (a_0+a_1)b_0 + (a_0-a_1)b_1 \, .</math>Because both <math>a_0</math> and <math>a_1</math> take the values <math>\pm 1</math>, then either <math>a_0 = a_1</math> or <math>a_0 = -a_1</math>. In the former case, the quantity <math>(a_0-a_1)b_1</math> must equal 0, while in the latter case, <math>(a_0+a_1)b_0 = 0</math>. So, one of the terms on the right-hand side of the above expression will vanish, and the other will equal <math>\pm 2</math>. Consequently, if the experiment is repeated over many trials, with Victor preparing new pairs of particles, the absolute value of the average of the combination <math>a_0b_0 + a_0b_1 + a_1b_0-a_1b_1</math> across all the trials will be less than or equal to 2. No ''single'' trial can measure this quantity, because Alice and Bob can only choose one measurement each, but on the assumption that the underlying properties exist, the average value of the sum is just the sum of the averages for each term. Using angle brackets to denote averages<math display="block">| \langle A_0B_0 \rangle + \langle A_0B_1 \rangle + \langle A_1B_0 \rangle - \langle A_1B_1 \rangle | \leq 2 \, .</math>
This is a Bell inequality, specifically, the [[CHSH inequality]].<ref name="mike-and-ike">{{Cite book|last1=Nielsen|first1=Michael A.|last2=Chuang|first2=Isaac L.|title=Quantum Computation and Quantum Information|author-link1=Michael Nielsen |author-link2=Isaac Chuang |title-link=Quantum Computation and Quantum Information |publisher=Cambridge University Press|location=Cambridge|year=2010|edition=2nd|oclc=844974180|isbn=978-1-107-00217-3}}</ref>{{Rp|115}} Its derivation here depends upon two assumptions: first, that the underlying physical properties <math>a_0, a_1, b_0,</math> and <math>b_1</math> exist independently of being observed or measured (sometimes called the assumption of ''realism''); and second, that Alice's choice of action cannot influence Bob's result or vice versa (often called the assumption of ''locality'').<ref name="mike-and-ike" />{{Rp|117}}

Quantum mechanics can violate the CHSH inequality, as follows. Victor prepares a pair of [[qubit]]s which he describes by the [[Bell state]]<math display="block">|\psi\rangle = \frac{|0\rangle \otimes |1\rangle - |1\rangle \otimes |0\rangle}{\sqrt{2}} ,</math>
where <math>|0\rangle</math> and <math>|1\rangle</math> are the eigenstates of one of the [[Pauli matrices]],<math display="block">\sigma_z = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}.</math>
Victor then passes the first qubit to Alice and the second to Bob. Alice and Bob's choices of possible [[measurement in quantum mechanics|measurements]] are also defined in terms of the Pauli matrices. Alice measures either of the two observables <math>\sigma_z</math> and <math>\sigma_x</math>:<math display="block">A_0 = \sigma_z,\ A_1 = \sigma_x = \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix};</math>
and Bob measures either of the two observables<math display="block">B_0 = -\frac{\sigma_x + \sigma_z}{\sqrt{2}},\ B_1 = \frac{\sigma_x - \sigma_z}{\sqrt{2}} .</math>
Victor can calculate the quantum expectation values for pairs of these observables using the [[Born rule]]:<math display="block">\langle A_0 \otimes B_0 \rangle = \frac{1}{\sqrt{2}}, \langle A_0 \otimes B_1 \rangle = \frac{1}{\sqrt{2}}, \langle A_1 \otimes B_0 \rangle = \frac{1}{\sqrt{2}}, \langle A_1 \otimes B_1 \rangle = -\frac{1}{\sqrt{2}} \, . </math>
While only one of these four measurements can be made in a single trial of the experiment, the sum<math display="block">\langle A_0 \otimes B_0 \rangle + \langle A_0 \otimes B_1 \rangle + \langle A_1 \otimes B_0 \rangle - \langle A_1 \otimes B_1 \rangle = 2\sqrt{2} </math>
gives the sum of the average values that Victor expects to find across multiple trials. This value exceeds the classical upper bound of 2 that was deduced from the hypothesis of local hidden variables.<ref name="mike-and-ike"/>{{Rp|116}} The value <math>2\sqrt{2}</math> is in fact the largest that quantum physics permits for this combination of expectation values, making it a [[Tsirelson's bound|Tsirelson bound]].<ref>{{Cite book |last=Rau |first=Jochen |url=https://www.worldcat.org/oclc/1256446911 |title=Quantum theory : an information processing approach |date=2021 |publisher=Oxford University Press |isbn=978-0-192-65027-6 |oclc=1256446911}}</ref>{{Rp|page=140}}

[[File:Chsh-illustration.png|thumb|An illustration of the CHSH game: the referee, Victor, sends a bit each to Alice and to Bob, and Alice and Bob each send a bit back to the referee.]]
The CHSH inequality can also be thought of as [[CHSH game|a ''game'' in which Alice and Bob try to coordinate their actions]].<ref>{{cite book|last1=Cleve |first1=R. |author-link1=Richard Cleve |last2=Hoyer |first2=P. |last3=Toner |first3=B. |last4=Watrous |first4=J. |author-link4=John Watrous (computer scientist) |year=2004 |chapter=Consequences and limits of nonlocal strategies |title=Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004. |pages=236–249 |publisher=[[IEEE]] |doi=10.1109/CCC.2004.1313847 |isbn=0-7695-2120-7 |oclc=55954993 |arxiv=quant-ph/0404076 |bibcode=2004quant.ph..4076C|s2cid=8077237 }}</ref><ref>{{Cite journal|last1=Barnum|first1=H.|last2=Beigi|first2=S.|last3=Boixo|first3=S.|last4=Elliott|first4=M. B.|last5=Wehner|first5=S.|date=2010-04-06|title=Local Quantum Measurement and No-Signaling Imply Quantum Correlations|journal=[[Physical Review Letters]]|language=en|volume=104|issue=14|pages=140401|arxiv=0910.3952|bibcode=2010PhRvL.104n0401B|doi=10.1103/PhysRevLett.104.140401|pmid=20481921|s2cid=17298392|issn=0031-9007}}</ref> Victor prepares two bits, <math>x</math> and <math>y</math>, independently and at random. He sends bit <math>x</math> to Alice and bit <math>y</math> to Bob. Alice and Bob win if they return answer bits <math>a</math> and <math>b</math> to Victor, satisfying
<math display="block">x y = a + b \mod 2 \, .</math>
Or, equivalently, Alice and Bob win if the [[logical AND]] of <math>x</math> and <math>y</math> is the [[logical XOR]] of <math>a</math> and <math>b</math>. Alice and Bob can agree upon any strategy they desire before the game, but they cannot communicate once the game begins. In any theory based on local hidden variables, Alice and Bob's probability of winning is no greater than <math>3/4</math>, regardless of what strategy they agree upon beforehand. However, if they share an entangled quantum state, their probability of winning can be as large as<math display="block">\frac{2+\sqrt{2}}{4} \approx 0.85 \, .</math>

