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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> Bell's inequality </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/15092/#Item_10" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" 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id="measure_theory">Measure theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/measurable+space">measurable space</a>, <a class="existingWikiWord" href="/nlab/show/measurable+locale">measurable locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measure">measure</a>, <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+measure+theory">geometric measure theory</a></p> </li> </ul> <h2 id="probability_theory">Probability theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+distribution">probability distribution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state">state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/states+in+AQFT+and+operator+algebra">in AQFT and operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entropy">entropy</a>, <a class="existingWikiWord" href="/nlab/show/relative+entropy">relative entropy</a></p> </li> </ul> <h2 id="information_geometry">Information geometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/information+geometry">information geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/information+metric">information metric</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wasserstein+metric">Wasserstein metric</a></p> </li> </ul> <h2 id="thermodynamics">Thermodynamics</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second+law+of+thermodynamics">second law of thermodynamics</a>, <a class="existingWikiWord" href="/nlab/show/generalized+second+law+of+theormodynamics">generalized second law</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ergodic+theory">ergodic theory</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Finetti%27s+theorem">de Finetti's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/law+of+large+numbers">law of large numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+extension+theorem">Kolmogorov extension theorem</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/machine+learning">machine learning</a>, <a class="existingWikiWord" href="/nlab/show/neural+networks">neural networks</a></li> </ul> </div></div> <h4 id="quantum_systems">Quantum systems</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>, <a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">dependent</a> <a class="existingWikiWord" href="/nlab/show/linear+type+theory">linear type theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a> in <a class="existingWikiWord" href="/nlab/show/quantum+information+theory+via+dagger-compact+categories">†-compact categories</a></p> <p><a class="existingWikiWord" href="/nlab/show/tensor+networks">tensor networks</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a></p> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a></p> </li> </ul> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+systems">quantum systems</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/parameterized+quantum+systems">parameterized</a>, <a class="existingWikiWord" href="/nlab/show/open+quantum+system">open</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+state+collapse">quantum state collapse</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+decoherence">quantum decoherence</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">quantum adiabatic theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/Berry+phases">Berry phases</a></p> <p><a class="existingWikiWord" href="/nlab/show/Dyson+formula">Dyson formula</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+many-body+physics">quantum many-body physics</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/functorial+quantum+field+theory">functorial quantum field theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-</a>)<a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+material">quantum material</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/topological+phases+of+matter">topological</a>) <a class="existingWikiWord" href="/nlab/show/phases+of+matter">phases of matter</a></p> </li> </ul> <p><br /></p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a> – <a class="existingWikiWord" href="/nlab/show/observables">observables</a> and <a class="existingWikiWord" href="/nlab/show/states">states</a></strong></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/states">states</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/quasi-state">quasi-state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a></p> <p><a class="existingWikiWord" href="/nlab/show/dimer">dimer</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">quantum state preparation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a>, <a class="existingWikiWord" href="/nlab/show/quantum+fluctuation">quantum fluctuation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+superposition">quantum superposition</a>, <a class="existingWikiWord" href="/nlab/show/quantum+interference">quantum interference</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a></p> <p><a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a></p> <p><a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superselection+sector">superselection sector</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> <p><a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coherent+quantum+state">coherent quantum state</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>, <a class="existingWikiWord" href="/nlab/show/excited+state">excited