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</li> <li id="toc-Free_will_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Free_will_theorem"> <div class="vector-toc-text"> 2.4 Free will theorem </div> </a> <ul id="toc-Free_will_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quasiclassical_entanglement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quasiclassical_entanglement"> <div class="vector-toc-text"> 2.5 Quasiclassical entanglement </div> </a> <ul id="toc-Quasiclassical_entanglement-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 3 History </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle History subsection </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Background" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Background"> <div class="vector-toc-text"> 3.1 Background </div> </a> <ul id="toc-Background-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bell's_publications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bell's_publications"> <div class="vector-toc-text"> 3.2 Bell's publications </div> </a> <ul id="toc-Bell's_publications-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Experiments" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Experiments"> <div class="vector-toc-text"> 4 Experiments </div> </a> <ul id="toc-Experiments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpretations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Interpretations"> <div class="vector-toc-text"> 5 Interpretations </div> </a> <button aria-controls="toc-Interpretations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Interpretations subsection </button> <ul id="toc-Interpretations-sublist" class="vector-toc-list"> <li id="toc-The_Copenhagen_interpretation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Copenhagen_interpretation"> <div class="vector-toc-text"> 5.1 The Copenhagen interpretation </div> </a> <ul id="toc-The_Copenhagen_interpretation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Many-worlds_interpretation_of_quantum_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Many-worlds_interpretation_of_quantum_mechanics"> <div class="vector-toc-text"> 5.2 Many-worlds interpretation of quantum mechanics </div> </a> <ul id="toc-Many-worlds_interpretation_of_quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-local_hidden_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-local_hidden_variables"> <div class="vector-toc-text"> 5.3 Non-local hidden variables </div> </a> <ul id="toc-Non-local_hidden_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Superdeterminism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Superdeterminism"> <div class="vector-toc-text"> 5.4 Superdeterminism </div> </a> <ul id="toc-Superdeterminism-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 6 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 7 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 8 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 9 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 10 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data-title="Няроўнасці Бела" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_de_Bell" title="Teorema de Bell – Catalan" lang="ca" hreflang="ca" data-title="Teorema de Bell" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bellova_nerovnost" title="Bellova nerovnost – Czech" lang="cs" hreflang="cs" data-title="Bellova nerovnost" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Bellsche_Ungleichung" title="Bellsche Ungleichung – German" lang="de" hreflang="de" data-title="Bellsche Ungleichung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_de_Bell" title="Teorema de Bell – Spanish" lang="es" hreflang="es" data-title="Teorema de Bell" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Teoremo_de_Bell" title="Teoremo de Bell – Esperanto" lang="eo" hreflang="eo" data-title="Teoremo de Bell" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%87_%D8%A8%D9%84" title="قضیه بل – Persian" lang="fa" hreflang="fa" data-title="قضیه بل" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/In%C3%A9galit%C3%A9s_de_Bell" title="Inégalités de Bell – French" lang="fr" hreflang="fr" data-title="Inégalités de Bell" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teorema_de_Bell" title="Teorema de Bell – Galician" lang="gl" hreflang="gl" data-title="Teorema de Bell" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B2%A8_%EB%B6%80%EB%93%B1%EC%8B%9D" title="벨 부등식 – Korean" lang="ko" hreflang="ko" data-title="벨 부등식" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema_di_Bell" title="Teorema di Bell – Italian" lang="it" hreflang="it" data-title="Teorema di Bell" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%91%D7%9C" title="משפט בל – Hebrew" lang="he" hreflang="he" data-title="משפט בל" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%91%D0%B5%D0%BB%D0%BB%D0%B8%D0%B9%D0%BD_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC" title="Беллийн теорем – Mongolian" lang="mn" hreflang="mn" data-title="Беллийн теорем" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stelling_van_Bell" title="Stelling van Bell – Dutch" lang="nl" hreflang="nl" data-title="Stelling van Bell" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%99%E3%83%AB%E3%81%AE%E4%B8%8D%E7%AD%89%E5%BC%8F" title="ベルの不等式 – Japanese" lang="ja" hreflang="ja" data-title="ベルの不等式" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bells_teorem" title="Bells teorem – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Bells teorem" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Bells_teorem" title="Bells teorem – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Bells teorem" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenie_Bella" title="Twierdzenie Bella – Polish" lang="pl" hreflang="pl" data-title="Twierdzenie Bella" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_de_Bell" title="Teorema de Bell – Portuguese" lang="pt" hreflang="pt" data-title="Teorema de Bell" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teorema_lui_Bell" title="Teorema lui Bell – Romanian" lang="ro" hreflang="ro" data-title="Teorema lui Bell" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D1%81%D1%82%D0%B2%D0%B0_%D0%91%D0%B5%D0%BB%D0%BB%D0%B0" title="Неравенства Белла – Russian" lang="ru" hreflang="ru" data-title="Неравенства Белла" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Bell%27s_theorem" title="Bell's theorem – Simple English" lang="en-simple" hreflang="en-simple" data-title="Bell's theorem" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Bellova_nerovnos%C5%A5" title="Bellova nerovnosť – Slovak" lang="sk" hreflang="sk" data-title="Bellova nerovnosť" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Bellin_teoreema" title="Bellin teoreema – Finnish" lang="fi" hreflang="fi" data-title="Bellin teoreema" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Bells_teorem" title="Bells teorem – Swedish" 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Theorem in physics</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Bell inequality" redirects here. For the related experiments, see <a href="/wiki/Bell_test" title="Bell test">Bell test</a>.</div> Bell's theorem is a term encompassing a number of closely related results in <a href="/wiki/Physics" title="Physics">physics</a>, all of which determine that <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> is incompatible with <a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">local hidden-variable theories</a>, given some basic assumptions about the nature of measurement. "Local" here refers to the <a href="/wiki/Principle_of_locality" title="Principle of locality">principle of locality</a>, the idea that a <a href="/wiki/Particle" title="Particle">particle</a> can only be influenced by its immediate surroundings, and that interactions mediated by <a href="/wiki/Field_(physics)" title="Field (physics)">physical fields</a> cannot propagate faster than the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>. "<a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden variables</a>" are supposed properties of quantum particles that are not included in quantum theory but nevertheless affect the outcome of experiments. In the words of physicist <a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">John Stewart Bell</a>, for whom this family of results is named, "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."<a href="#cite_note-1">[1]</a> The first such result was introduced by Bell in 1964, building upon the <a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">Einstein–Podolsky–Rosen paradox</a>, which had called attention to the phenomenon of <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">quantum entanglement</a>. 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 <a href="/wiki/Inequality_(mathematics)" title="Inequality (mathematics)">inequality</a>. Multiple variations on Bell's theorem were put forward in the following years, 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 <a href="/wiki/John_Clauser" title="John Clauser">John Clauser</a> and <a href="/wiki/Stuart_Freedman" title="Stuart Freedman">Stuart Freedman</a>.<a href="#cite_note-2">[2]</a> More advanced experiments, known collectively as <a href="/wiki/Bell_test" title="Bell test">Bell tests</a>, 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.<a href="#cite_note-NAT-20180509-3">[3]</a><a href="#cite_note-4">[4]</a> The exact nature of the assumptions required to prove a Bell-type constraint on correlations has been debated by physicists and by <a href="/wiki/Philosophy_of_physics" title="Philosophy of physics">philosophers</a>. While the significance of Bell's theorem is not in doubt, different <a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">interpretations of quantum mechanics</a> disagree about what exactly it implies. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Theorem">Theorem</h2>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=1" title="Edit section: Theorem">edit</a>]</div> There are many variations on the basic idea, some employing stronger mathematical assumptions than others.<a href="#cite_note-Stanford-5">[5]</a> 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.<a href="#cite_note-mike-and-ike-6">[6]</a> Hypothetical characters <a href="/wiki/Alice_and_Bob" title="Alice and Bob">Alice and Bob</a> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c98dbd929292d464de6942e95db306b8b8a6fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{0}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{1}}">. Both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c98dbd929292d464de6942e95db306b8b8a6fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{0}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{1}}"> are binary measurements: the result of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c98dbd929292d464de6942e95db306b8b8a6fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{0}}"> is either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}">, and likewise for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{1}}">. When Bob receives his particle, he chooses one of two measurements, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03a3f39a56ba486e7c6ec89b99f5ae2a21fa75b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle B_{0}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fa091eb428443c9c5c5fcf32a69d3665c89e00c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle B_{1}}">, 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c98dbd929292d464de6942e95db306b8b8a6fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{0}}"> and obtains the result <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}">, then the particle she received carried a value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}"> for a property <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}">.<a href="#cite_note-9">[note 1]</a> Consider the combination<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}b_{0}+a_{0}b_{1}+a_{1}b_{0}-a_{1}b_{1}=(a_{0}+a_{1})b_{0}+(a_{0}-a_{1})b_{1}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}b_{0}+a_{0}b_{1}+a_{1}b_{0}-a_{1}b_{1}=(a_{0}+a_{1})b_{0}+(a_{0}-a_{1})b_{1}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/246a8d0258e71fe232ab8a60ef20657da76abca7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.377ex; height:2.843ex;" alt="{\displaystyle a_{0}b_{0}+a_{0}b_{1}+a_{1}b_{0}-a_{1}b_{1}=(a_{0}+a_{1})b_{0}+(a_{0}-a_{1})b_{1}\,.