==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>

==History==
===Background===
{{main|EPR paradox|History of quantum mechanics}}

The question of whether quantum mechanics can be "completed" by hidden variables dates to the early years of quantum theory. In his [[Mathematical Foundations of Quantum Mechanics|1932 textbook on quantum mechanics]], the Hungarian-born polymath [[John von Neumann]] presented what he claimed to be a proof that there could be no "hidden parameters". The validity and definitiveness of von Neumann's proof were questioned by [[Hans Reichenbach]], in more detail by [[Grete Hermann]], and possibly in conversation though not in print by Albert Einstein.{{refn|group=note|See Reichenbach<ref>{{cite book|first=Hans |last=Reichenbach |author-link=Hans Reichenbach |title=Philosophic Foundations of Quantum Mechanics |year=1944 |publisher=University of California Press |page=14 |oclc=872622725}}</ref> and Jammer,<ref name="jammer1974">{{cite book|last=Jammer|first=Max|title=The Philosophy of Quantum Mechanics|publisher=John Wiley and Sons|year=1974|isbn=0-471-43958-4|author-link=Max Jammer}}</ref>{{Rp|276}} Mermin and Schack,<ref>{{cite journal|title=Homer nodded: von Neumann's surprising oversight |journal=[[Foundations of Physics]] |volume=48 |issue=9 |pages=1007–1020 |year=2018 |arxiv=1805.10311 |last1=Mermin |first1=N. David |last2=Schack |first2=Rüdiger |author-link1=N. David Mermin|doi=10.1007/s10701-018-0197-5 |bibcode=2018FoPh...48.1007M |s2cid=118951033 }}</ref> and for Einstein's remarks, Clauser and Shimony<ref>{{cite journal | last1 = Clauser | first1 = J. F. | last2 = Shimony | first2 = A. | title = Bell's theorem: Experimental tests and implications | url = http://www.physics.oregonstate.edu/~ostroveo/COURSES/ph651/Supplements_Phys651/RPP1978_Bell.pdf | journal = Reports on Progress in Physics | volume = 41 | issue = 12 | pages = 1881–1927 | year = 1978 | doi = 10.1088/0034-4885/41/12/002 | bibcode = 1978RPPh...41.1881C | citeseerx = 10.1.1.482.4728 | s2cid = 250885175 | access-date = 2017-10-28 | archive-date = 2017-09-23 | archive-url = https://web.archive.org/web/20170923004338/http://physics.oregonstate.edu/~ostroveo/COURSES/ph651/Supplements_Phys651/RPP1978_Bell.pdf | url-status = live }}</ref> and Wick.<ref name=":1"/>{{Rp|286}}}} ([[Simon B. Kochen|Simon Kochen]] and [[Ernst Specker]] rejected von Neumann's key assumption as early as 1961, but did not publish a criticism of it until 1967.<ref>{{Cite book |author-first1=John |author-last1=Conway |author-link1=John Horton Conway |author-first2=Simon |author-last2=Kochen |author-link2=Simon B. Kochen |chapter=The Geometry of the Quantum Paradoxes |pages=257–269 |title=Quantum [Un]speakables: From Bell to Quantum Information |date=2002 |publisher=Springer |editor-first1=Reinhold A. |editor-last1=Bertlmann |editor-link1=Reinhold Bertlmann |editor-first2=Anton |editor-last2=Zeilinger |editor-link2=Anton Zeilinger |isbn=3-540-42756-2 |location=Berlin |oclc=49404213}}</ref>)

Einstein argued persistently that quantum mechanics could not be a complete theory. His preferred argument relied on a principle of locality:
:Consider a mechanical system constituted of two partial systems ''A'' and ''B'' which have interaction with each other only during limited time. Let the ψ function before their interaction be given. Then the [[Schrödinger equation]] will furnish the ψ function after their interaction has taken place. Let us now determine the physical condition of the partial system ''A'' as completely as possible by measurements. Then the quantum mechanics allows us to determine the ψ function of the partial system ''B'' from the measurements made, and from the ψ function of the total system. This determination, however, gives a result which depends upon ''which'' of the determining magnitudes specifying the condition of ''A'' has been measured (for instance coordinates ''or'' momenta). Since there can be only ''one'' physical condition of ''B'' after the interaction and which can reasonably not be considered as dependent on the particular measurement we perform on the system ''A'' separated from ''B'' it may be concluded that the ψ function is not unambiguously coordinated with the physical condition. This coordination of several ψ functions with the same physical condition of system ''B'' shows again that the ψ function cannot be interpreted as a (complete) description of a physical condition of a unit system.<ref>{{cite journal|first=Albert |last=Einstein |author-link=Albert Einstein |title=Physics and reality |journal=Journal of the Franklin Institute |volume=221 |number=3 |date=March 1936 |pages=349–382 |doi=10.1016/S0016-0032(36)91047-5 |bibcode=1936FrInJ.221..349E}}</ref>

The EPR thought experiment is similar, also considering two separated systems ''A'' and ''B'' described by a joint wave function. However, the EPR paper adds the idea later known as the EPR criterion of reality, according to which the ability to predict with probability 1 the outcome of a measurement upon ''B'' implies the existence of an "element of reality" within ''B''.<ref>{{cite journal|first1=Nicholas |last1=Harrigan |first2=Robert W. |last2=Spekkens |title=Einstein, incompleteness, and the epistemic view of quantum states |journal=[[Foundations of Physics]] |volume=40 |issue=2 |pages=125 |year=2010 |doi=10.1007/s10701-009-9347-0 |arxiv=0706.2661|bibcode=2010FoPh...40..125H |s2cid=32755624 }}</ref>

In 1951, [[David Bohm]] proposed a variant of the EPR thought experiment in which the measurements have discrete ranges of possible outcomes, unlike the position and momentum measurements considered by EPR.<ref>{{cite book|last=Bohm |first=David |author-link=David Bohm |year=1989 |orig-date=1951 |title=Quantum Theory |publisher=Prentice-Hall |edition=Dover reprint |isbn=978-0-486-65969-5 |oclc=1103789975 |pages=614–623}}</ref> The year before, [[Chien-Shiung Wu]] and Irving Shaknov had successfully measured polarizations of photons produced in entangled pairs, thereby making the Bohm version of the EPR thought experiment practically feasible.<ref>{{cite journal |last1=Wu |first1=C.-S. |author-link=Chien-Shiung Wu |last2=Shaknov |first2=I. |year=1950 |title=The Angular Correlation of Scattered Annihilation Radiation |journal=[[Physical Review]] |volume=77 |issue=1 |pages=136 |bibcode=1950PhRv...77..136W |doi=10.1103/PhysRev.77.136}}</ref>