state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>, <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+amplitude">vacuum amplitude</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+fluctuation">vacuum fluctuation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+polarization">vacuum polarization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/thermal+vacuum">thermal vacuum</a>, <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/false+vacuum">false vacuum</a>, <a class="existingWikiWord" href="/nlab/show/tachyon">tachyon</a>, <a class="existingWikiWord" href="/nlab/show/Coleman-De+Luccia+instanton">Coleman-De Luccia instanton</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theta+vacuum">theta vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/non-geometric+string+theory+vacuum">non-geometric string theory vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entangled+state">entangled state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix+product+state">matrix product state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tree+tensor+network+state">tree tensor network state</a></p> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/observables">observables</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>, <a class="existingWikiWord" href="/nlab/show/beable">beable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator (in geometric quantization)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+effect">quantum effect</a>, <a class="existingWikiWord" href="/nlab/show/effect+algebra">effect algebra</a></p> </li> <li> <p>in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+observable">linear observable</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+observable">regular observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>, <a class="existingWikiWord" href="/nlab/show/retarded+product">retarded product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra+of+observables">interacting field algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorems">theorems</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nuiten%27s+lemma">Nuiten's lemma</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner%27s+theorem">Wigner's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> </ul> </li> </ul> </div> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information+via+dagger-compact+categories">quantum information via dagger-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+channel">quantum channel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+teleportation">quantum teleportation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+entanglement+entropy">topological entanglement entropy</a></p> </li> </ul> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adiabatic+quantum+computation">adiabatic quantum computation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measurement-based+quantum+computation">measurement-based quantum computation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+gate">quantum gate</a>, <a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+programming+language">quantum programming language</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+error+correction">quantum error correction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/HaPPY+code">HaPPY code</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Majorana+dimer+code">Majorana dimer code</a></p> </li> </ul> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/qbit">qbit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/spin+resonance+qbit">spin resonance qbit</a></li> </ul> <p>quantum algorithms:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grover%27s+algorithm">Grover's algorithm</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Shor%27s+algorithm">Shor's algorithm</a></p> </li> </ul> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+sensing">quantum sensing</a></strong></p> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+communication">quantum communication</a></strong></p> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#Statement'>Statement</a></li> <ul> <li><a href='#OriginalFormulation'>Original formulation</a></li> <ul> <li><a href='#quantum_mechanical_violations'>Quantum mechanical violations</a></li> <li><a href='#locality'>Locality</a></li> <li><a href='#wigners_derivation'>Wigner’s derivation</a></li> <li><a href='#violations_and_geometry'>Violations and geometry</a></li> </ul> <li><a href='#CompactReformulation'>Compact reformulation</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#in_quantum_field_theory'>In quantum field theory</a></li> <li><a href='#ReferencesProbabilisticOpposition'>Probabilistic opposition</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a>/<a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a>, What came to be called <em>Bell’s inequality</em> (<a href="#Bell64">Bell 1964</a>) is an <a class="existingWikiWord" href="/nlab/show/inequality">inequality</a> satisfied by the three pairwise <a class="existingWikiWord" href="/nlab/show/correlation+functions">correlation functions</a> between three <a class="existingWikiWord" href="/nlab/show/random+variables">random variables</a> defined on one and the same classical <a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a>. As such, it is an elementary statement about classical probability theory which as been argued (<a href="#Pitowsky89a">Pitowsky 1989a</a>) to have been known already to <a class="existingWikiWord" href="/nlab/show/Boole+--+The+Laws+of+Thought">Boole (1854)</a>.