}">Because both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf42ecda092975c9c69dae84e16182ba5fe2e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}"> take the values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfeaa85da53ad1947d8000926cfea33827ef1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 1}">, then either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}=a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}=a_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc49820ff0c3e8fbe0e51cd2562682d54ef84e31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.667ex; height:2.009ex;" alt="{\displaystyle a_{0}=a_{1}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}=-a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}=-a_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b297172b3885d08ce24baafd2e8afad5eb170a0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.475ex; height:2.343ex;" alt="{\displaystyle a_{0}=-a_{1}}">. In the former case, the quantity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{0}-a_{1})b_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{0}-a_{1})b_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2161443f541ad370ddbd3201fe6c7afc4caaed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.27ex; height:2.843ex;" alt="{\displaystyle (a_{0}-a_{1})b_{1}}"> must equal 0, while in the latter case, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{0}+a_{1})b_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{0}+a_{1})b_{0}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47a9a4aa56fc083664c66c1e7dcc8f99e38312a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.531ex; height:2.843ex;" alt="{\displaystyle (a_{0}+a_{1})b_{0}=0}">. So, one of the terms on the right-hand side of the above expression will vanish, and the other will equal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5535e5dda77acfaf4c88d62923ff40b40175c91d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 2}">. 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}b_{0}+a_{0}b_{1}+a_{1}b_{0}-a_{1}b_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}b_{0}+a_{0}b_{1}+a_{1}b_{0}-a_{1}b_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9927e97001497cd2d6ab68269e548dd18da1f758" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.865ex; height:2.509ex;" alt="{\displaystyle a_{0}b_{0}+a_{0}b_{1}+a_{1}b_{0}-a_{1}b_{1}}"> 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" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\langle A_{0}B_{0}\rangle +\langle A_{0}B_{1}\rangle +\langle A_{1}B_{0}\rangle -\langle A_{1}B_{1}\rangle |\leq 2\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\langle A_{0}B_{0}\rangle +\langle A_{0}B_{1}\rangle +\langle A_{1}B_{0}\rangle -\langle A_{1}B_{1}\rangle |\leq 2\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bf31903c9d7582f47999568fce8372fab2e6709" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.809ex; height:2.843ex;" alt="{\displaystyle |\langle A_{0}B_{0}\rangle +\langle A_{0}B_{1}\rangle +\langle A_{1}B_{0}\rangle -\langle A_{1}B_{1}\rangle |\leq 2\,.}"> This is a Bell inequality, specifically, the <a href="/wiki/CHSH_inequality" title="CHSH inequality">CHSH inequality</a>.<a href="#cite_note-mike-and-ike-6">[6]</a>: 115  Its derivation here depends upon two assumptions: first, that the underlying physical properties <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0},a_{1},b_{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0},a_{1},b_{0},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbcd5e64bf1fe7b2acd839da3ebea05346dd5e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.335ex; height:2.509ex;" alt="{\displaystyle a_{0},a_{1},b_{0},}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af2720c91be489f57ecde4bb651b95e113d0144" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{1}}"> 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).<a href="#cite_note-mike-and-ike-6">[6]</a>: 117  Quantum mechanics can violate the CHSH inequality, as follows. Victor prepares a pair of <a href="/wiki/Qubit" title="Qubit">qubits</a> which he describes by the <a href="/wiki/Bell_state" title="Bell state">Bell state</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle ={\frac {|0\rangle \otimes |1\rangle -|1\rangle \otimes |0\rangle }{\sqrt {2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle ={\frac {|0\rangle \otimes |1\rangle -|1\rangle \otimes |0\rangle }{\sqrt {2}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b7bb3761ba3a176c62bd22d4a450a2f05a280f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.023ex; height:6.676ex;" alt="{\displaystyle |\psi \rangle ={\frac {|0\rangle \otimes |1\rangle -|1\rangle \otimes |0\rangle }{\sqrt {2}}},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |0\rangle }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |1\rangle }"> are the eigenstates of one of the <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a>,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010b16a45c3270ca643e63277926efebbc5b7032" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.703ex; height:6.176ex;" alt="{\displaystyle \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}"> Victor then passes the first qubit to Alice and the second to Bob. Alice and Bob's choices of possible <a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">measurements</a> are also defined in terms of the Pauli matrices. Alice measures either of the two observables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ea9b1af0a90b25f6dcdc140820ff2428fd92c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.329ex; height:2.009ex;" alt="{\displaystyle \sigma _{z}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf4ab08fe163a6c495cad6f4d67653287c4044c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \sigma _{x}}">:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{0}=\sigma _{z},\ A_{1}=\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}=\sigma _{z},\ A_{1}=\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc419ea0cddb4765a0c19bd807db562705db344" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.801ex; height:6.176ex;" alt="{\displaystyle A_{0}=\sigma _{z},\ A_{1}=\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}};}"> and Bob measures either of the two observables<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{0}=-{\frac {\sigma _{x}+\sigma _{z}}{\sqrt {2}}},\ B_{1}={\frac {\sigma _{x}-\sigma _{z}}{\sqrt {2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{0}=-{\frac {\sigma _{x}+\sigma _{z}}{\sqrt {2}}},\ B_{1}={\frac {\sigma _{x}-\sigma _{z}}{\sqrt {2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/403916c2c1364a392c6b591dd4f82d4e50bc3a76" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.914ex; height:6.009ex;" alt="{\displaystyle B_{0}=-{\frac {\sigma _{x}+\sigma _{z}}{\sqrt {2}}},\ B_{1}={\frac {\sigma _{x}-\sigma _{z}}{\sqrt {2}}}.}"> Victor can calculate the quantum expectation values for pairs of these observables using the <a href="/wiki/Born_rule" title="Born rule">Born rule</a>:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>,</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>,</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>,</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \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}}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a93276e6888931273c9e73d11cbb94791d9267b1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:75.137ex; height:6.176ex;" alt="{\displaystyle \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}}}\,.}"> While only one of these four measurements can be made in a single trial of the experiment, the sum<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \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}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d86aa90478bc9ea3420fc90694fc9baf26527db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.942ex; height:3.176ex;" alt="{\displaystyle \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}}}"> 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.<a href="#cite_note-mike-and-ike-6">[6]</a>: 116  The value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\sqrt {2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e70f0d473d5c0f14cd3c10c139f51295788fb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:3.009ex;" alt="{\displaystyle 2{\sqrt {2}}}"> is in fact the largest that quantum physics permits for this combination of expectation values, making it a <a href="/wiki/Tsirelson%27s_bound" title="Tsirelson's bound">Tsirelson bound</a>.<a href="#cite_note-10">[9]</a>: 140  <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Chsh-illustration.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Chsh-illustration.png/220px-Chsh-illustration.png" decoding="async" width="220" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Chsh-illustration.png/330px-Chsh-illustration.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Chsh-illustration.png/440px-Chsh-illustration.png 2x" data-file-width="3560" data-file-height="2781" /></a><figcaption>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.</figcaption></figure> The CHSH inequality can also be thought of as <a href="/wiki/CHSH_game" class="mw-redirect" title="CHSH game">a game in which Alice and Bob try to coordinate their actions</a>.<a href="#cite_note-11">[10]</a><a href="#cite_note-12">[11]</a> Victor prepares two bits, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">, independently and at random. He sends bit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> to Alice and bit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> to Bob. Alice and Bob win if they return answer bits <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> to Victor, satisfying <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy=a+b\mod 2\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mn>2</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy=a+b\mod 2\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/119a4dbd9e7831b80c0f41c2f571bcca91183569" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.11ex; height:2.509ex;" alt="{\displaystyle xy=a+b\mod 2\,.}"> Or, equivalently, Alice and Bob win if the <a href="/wiki/Logical_AND" class="mw-redirect" title="Logical AND">logical AND</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> is the <a href="/wiki/Logical_XOR" class="mw-redirect" title="Logical XOR">logical XOR</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">. 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3/4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3/4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1a50dd3c11ada865259716b347224485ab66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle 3/4}">, 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" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2+{\sqrt {2}}}{4}}\approx 0.85\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>≈</mo> <mn>0.85</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2+{\sqrt {2}}}{4}}\approx 0.85\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85c7adb09c920ca1d16105d4cff561b1bcac998f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.204ex; height:5.843ex;" alt="{\displaystyle {\frac {2+{\sqrt {2}}}{4}}\approx 0.85\,.}"> <div class="mw-heading mw-heading2"><h2 id="Variations_and_related_results">Variations and related results</h2>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=2" title="Edit section: Variations and related results">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Bell_(1964)">Bell (1964)</h3>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=3" title="Edit section: Bell (1964)">edit</a>]</div> Bell's 1964 paper points out that under restricted conditions, local hidden-variable models can reproduce the predictions of quantum mechanics. He then demonstrates that this cannot hold true in general.<a href="#cite_note-Bell1964-13">[12]</a> Bell considers a refinement by <a href="/wiki/David_Bohm" title="David Bohm">David Bohm</a> 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 <a href="/wiki/Singlet_state" title="Singlet state">spin singlet state</a> (which is an example of an entangled state). The particles then move apart in opposite directions. Each particle is measured by a <a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach device</a>, a measuring instrument that can be oriented in different directions and that reports one of two possible outcomes, representable by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}">. The configuration of each measuring instrument is represented by a unit <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a>, and the quantum-mechanical prediction for the <a href="/wiki/Quantum_correlation" title="Quantum correlation">correlation</a> between two detectors with settings <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"> is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P({\vec {a}},{\vec {b}})=-{\vec {a}}\cdot {\vec {b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P({\vec {a}},{\vec {b}})=-{\vec {a}}\cdot {\vec {b}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6201b312ce68f6e2b93370bf5c598e0b6d13d103" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.469ex; height:3.343ex;" alt="{\displaystyle P({\vec {a}},{\vec {b}})=-{\vec {a}}\cdot {\vec {b}}.