By the late 1940s, the mathematician [[George Mackey]] had grown interested in the foundations of quantum physics, and in 1957 he drew up a list of postulates that he took to be a precise definition of quantum mechanics.<ref>{{Cite journal |last=Mackey |first=George W. |author-link=George Mackey |title=Quantum Mechanics and Hilbert Space |journal=[[The American Mathematical Monthly]] |year=1957 |volume=64 |number=8P2 |pages=45–57 |doi=10.1080/00029890.1957.11989120 |jstor=2308516}}</ref> Mackey conjectured that one of the postulates was redundant, and shortly thereafter, [[Andrew M. Gleason]] proved that it was indeed deducible from the other postulates.<ref name="gleason1957">{{cite journal|first=Andrew M.|author-link=Andrew M. Gleason|year = 1957|title = Measures on the closed subspaces of a Hilbert space|url = http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1957/6/56050|journal = [[Indiana University Mathematics Journal]]|volume = 6|issue=4|pages = 885–893|doi=10.1512/iumj.1957.6.56050|mr=0096113|last = Gleason|doi-access = free}}</ref><ref name="chernoff2009">{{Cite journal|last=Chernoff |first=Paul R. |author-link=Paul Chernoff |title=Andy Gleason and Quantum Mechanics |journal=[[Notices of the AMS]] |volume=56 |number=10 |pages=1253–1259 |url=https://www.ams.org/notices/200910/rtx091001236p.pdf}}</ref> [[Gleason's theorem]] provided an argument that a broad class of hidden-variable theories are incompatible with quantum mechanics.{{refn|group=note|A hidden-variable theory that is [[determinism|deterministic]] implies that the probability of a given outcome is ''always'' either 0 or 1. For example, a Stern–Gerlach measurement on a [[Spin (physics)|spin-1]] atom will report that the atom's angular momentum along the chosen axis is one of three possible values, which can be designated <math>-</math>, <math>0</math> and <math>+</math>. In a deterministic hidden-variable theory, there exists an underlying physical property that fixes the result found in the measurement. Conditional on the value of the underlying physical property, any given outcome (for example, a result of <math>+</math>) must be either impossible or guaranteed. But Gleason's theorem implies that there can be no such deterministic probability measure, because it proves that any probability measure must take the form of a mapping <math>u \to \langle \rho u, u \rangle</math> for some density operator <math>\rho</math>. This mapping is continuous on the [[unit sphere]] of the Hilbert space, and since this unit sphere is [[Connected (topology)|connected]], no continuous probability measure on it can be deterministic.<ref name="wilce2017">{{cite book|last=Wilce |first=A. |year=2017 |chapter-url=https://plato.stanford.edu/entries/qt-quantlog/ |chapter=Quantum Logic and Probability Theory |title=Stanford Encyclopedia of Philosophy |title-link=Stanford Encyclopedia of Philosophy|publisher=Metaphysics Research Lab, Stanford University }}</ref>{{rp|§1.3}}}} More specifically, Gleason's theorem rules out hidden-variable models that are "noncontextual". Any hidden-variable model for quantum mechanics must, in order to avoid the implications of Gleason's theorem, involve hidden variables that are not properties belonging to the measured system alone but also dependent upon the external context in which the measurement is made. This type of dependence is often seen as contrived or undesirable; in some settings, it is inconsistent with [[special relativity]].<ref name = "ND Mermin 1993-07">{{cite journal | last = Mermin |first = N. David |author-link=N. David Mermin |title = Hidden Variables and the Two Theorems of John Bell | journal = [[Reviews of Modern Physics]] | volume = 65 |pages = 803–815 | number = 3| date = July 1993  | url = http://cqi.inf.usi.ch/qic/Mermin1993.pdf |arxiv=1802.10119 |doi = 10.1103/RevModPhys.65.803 |bibcode = 1993RvMP...65..803M |s2cid = 119546199 }}</ref><ref>{{Cite journal|last=Shimony |first=Abner |author-link=Abner Shimony |title=Contextual Hidden Variable Theories and Bell's Inequalities |journal=[[British Journal for the Philosophy of Science]] |year=1984 |volume=35 |number=1 |pages=25–45 |doi=10.1093/bjps/35.1.25}}</ref> The Kochen–Specker theorem refines this statement by constructing a specific finite subset of rays on which no such probability measure can be defined.<ref name="ND Mermin 1993-07" /><ref>{{Cite journal|last=Peres|first=Asher|author-link=Asher Peres|date=1991|title=Two simple proofs of the Kochen-Specker theorem|url=http://stacks.iop.org/0305-4470/24/i=4/a=003|journal=[[Journal of Physics A: Mathematical and General]]|language=en|volume=24|issue=4|pages=L175–L178|doi=10.1088/0305-4470/24/4/003|issn=0305-4470|bibcode=1991JPhA...24L.175P}}</ref>

[[Tsung-Dao Lee]] came close to deriving Bell's theorem in 1960. He considered events where two [[kaon]]s were produced traveling in opposite directions, and came to the conclusion that hidden variables could not explain the correlations that could be obtained in such situations. However, complications arose due to the fact that kaons decay, and he did not go so far as to deduce a Bell-type inequality.<ref name="jammer1974"/>{{Rp|308}}

===Bell's publications===
Bell chose to publish his theorem in a comparatively obscure journal because it did not require [[page charge]]s, in fact paying the authors who published there at the time. Because the journal did not provide free reprints of articles for the authors to distribute, however, Bell had to spend the money he received to buy copies that he could send to other physicists.<ref name=":0">{{Cite book|last=Whitaker|first=Andrew|url=https://books.google.com/books?id=3Rg9DAAAQBAJ&q=fizika|title=John Stewart Bell and Twentieth Century Physics: Vision and Integrity|date=2016|publisher=Oxford University Press|isbn=978-0-19-874299-9|language=en}}</ref> While the articles printed in the journal themselves listed the publication's name simply as ''Physics'', the covers carried the trilingual version ''[[Physics Physique Физика]]'' to reflect that it would print articles in English, French and Russian.<ref name=":1">{{cite book|last=Wick|first=David|chapter=Bell's Theorem |pages=92–100 |year=1995|title=The Infamous Boundary: Seven Decades of Heresy in Quantum Physics |publisher=Springer |location=New York|doi=10.1007/978-1-4612-4030-3_11|isbn=978-0-387-94726-6}}</ref>{{Rp|92–100, 289}}