</p> <p>The point of the argument by <a href="#Bell64">Bell 1964</a> was to highlight that when taking these three random variables to be the results of <a class="existingWikiWord" href="/nlab/show/quantum+measurements">quantum measurements</a> of the <a class="existingWikiWord" href="/nlab/show/spin">spin</a> of an <a class="existingWikiWord" href="/nlab/show/electron">electron</a> along three pairwise non-orthogonal axes (as in the <a class="existingWikiWord" href="/nlab/show/Stern-Gerlach+experiment">Stern-Gerlach experiment</a>) then <a class="existingWikiWord" href="/nlab/show/quantum+theory">quantum theory</a> predicts that this inequality is <em>violated</em> – implying that there is no single classical <a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a> (called a <em><a class="existingWikiWord" href="/nlab/show/hidden+variable+theory">hidden variable</a></em> in the context of <a class="existingWikiWord" href="/nlab/show/interpretations+of+quantum+mechanics">interpretations of quantum mechanics</a>) on which these three <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a>-results are jointly <a class="existingWikiWord" href="/nlab/show/random+variables">random variables</a>.</p> <p>A number of <a class="existingWikiWord" href="/nlab/show/experiments">experiments</a> have sought to check Bell’s inequalities in <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> (“<a class="existingWikiWord" href="/nlab/show/Bell+tests">Bell tests</a>”) and all claim to have verified that it is indeed violated in nature (see <a href="#Aspect15">Aspect 2015</a>), as predicted by <a class="existingWikiWord" href="/nlab/show/quantum+theory">quantum theory</a>.</p> <p>Bell’s inequality has been and is receiving an enormous amount of attention, first in discussions of <a class="existingWikiWord" href="/nlab/show/interpretations+of+quantum+mechanics">interpretations of quantum mechanics</a>, but more recently and more concretely also in the context of <a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a>.</p> <h2 id="Statement">Statement</h2> <h3 id="OriginalFormulation">Original formulation</h3> <blockquote> <p>The following is fairly verbatim recap of the original argument in <a href="#Bell64">Bell 1964</a>. For a streamlined re-statement see further <a href="#CompactReformulation">below</a>.</p> </blockquote> <p>Let us denote the result <em>A</em> of a measurement that is determined by a unit vector, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math>, and some parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">A(\vec{a},\lambda)=\pm 1</annotation></semantics></math> where we further suppose that the outcome of the measurement is either +1 or -1. Likewise, we may do the same for the result <em>B</em> of a second measurement, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(\vec{b},\lambda)</annotation></semantics></math>. We further make the vital assumption that the result <em>B</em> does not depend on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math> and likewise <em>A</em> does not depend on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{b}</annotation></semantics></math>.</p> <p>Before proceeding, we should note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> here plays the role of a “hidden” parameter or variable. We say it is “hidden” because its precise nature is not known. However, it is still a very real parameter with a probability distribution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(\lambda)</annotation></semantics></math>. The expectation value of the product of the two measurements is</p> <div class="maruku-equation" id="eq:AnExpectationValue"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mo>∫</mo><mi>d</mi><mi>λ</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>A</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>B</mi><mo stretchy="false">(</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> P(\vec{a},\vec{b})=\int d\lambda\rho(\lambda)A(\vec{a},\lambda)B(\vec{b},\lambda). </annotation></semantics></math></div> <p>Because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> is a normalized probability distribution,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∫</mo><mi>d</mi><mi>λ</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex"> \int d\lambda \rho(\lambda) = 1 </annotation></semantics></math></div> <p>and because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">A(\vec{a},\lambda)=\pm 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">B(\vec{b},\lambda)=\pm 1</annotation></semantics></math>, P cannot be less than -1. It can be equal to -1 at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>=</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{a}=\vec{b}</annotation></semantics></math> only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>B</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">A(\vec{a},\lambda)=\pm 1 = -B(\vec{a},\lambda)=\pm 1</annotation></semantics></math> except at a set of points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> of zero probability. Thus we can write <a class="maruku-eqref" href="#eq:AnExpectationValue">(1)</a> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>∫</mo><mi>d</mi><mi>λ</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>A</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>A</mi><mo stretchy="false">(</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> P(\vec{a},\vec{b})=-\int d\lambda\rho(\lambda)A(\vec{a},\lambda)A(\vec{b},\lambda). </annotation></semantics></math></div> <p>If we introduce a third unit vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{c}</annotation></semantics></math> we can find the difference between the