}"> In particular, if the orientation of the two detectors is the same (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}={\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}={\vec {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e92f9a9f48bb9c33563d9bd50137e4433db1efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.422ex; height:2.843ex;" alt="{\displaystyle {\vec {a}}={\vec {b}}}">), then the outcome of one measurement is certain to be the negative of the outcome of the other, giving <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P({\vec {a}},{\vec {a}})=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P({\vec {a}},{\vec {a}})=-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43a4b4f7e706b6af5dc6f65c5c9f8b7b99bcab66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.117ex; height:2.843ex;" alt="{\displaystyle P({\vec {a}},{\vec {a}})=-1}">. And if the orientations of the two detectors are orthogonal (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}\cdot {\vec {b}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}\cdot {\vec {b}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f232b9d764e121872b39a9530bc9237c5baf276f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.264ex; height:2.843ex;" alt="{\displaystyle {\vec {a}}\cdot {\vec {b}}=0}">), then the outcomes are uncorrelated, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P({\vec {a}},{\vec {b}})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P({\vec {a}},{\vec {b}})=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7abe176aa1d609f825b86fe38a7bd047c046939" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.173ex; height:3.343ex;" alt="{\displaystyle P({\vec {a}},{\vec {b}})=0}">. 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }">:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P({\vec {a}},{\vec {b}})=\int d\lambda \,\rho (\lambda )A({\vec {a}},\lambda )B({\vec {b}},\lambda ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mi>d</mi> <mi>λ</mi> <mspace width="thinmathspace" /> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P({\vec {a}},{\vec {b}})=\int d\lambda \,\rho (\lambda )A({\vec {a}},\lambda )B({\vec {b}},\lambda ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffada964765bb25f2399bb23ae836916bcc3e83e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.791ex; height:5.676ex;" alt="{\displaystyle P({\vec {a}},{\vec {b}})=\int d\lambda \,\rho (\lambda )A({\vec {a}},\lambda )B({\vec {b}},\lambda ),}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85a737d6c810a0dea4b252efe90f371bbf0d8dc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.366ex; height:2.843ex;" alt="{\displaystyle \rho (\lambda )}"> is a <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a>. The two functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A({\vec {a}},\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A({\vec {a}},\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f0c06fbf3fa121791df6a0ea0edbd92841439c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.171ex; height:2.843ex;" alt="{\displaystyle A({\vec {a}},\lambda )}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B({\vec {b}},\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B({\vec {b}},\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5ef27e9126ea1564e438ecd87ced396088f3f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.056ex; height:3.343ex;" alt="{\displaystyle B({\vec {b}},\lambda )}"> provide the responses of the two detectors given the orientation vectors and the hidden variable:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A({\vec {a}},\lambda )=\pm 1,\,B({\vec {b}},\lambda )=\pm 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mi>B</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>±</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A({\vec {a}},\lambda )=\pm 1,\,B({\vec {b}},\lambda )=\pm 1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eb2327aafff91d2822e01dcdead6d9ad38a3b7c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.434ex; height:3.343ex;" alt="{\displaystyle A({\vec {a}},\lambda )=\pm 1,\,B({\vec {b}},\lambda )=\pm 1.}"> Crucially, the outcome of detector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> does not depend upon <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}">, and likewise the outcome of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> does not depend upon <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}">, 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"> or to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965bd8710781b710cbfdb79da0b4e3b097bef506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {c}}}">. Bell proves that the difference in correlation between these two choices of detector setting must satisfy the inequality<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |P({\vec {a}},{\vec {b}})-P({\vec {a}},{\vec {c}})|\leq 1+P({\vec {b}},{\vec {c}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1</mn> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |P({\vec {a}},{\vec {b}})-P({\vec {a}},{\vec {c}})|\leq 1+P({\vec {b}},{\vec {c}}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e9fe5d7f8ad0c96200402b7e82f4262e6817c18" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.742ex; height:3.343ex;" alt="{\displaystyle |P({\vec {a}},{\vec {b}})-P({\vec {a}},{\vec {c}})|\leq 1+P({\vec {b}},{\vec {c}}).}"> However, it is easy to find situations where quantum mechanics violates the Bell inequality.<a href="#cite_note-14">[13]</a>: 425–426  For example, let the vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}"> be orthogonal, and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965bd8710781b710cbfdb79da0b4e3b097bef506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {c}}}"> lie in their plane at a 45° angle from both of them. Then<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P({\vec {a}},{\vec {b}})=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P({\vec {a}},{\vec {b}})=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb449f4ab46ba39b552c0fe1e178b13f2aabff32" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.82ex; height:3.343ex;" alt="{\displaystyle P({\vec {a}},{\vec {b}})=0,}"> while <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P({\vec {a}},{\vec {c}})=P({\vec {b}},{\vec {c}})=-{\frac {\sqrt {2}}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P({\vec {a}},{\vec {c}})=P({\vec {b}},{\vec {c}})=-{\frac {\sqrt {2}}{2}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1a31a36f610ec0db26d14658c89644bccfdb64d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.534ex; height:5.843ex;" alt="{\displaystyle P({\vec {a}},{\vec {c}})=P({\vec {b}},{\vec {c}})=-{\frac {\sqrt {2}}{2}},}"> but <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sqrt {2}}{2}}\nleq 1-{\frac {\sqrt {2}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>≰</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {2}}{2}}\nleq 1-{\frac {\sqrt {2}}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3de88a1464a1e8c2426d6b6c637257d89d73ff7d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.617ex; height:5.843ex;" alt="{\displaystyle {\frac {\sqrt {2}}{2}}\nleq 1-{\frac {\sqrt {2}}{2}}.}"> Therefore, there is no local hidden-variable model that can reproduce the predictions of quantum mechanics for all choices of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546e6615827e17295718741fd0b86f639a947f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9ef58be7103eb0b2bfcb460df23430f6a36216" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {c}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {c}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91f7718a5c0bf2cc1edcacefa32a5f3b79db66be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.87ex; height:2.343ex;" alt="{\displaystyle {\vec {c}}.}"> Experimental results contradict the classical curves and match the curve predicted by quantum mechanics as long as experimental shortcomings are accounted for.<a href="#cite_note-Stanford-5">[5]</a> Bell's 1964 theorem requires the possibility of perfect anti-correlations: the ability to make a probability-1 prediction about the result from the second detector, knowing the result from the first. This is related to 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."<a href="#cite_note-EPR-15">[14]</a> <div class="mw-heading mw-heading3"><h3 id="GHZ–Mermin_(1990)">GHZ–Mermin (1990)</h3>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=4" title="Edit section: GHZ–Mermin (1990)">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/GHZ_experiment" class="mw-redirect" title="GHZ experiment">GHZ experiment</a></div> <a href="/wiki/Daniel_Greenberger" title="Daniel Greenberger">Daniel Greenberger</a>, <a href="/wiki/Michael_Horne_(physicist)" title="Michael Horne (physicist)">Michael A. Horne</a>, and <a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Anton Zeilinger</a> presented a four-particle thought experiment in 1990, which <a href="/wiki/N._David_Mermin" title="N. David Mermin">David Mermin</a> then simplified to use only three particles.<a href="#cite_note-GHZ1990-16">[15]</a><a href="#cite_note-mermin1990-17">[16]</a> In this thought experiment, Victor generates a set of three spin-1/2 particles described by the quantum state<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle ={\frac {1}{\sqrt {2}}}(|000\rangle -|111\rangle )\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>000</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>111</mn> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle ={\frac {1}{\sqrt {2}}}(|000\rangle -|111\rangle )\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36983c34aa2c4216270aa274e6e03c1cd79c3db8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.859ex; height:6.176ex;" alt="{\displaystyle |\psi \rangle ={\frac {1}{\sqrt {2}}}(|000\rangle -|111\rangle )\,,}"> where as above, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |0\rangle }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |1\rangle }"> are the eigenvectors of the Pauli matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ea9b1af0a90b25f6dcdc140820ff2428fd92c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.329ex; height:2.009ex;" alt="{\displaystyle \sigma _{z}}">. Victor then sends a particle each to Alice, Bob, and Charlie, who wait at widely separated locations. Alice measures either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf4ab08fe163a6c495cad6f4d67653287c4044c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \sigma _{x}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b22d96fceb022e70169a37383278199a26b3534a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.377ex; height:2.343ex;" alt="{\displaystyle \sigma _{y}}"> on her particle, and so do Bob and Charlie. The result of each measurement is either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}">. Applying the Born rule to the three-qubit state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }">, Victor predicts that whenever the three measurements include one <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf4ab08fe163a6c495cad6f4d67653287c4044c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \sigma _{x}}"> and two <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b22d96fceb022e70169a37383278199a26b3534a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.377ex; height:2.343ex;" alt="{\displaystyle \sigma _{y}}">'s, the product of the outcomes will always be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}">. This follows because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"> is an eigenvector of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}\otimes \sigma _{y}\otimes \sigma _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}\otimes \sigma _{y}\otimes \sigma _{y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/828faf89bce207443464069d236965fa3e205bf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.934ex; height:2.676ex;" alt="{\displaystyle \sigma _{x}\otimes \sigma _{y}\otimes \sigma _{y}}"> with eigenvalue <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}">, and likewise for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}\otimes \sigma _{x}\otimes \sigma _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{y}\otimes \sigma _{x}\otimes \sigma _{y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c16e29de4bf285c1dbb5ed87bc21cf51a1a231a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.934ex; height:2.676ex;" alt="{\displaystyle \sigma _{y}\otimes \sigma _{x}\otimes \sigma _{y}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}\otimes \sigma _{y}\otimes \sigma _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{y}\otimes \sigma _{y}\otimes \sigma _{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546fd7228896f9238c52ae68acbc0c9c14d4aec5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.934ex; height:2.676ex;" alt="{\displaystyle \sigma _{y}\otimes \sigma _{y}\otimes \sigma _{x}}">. Therefore, knowing Alice's result for a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf4ab08fe163a6c495cad6f4d67653287c4044c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \sigma _{x}}"> measurement and Bob's result for a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b22d96fceb022e70169a37383278199a26b3534a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.377ex; height:2.343ex;" alt="{\displaystyle \sigma _{y}}"> measurement, Victor can predict with probability 1 what result Charlie will return for a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b22d96fceb022e70169a37383278199a26b3534a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.377ex; height:2.343ex;" alt="{\displaystyle \sigma _{y}}"> measurement. According to the EPR criterion of reality, there would be an "element of reality" corresponding to the outcome of a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b22d96fceb022e70169a37383278199a26b3534a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.377ex; height:2.343ex;" alt="{\displaystyle \sigma _{y}}"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf4ab08fe163a6c495cad6f4d67653287c4044c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \sigma _{x}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b22d96fceb022e70169a37383278199a26b3534a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.377ex; height:2.343ex;" alt="{\displaystyle \sigma _{y}}"> measurement upon it. The set of all three particles would then be described by the instruction set<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{x},a_{y},b_{x},b_{y},c_{x},c_{y})\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{x},a_{y},b_{x},b_{y},c_{x},c_{y})\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace2ae772546cd2e1311b63c8bec54ab3f633db3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.147ex; height:3.009ex;" alt="{\displaystyle (a_{x},a_{y},b_{x},b_{y},c_{x},c_{y})\,,}"> with each entry being either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}">, and each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf4ab08fe163a6c495cad6f4d67653287c4044c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \sigma _{x}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b22d96fceb022e70169a37383278199a26b3534a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.377ex; height:2.343ex;" alt="{\displaystyle \sigma _{y}}"> measurement simply returning the appropriate value. If Alice, Bob, and Charlie all perform the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf4ab08fe163a6c495cad6f4d67653287c4044c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \sigma _{x}}"> measurement, then the product of their results would be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{x}b_{x}c_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{x}b_{x}c_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/169883bb9bb3755ce5dae36f50513539793355e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.752ex; height:2.509ex;" alt="{\displaystyle a_{x}b_{x}c_{x}}">. This value can be deduced from<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (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}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (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}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/815dc41c5e63205db5eadb16e9a800bec29257be" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:52.075ex; height:3.176ex;" alt="{\displaystyle (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}\,,}"> because the square of either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">. Each factor in parentheses equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}">, so<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{x}b_{x}c_{x}=+1\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo>+</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{x}b_{x}c_{x}=+1\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a18ac85063dc6de93b1ffc124551b708d813a7e1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.855ex; height:2.509ex;" alt="{\displaystyle a_{x}b_{x}c_{x}=+1\,,}"> and the product of Alice, Bob, and Charlie's results will be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}"> with probability unity. But this is inconsistent with quantum physics: Victor can predict using the state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"> that the measurement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}\otimes \sigma _{x}\otimes \sigma _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}\otimes \sigma _{x}\otimes \sigma _{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1aae5e1a37691678a3aed985ece56afe85d01e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.18ex; height:2.343ex;" alt="{\displaystyle \sigma _{x}\otimes \sigma _{x}\otimes \sigma _{x}}"> will instead yield <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"> 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.<a href="#cite_note-Brassard_2004-18">[17]</a> In it, Alice, Bob, and Charlie receive bits <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}"> from Victor, promised to always have an even number of ones, that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\oplus y\oplus z=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⊕</mo> <mi>y</mi> <mo>⊕</mo> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\oplus y\oplus z=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4824e314aec4a0ee773966de6080b517eb058a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.515ex; height:2.509ex;" alt="{\displaystyle x\oplus y\oplus z=0}">, and send him back bits <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}">. They win the game if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}"> have an odd number of ones for all inputs except <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y=z=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>=</mo> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y=z=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/806816016dd7dfb90cb1cd0560506c14a29e2535" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.031ex; height:2.509ex;" alt="{\displaystyle x=y=z=0}">, when they need to have an even number of ones. That is, they win the game iff <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\oplus b\oplus c=x\lor y\lor z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⊕</mo> <mi>b</mi> <mo>⊕</mo> <mi>c</mi> <mo>=</mo> <mi>x</mi> <mo>∨</mo> <mi>y</mi> <mo>∨</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\oplus b\oplus c=x\lor y\lor z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9bb2737a2c4c599f1a7f5eac66615d9ba522525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.752ex; height:2.509ex;" alt="{\displaystyle a\oplus b\oplus c=x\lor y\lor z}">. 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 <a href="/wiki/Quantum_pseudo-telepathy" title="Quantum pseudo-telepathy">quantum pseudo-telepathy</a>. <div class="mw-heading mw-heading3"><h3 id="Kochen–Specker_theorem_(1967)">Kochen–Specker theorem (1967)</h3>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=5" title="Edit section: Kochen–Specker theorem (1967)">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Kochen%E2%80%93Specker_theorem" title="Kochen–Specker theorem">Kochen–Specker theorem</a></div> In quantum theory, orthonormal bases for a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> 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.<a href="#cite_note-23">[note 2]</a> Suppose that a hidden variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> exists, so that knowing the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> would imply certainty about the outcome of any measurement. Given a value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }">, 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> can be controlling the measurement outcomes.<a href="#cite_note-24">[22]</a>: 196–201  <div class="mw-heading mw-heading3"><h3 id="Free_will_theorem">Free will theorem</h3>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=6" title="Edit section: Free will theorem">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Free_will_theorem" title="Free will theorem">Free will theorem</a></div> 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,<a href="#cite_note-25">[23]</a> Heywood and Redhead,<a href="#cite_note-26">[24]</a> Stairs,<a href="#cite_note-27">[25]</a> and Brown and Svetlichny.<a href="#cite_note-28">[26]</a> 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.<a href="#cite_note-29">[27]</a> In other words, by this criterion, a hidden variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> 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 <a href="/wiki/John_Horton_Conway" title="John Horton Conway">John Conway</a> and <a href="/wiki/Simon_B._Kochen" title="Simon B. Kochen">Simon Kochen</a> under the name of the <a href="/wiki/Free_will_theorem" title="Free will theorem">free will theorem</a>.<a href="#cite_note-30">[28]</a><a href="#cite_note-31">[29]</a><a href="#cite_note-32">[30]</a> The Conway–Kochen theorem uses a pair of entangled <a href="/wiki/Qutrit" title="Qutrit">qutrits</a> and a Kochen–Specker configuration discovered by <a href="/wiki/Asher_Peres" title="Asher Peres">Asher Peres</a>.<a href="#cite_note-33">[31]</a> <div class="mw-heading mw-heading3"><h3 id="Quasiclassical_entanglement">Quasiclassical entanglement</h3>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=7" title="Edit section: Quasiclassical entanglement">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Spekkens_toy_model" title="Spekkens toy model">Spekkens toy model</a> and <a href="/wiki/Werner_state" title="Werner state">Werner state</a></div> 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, <a href="/wiki/Reinhard_F._Werner" title="Reinhard F. Werner">Reinhard Werner</a> introduced what are now called <a href="/wiki/Werner_state" title="Werner state">Werner states</a>, joint quantum states for a pair of systems that yield EPR-type correlations but also admit a hidden-variable model.<a href="#cite_note-34">[32]</a> Werner states are bipartite quantum states that are invariant under <a href="/wiki/Unitarity_(physics)" title="Unitarity (physics)">unitaries</a> of symmetric <a href="/wiki/Kronecker_product" title="Kronecker product">tensor-product</a> form: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{AB}=(U\otimes U)\rho _{AB}(U^{\dagger }\otimes U^{\dagger }).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo>⊗</mo> <mi>U</mi> <mo stretchy="false">)</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>⊗</mo> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{AB}=(U\otimes U)\rho _{AB}(U^{\dagger }\otimes U^{\dagger }).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab52b8c0f08b7ccaac64c309b50ddd269644fe38" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.045ex; height:3.176ex;" alt="{\displaystyle \rho _{AB}=(U\otimes U)\rho _{AB}(U^{\dagger }\otimes U^{\dagger }).}"> In 2004, <a href="/wiki/Robert_Spekkens" title="Robert Spekkens">Robert Spekkens</a> introduced a <a href="/wiki/Spekkens_toy_model" title="Spekkens toy model">toy model</a> 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 <a href="/wiki/Monogamy_of_entanglement" title="Monogamy of entanglement">monogamy</a>, but by construction, the toy model can never violate a Bell inequality.<a href="#cite_note-35">[33]</a><a href="#cite_note-36">[34]</a> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=8" title="Edit section: History">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Background">Background</h3>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=9" title="Edit section: Background">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/EPR_paradox" class="mw-redirect" title="EPR paradox">EPR paradox</a> and <a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History of quantum mechanics</a></div> The question of whether quantum mechanics can be "completed" by hidden variables dates to the early years of quantum theory. In his <a href="/wiki/Mathematical_Foundations_of_Quantum_Mechanics" title="Mathematical Foundations of Quantum Mechanics">1932 textbook on quantum mechanics</a>, the Hungarian-born polymath <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a> 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 <a href="/wiki/Hans_Reichenbach" title="Hans Reichenbach">Hans Reichenbach</a>, in more detail by <a href="/wiki/Grete_Hermann" title="Grete Hermann">Grete Hermann</a>, and possibly in conversation though not in print by Albert Einstein.<a href="#cite_note-42">[note 3]</a> (<a href="/wiki/Simon_B._Kochen" title="Simon B. Kochen">Simon Kochen</a> and <a href="/wiki/Ernst_Specker" title="Ernst Specker">Ernst Specker</a> rejected von Neumann's key assumption as early as 1961, but did not publish a criticism of it until 1967.<a href="#cite_note-43">[40]</a>) Einstein argued persistently that quantum mechanics could not be a complete theory. His preferred argument relied on a principle of locality: <dl><dd>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 <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> 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.<a href="#cite_note-44">[41]</a></dd></dl> 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.<a href="#cite_note-45">[42]</a> In 1951, <a href="/wiki/David_Bohm" title="David Bohm">David Bohm</a> 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.<a href="#cite_note-46">[43]</a> The year before, <a href="/wiki/Chien-Shiung_Wu" title="Chien-Shiung Wu">Chien-Shiung Wu</a> 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.<a href="#cite_note-47">[44]</a> By the late 1940s, the mathematician <a href="/wiki/George_Mackey" title="George Mackey">George Mackey</a> 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.