Prior to proving his 1964 result, Bell also proved a result equivalent to the Kochen–Specker theorem (hence the latter is sometimes also known as the Bell–Kochen–Specker or Bell–KS theorem). However, publication of this theorem was inadvertently delayed until 1966.<ref name="ND Mermin 1993-07" /><ref name="Bell1966">{{cite journal | last1 = Bell | first1 = J. S. | title = On the problem of hidden variables in quantum mechanics | journal = Reviews of Modern Physics | volume = 38 | issue = 3 | pages = 447–452 | year = 1966 | doi = 10.1103/revmodphys.38.447 |bibcode = 1966RvMP...38..447B | osti = 1444158 }}</ref> In that paper, Bell argued that because an explanation of quantum phenomena in terms of hidden variables would require nonlocality, the EPR paradox "is resolved in the way which Einstein would have liked least."<ref name="Bell1966"/>

==Experiments==
[[Image:Bell-test-photon-analyer.png|thumb|upright=2|'''Scheme of a "two-channel" Bell test'''<br />The source S produces pairs of "photons", sent in opposite directions. Each photon encounters a two-channel polariser whose orientation (a or b) can be set by the experimenter. Emerging signals from each channel are detected and coincidences of four types (++, −−, +− and −+) counted by the coincidence monitor.]]
{{main|Bell test}}

In 1967, the unusual title ''Physics Physique Физика'' caught the attention of [[John Clauser]], who then discovered Bell's paper and began to consider how to perform a [[Bell test]] in the laboratory.<ref>{{Cite web|url=https://www.scientificamerican.com/article/how-the-hippies-saved-physics-science-counterculture-and-quantum-revival-excerpt/|title=How the Hippies Saved Physics: Science, Counterculture, and the Quantum Revival [Excerpt]|last=Kaiser|first=David|author-link=David Kaiser (physicist)|date=2012-01-30|website=[[Scientific American]]|language=en|access-date=2020-02-11}}</ref> Clauser and [[Stuart Freedman]] would go on to perform a Bell test in 1972.<ref>{{cite journal|last1=Freedman|first1=S. J.|author-link=Stuart Freedman|last2=Clauser|first2=J. F.|author-link2=John Clauser|year=1972|title=Experimental test of local hidden-variable theories|url=https://www.rpi.edu/dept/phys/Courses/PHYS4100/S06/BellsInequ1972.pdf|journal=[[Physical Review Letters]]|volume=28|issue=938|pages=938–941|bibcode=1972PhRvL..28..938F|doi=10.1103/PhysRevLett.28.938}}</ref><ref>{{cite thesis|url=https://escholarship.org/content/qt2f18n5nk/qt2f18n5nk.pdf?t=p2au19 |title=Experimental test of local hidden-variable theories |first=Stuart Jay |last=Freedman |date=1972-05-05 |type=PhD |publisher=University of California, Berkeley}}</ref> This was only a limited test, because the choice of detector settings was made before the photons had left the source. In 1982, [[Alain Aspect]] and collaborators performed the [[Aspect's experiment|first Bell test]] to remove this limitation.<ref>{{cite journal |first1=Alain |last1=Aspect |author-link1=Alain Aspect |first2=Jean |last2=Dalibard |first3=Gérard |last3=Roger |year=1982 |title=Experimental Test of Bell's Inequalities Using Time-Varying Analyzers |journal=[[Physical Review Letters]] |volume=49 |issue=25 |pages=1804–7 |doi=10.1103/PhysRevLett.49.1804|bibcode = 1982PhRvL..49.1804A|doi-access=free }}</ref> This began a trend of progressively more stringent Bell tests. The GHZ thought experiment was implemented in practice, using entangled triplets of photons, in 2000.<ref name="GHZ2000">{{cite journal |first1=Jian-Wei |last1=Pan |first2=D. |last2=Bouwmeester |first3=M. |last3=Daniell |first4=H. |last4=Weinfurter |first5=A. |last5=Zeilinger |author-link5=Anton Zeilinger |year=2000 |title=Experimental test of quantum nonlocality in three-photon GHZ entanglement |journal=[[Nature (journal)|Nature]] |volume=403 |issue=6769 |pages=515–519 |bibcode=2000Natur.403..515P |doi=10.1038/35000514 |pmid=10676953|s2cid=4309261 }}</ref> By 2002, testing the CHSH inequality was feasible in undergraduate laboratory courses.<ref>{{cite journal|title=Entangled photons, nonlocality, and Bell inequalities in the undergraduate laboratory |first1=Dietrich |last1=Dehlinger |first2=M. W. |last2=Mitchell |journal=[[American Journal of Physics]] |volume=70 |pages=903–910 |year=2002 |issue=9 |doi=10.1119/1.1498860|arxiv=quant-ph/0205171 |bibcode=2002AmJPh..70..903D |s2cid=49487096 }}</ref>

In Bell tests, there may be problems of experimental design or set-up that affect the validity of the experimental findings. These problems are often referred to as "loopholes". The purpose of the experiment is to test whether nature can be described by [[local hidden-variable theory]], which would contradict the predictions of quantum mechanics.

The most prevalent loopholes in real experiments are the ''detection'' and ''locality'' loopholes.<ref name=larsson14>{{cite journal |last1=Larsson |first1=Jan-Åke |title=Loopholes in Bell inequality tests of local realism |journal=Journal of Physics A: Mathematical and Theoretical |date=2014 |volume=47 |issue=42 |page=424003 |doi=10.1088/1751-8113/47/42/424003 |arxiv=1407.0363 |bibcode=2014JPhA...47P4003L |s2cid=40332044 }}</ref> The detection loophole is opened when a small fraction of the particles (usually photons) are detected in the experiment, making it possible to explain the data with local hidden variables by assuming that the detected particles are an unrepresentative sample. The locality loophole is opened when the detections are not done with a [[Spacetime#Spacetime interval|spacelike separation]], making it possible for the result of one measurement to influence the other without contradicting relativity. In some experiments there may be additional defects that make local-hidden-variable explanations of Bell test violations possible.<ref>{{cite journal|first1=I. |last1=Gerhardt |first2=Q. |last2=Liu |first3=A. |last3=Lamas-Linares |first4=J. |last4=Skaar |first5=V. |last5=Scarani |first6=V. |last6=Makarov |first7=C. |last7=Kurtsiefer |display-authors=5|year=2011 |title=Experimentally faking the violation of Bell's inequalities |journal=[[Physical Review Letters]] |volume=107 |issue=17 |page=170404 |arxiv=1106.3224 |doi=10.1103/PhysRevLett.107.170404 |bibcode=2011PhRvL.107q0404G |pmid=22107491|s2cid=16306493 }}</ref>