correlation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math> to the two other unit vectors,</p> <div class="maruku-equation" id="eq:DifferenceOfCorrelations"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>∫</mo><mi>d</mi><mi>λ</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>A</mi><mo stretchy="false">(</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>−</mo><mi>A</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>A</mi><mo stretchy="false">(</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> P(\vec{a},\vec{b})-P(\vec{a},\vec{c})=-\int d\lambda\rho(\lambda)[A(\vec{a},\lambda)A(\vec{b},\lambda)-A(\vec{a},\lambda)A(\vec{c},\lambda)]. </annotation></semantics></math></div> <p>Rearranging this we may write <a class="maruku-eqref" href="#eq:DifferenceOfCorrelations">(2)</a> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>∫</mo><mi>d</mi><mi>λ</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>A</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>A</mi><mo stretchy="false">(</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>A</mi><mo stretchy="false">(</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>A</mi><mo stretchy="false">(</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> P(\vec{a},\vec{b})-P(\vec{a},\vec{c})=-\int d\lambda\rho(\lambda)A(\vec{a},\lambda)A(\vec{b},\lambda)[A(\vec{b},\lambda)A(\vec{c},\lambda)-1]. </annotation></semantics></math></div> <p>Given the limitations we have placed on the value of <em>A</em>, we may write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>≤</mo><mo>∫</mo><mi>d</mi><mi>λ</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><mi>A</mi><mo stretchy="false">(</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>A</mi><mo stretchy="false">(</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> |P(\vec{a},\vec{b})-P(\vec{a},\vec{c})| \le \int d\lambda\rho(\lambda)[1-A(\vec{b},\lambda)A(\vec{c},\lambda)]. </annotation></semantics></math></div> <p>But the second term on the right is simply <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(\vec{b},\vec{c})</annotation></semantics></math> and thus</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo>≥</mo><mo stretchy="false">|</mo><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex"> 1 + P(\vec{b},\vec{c}) \ge |P(\vec{a},\vec{b})-P(\vec{a},\vec{c})| </annotation></semantics></math></div> <p>which is the original form of Bell’s inequality. Note that this may be written in terms of correlation coefficients,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>+</mo><mi>C</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≥</mo><mo stretchy="false">|</mo><mi>C</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mi>C</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex"> 1 + C(b,c) \ge |C(a,b)-C(a,c)| </annotation></semantics></math></div> <p>where <em>a</em>, <em>b</em>, and <em>c</em> are now settings on the measurement apparatus.</p> <h4 id="quantum_mechanical_violations">Quantum mechanical violations</h4> <p>The original derivation of Bell’s inequalities involved the use of a <a class="existingWikiWord" href="/nlab/show/Stern-Gerlach+device">Stern-Gerlach device</a> that measures spin along an axis. Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\sigma_{1}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\sigma_{2}</annotation></semantics></math> are spins. The result, <em>A</em>, of measuring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>1</mn></msub><mo>⋅</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\sigma_{1}\cdot\vec{a}</annotation></semantics></math> is then interpreted as being entirely determined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>. Likewise for <em>B</em> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>2</mn></msub><mo>⋅</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\sigma_{2}\cdot\vec{b}</annotation></semantics></math>. It is also important to remember that the result <em>B</em> does not depend on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math> and likewise <em>A</em> does not depend on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{b}</annotation></semantics></math>.</p> <p>For a singlet state (that is a state with total spin of zero), the quantum mechanical expectation value of measurements along two different axes (see the Wigner derivation below for a more intuitive explanation of the physical nature of this) is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>σ</mi> <mn>1</mn></msub><mo>⋅</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><msub><mi>σ</mi> <mn>2</mn></msub><mo>⋅</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">⟩</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>⋅</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>.</mo></mrow><annotation encoding="application/x-tex"> \langle\sigma_{1}\cdot\vec{a},\sigma_{2}\cdot\vec{b}\rangle = - \vec{a}\cdot\vec{b}. </annotation></semantics></math></div> <p>In theory this ought to equal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(\vec{a},\vec{b})</annotation></semantics></math> but in practice it does not. It is important to remember that we are using <em>classical</em> reasoning throughout our derivations of the various forms of Bell’s inequalities.</p> <p>The setup envisioned here consists of pairs of spin-1/2 particles produced in singlet states that then each pass through separate Stern-Gerlach (SG) devices. Since they are in singlet states, if we measured the first particle of a pair to be aligned with a given axis, say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math>, then the second should be measured to be anti-aligned with that same axis, giving a total spin of zero.