<a href="#cite_note-48">[45]</a> Mackey conjectured that one of the postulates was redundant, and shortly thereafter, <a href="/wiki/Andrew_M._Gleason" title="Andrew M. Gleason">Andrew M. Gleason</a> proved that it was indeed deducible from the other postulates.<a href="#cite_note-gleason1957-49">[46]</a><a href="#cite_note-chernoff2009-50">[47]</a> <a href="/wiki/Gleason%27s_theorem" title="Gleason's theorem">Gleason's theorem</a> provided an argument that a broad class of hidden-variable theories are incompatible with quantum mechanics.<a href="#cite_note-52">[note 4]</a> 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 <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>.<a href="#cite_note-ND_Mermin_1993-07-53">[49]</a><a href="#cite_note-54">[50]</a> The Kochen–Specker theorem refines this statement by constructing a specific finite subset of rays on which no such probability measure can be defined.<a href="#cite_note-ND_Mermin_1993-07-53">[49]</a><a href="#cite_note-55">[51]</a> <a href="/wiki/Tsung-Dao_Lee" title="Tsung-Dao Lee">Tsung-Dao Lee</a> came close to deriving Bell's theorem in 1960. He considered events where two <a href="/wiki/Kaon" title="Kaon">kaons</a> 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.<a href="#cite_note-jammer1974-38">[36]</a>: 308  <div class="mw-heading mw-heading3"><h3 id="Bell's_publications">Bell's publications</h3>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=10" title="Edit section: Bell's publications">edit</a>]</div> Bell chose to publish his theorem in a comparatively obscure journal because it did not require <a href="/wiki/Page_charge" class="mw-redirect" title="Page charge">page charges</a>, 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.<a href="#cite_note-:0-56">[52]</a> While the articles printed in the journal themselves listed the publication's name simply as Physics, the covers carried the trilingual version <a href="/wiki/Physics_Physique_%D0%A4%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0" title="Physics Physique Физика">Physics Physique Физика</a> to reflect that it would print articles in English, French and Russian.<a href="#cite_note-:1-41">[39]</a>: 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.<a href="#cite_note-ND_Mermin_1993-07-53">[49]</a><a href="#cite_note-Bell1966-57">[53]</a> 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."<a href="#cite_note-Bell1966-57">[53]</a> <div class="mw-heading mw-heading2"><h2 id="Experiments">Experiments</h2>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=11" title="Edit section: Experiments">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bell-test-photon-analyer.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Bell-test-photon-analyer.png/440px-Bell-test-photon-analyer.png" decoding="async" width="440" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Bell-test-photon-analyer.png/660px-Bell-test-photon-analyer.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Bell-test-photon-analyer.png/880px-Bell-test-photon-analyer.png 2x" data-file-width="1232" data-file-height="544" /></a><figcaption>Scheme of a "two-channel" Bell test 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.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Bell_test" title="Bell test">Bell test</a></div> In 1967, the unusual title Physics Physique Физика caught the attention of <a href="/wiki/John_Clauser" title="John Clauser">John Clauser</a>, who then discovered Bell's paper and began to consider how to perform a <a href="/wiki/Bell_test" title="Bell test">Bell test</a> in the laboratory.<a href="#cite_note-58">[54]</a> Clauser and <a href="/wiki/Stuart_Freedman" title="Stuart Freedman">Stuart Freedman</a> would go on to perform a Bell test in 1972.<a href="#cite_note-59">[55]</a><a href="#cite_note-60">[56]</a> This was only a limited test, because the choice of detector settings was made before the photons had left the source. In 1982, <a href="/wiki/Alain_Aspect" title="Alain Aspect">Alain Aspect</a> and collaborators performed the <a href="/wiki/Aspect%27s_experiment" title="Aspect's experiment">first Bell test</a> to remove this limitation.<a href="#cite_note-61">[57]</a> 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.<a href="#cite_note-GHZ2000-62">[58]</a> By 2002, testing the CHSH inequality was feasible in undergraduate laboratory courses.<a href="#cite_note-63">[59]</a> 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 <a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">local hidden-variable theory</a>, which would contradict the predictions of quantum mechanics. The most prevalent loopholes in real experiments are the detection and locality loopholes.<a href="#cite_note-larsson14-64">[60]</a> 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 <a href="/wiki/Spacetime#Spacetime_interval" title="Spacetime">spacelike separation</a>, 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.<a href="#cite_note-65">[61]</a> 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.<a href="#cite_note-66">[62]</a><a href="#cite_note-NYT-20151021-67">[63]</a><a href="#cite_note-NTR-20151021-68">[64]</a><a href="#cite_note-PRL115-250402-69">[65]</a><a href="#cite_note-PRL115-250401-70">[66]</a> Regarding these results, <a href="/wiki/Alain_Aspect" title="Alain Aspect">Alain Aspect</a> 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."<a href="#cite_note-71">[67]</a> These efforts to experimentally validate violations of the Bell inequalities would later result in Clauser, Aspect, and <a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Anton Zeilinger</a> being awarded the 2022 <a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a>.<a href="#cite_note-72">[68]</a> <div class="mw-heading mw-heading2"><h2 id="Interpretations">Interpretations</h2>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=12" title="Edit section: Interpretations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations of quantum mechanics</a></div> 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."<a href="#cite_note-73">[69]</a> <div class="mw-heading mw-heading3"><h3 id="The_Copenhagen_interpretation">The Copenhagen interpretation</h3>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=13" title="Edit section: The Copenhagen interpretation">edit</a>]</div> <a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen-type interpretations</a> generally take the violation of Bell inequalities as grounds to reject the assumption often called <a href="/wiki/Counterfactual_definiteness" title="Counterfactual definiteness">counterfactual definiteness</a> or "realism", which is not necessarily the same as abandoning realism in a broader philosophical sense.<a href="#cite_note-74">[70]</a><a href="#cite_note-75">[71]</a> For example, <a href="/wiki/Roland_Omn%C3%A8s" title="Roland Omnès">Roland Omnès</a> 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".<a href="#cite_note-omnes-76">[72]</a>: 531  Likewise, <a href="/wiki/Rudolf_Peierls" title="Rudolf Peierls">Rudolf Peierls</a> 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".<a href="#cite_note-77">[73]</a><a href="#cite_note-78">[74]</a> This is also the route taken by interpretations that descend from the Copenhagen tradition, such as <a href="/wiki/Consistent_histories" title="Consistent histories">consistent histories</a> (often advertised as "Copenhagen done right"),<a href="#cite_note-79">[75]</a>: <span title="Page / location: 2839
Quotation: "CQT most definitely opts for retaining locality (EPR2) and rejecting classical realism (EPR1)"" class="tooltip tooltip-dashed" style="border-bottom: 1px dashed;">2839  as well as <a href="/wiki/QBism" class="mw-redirect" title="QBism">QBism</a>.<a href="#cite_note-80">[76]</a> <div class="mw-heading mw-heading3"><h3 id="Many-worlds_interpretation_of_quantum_mechanics">Many-worlds interpretation of quantum mechanics</h3>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=14" title="Edit section: Many-worlds interpretation of quantum mechanics">edit</a>]</div> The <a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds interpretation</a>, also known as the <a href="/wiki/Hugh_Everett_III" title="Hugh Everett III">Everett</a> interpretation, is dynamically local, meaning that it does not call for <a href="/wiki/Action_at_a_distance" title="Action at a distance">action at a distance</a>,<a href="#cite_note-BrownTimpson-81">[77]</a>: 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.<a href="#cite_note-82">[78]</a> 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.<a href="#cite_note-BrownTimpson-81">[77]</a>: <span title="Page / location: 28
Quotation: "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"" class="tooltip tooltip-dashed" style="border-bottom: 1px dashed;">28  <div class="mw-heading mw-heading3"><h3 id="Non-local_hidden_variables">Non-local hidden variables</h3>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=15" title="Edit section: Non-local hidden variables">edit</a>]</div> Most advocates of the hidden-variables idea believe that experiments have ruled out local hidden variables.<a href="#cite_note-85">[note 5]</a> They are ready to give up locality, explaining the violation of Bell's inequality by means of a non-local <a href="/wiki/Hidden_variable_theory" class="mw-redirect" title="Hidden variable theory">hidden variable theory</a>, in which the particles exchange information about their states. This is the basis of the <a href="/wiki/Bohm_interpretation" class="mw-redirect" title="Bohm interpretation">Bohm interpretation</a> 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.<a href="#cite_note-86">[81]</a> A 2007 experiment ruled out a large class of non-Bohmian non-local hidden variable theories, though not Bohmian mechanics itself.<a href="#cite_note-87">[82]</a> The <a href="/wiki/Transactional_interpretation" title="Transactional interpretation">transactional interpretation</a>, which postulates waves traveling both backwards and forwards in time, is likewise non-local.<a href="#cite_note-88">[83]</a> <div class="mw-heading mw-heading3"><h3 id="Superdeterminism">Superdeterminism</h3>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=16" title="Edit section: Superdeterminism">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></div> 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 "<a href="/wiki/Free_will" title="Free will">free will</a>" 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.<a href="#cite_note-larsson14-64">[60]</a> A few advocates of deterministic models have not given up on local hidden variables. For example, <a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">Gerard 't Hooft</a> has argued that superdeterminism cannot be dismissed.<a href="#cite_note-89">[84]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=17" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1259569809">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/25px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="25" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/37px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/49px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a><a href="/wiki/Portal:Physics" title="Portal:Physics">physics portal</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Einstein%27s_thought_experiments" title="Einstein's thought experiments">Einstein's thought experiments</a></li> <li><a href="/wiki/Epistemological_Letters" title="Epistemological Letters">Epistemological Letters</a></li> <li><a href="/wiki/Fundamental_Fysiks_Group" title="Fundamental Fysiks Group">Fundamental Fysiks Group</a></li> <li><a href="/wiki/Leggett_inequality" title="Leggett inequality">Leggett inequality</a></li> <li><a href="/wiki/Leggett%E2%80%93Garg_inequality" title="Leggett–Garg inequality">Leggett–Garg inequality</a></li> <li><a href="/wiki/Mermin%27s_device" title="Mermin's device">Mermin's device</a></li> <li><a href="/wiki/Mott_problem" title="Mott problem">Mott problem</a></li> <li><a href="/wiki/PBR_theorem" class="mw-redirect" title="PBR theorem">PBR theorem</a></li> <li><a href="/wiki/Quantum_contextuality" title="Quantum contextuality">Quantum contextuality</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Quantum nonlocality</a></li> <li><a href="/wiki/Renninger_negative-result_experiment" title="Renninger negative-result experiment">Renninger negative-result experiment</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=18" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-9"><a href="#cite_ref-9">^</a> 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.