Although both the locality and detection loopholes had been closed in different experiments, a long-standing challenge was to close both simultaneously in the same experiment. This was finally achieved in three experiments in 2015.<ref>{{cite journal|title=Quantum 'spookiness' passes toughest test yet|journal=[[Nature News]] |date=27 August 2015|first=Zeeya|last=Merali|volume=525 |issue=7567|pages=14–15|doi=10.1038/nature.2015.18255 |pmid=26333448|bibcode=2015Natur.525...14M |s2cid=4409566|doi-access=free}}</ref><ref name="NYT-20151021">{{cite news |last=Markoff |first=Jack |title=Sorry, Einstein. Quantum Study Suggests 'Spooky Action' Is Real. |url=https://www.nytimes.com/2015/10/22/science/quantum-theory-experiment-said-to-prove-spooky-interactions.html |date=21 October 2015 |work=[[New York Times]] |accessdate=21 October 2015 }}</ref><ref name="NTR-20151021">{{cite journal |author=Hensen, B. |title=Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres |date=21 October 2015 |journal=[[Nature (journal)|Nature]] |doi=10.1038/nature15759 |display-authors=etal |volume=526 |issue=7575 |pages=682–686 |bibcode=2015Natur.526..682H |pmid=26503041|arxiv=1508.05949 |s2cid=205246446 }}</ref><ref name="PRL115-250402">{{cite journal |last=Shalm |first=L. K. |title=Strong Loophole-Free Test of Local Realism|date=16 December 2015|journal=[[Physical Review Letters]] |display-authors=etal |volume=115|issue=25|page= 250402| doi=10.1103/PhysRevLett.115.250402 |bibcode=2015PhRvL.115y0402S |pmid=26722906|pmc=5815856|arxiv=1511.03189}}</ref><ref name="PRL115-250401">{{cite journal |last=Giustina |first=M. |title=Significant-Loophole-Free Test of Bell's Theorem with Entangled Photons|date=16 December 2015|journal=[[Physical Review Letters]] |display-authors=etal |volume=115|issue=25|page= 250401| doi=10.1103/PhysRevLett.115.250401 |pmid=26722905|arxiv=1511.03190|bibcode=2015PhRvL.115y0401G|s2cid=13789503}}</ref>
Regarding these results, [[Alain Aspect]] writes that "no experiment ... can be said to be totally loophole-free," but he says the experiments "remove the last doubts that we should renounce" local hidden variables, and refers to examples of remaining loopholes as being "far fetched" and "foreign to the usual way of reasoning in physics."<ref>{{cite journal |last=Aspect |first=Alain |author-link=Alain Aspect |date=December 16, 2015 |title=Closing the Door on Einstein and Bohr's Quantum Debate |journal=[[Physics (magazine)|Physics]] |volume=8 |pages=123 |bibcode=2015PhyOJ...8..123A |doi=10.1103/Physics.8.123 |doi-access=free}}</ref>

These efforts to experimentally validate violations of the Bell inequalities would later result in Clauser, Aspect, and [[Anton Zeilinger]] being awarded the 2022 [[Nobel Prize in Physics]].<ref>{{Cite news |last1=Ahlander |first1=Johan |last2=Burger |first2=Ludwig |last3=Pollard |first3=Niklas |date=2022-10-04 |title=Nobel physics prize goes to sleuths of 'spooky' quantum science |language=en |work=Reuters |url=https://www.reuters.com/world/aspect-clauser-zeilinger-win-2022-nobel-prize-physics-2022-10-04/ |access-date=2022-10-04}}</ref>

== Interpretations ==
{{main|Interpretations of quantum mechanics}}
Reactions to Bell's theorem have been many and varied. Maximilian Schlosshauer, Johannes Kofler, and Zeilinger write that Bell inequalities provide "a wonderful example of how we can have a rigorous theoretical result tested by numerous experiments, and yet disagree about the implications."<ref>{{Cite journal| arxiv=1301.1069
 | title=A Snapshot of Foundational Attitudes Toward Quantum Mechanics
 | journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics
 | volume=44
 | issue=3
 | pages=222–230
 | date=2013-01-06
 | last1=Schlosshauer
 | first1=Maximilian
 | last2=Kofler
 | first2=Johannes
 | last3=Zeilinger
 | first3=Anton
 | author-link3=Anton Zeilinger
 | doi=10.1016/j.shpsb.2013.04.004
 | bibcode=2013SHPMP..44..222S
 | s2cid=55537196
 }}</ref>

=== The Copenhagen interpretation ===
[[Copenhagen interpretation|Copenhagen-type interpretations]] generally take the violation of Bell inequalities as grounds to reject the assumption often called [[counterfactual definiteness]] or "realism", which is not necessarily the same as abandoning realism in a broader philosophical sense.<ref>{{Cite journal|last=Werner|first=Reinhard F. |author-link=Reinhard F. Werner |date=2014-10-24|title=Comment on 'What Bell did'|journal=[[Journal of Physics A: Mathematical and Theoretical]]|volume=47|issue=42|pages=424011|doi=10.1088/1751-8113/47/42/424011|issn=1751-8113 |bibcode=2014JPhA...47P4011W|s2cid=122180759 }}</ref><ref>{{cite book|last=Żukowski|first=Marek|title=Quantum &#91;Un&#93;Speakables II |chapter=Bell's Theorem Tells Us Not What Quantum Mechanics is, but What Quantum Mechanics is Not |date=2017|series=The Frontiers Collection|pages=175–185|editor-last=Bertlmann|editor-first=Reinhold|place=Cham|publisher=Springer International Publishing|doi=10.1007/978-3-319-38987-5_10|isbn=978-3-319-38985-1|editor2-last=Zeilinger|editor2-first=Anton |editor-link2=Anton Zeilinger |arxiv=1501.05640|s2cid=119214547}}</ref> For example, [[Roland Omnès]] argues for the rejection of hidden variables and concludes that "quantum mechanics is probably as realistic as any theory of its scope and maturity ever will be".<ref name="omnes">{{cite book|first=R. |last=Omnès |author-link=Roland Omnès |title=The Interpretation of Quantum Mechanics |publisher=Princeton University Press |year=1994 |isbn=978-0-691-03669-4 |oclc=439453957 }}</ref>{{Rp|531}} Likewise, [[Rudolf Peierls]] took the message of Bell's theorem to be that, because the premise of locality is physically reasonable, "hidden variables cannot be introduced without abandoning some of the results of quantum mechanics".<ref>{{cite book|last=Peierls |first=Rudolf |author-link=Rudolf Peierls |title=Surprises in Theoretical Physics |pages=26–29 |publisher=Princeton University Press |year=1979 |isbn=0-691-08241-3}}</ref><ref>{{cite journal|last=Mermin |first=N. D. |author-link=N. David Mermin |title=What Do These Correlations Know About Reality? Nonlocality and the Absurd |journal=[[Foundations of Physics]] |volume=29 |year=1999 |issue=4 |pages=571–587 |arxiv=quant-ph/9807055 |bibcode=1998quant.ph..7055M |doi=10.1023/A:1018864225930}}</ref>