</p> <p>In practice we are dealing with <em>beams</em> of particles and thus we can never be absolutely certain that correlated pairs are measured simultaneously and so we ultimately are making statistical predictions. Nevertheless, in a given sample consisting of a large-enough number of randomly distributed spin-1/2 particles, we can be certain that, for example, a definite number are aligned with an axis <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math> while a definite number are aligned with an axis <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{b}</annotation></semantics></math>.</p> <p>Now take an individual particle and suppose that, for this particle, if we measured <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>⋅</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\sigma\cdot\vec{a}</annotation></semantics></math> we would obtain a +1 with certainty (meaning it is aligned with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math>) but if we instead chose to measure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>⋅</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\sigma\cdot\vec{b}</annotation></semantics></math> we would obtain a -1 with certainty (meaning it is anti-aligned with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{b}</annotation></semantics></math>). Notationally we refer to such a particle as belonging to type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\vec{a}+,\vec{b}-)</annotation></semantics></math>. Clearly for a given pair of particles in a singlet state, if particle 1 is of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\vec{a}+,\vec{b}-)</annotation></semantics></math>, then particle 2 must be of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\vec{a}-,\vec{b}+)</annotation></semantics></math>.</p> <h4 id="locality">Locality</h4> <p>For beams of correlated particles measuring along only two axes, we should expect to get a roughly evenly balanced distribution of types as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mtext> Particle 1 </mtext></mtd> <mtd></mtd> <mtd><mtext> Particle 2 </mtext></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>↔</mo></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>↔</mo></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>↔</mo></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>↔</mo></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \text{ Particle 1 } & & \text{ Particle 2 } \\ (\vec{a}+,\vec{b}-) & \leftrightarrow & (\vec{a}-,\vec{b}+) \\ (\vec{a}+,\vec{b}+) & \leftrightarrow & (\vec{a}-,\vec{b}-) \\ (\vec{a}-,\vec{b}-) & \leftrightarrow & (\vec{a}+,\vec{b}+) \\ (\vec{a}-,\vec{b}+) & \leftrightarrow & (\vec{a}+,\vec{b}-) } </annotation></semantics></math></div> <p>There is a very important assumption implied here. Suppose a particular pair belongs to the first grouping, that is if an observer <em>A</em> decides to measure the spin along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math> for particle 1, he or she <em>necessarily</em> obtains a plus sign (corresponding to it being aligned with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math>) <em>regardless</em> of any measurement observer <em>B</em> may make on particle 2. This is the principle of locality: <em>A</em>’s result is predetermined independently of <em>B</em>’s choice of what to measure.</p> <h4 id="wigners_derivation">Wigner’s derivation</h4> <p>Now suppose we introduce a third axis, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{c}</annotation></semantics></math>, so that we can have, for example, particles of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\vec{a}+,\vec{b}+,\vec{c}-)</annotation></semantics></math> corresponding to being aligned if measured on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{b}</annotation></semantics></math> and anti-aligned on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{c}</annotation></semantics></math>. Further let us “count” the pairs that fall into the various groupings and label the populations as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mtext> Population </mtext></mtd> <mtd><mtext> Particle 1 </mtext></mtd> <mtd><mtext> Particle 2 </mtext></mtd></mtr> <mtr><mtd><msub><mi>N</mi> <mn>1</mn></msub></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>N</mi> <mn>2</mn></msub></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>N</mi> <mn>3</mn></msub></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>N</mi> <mn>4</mn></msub></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>N</mi> <mn>5</mn></msub></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>N</mi> <mn>6</mn></msub></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>N</mi> <mn>7</mn></msub></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>N</mi> <mn>8</mn></msub></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>,</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \text{ Population } & \text{ Particle 1 } & \text{ Particle 2 } \\ N_{1} & (\vec{a}+,\vec{b}+, \vec{c}+) & (\vec{a}-,\vec{b}-,\vec{c}-) \\ N_{2} & (\vec{a}+,\vec{b}+, \vec{c}-) & (\vec{a}-,\vec{b}-,\vec{c}+) \\ N_{3} & (\vec{a}+,\vec{b}-, \vec{c}+) & (\vec{a}-,\vec{b}+,\vec{c}-) \\ N_{4} & (\vec{a}+,\vec{b}-, \vec{c}-) & (\vec{a}-,\vec{b}+,\vec{c}+) \\ N_{5} & (\vec{a}-,\vec{b}+, \vec{c}+) & (\vec{a}+,\vec{b}-,\vec{c}-) \\ N_{6} & (\vec{a}-,\vec{b}+, \vec{c}-) & (\vec{a}+,\vec{b}-,\vec{c}+) \\ N_{7} & (\vec{a}-,\vec{b}-, \vec{c}+) & (\vec{a}+,\vec{b}+,\vec{c}-) \\ N_{8} & (\vec{a}-,\vec{b}-, \vec{c}-) & (\vec{a}+,\vec{b}+,\vec{c}+) } </annotation></semantics></math></div> <p>Let’s