<a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a> </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> In more detail, as developed by <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a>,<a href="#cite_note-19">[18]</a> <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>,<a href="#cite_note-20">[19]</a> <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a>,<a href="#cite_note-21">[20]</a> and <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a>,<a href="#cite_note-22">[21]</a> the state of a quantum mechanical system is a vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"> belonging to a (<a href="/wiki/Separable_space" title="Separable space">separable</a>) Hilbert space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}">. Physical quantities of interest — position, momentum, energy, spin — are represented by "observables", which are <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">self-adjoint</a> linear <a href="/wiki/Operator_(physics)" title="Operator (physics)">operators</a> acting on the Hilbert space. When an observable is measured, the result will be one of its eigenvalues with probability given by the <a href="/wiki/Born_rule" title="Born rule">Born rule</a>: in the simplest case the eigenvalue <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"> is non-degenerate and the probability is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\langle \eta |\psi \rangle |^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\langle \eta |\psi \rangle |^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38b0862b588e2ecc8975dc68c457e5a17844f4cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.487ex; height:3.343ex;" alt="{\displaystyle |\langle \eta |\psi \rangle |^{2}}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\eta \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>η</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\eta \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1731d8f806cedf7f01c6012042f4ccae2a230a39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.721ex; height:2.843ex;" alt="{\displaystyle |\eta \rangle }"> is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \psi |P_{\eta }\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> </msub> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi |P_{\eta }\psi \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18d4db25091fa5dc2f8109bd81a9e6a121bda05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.034ex; height:3.009ex;" alt="{\displaystyle \langle \psi |P_{\eta }\psi \rangle }">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\eta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\eta }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/678d243a7593224cc7a342b2ac662b509f75d075" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.551ex; height:2.843ex;" alt="{\displaystyle P_{\eta }}"> is the projector onto its associated eigenspace. For the purposes of this discussion, we can take the eigenvalues to be non-degenerate. </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> See Reichenbach<a href="#cite_note-37">[35]</a> and Jammer,<a href="#cite_note-jammer1974-38">[36]</a>: 276  Mermin and Schack,<a href="#cite_note-39">[37]</a> and for Einstein's remarks, Clauser and Shimony<a href="#cite_note-40">[38]</a> and Wick.<a href="#cite_note-:1-41">[39]</a>: 286  </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> A hidden-variable theory that is <a href="/wiki/Determinism" title="Determinism">deterministic</a> implies that the probability of a given outcome is always either 0 or 1. For example, a Stern–Gerlach measurement on a <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin-1</a> atom will report that the atom's angular momentum along the chosen axis is one of three possible values, which can be designated <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}">. 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}">) 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\to \langle \rho u,u\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">→</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ρ</mi> <mi>u</mi> <mo>,</mo> <mi>u</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\to \langle \rho u,u\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f9dd8a2ca37be0af5c3c49b5036aefba0be8209" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.648ex; height:2.843ex;" alt="{\displaystyle u\to \langle \rho u,u\rangle }"> for some density operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }">. This mapping is continuous on the <a href="/wiki/Unit_sphere" title="Unit sphere">unit sphere</a> of the Hilbert space, and since this unit sphere is <a href="/wiki/Connected_(topology)" class="mw-redirect" title="Connected (topology)">connected</a>, no continuous probability measure on it can be deterministic.<a href="#cite_note-wilce2017-51">[48]</a>: §1.3  </li> <li id="cite_note-85"><a href="#cite_ref-85">^</a> <a href="/wiki/E._T._Jaynes" class="mw-redirect" title="E. T. Jaynes">E. T. Jaynes</a> was one exception,<a href="#cite_note-E.T._Jaynes_1989-83">[79]</a> but Jaynes' arguments have not generally been found persuasive.<a href="#cite_note-Gill2002-84">[80]</a> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=19" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBell1987" class="citation book cs1"><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell, John S.</a> (1987). 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Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004. <a href="/wiki/IEEE" class="mw-redirect" title="IEEE">IEEE</a>. pp. 236–249. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0404076">quant-ph/0404076</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004quant.ph..4076C">2004quant.ph..4076C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FCCC.2004.1313847">10.1109/CCC.2004.1313847</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7695-2120-7" title="Special:BookSources/0-7695-2120-7"><bdi>0-7695-2120-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/55954993">55954993</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8077237">8077237</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Consequences+and+limits+of+nonlocal+strategies&rft.btitle=Proceedings.+19th+IEEE+Annual+Conference+on+Computational+Complexity%2C+2004.&rft.pages=236-249&rft.pub=IEEE&rft.date=2004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A8077237%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2004quant.ph..4076C&rft_id=info%3Aarxiv%2Fquant-ph%2F0404076&rft_id=info%3Adoi%2F10.1109%2FCCC.2004.1313847&rft_id=info%3Aoclcnum%2F55954993&rft.isbn=0-7695-2120-7&rft.aulast=Cleve&rft.aufirst=R.&rft.au=Hoyer%2C+P.&rft.au=Toner%2C+B.&rft.au=Watrous%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarnumBeigiBoixoElliott2010" class="citation journal cs1">Barnum, H.; Beigi, S.; Boixo, S.; Elliott, M. 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Mathematische Grundlagen der Quantenmechanik. Berlin: Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematische+Grundlagen+der+Quantenmechanik&rft.place=Berlin&rft.pub=Springer&rft.date=1932&rft.aulast=von+Neumann&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"> English translation: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Mathematical_Foundations_of_Quantum_Mechanics" title="Mathematical Foundations of Quantum Mechanics">Mathematical Foundations of Quantum Mechanics</a>. Translated by <a href="/wiki/Robert_T._Beyer" title="Robert T. 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P.</a> Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-60269-1" title="Special:BookSources/978-0-486-60269-1"><bdi>978-0-486-60269-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Groups+and+Quantum+Mechanics&rft.pub=Dover&rft.date=1950&rft.isbn=978-0-486-60269-1&rft.aulast=Weyl&rft.aufirst=Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"> Translated from the German <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1">Gruppentheorie und Quantenmechanik (2nd ed.). <a href="/w/index.php?title=S._Hirzel_Verlag&action=edit&redlink=1" class="new" title="S. Hirzel Verlag (page does not exist)">S. Hirzel Verlag</a> [<a href="https://de.wikipedia.org/wiki/S._Hirzel_Verlag" class="extiw" title="de:S. Hirzel Verlag">de</a>]. 1931.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gruppentheorie+und+Quantenmechanik&rft.edition=2nd&rft.pub=S.+Hirzel+Verlag%3Cspan+class%3D%22noprint%22+style%3D%22font-size%3A85%25%3B+font-style%3A+normal%3B+%22%3E+%26%2391%3Bde%26%2393%3B%3C%2Fspan%3E&rft.date=1931&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"> </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeres1993" class="citation book cs1"><a href="/wiki/Asher_Peres" title="Asher Peres">Peres, Asher</a> (1993). <a href="/wiki/Quantum_Theory:_Concepts_and_Methods" title="Quantum Theory: Concepts and Methods">Quantum Theory: Concepts and Methods</a>. <a href="/wiki/Kluwer" class="mw-redirect" title="Kluwer">Kluwer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-2549-4" title="Special:BookSources/0-7923-2549-4"><bdi>0-7923-2549-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/28854083">28854083</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Theory%3A+Concepts+and+Methods&rft.pub=Kluwer&rft.date=1993&rft_id=info%3Aoclcnum%2F28854083&rft.isbn=0-7923-2549-4&rft.aulast=Peres&rft.aufirst=Asher&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"> </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRedheadBrown1991" class="citation journal cs1"><a href="/wiki/Michael_Redhead" title="Michael Redhead">Redhead, Michael</a>; <a href="/wiki/Harvey_R._Brown" title="Harvey R. Brown">Brown, Harvey</a> (1991-07-01). "Nonlocality in Quantum Mechanics". <a href="/wiki/Aristotelian_Society" title="Aristotelian Society">Proceedings of the Aristotelian Society, Supplementary Volumes</a>. 65 (1): 119–160. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Faristoteliansupp%2F65.1.119">10.1093/aristoteliansupp/65.1.119</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0309-7013">0309-7013</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/4106773">4106773</a>. <q>A similar approach was arrived at independently by Simon Kochen, although never published (private communication).</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Aristotelian+Society%2C+Supplementary+Volumes&rft.atitle=Nonlocality+in+Quantum+Mechanics&rft.volume=65&rft.issue=1&rft.pages=119-160&rft.date=1991-07-01&rft.issn=0309-7013&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F4106773%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1093%2Faristoteliansupp%2F65.1.119&rft.aulast=Redhead&rft.aufirst=Michael&rft.au=Brown%2C+Harvey&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeywoodRedhead1983" class="citation journal cs1">Heywood, Peter; <a href="/wiki/Michael_Redhead" title="Michael Redhead">Redhead, Michael L. 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Retrieved 2011-10-18.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Clearing+up+Mysteries+%E2%80%94+the+Original+Goal&rft.btitle=Maximum+Entropy+and+Bayesian+Methods&rft.pages=1-27&rft.date=1989&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.46.1264%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1007%2F978-94-015-7860-8_1&rft.isbn=978-90-481-4044-2&rft.aulast=Jaynes&rft.aufirst=E.+T.&rft_id=http%3A%2F%2Fbayes.wustl.edu%2Fetj%2Farticles%2Fcmystery.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"> </li> <li id="cite_note-Gill2002-84"><a href="#cite_ref-Gill2002_84-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGill2002" class="citation book cs1">Gill, Richard D. (2002). "Time, Finite Statistics, and Bell's Fifth Position". Proceedings of the Conference Foundations of Probability and Physics - 2 : Växjö (Soland), Sweden, June 2-7, 2002. Vol. 5. Växjö University Press. pp. 179–206. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0301059">quant-ph/0301059</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Time%2C+Finite+Statistics%2C+and+Bell%27s+Fifth+Position&rft.btitle=Proceedings+of+the+Conference+Foundations+of+Probability+and+Physics+-+2+%3A+V%C3%A4xj%C3%B6+%28Soland%29%2C+Sweden%2C+June+2-7%2C+2002&rft.pages=179-206&rft.pub=V%C3%A4xj%C3%B6+University+Press&rft.date=2002&rft_id=info%3Aarxiv%2Fquant-ph%2F0301059&rft.aulast=Gill&rft.aufirst=Richard+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"> </li> <li id="cite_note-86"><a href="#cite_ref-86">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWoodSpekkens2015" class="citation journal cs1">Wood, Christopher J.; <a href="/wiki/Robert_Spekkens" title="Robert Spekkens">Spekkens, Robert W.</a> (2015-03-03). <a rel="nofollow" class="external text" href="https://iopscience.iop.org/article/10.1088/1367-2630/17/3/033002">"The lesson of causal discovery algorithms for quantum correlations: causal explanations of Bell-inequality violations require fine-tuning"</a>. <a href="/wiki/New_Journal_of_Physics" title="New Journal of Physics">New Journal of Physics</a>. 17 (3): 033002. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1208.4119">1208.4119</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015NJPh...17c3002W">2015NJPh...17c3002W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1367-2630%2F17%2F3%2F033002">10.1088/1367-2630/17/3/033002</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1367-2630">1367-2630</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118518558">118518558</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=New+Journal+of+Physics&rft.atitle=The+lesson+of+causal+discovery+algorithms+for+quantum+correlations%3A+causal+explanations+of+Bell-inequality+violations+require+fine-tuning&rft.volume=17&rft.issue=3&rft.pages=033002&rft.date=2015-03-03&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118518558%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2015NJPh...17c3002W&rft_id=info%3Aarxiv%2F1208.4119&rft.issn=1367-2630&rft_id=info%3Adoi%2F10.1088%2F1367-2630%2F17%2F3%2F033002&rft.aulast=Wood&rft.aufirst=Christopher+J.