This is also the route taken by interpretations that descend from the Copenhagen tradition, such as [[consistent histories]] (often advertised as "Copenhagen done right"),<ref>{{Cite journal|last=Hohenberg|first=P. C.|author-link=Pierre Hohenberg|date=2010-10-05|title=Colloquium : An introduction to consistent quantum theory|journal=[[Reviews of Modern Physics]] |language=en |volume=82 |issue=4 |pages=2835–2844 |arxiv=0909.2359 |doi=10.1103/RevModPhys.82.2835 |issn=0034-6861 |bibcode=2010RvMP...82.2835H|s2cid=20551033}}</ref>{{rp|2839|q=CQT most definitely opts for retaining locality (EPR2) and rejecting classical realism (EPR1)}} as well as [[QBism]].<ref>{{Cite book|chapter-url=https://plato.stanford.edu/entries/quantum-bayesian/|title=[[Stanford Encyclopedia of Philosophy]] |last=Healey|first=Richard|publisher=Metaphysics Research Lab, Stanford University|year=2016|editor-last=Zalta|editor-first=Edward N.|chapter=Quantum-Bayesian and Pragmatist Views of Quantum Theory|access-date=2021-09-16|archive-date=2021-08-17|archive-url=https://web.archive.org/web/20210817204745/https://plato.stanford.edu/entries/quantum-bayesian/|url-status=live}}</ref>

=== Many-worlds interpretation of quantum mechanics ===

The [[Many-worlds interpretation]], also known as the [[Hugh Everett III|Everett]] interpretation, is dynamically local, meaning that it does not call for [[action at a distance]],<ref name=BrownTimpson/>{{rp|17}} and deterministic, because it consists of the unitary part of quantum mechanics without collapse. It can generate correlations that violate a Bell inequality because it violates an implicit assumption by Bell that measurements have a single outcome. In fact, Bell's theorem can be proven in the Many-Worlds framework from the assumption that a measurement has a single outcome. Therefore, a violation of a Bell inequality can be interpreted as a demonstration that measurements have multiple outcomes.<ref>{{cite journal |first1=David |last1=Deutsch |author-link1=David Deutsch |first2=Patrick |last2=Hayden |author-link2=Patrick Hayden (scientist) |title=Information flow in entangled quantum systems |journal=[[Proceedings of the Royal Society A]] |date=2000 |volume=456 |issue=1999 |pages=1759–1774 |doi=10.1098/rspa.2000.0585|arxiv=quant-ph/9906007|bibcode=2000RSPSA.456.1759D |s2cid=13998168 }}</ref>

The explanation it provides for the Bell correlations is that when Alice and Bob make their measurements, they split into local branches. From the point of view of each copy of Alice, there are multiple copies of Bob experiencing different results, so Bob cannot have a definite result, and the same is true from the point of view of each copy of Bob. They will obtain a mutually well-defined result only when their future light cones overlap. At this point we can say that the Bell correlation starts existing, but it was produced by a purely local mechanism. Therefore, the violation of a Bell inequality cannot be interpreted as a proof of non-locality.<ref name=BrownTimpson>{{Cite book|first1=Harvey R. |last1=Brown |author-link1=Harvey R. Brown |first2 = Christopher G. |last2=Timpson|chapter=Bell on Bell's Theorem: The Changing Face of Nonlocality|title=Quantum Nonlocality and Reality: 50 years of Bell's theorem |editor-first1=Mary |editor-last1=Bell |editor-first2=Shan |editor-last2=Gao |publisher=Cambridge University Press|year=2016|pages = 91–123|arxiv=1501.03521|doi=10.1017/CBO9781316219393.008|isbn = 9781316219393|s2cid = 118686956}}</ref>{{rp|28|q=In our discussion of locality in the Everett interpretation we have sought to provide a constructive example illustrating precisely how a theory can be dynamically local, whilst violating local causality}}

=== Non-local hidden variables ===

Most advocates of the hidden-variables idea believe that experiments have ruled out local hidden variables.{{refn|group=note|[[E. T. Jaynes]] was one exception,<ref name="E.T. Jaynes 1989">{{Cite book |year=1989 |last1=Jaynes |first1=E. T. |title=Maximum Entropy and Bayesian Methods |chapter=Clearing up Mysteries — the Original Goal |pages=1–27 |url=http://bayes.wustl.edu/etj/articles/cmystery.pdf |doi=10.1007/978-94-015-7860-8_1 |isbn=978-90-481-4044-2 |citeseerx=10.1.1.46.1264 |access-date=2011-10-18 |archive-date=2011-10-28 |archive-url=https://web.archive.org/web/20111028131916/http://bayes.wustl.edu/etj/articles/cmystery.pdf |url-status=live }}</ref> but Jaynes' arguments have not generally been found persuasive.<ref name="Gill2002">{{cite book|chapter=Time, Finite Statistics, and Bell's Fifth Position|first=Richard D.|last=Gill|pages=179–206|title=Proceedings of the Conference Foundations of Probability and Physics - 2 : Växjö (Soland), Sweden, June 2-7, 2002 |volume=5|publisher=Växjö University Press|date=2002|arxiv=quant-ph/0301059 }}</ref>}} They are ready to give up locality, explaining the violation of Bell's inequality by means of a non-local [[hidden variable theory]], in which the particles exchange information about their states. This is the basis of the [[Bohm interpretation]] of quantum mechanics, which requires that all particles in the universe be able to instantaneously exchange information with all others. One challenge for non-local hidden variable theories is to explain why this instantaneous communication can exist at the level of the hidden variables, but it cannot be used to send signals.<ref>{{Cite journal |last1=Wood |first1=Christopher J. |last2=Spekkens |first2=Robert W. |author-link2=Robert Spekkens |date=2015-03-03 |title=The lesson of causal discovery algorithms for quantum correlations: causal explanations of Bell-inequality violations require fine-tuning |url=https://iopscience.iop.org/article/10.1088/1367-2630/17/3/033002 |journal=[[New Journal of Physics]] |volume=17 |issue=3 |pages=033002 |arxiv=1208.4119 |bibcode=2015NJPh...17c3002W |doi=10.1088/1367-2630/17/3/033002 |s2cid=118518558 |issn=1367-2630}}</ref> A 2007 experiment ruled out a large class of non-Bohmian non-local hidden variable theories, though not Bohmian mechanics itself.<ref>{{cite journal |doi=10.1038/nature05677 |title=An experimental test of non-local realism |year=2007 |last1=Gröblacher |first1=Simon |last2=Paterek |first2=Tomasz |last3=Kaltenbaek |first3=Rainer |last4=Brukner |first4=Časlav |author-link4=Časlav Brukner |last5=Żukowski |first5=Marek |last6=Aspelmeyer |first6=Markus |last7=Zeilinger |first7=Anton |author-link7=Anton Zeilinger |journal=[[Nature (journal)|Nature]] |volume=446 |issue=7138 |pages=871–5 |pmid=17443179|bibcode = 2007Natur.446..871G | arxiv= 0704.2529 |s2cid=4412358 }}</ref>