suppose that observer <em>A</em> finds particle 1 is aligned with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math>, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo></mrow><annotation encoding="application/x-tex">\vec{a}+</annotation></semantics></math>, and that observer <em>B</em> finds particle 2 is aligned with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{b}</annotation></semantics></math>, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo></mrow><annotation encoding="application/x-tex">\vec{b}+</annotation></semantics></math>. From the above table it is clear that the pair belong to either population 3 or 4. Note that because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">N_{i}</annotation></semantics></math> is positive semi-definite we must be able to construct relations like, for instance,</p> <div class="maruku-equation" id="eq:APositivityCondition"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>3</mn></msub><mo>+</mo><msub><mi>N</mi> <mn>4</mn></msub><mo>≤</mo><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>3</mn></msub><mo>+</mo><msub><mi>N</mi> <mn>7</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>4</mn></msub><mo>+</mo><msub><mi>N</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> N_{3} + N_{4} \le (N_{3} + N_{7}) + (N_{4} + N_{2}). </annotation></semantics></math></div> <p>Now let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>;</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(\vec{a}+;\vec{b}+)</annotation></semantics></math> be the probability that, in a random selection, <em>A</em> finds particle 1 to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo></mrow><annotation encoding="application/x-tex">\vec{a}+</annotation></semantics></math> and <em>B</em> finds particle 2 to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo></mrow><annotation encoding="application/x-tex">\vec{b}+</annotation></semantics></math>. In terms of populations, we have</p> <div class="maruku-equation" id="eq:Populations1"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>;</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>3</mn></msub><mo>+</mo><msub><mi>N</mi> <mn>4</mn></msub><mo stretchy="false">)</mo></mrow><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi> <mn>8</mn></munderover><msub><mi>N</mi> <mi>i</mi></msub></mrow></mfrac><mo>.</mo></mrow><annotation encoding="application/x-tex"> P(\vec{a}+;\vec{b}+) = \frac{(N_{3} + N_{4})}{\sum_{i}^{8}N_{i}}. </annotation></semantics></math></div> <p>Similarly we have</p> <div class="maruku-equation" id="eq:Populations2"><span class="maruku-eq-number">(5)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>;</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>N</mi> <mn>4</mn></msub><mo stretchy="false">)</mo></mrow><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi> <mn>8</mn></munderover><msub><mi>N</mi> <mi>i</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex"> P(\vec{a}+;\vec{c}+) = \frac{(N_{2} + N_{4})}{\sum_{i}^{8}N_{i}} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation" id="eq:Populations3"><span class="maruku-eq-number">(6)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>;</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>3</mn></msub><mo>+</mo><msub><mi>N</mi> <mn>7</mn></msub><mo stretchy="false">)</mo></mrow><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi> <mn>8</mn></munderover><msub><mi>N</mi> <mi>i</mi></msub></mrow></mfrac><mo>.</mo></mrow><annotation encoding="application/x-tex"> P(\vec{c}+;\vec{b}+) = \frac{(N_{3} + N_{7})}{\sum_{i}^{8}N_{i}}. </annotation></semantics></math></div> <p>The positivity condition <a class="maruku-eqref" href="#eq:APositivityCondition">(3)</a> then becomes</p> <div class="maruku-equation" id="eq:WignerFormOfBellInequality"><span class="maruku-eq-number">(7)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>;</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo><mo>≤</mo><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>;</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>;</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> P(\vec{a}+;\vec{b}+) \le P(\vec{a}+;\vec{c}+) + P(\vec{c}+;\vec{b}+). </annotation></semantics></math></div> <p>This is Wigner’s form of Bell’s inequality.</p> <h4 id="violations_and_geometry">Violations and geometry</h4> <p>As we mentioned before, we have used purely classical reasoning to derive the two forms of Bell’s inequality that we have thusfar encountered. Recall that the context within which the above were derived was the Stern-Gerlach experiment are we are measuring along axes of the magnetic field. As such, there are angles between these various axes. Thus the quantum mechanically-derived probabilities corresponding to <a class="maruku-eqref" href="#eq:Populations1">(4)</a>, <a class="maruku-eqref" href="#eq:Populations2">(5)</a>, and <a class="maruku-eqref" href="#eq:Populations3">(6)</a> are</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>;</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>sin</mi> <mn>2</mn></msup><mrow><mo>(</mo><mfrac><mrow><msub><mi>θ</mi> <mi>ab</mi></msub></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mo>,</mo></mrow><annotation encoding="application/x-tex"> P(\vec{a}+;\vec{b}+) = \frac{1}{2}sin^{2}\left(\frac{\theta_{ab}}{2}\right), </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mover><mi>a</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>;</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>sin</mi> <mn>2</mn></msup><mrow><mo>(</mo><mfrac><mrow><msub><mi>θ</mi> <mi>ac</mi></msub></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mo>,</mo></mrow><annotation