&rft.au=Spekkens%2C+Robert+W.&rft_id=https%3A%2F%2Fiopscience.iop.org%2Farticle%2F10.1088%2F1367-2630%2F17%2F3%2F033002&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"> </li> <li id="cite_note-87"><a href="#cite_ref-87">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGröblacherPaterekKaltenbaekBrukner2007" class="citation journal cs1">Gröblacher, Simon; Paterek, Tomasz; Kaltenbaek, Rainer; <a href="/wiki/%C4%8Caslav_Brukner" title="Časlav Brukner">Brukner, Časlav</a>; Żukowski, Marek; Aspelmeyer, Markus; <a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Zeilinger, Anton</a> (2007). "An experimental test of non-local realism". <a href="/wiki/Nature_(journal)" title="Nature (journal)">Nature</a>. 446 (7138): 871–5. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0704.2529">0704.2529</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007Natur.446..871G">2007Natur.446..871G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature05677">10.1038/nature05677</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17443179">17443179</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4412358">4412358</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=An+experimental+test+of+non-local+realism&rft.volume=446&rft.issue=7138&rft.pages=871-5&rft.date=2007&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4412358%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2007Natur.446..871G&rft_id=info%3Aarxiv%2F0704.2529&rft_id=info%3Apmid%2F17443179&rft_id=info%3Adoi%2F10.1038%2Fnature05677&rft.aulast=Gr%C3%B6blacher&rft.aufirst=Simon&rft.au=Paterek%2C+Tomasz&rft.au=Kaltenbaek%2C+Rainer&rft.au=Brukner%2C+%C4%8Caslav&rft.au=%C5%BBukowski%2C+Marek&rft.au=Aspelmeyer%2C+Markus&rft.au=Zeilinger%2C+Anton&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"> </li> <li id="cite_note-88"><a href="#cite_ref-88">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKastner2010" class="citation journal cs1">Kastner, Ruth E. (May 2010). <a rel="nofollow" class="external text" href="https://linkinghub.elsevier.com/retrieve/pii/S135521981000002X">"The quantum liar experiment in Cramer's transactional interpretation"</a>. <a href="/wiki/Studies_in_History_and_Philosophy_of_Science_Part_B:_Studies_in_History_and_Philosophy_of_Modern_Physics" class="mw-redirect" title="Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics">Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics</a>. 41 (2): 86–92. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0906.1626">0906.1626</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010SHPMP..41...86K">2010SHPMP..41...86K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.shpsb.2010.01.001">10.1016/j.shpsb.2010.01.001</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16242184">16242184</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180624053010/https://linkinghub.elsevier.com/retrieve/pii/S135521981000002X">Archived</a> from the original on 2018-06-24. Retrieved 2021-09-16.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Studies+in+History+and+Philosophy+of+Science+Part+B%3A+Studies+in+History+and+Philosophy+of+Modern+Physics&rft.atitle=The+quantum+liar+experiment+in+Cramer%27s+transactional+interpretation&rft.volume=41&rft.issue=2&rft.pages=86-92&rft.date=2010-05&rft_id=info%3Aarxiv%2F0906.1626&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16242184%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.shpsb.2010.01.001&rft_id=info%3Abibcode%2F2010SHPMP..41...86K&rft.aulast=Kastner&rft.aufirst=Ruth+E.&rft_id=https%3A%2F%2Flinkinghub.elsevier.com%2Fretrieve%2Fpii%2FS135521981000002X&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"> </li> <li id="cite_note-89"><a href="#cite_ref-89">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREF't_Hooft2016" class="citation book cs1"><a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">'t Hooft, Gerard</a> (2016). <a rel="nofollow" class="external text" href="http://www.oapen.org/search?identifier=1002003">The Cellular Automaton Interpretation of Quantum Mechanics</a>. Fundamental Theories of Physics. Vol. 185. Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-41285-6">10.1007/978-3-319-41285-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-41284-9" title="Special:BookSources/978-3-319-41284-9"><bdi>978-3-319-41284-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/951761277">951761277</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7779840">7779840</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211229062338/https://library.oapen.org/handle/20.500.12657/27994">Archived</a> from the original on 2021-12-29. Retrieved 2020-08-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Cellular+Automaton+Interpretation+of+Quantum+Mechanics&rft.series=Fundamental+Theories+of+Physics&rft.pub=Springer&rft.date=2016&rft_id=info%3Aoclcnum%2F951761277&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7779840%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-319-41285-6&rft.isbn=978-3-319-41284-9&rft.aulast=%27t+Hooft&rft.aufirst=Gerard&rft_id=http%3A%2F%2Fwww.oapen.org%2Fsearch%3Fidentifier%3D1002003&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Bell%27s_theorem&action=edit&section=20" title="Edit section: Further reading">edit</a>]</div> The following are intended for general audiences. <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAczel2001" class="citation book cs1"><a href="/wiki/Amir_Aczel" title="Amir Aczel">Aczel, Amir D.</a> (2001). Entanglement: The greatest mystery in physics. New York: Four Walls Eight Windows.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Entanglement%3A+The+greatest+mystery+in+physics&rft.place=New+York&rft.pub=Four+Walls+Eight+Windows&rft.date=2001&rft.aulast=Aczel&rft.aufirst=Amir+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAfriatSelleri1999" class="citation book cs1">Afriat, A.; Selleri, F. (1999). The Einstein, Podolsky and Rosen Paradox. New York and London: Plenum Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Einstein%2C+Podolsky+and+Rosen+Paradox&rft.place=New+York+and+London&rft.pub=Plenum+Press&rft.date=1999&rft.aulast=Afriat&rft.aufirst=A.&rft.au=Selleri%2C+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaggott1992" class="citation book cs1"><a href="/wiki/Jim_Baggott" title="Jim Baggott">Baggott, J.</a> (1992). The Meaning of Quantum Theory. Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Meaning+of+Quantum+Theory&rft.pub=Oxford+University+Press&rft.date=1992&rft.aulast=Baggott&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGilder2008" class="citation book cs1">Gilder, Louisa (2008). The Age of Entanglement: When Quantum Physics Was Reborn. New York: Alfred A. Knopf.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Age+of+Entanglement%3A+When+Quantum+Physics+Was+Reborn&rft.place=New+York&rft.pub=Alfred+A.+Knopf&rft.date=2008&rft.aulast=Gilder&rft.aufirst=Louisa&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreene2004" class="citation book cs1"><a href="/wiki/Brian_Greene" title="Brian Greene">Greene, Brian</a> (2004). <a href="/wiki/The_Fabric_of_the_Cosmos" title="The Fabric of the Cosmos">The Fabric of the Cosmos</a>. Vintage. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-375-72720-5" title="Special:BookSources/0-375-72720-5"><bdi>0-375-72720-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Fabric+of+the+Cosmos&rft.pub=Vintage&rft.date=2004&rft.isbn=0-375-72720-5&rft.aulast=Greene&rft.aufirst=Brian&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMermin1981" class="citation journal cs1"><a href="/wiki/N._David_Mermin" title="N. David Mermin">Mermin, N. David</a> (1981). "Bringing home the atomic world: Quantum mysteries for anybody". American Journal of Physics. 49 (10): 940–943. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1981AmJPh..49..940M">1981AmJPh..49..940M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.12594">10.1119/1.12594</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122724592">122724592</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Bringing+home+the+atomic+world%3A+Quantum+mysteries+for+anybody&rft.volume=49&rft.issue=10&rft.pages=940-943&rft.date=1981&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122724592%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1119%2F1.12594&rft_id=info%3Abibcode%2F1981AmJPh..49..940M&rft.aulast=Mermin&rft.aufirst=N.+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMermin1985" class="citation journal cs1">Mermin, N. David (April 1985). "Is the moon there when nobody looks? Reality and the quantum theory". Physics Today. 38 (4): 38–47. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1985PhT....38d..38M">1985PhT....38d..38M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.880968">10.1063/1.880968</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=Is+the+moon+there+when+nobody+looks%3F+Reality+and+the+quantum+theory&rft.volume=38&rft.issue=4&rft.pages=38-47&rft.date=1985-04&rft_id=info%3Adoi%2F10.1063%2F1.880968&rft_id=info%3Abibcode%2F1985PhT....38d..38M&rft.aulast=Mermin&rft.aufirst=N.+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"></li></ul> The following are more technically oriented. <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 60em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAspect1981" class="citation journal cs1"><a href="/wiki/Alain_Aspect" title="Alain Aspect">Aspect, A.</a>; et al. (1981). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevlett.47.460">"Experimental Tests of Realistic Local Theories via Bell's Theorem"</a>. Phys. Rev. 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MIT Press. pp. 60–65. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-01506-6" title="Special:BookSources/978-0-262-01506-6"><bdi>978-0-262-01506-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=4.4+EPR+Paradox+and+Bell%27s+Theorem&rft.btitle=Quantum+Computing%3A+A+Gentle+Introduction&rft.pages=60-65&rft.pub=MIT+Press&rft.date=2011-03-04&rft.isbn=978-0-262-01506-6&rft.aulast=Rieffel&rft.aufirst=Eleanor+G.&rft.au=Polak%2C+Wolfgang+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSulcs2003" class="citation journal cs1">Sulcs, S. (2003). "The Nature of Light and Twentieth Century Experimental Physics". Foundations of Science. 8 (4): 365–391. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1026323203487">10.1023/A:1026323203487</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118769677">118769677</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Science&rft.atitle=The+Nature+of+Light+and+Twentieth+Century+Experimental+Physics&rft.volume=8&rft.issue=4&rft.pages=365-391&rft.date=2003&rft_id=info%3Adoi%2F10.1023%2FA%3A1026323203487&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118769677%23id-name%3DS2CID&rft.aulast=Sulcs&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_Fraassen1991" class="citation book cs1"><a href="/wiki/Bas_van_Fraassen" title="Bas van Fraassen">van Fraassen, B. C.</a> (1991). Quantum Mechanics: An Empiricist View. Clarendon Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-198-24861-3" title="Special:BookSources/978-0-198-24861-3"><bdi>978-0-198-24861-3</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/22906474">22906474</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics%3A+An+Empiricist+View&rft.pub=Clarendon+Press&rft.date=1991&rft_id=info%3Aoclcnum%2F22906474&rft.isbn=978-0-198-24861-3&rft.aulast=van+Fraassen&rft.aufirst=B.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhartonArgaman2020" class="citation journal cs1">Wharton, K. B.; Argaman, N. (2020-05-18). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/RevModPhys.92.021002">"Colloquium : Bell's theorem and locally mediated reformulations of quantum mechanics"</a>. 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Oppenheimer Lecture</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="http://www.iep.utm.edu/epr">"Bell's theorem"</a>. <a href="/wiki/Internet_Encyclopedia_of_Philosophy" title="Internet Encyclopedia of Philosophy">Internet Encyclopedia of Philosophy</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bell%27s+theorem&rft.btitle=Internet+Encyclopedia+of+Philosophy&rft_id=http%3A%2F%2Fwww.iep.utm.edu%2Fepr&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Bell_inequalities">"Bell inequalities"</a>. <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>. <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>. 2001 [1994].