The [[transactional interpretation]], which postulates waves traveling both backwards and forwards in time, is likewise non-local.<ref>{{Cite journal|last=Kastner|first=Ruth E.|date=May 2010|title=The quantum liar experiment in Cramer's transactional interpretation|url=https://linkinghub.elsevier.com/retrieve/pii/S135521981000002X|journal=[[Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics]] |language=en|volume=41|issue=2|pages=86–92|arxiv=0906.1626|bibcode=2010SHPMP..41...86K|doi=10.1016/j.shpsb.2010.01.001|s2cid=16242184|access-date=2021-09-16|archive-date=2018-06-24|archive-url=https://web.archive.org/web/20180624053010/https://linkinghub.elsevier.com/retrieve/pii/S135521981000002X|url-status=live}}</ref>

===Superdeterminism===
{{Main|Superdeterminism}}
A necessary assumption to derive Bell's theorem is that the hidden variables are not correlated with the measurement settings. This assumption has been justified on the grounds that the experimenter has "[[free will]]" to choose the settings, and that it is necessary to do science in the first place. A (hypothetical) theory where the choice of measurement is necessarily correlated with the system being measured is known as ''superdeterministic''.<ref name=larsson14/>

A few advocates of deterministic models have not given up on local hidden variables. For example, [[Gerard 't Hooft]] has argued that superdeterminism cannot be dismissed.<ref>{{cite book |last='t Hooft |first=Gerard |author-link=Gerard 't Hooft |title=The Cellular Automaton Interpretation of Quantum Mechanics |volume=185 |publisher=Springer |year=2016 |doi=10.1007/978-3-319-41285-6 |isbn=978-3-319-41284-9 |oclc=951761277 |series=Fundamental Theories of Physics |s2cid=7779840 |url=http://www.oapen.org/search?identifier=1002003 |access-date=2020-08-27 |archive-date=2021-12-29 |archive-url=https://web.archive.org/web/20211229062338/https://library.oapen.org/handle/20.500.12657/27994 |url-status=live }}</ref>

==See also==
{{portal|physics}}
{{cols|colwidth=20em}}
* [[Buscemi nonlocality]]
* [[Einstein's thought experiments]]
* ''[[Epistemological Letters]]''
* [[Fundamental Fysiks Group]]
* [[Leggett inequality]]
* [[Leggett–Garg inequality]]
* [[Mermin's device]]
* [[Mott problem]]
* [[PBR theorem]]
* [[Quantum contextuality]]
* [[Quantum nonlocality]]
* [[Renninger negative-result experiment]]
{{colend}}

==Notes==
{{reflist|group=note}}

==References==
{{Reflist}}

== Further reading ==
The following are intended for general audiences.
* {{cite book|first1=A. |last1=Afriat |first2=F. |last2=Selleri |title=The Einstein, Podolsky and Rosen Paradox |publisher=Plenum Press |location=New York and London |year=1999}}
* {{cite book|first1=J. |last1=Baggott |author-link=Jim Baggott |title=The Meaning of Quantum Theory |publisher=Oxford University Press |year=1992}}
* {{cite book|first=Louisa |last=Gilder |title=The Age of Entanglement: When Quantum Physics Was Reborn |location=New York |publisher=Alfred A. Knopf |year=2008}}
* {{cite book|first=Brian |last=Greene |author-link=Brian Greene |title=The Fabric of the Cosmos |title-link=The Fabric of the Cosmos |publisher=Vintage |year=2004 |isbn=0-375-72720-5}}
* {{cite journal |doi=10.1119/1.12594 |title=Bringing home the atomic world: Quantum mysteries for anybody |year=1981 |last1=Mermin |first1=N. David |author-link=N. David Mermin |s2cid=122724592 |journal=American Journal of Physics |volume=49 |issue=10 |pages=940–943|bibcode = 1981AmJPh..49..940M }}
* {{cite journal|first=N. David |last=Mermin |author-link= |title=Is the moon there when nobody looks? Reality and the quantum theory |journal=Physics Today |date=April 1985 |volume=38 |issue=4 |pages=38–47 |doi=10.1063/1.880968 |bibcode=1985PhT....38d..38M}}
* {{cite book |first=Valerio |last=Scarani |title=Bell Nonlocality |year=2019 |url=https://academic.oup.com/book/36528 |isbn=9780191830327 |publisher=[[Oxford University Press]]}}

The following are more technically oriented.
{{Refbegin|colwidth=60em}}
* {{cite journal | last1 = Aspect | first1 = A. | author-link=Alain Aspect | title = Experimental Tests of Realistic Local Theories via Bell's Theorem | journal = Phys. Rev. Lett. | volume = 47 | pages = 460–463 | issue = 7| year = 1981 |display-authors=etal | doi=10.1103/physrevlett.47.460|bibcode = 1981PhRvL..47..460A | doi-access = free }}
* {{cite journal | last1 = Aspect | first1 = A. | title = Experimental Realization of Einstein–Podolsky–Rosen–Bohm Gedankenexperiment: A New Violation of Bell's Inequalities | journal = Phys. Rev. Lett. | volume = 49 | issue = 2 | pages = 91–94 | year = 1982 |display-authors=etal | doi=10.1103/physrevlett.49.91|bibcode = 1982PhRvL..49...91A | doi-access = free }}
* {{cite journal | last1 = Aspect | first1 = A. | last2 = Grangier | first2 = P. | title = About resonant scattering and other hypothetical effects in the Orsay atomic-cascade experiment tests of Bell inequalities: a discussion and some new experimental data | journal = Lettere al Nuovo Cimento | volume = 43 | issue = 8 | pages = 345–348 | year = 1985 | doi=10.1007/bf02746964| s2cid = 120840672 }}
* {{cite book|first=J. S. |last=Bell |author-link=John Stewart Bell |chapter=Introduction to the hidden variable question |title=Proceedings of the International School of Physics 'Enrico Fermi', Course IL, Foundations of Quantum Mechanics |year=1971 |pages=171–81}}
* {{cite book|first=J. S. |last=Bell |chapter=Bertlmann's Socks and the Nature of Reality |title=Speakable and Unspeakable in Quantum Mechanics |publisher=Cambridge University Press |pages=139–158 |year=2004}}
* {{cite journal | last1 = D'Espagnat | first1 = B. | author-link=Bernard d'Espagnat | year = 1979 | title = The Quantum Theory and Reality | url = http://www.sciam.com/media/pdf/197911_0158.pdf | journal = Scientific American | volume = 241 | issue = 5 | pages = 158–181 | doi = 10.1038/scientificamerican1179-158 | bibcode = 1979SciAm.241e.158D | access-date = 2009-03-18 | archive-date = 2009-03-27 | archive-url = https://web.archive.org/web/20090327023619/http://www.sciam.com/media/pdf/197911_0158.pdf | url-status = live }}
* {{cite journal | last1 = Fry | first1 = E. S. | last2 = Walther | first2 = T. | last3 = Li | first3 = S. | title = Proposal for a loophole-free test of the Bell inequalities | url = http://oaktrust.library.tamu.edu/bitstream/1969.1/126533/1/PhysRevA.52.4381.pdf | journal = Phys. Rev. A | volume = 52 | issue = 6 | pages = 4381–4395 | year = 1995 | doi = 10.1103/physreva.52.4381 | pmid = 9912775 | bibcode = 1995PhRvA..52.4381F | hdl = 1969.1/126533 | hdl-access = free | access-date = 2018-03-19 | archive-date = 2021-12-29 | archive-url = https://web.archive.org/web/20211229062332/http://oaktrust.library.tamu.edu/bitstream/handle/1969.1/126533/PhysRevA.52.4381.pdf;jsessionid=50AFAA1E4F54828C672C8FF01C56B167?sequence=1 | url-status = live }}
* {{cite book|first1=E. S. |last1=Fry |first2=T. |last2=Walther |chapter=Atom based tests of the Bell Inequalities — the legacy of John Bell continues |pages=103–117 |title=Quantum [Un]speakables |editor-first1=R. A. |editor-last1=Bertlmann |editor-first2=A. |editor-last2=Zeilinger |editor-link2=Anton Zeilinger |publisher=Springer |location=Berlin-Heidelberg-New York |year=2002}}
* {{cite journal | last1 = Goldstein | first1 = Sheldon | display-authors = etal   | year = 2011| title = Bell's theorem | journal = [[Scholarpedia]] | volume = 6 | issue = 10| page = 8378 | doi = 10.4249/scholarpedia.8378 |bibcode = 2011SchpJ...6.8378G | doi-access = free }}
* {{cite book|first=R. B. |last=Griffiths |title=Consistent Quantum Theory |publisher=Cambridge University Press |year=2001 |isbn=978-0-521-80349-6 |oclc=1180958776}}
* {{cite journal | last1 = Hardy | first1 = L. | author-link = Lucien Hardy | s2cid = 11839894 | year = 1993 | title = Nonlocality for 2 particles without inequalities for almost all entangled states | journal = Physical Review Letters | volume = 71 | issue = 11| pages = 1665–1668 | doi=10.1103/physrevlett.71.1665|bibcode = 1993PhRvL..71.1665H | pmid=10054467}}
* {{cite journal | last1 = Matsukevich | first1 = D. N. | last2 = Maunz | first2 = P. | last3 = Moehring | first3 = D. L. | last4 = Olmschenk | first4 = S. | last5 = Monroe | first5 = C. | year = 2008 | title = Bell Inequality Violation with Two Remote Atomic Qubits | journal = Phys. Rev. Lett. | volume = 100 | issue = 15| page = 150404 | doi=10.1103/physrevlett.100.150404|arxiv = 0801.2184 |bibcode = 2008PhRvL.100o0404M | pmid=18518088| s2cid = 11536757 }}
* {{Cite book|title-link= Quantum Computing: A Gentle Introduction |title=Quantum Computing: A Gentle Introduction|last1=Rieffel|first1=Eleanor G.|last2=Polak|first2=Wolfgang H.|date=4 March 2011|publisher=MIT Press|isbn=978-0-262-01506-6|language=en|author-link=Eleanor Rieffel |chapter=4.4 EPR Paradox and Bell's Theorem |pages=60–65}}
* {{cite journal | last1 = Sulcs | first1 = S. | year = 2003 | title = The Nature of Light and Twentieth Century Experimental Physics | doi = 10.1023/A:1026323203487 | journal = Foundations of Science | volume = 8 | issue = 4| pages = 365–391 | s2cid = 118769677 }}
* {{cite book|first=B. C. |last=van Fraassen |author-link=Bas van Fraassen |title=Quantum Mechanics: An Empiricist View |publisher=Clarendon Press |year=1991 |isbn=978-0-198-24861-3 |oclc=22906474}}
* {{Cite journal |last1=Wharton |first1=K. B. |last2=Argaman |first2=N. |date=2020-05-18 |title=Colloquium : Bell's theorem and locally mediated reformulations of quantum mechanics |url=https://link.aps.org/doi/10.1103/RevModPhys.92.021002 |journal=Reviews of Modern Physics |language=en |volume=92 |issue=2 |page=021002 |doi=10.1103/RevModPhys.92.021002 |issn=0034-6861|arxiv=1906.04313 |bibcode=2020RvMP...92b1002W }}
{{Refend}}

==External links==
{{wikibooks |Quantum_Mechanics }}
{{wikiversity |Bell's_theorem }}
{{Commons category|Bell's theorem}}
* [https://www.youtube.com/watch?v=ta09WXiUqcQ Mermin: Spooky Actions At A Distance? Oppenheimer Lecture].
* {{cite IEP |url-id=epr |title=Bell's theorem}}
* {{springer|title=Bell inequalities|id=p/b110230 |mode=cs1}}

{{Quantum mechanics topics}}
{{Quantum information}}
{{Authority control}}

[[Category:Quantum information science]]
[[Category:Quantum measurement]]
[[Category:Theorems in quantum mechanics]]
[[Category:Hidden variable theory]]
[[Category:Inequalities (mathematics)]]
[[Category:1964 introductions]]
[[Category:No-go theorems]]