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bell+inequalities&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DBell_inequalities&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABell%27s+theorem" class="Z3988"></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline 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title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics_topics" title="Template talk:Quantum mechanics topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics_topics" title="Special:EditPage/Template:Quantum mechanics topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_mechanics" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a> <ul><li><a href="/wiki/Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Born_rule" title="Born rule">Born rule</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a> <ul><li><a href="/wiki/Ground_state" title="Ground state">Ground state</a></li> <li><a href="/wiki/Excited_state" title="Excited state">Excited state</a></li> <li><a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a href="/wiki/Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a> <ul><li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat_in_popular_culture" title="Schrödinger's cat in popular culture">in popular culture</a></li></ul></li> <li><a href="/wiki/Wigner%27s_friend" title="Wigner's friend">Wigner's friend</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Quantum_mechanics" title="Category:Quantum mechanics">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Quantum_information_science" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_information" title="Template:Quantum information"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_information" title="Template talk:Quantum information"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_information" title="Special:EditPage/Template:Quantum information"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_information_science" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/DiVincenzo%27s_criteria" title="DiVincenzo's criteria">DiVincenzo's criteria</a></li> <li><a href="/wiki/Noisy_intermediate-scale_quantum_era" title="Noisy intermediate-scale quantum era">NISQ era</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">timeline</a></li></ul></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulation</a></li> <li><a href="/wiki/Qubit" title="Qubit">Qubit</a> <ul><li><a href="/wiki/Physical_and_logical_qubits" title="Physical and logical qubits">physical vs. logical</a></li></ul></li> <li><a href="/wiki/List_of_quantum_processors" title="List of quantum processors">Quantum processors</a> <ul><li><a href="/wiki/Cloud-based_quantum_computing" title="Cloud-based quantum computing">cloud-based</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Bell's</a></li> <li><a href="/wiki/Eastin%E2%80%93Knill_theorem" title="Eastin–Knill theorem">Eastin–Knill</a></li> <li><a href="/wiki/Gleason%27s_theorem" title="Gleason's theorem">Gleason's</a></li> <li><a href="/wiki/Gottesman%E2%80%93Knill_theorem" title="Gottesman–Knill theorem">Gottesman–Knill</a></li> <li><a href="/wiki/Holevo%27s_theorem" title="Holevo's theorem">Holevo's</a></li> <li><a href="/wiki/No-broadcasting_theorem" title="No-broadcasting theorem">No-broadcasting</a></li> <li><a href="/wiki/No-cloning_theorem" title="No-cloning theorem">No-cloning</a></li> <li><a href="/wiki/No-communication_theorem" title="No-communication theorem">No-communication</a></li> <li><a href="/wiki/No-deleting_theorem" title="No-deleting theorem">No-deleting</a></li> <li><a href="/wiki/No-hiding_theorem" title="No-hiding theorem">No-hiding</a></li> <li><a href="/wiki/No-teleportation_theorem" title="No-teleportation theorem">No-teleportation</a></li> <li><a href="/wiki/PBR_theorem" class="mw-redirect" title="PBR theorem">PBR</a></li> <li><a href="/wiki/Quantum_speed_limit_theorems" class="mw-redirect" title="Quantum speed limit theorems">Quantum speed limit</a></li> <li><a href="/wiki/Threshold_theorem" title="Threshold theorem">Threshold</a></li> <li><a href="/wiki/Solovay%E2%80%93Kitaev_theorem" title="Solovay–Kitaev theorem">Solovay–Kitaev</a></li> <li><a href="/wiki/Schr%C3%B6dinger%E2%80%93HJW_theorem" title="Schrödinger–HJW theorem">Purification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum communication</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_capacity" title="Classical capacity">Classical capacity</a> <ul><li><a href="/wiki/Entanglement-assisted_classical_capacity" title="Entanglement-assisted classical capacity">entanglement-assisted</a></li> <li><a href="/wiki/Quantum_capacity" title="Quantum capacity">quantum capacity</a></li></ul></li> <li><a href="/wiki/Entanglement_distillation" title="Entanglement distillation">Entanglement distillation</a></li> <li><a href="/wiki/Monogamy_of_entanglement" title="Monogamy of entanglement">Monogamy of entanglement</a></li> <li><a href="/wiki/LOCC" title="LOCC">LOCC</a></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a> <ul><li><a href="/wiki/Quantum_network" title="Quantum network">quantum network</a></li></ul></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a> <ul><li><a href="/wiki/Quantum_gate_teleportation" title="Quantum gate teleportation">quantum gate teleportation</a></li></ul></li> <li><a href="/wiki/Superdense_coding" title="Superdense coding">Superdense coding</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Quantum_cryptography" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">Post-quantum cryptography</a></li> <li><a href="/wiki/Quantum_coin_flipping" title="Quantum coin flipping">Quantum coin flipping</a></li> <li><a href="/wiki/Quantum_money" title="Quantum money">Quantum money</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a> <ul><li><a href="/wiki/BB84" title="BB84">BB84</a></li> <li><a href="/wiki/SARG04" title="SARG04">SARG04</a></li> <li><a href="/wiki/List_of_quantum_key_distribution_protocols" title="List of quantum key distribution protocols">other protocols</a></li></ul></li> <li><a href="/wiki/Quantum_secret_sharing" title="Quantum secret sharing">Quantum secret sharing</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amplitude_amplification" title="Amplitude amplification">Amplitude amplification</a></li> <li><a href="/wiki/Bernstein%E2%80%93Vazirani_algorithm" title="Bernstein–Vazirani algorithm">Bernstein–Vazirani</a></li> <li><a href="/wiki/BHT_algorithm" title="BHT algorithm">BHT</a></li> <li><a href="/wiki/Boson_sampling" title="Boson sampling">Boson sampling</a></li> <li><a href="/wiki/Deutsch%E2%80%93Jozsa_algorithm" title="Deutsch–Jozsa algorithm">Deutsch–Jozsa</a></li> <li><a href="/wiki/Grover%27s_algorithm" title="Grover's algorithm">Grover's</a></li> <li><a href="/wiki/HHL_algorithm" title="HHL algorithm">HHL</a></li> <li><a href="/wiki/Hidden_subgroup_problem" title="Hidden subgroup problem">Hidden subgroup</a></li> <li><a href="/wiki/Quantum_annealing" title="Quantum annealing">Quantum annealing</a></li> <li><a href="/wiki/Quantum_counting_algorithm" title="Quantum counting algorithm">Quantum counting</a></li> <li><a href="/wiki/Quantum_Fourier_transform" title="Quantum Fourier transform">Quantum Fourier transform</a></li> <li><a href="/wiki/Quantum_optimization_algorithms" title="Quantum optimization algorithms">Quantum optimization</a></li> <li><a href="/wiki/Quantum_phase_estimation_algorithm" title="Quantum phase estimation algorithm">Quantum phase estimation</a></li> <li><a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's</a></li> <li><a href="/wiki/Simon%27s_problem" title="Simon's problem">Simon's</a></li> <li><a href="/wiki/Variational_quantum_eigensolver" title="Variational quantum eigensolver">VQE</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/BQP" title="BQP">BQP</a></li> <li><a href="/wiki/Exact_quantum_polynomial_time" title="Exact quantum polynomial time">EQP</a></li> <li><a href="/wiki/QIP_(complexity)" title="QIP (complexity)">QIP</a></li> <li><a href="/wiki/QMA" title="QMA">QMA</a></li> <li><a href="/wiki/PostBQP" title="PostBQP">PostBQP</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum processor benchmarks</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_supremacy" title="Quantum supremacy">Quantum supremacy</a></li> <li><a href="/wiki/Quantum_volume" title="Quantum volume">Quantum volume</a></li> <li><a href="/wiki/Randomized_benchmarking" title="Randomized benchmarking">Randomized benchmarking</a> <ul><li><a href="/wiki/Cross-entropy_benchmarking" title="Cross-entropy benchmarking">XEB</a></li></ul></li> <li><a href="/wiki/Relaxation_(NMR)" title="Relaxation (NMR)">Relaxation times</a> <ul><li><a href="/wiki/Spin%E2%80%93lattice_relaxation" title="Spin–lattice relaxation">T1</a></li> <li><a href="/wiki/Spin%E2%80%93spin_relaxation" title="Spin–spin relaxation">T2</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum <a href="/wiki/Model_of_computation" title="Model of computation">computing models</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adiabatic_quantum_computation" title="Adiabatic quantum computation">Adiabatic quantum computation</a></li> <li><a href="/wiki/Continuous-variable_quantum_information" title="Continuous-variable quantum information">Continuous-variable quantum information</a></li> <li><a href="/wiki/One-way_quantum_computer" title="One-way quantum computer">One-way quantum computer</a> <ul><li><a href="/wiki/Cluster_state" title="Cluster state">cluster state</a></li></ul></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a> <ul><li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">quantum logic gate</a></li></ul></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a> <ul><li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">quantum neural network</a></li></ul></li> <li><a href="/wiki/Quantum_Turing_machine" title="Quantum Turing machine">Quantum Turing machine</a></li> <li><a href="/wiki/Topological_quantum_computer" title="Topological quantum computer">Topological quantum computer</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Codes <ul><li><a href="/wiki/CSS_code" title="CSS code">CSS</a></li> <li><a href="/wiki/Quantum_convolutional_code" title="Quantum convolutional code">quantum convolutional</a></li> <li><a href="/wiki/Stabilizer_code" title="Stabilizer code">stabilizer</a></li> <li><a href="/wiki/Shor_code" class="mw-redirect" title="Shor code">Shor</a></li> <li><a href="/wiki/Bacon%E2%80%93Shor_code" title="Bacon–Shor code">Bacon–Shor</a></li> <li><a href="/wiki/Steane_code" title="Steane code">Steane</a></li> <li><a href="/wiki/Toric_code" title="Toric code">Toric</a></li> <li><a href="/wiki/Gnu_code" title="Gnu code">gnu</a></li></ul></li> <li><a href="/wiki/Entanglement-assisted_stabilizer_formalism" title="Entanglement-assisted stabilizer formalism">Entanglement-assisted</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Physical implementations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cavity_quantum_electrodynamics" title="Cavity quantum electrodynamics">Cavity QED</a></li> <li><a href="/wiki/Circuit_quantum_electrodynamics" title="Circuit quantum electrodynamics">Circuit QED</a></li> <li><a href="/wiki/Linear_optical_quantum_computing" title="Linear optical quantum computing">Linear optical QC</a></li> <li><a href="/wiki/KLM_protocol" title="KLM protocol">KLM protocol</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Ultracold_atom" title="Ultracold atom">Ultracold atoms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Neutral_atom_quantum_computer" title="Neutral atom quantum computer">Neutral atom QC</a></li> <li><a href="/wiki/Trapped-ion_quantum_computer" title="Trapped-ion quantum computer">Trapped-ion QC</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Spin_(physics)" title="Spin (physics)">Spin</a>-based</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kane_quantum_computer" title="Kane quantum computer">Kane QC</a></li> <li><a href="/wiki/Spin_qubit_quantum_computer" title="Spin qubit quantum computer">Spin qubit QC</a></li> <li><a href="/wiki/Nitrogen-vacancy_center" title="Nitrogen-vacancy center">NV center</a></li> <li><a href="/wiki/Nuclear_magnetic_resonance_quantum_computer" title="Nuclear magnetic resonance quantum computer">NMR QC</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Superconducting_quantum_computing" title="Superconducting quantum computing">Superconducting</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Charge_qubit" title="Charge qubit">Charge qubit</a></li> <li><a href="/wiki/Flux_qubit" title="Flux qubit">Flux qubit</a></li> <li><a href="/wiki/Phase_qubit" title="Phase qubit">Phase qubit</a></li> <li><a href="/wiki/Transmon" title="Transmon">Transmon</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/OpenQASM" title="OpenQASM">OpenQASM</a>–<a href="/wiki/Qiskit" title="Qiskit">Qiskit</a>–<a href="/wiki/IBM_Quantum_Experience" class="mw-redirect" title="IBM Quantum Experience">IBM QX</a></li> <li><a href="/wiki/Quil_(instruction_set_architecture)" title="Quil (instruction set architecture)">Quil</a>–<a href="/wiki/Rigetti_Computing" title="Rigetti Computing">Forest/Rigetti QCS</a></li> <li><a href="/wiki/Cirq" title="Cirq">Cirq</a></li> <li><a href="/wiki/Q_Sharp" title="Q Sharp">Q#</a></li> <li><a href="/wiki/Libquantum" title="Libquantum">libquantum</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">many others...</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Quantum_information_science" title="Category:Quantum information science">Quantum information science</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/16px-Symbol_template_class_pink.svg.png" 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