encoding="application/x-tex"> P(\vec{a}+;\vec{c}+) = \frac{1}{2}sin^{2}\left(\frac{\theta_{ac}}{2}\right), </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo>;</mo><mover><mi>b</mi><mo stretchy="false">→</mo></mover><mo>+</mo><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>sin</mi> <mn>2</mn></msup><mrow><mo>(</mo><mfrac><mrow><msub><mi>θ</mi> <mi>cb</mi></msub></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mo>,</mo></mrow><annotation encoding="application/x-tex"> P(\vec{c}+;\vec{b}+) = \frac{1}{2}sin^{2}\left(\frac{\theta_{cb}}{2}\right), </annotation></semantics></math></div> <p>respectively. Bell’s inequality, <a class="maruku-eqref" href="#eq:WignerFormOfBellInequality">(7)</a>, then becomes</p> <div class="maruku-equation" id="eq:AnotherFormOfBellInequality"><span class="maruku-eq-number">(8)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>sin</mi> <mn>2</mn></msup><mrow><mo>(</mo><mfrac><mrow><msub><mi>θ</mi> <mi>ab</mi></msub></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mo>≤</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>sin</mi> <mn>2</mn></msup><mrow><mo>(</mo><mfrac><mrow><msub><mi>θ</mi> <mi>ac</mi></msub></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>sin</mi> <mn>2</mn></msup><mrow><mo>(</mo><mfrac><mrow><msub><mi>θ</mi> <mi>cb</mi></msub></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{1}{2}sin^{2}\left(\frac{\theta_{ab}}{2}\right) \le \frac{1}{2}sin^{2}\left(\frac{\theta_{ac}}{2}\right) + \frac{1}{2}sin^{2}\left(\frac{\theta_{cb}}{2}\right). </annotation></semantics></math></div> <p>From a geometric point of view, this inequality is not always possible. For example, suppose, for simplicity that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{b}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{c}</annotation></semantics></math> lie in a plane and suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{c}</annotation></semantics></math> bisects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{a}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>b</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{b}</annotation></semantics></math>, i.e.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>θ</mi> <mi>ab</mi></msub><mo>=</mo><mn>2</mn><mi>θ</mi></mtd> <mtd><mtext> and </mtext></mtd> <mtd><msub><mi>θ</mi> <mi>ac</mi></msub><mo>=</mo><msub><mi>θ</mi> <mi>cb</mi></msub><mo>=</mo><mi>θ</mi><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \theta_{ab} = 2\theta & \text{ and } & \theta_{ac}=\theta_{cb}=\theta. } </annotation></semantics></math></div> <p>Then <a class="maruku-eqref" href="#eq:AnotherFormOfBellInequality">(8)</a> is violated for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo><</mo><mi>θ</mi><mo><</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">0 \lt \theta \lt \frac{\pi}{2}</annotation></semantics></math>. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>=</mo><mfrac><mi>π</mi><mn>4</mn></mfrac></mrow><annotation encoding="application/x-tex">\theta = \frac{\pi}{4}</annotation></semantics></math>, <a class="maruku-eqref" href="#eq:AnotherFormOfBellInequality">(8)</a> would become <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0.500</mn><mo>≤</mo><mn>0.292</mn></mrow><annotation encoding="application/x-tex">0.500 \le 0.292</annotation></semantics></math> which is absurd!</p> <h3 id="CompactReformulation">Compact reformulation</h3> <p>A transparent and compact way to derive the actual <a class="existingWikiWord" href="/nlab/show/inequality">inequality</a> of <a href="#Bell64">Bell 1964</a> (adjusting the original argument only slightly for mathematical elegance) is reviewed in <a href="#Khrennikov08">Khrennikov 2008, §10.1</a>, which we broadly follow:</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p>Given</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Λ</mi><mo>,</mo><mi>d</mi><mi>ρ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Lambda, d\rho)</annotation></semantics></math> with</p> </li> <li> <p>three <a class="existingWikiWord" href="/nlab/show/random+variables">random variables</a> taking values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo>±</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\pm 1\}</annotation></semantics></math> (regarded inside the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>):</p> <div class="maruku-equation" id="eq:TheRandomVariables"><span class="maruku-eq-number">(9)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mo stretchy="false">{</mo><mo>±</mo><mn>1</mn><mo stretchy="false">}</mo><mo>↪</mo><mi>ℝ</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>i</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> S_i \;\colon\; X \longrightarrow \{\pm 1\} \hookrightarrow \mathbb{R} \,, \;\;\;\; i\,\in\, \{1,2,3\} </annotation></semantics></math></div></li> </ol> <p>then the <a class="existingWikiWord" href="/nlab/show/correlation+functions">correlation functions</a></p> <div class="maruku-equation" id="eq:TheCorrelations"><span class="maruku-eq-number">(10)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>S</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><msub><mi>S</mi> <mi>j</mi></msub><mo stretchy="false">⟩</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mo>∫</mo> <mi>Λ</mi></msub><mspace width="thickmathspace"></mspace><msub><mi>S</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>S</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi>d</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \langle S_{i} \, S_j\rangle \;\coloneqq\; \int_{\Lambda} \; S_i(\lambda) \, S_j(\lambda) \; d\rho(\lambda) </annotation></semantics></math></div> <p>satisfy this <a class="existingWikiWord" href="/nlab/show/inequality">inequality</a>:</p> <div class="maruku-equation" id="eq:TheInequality"><span class="maruku-eq-number">(11)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">|</mo><mo stretchy="false">⟨</mo><msub><mi>S</mi> <mn>1</mn></msub><msub><mi>S</mi> <mn>2</mn></msub><mo stretchy="false">⟩</mo><mo>−</mo><mo stretchy="false">⟨</mo><msub><mi>S</mi> <mn>3</mn></msub><msub><mi>S</mi> <mn>2</mn></msub><mo stretchy="false">⟩</mo><mo maxsize="1.2em" minsize="1.2em">|</mo><mspace width="thickmathspace"></mspace><mo>≤</mo><mspace width="thickmathspace"></mspace><mn>1</mn><mo>−</mo><mo stretchy="false">⟨</mo><msub><mi>S</mi> <mn>1</mn></msub><msub><mi>S</mi> <mn>3</mn></msub><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \big\vert \langle S_1 S_2\rangle - \langle S_3 S_2\rangle \big\vert \;\leq\; 1 - \langle S_1 S_3\rangle \,. </annotation></semantics></math></div> <p>(where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>|</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>|</mo></mrow></mrow><annotation encoding="application/x-tex">\left\vert-\right\vert</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/absolute+value">absolute value</a>)</p> </div> <div class='proof'> <h6>Proof</h6> <p>Recall that the <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a> of a random variable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>Λ</mi><mo>⟶</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">P \,\colon\, \Lambda \longrightarrow \mathbb{R}</annotation></semantics></math> is given by its <a class="existingWikiWord" href="/nlab/show/Lebesgue+integral">Lebesgue integral</a> against the <a class="existingWikiWord" href="/nlab/show/probability+measure">probability measure</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>P</mi><mo stretchy="false">⟩</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mo>∫</mo> <mi>Λ</mi></msub><mi>P</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>d</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \langle P \rangle \;\coloneqq\; \int_\Lambda P(\lambda) \, d\rho(\lambda) \,, </annotation></semantics></math></div> <p>and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>ρ</mi></mrow><annotation encoding="application/x-tex">d\rho</annotation></semantics></math> being a <a class="existingWikiWord" href="/nlab/show/probability+measure">probability measure</a> implies the normalization</p> <div class="maruku-equation" id="eq:NormalizationOfProbability"><span class="maruku-eq-number">(12)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mn>1</mn><mo stretchy="false">⟩</mo><mspace width="thickmathspace"></mspace><mo>≡</mo><mspace width="thickmathspace"></mspace><msub><mo>∫</mo> <mi>Λ</mi></msub><mn>1</mn><mspace width="thinmathspace"></mspace><mi>d</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>1</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \langle 1 \rangle \;\equiv\; \int_\Lambda 1 \, d\rho(\lambda) \;=\; 1 \,. </annotation></semantics></math></div> <p>Moreover, the assumption <a class="maruku-eqref" href="#eq:TheRandomVariables">(9)</a> that the random variables <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math> take values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo>±</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\pm 1\}</annotation></semantics></math> immediately implies for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mspace width="thinmathspace"></mspace><mi>in</mi><mspace width="thinmathspace"></mspace><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">i,j \,in\, \{1,2,3\}</annotation></semantics></math> that</p> <div class="maruku-equation" id="eq:IdempotencyOfTheRandomVariables"><span class="maruku-eq-number">(13)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>⋅</mo><msub><mi>S</mi> <mi>i</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mn>1</mn><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mtext>i.e.</mtext><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><munder><mo>∀</mo><mrow><mi>λ</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>Λ</mi></mrow></munder><msub><mi>S</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>S</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mo>±</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mn>1</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \big( S_i \cdot S_i \big) \,=\, 1 \,, \;\;\;\; \text{i.e.} \;\;\; \underset{\lambda \,\in\, \Lambda}{\forall} S_i(\lambda) \, S_i(\lambda) \,=\, (\pm 1)^2 \,=\, 1 \,. </annotation></semantics></math></div> <p>Together this implies – by repeatedly using the <a class="existingWikiWord" href="/nlab/show/Cauchy-Schwarz+inequality">Cauchy-Schwarz inequality</a> – the bounds:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">|</mo><mo stretchy="false">⟨</mo><msub><mi>S</mi> <mi>i</mi></msub><mo stretchy="false">⟩</mo><mo maxsize="1.2em" minsize="1.2em">|</mo><mspace width="thickmathspace"></mspace><mo>≤</mo><mspace width="thickmathspace"></mspace><mn>1</mn><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">|</mo><mo stretchy="false">⟨</mo><msub><mi>S</mi> <mi>i</mi></msub><msub><mi>S</mi> <mi>j</mi></msub><mo stretchy="false">⟩</mo><mo maxsize="1.2em" minsize="1.2em">|</mo><mspace width="thickmathspace"></mspace><mo>≤</mo><mspace width="thickmathspace"></mspace><mn>1</mn></mrow><annotation encoding="application/x-tex"> \big\vert \langle S_i \rangle \big\vert \;\leq\; 1 \,, \;\;\;\;\;\;\; \big\vert \langle S_i S_j\rangle \big\vert \;\leq\; 1 </annotation></semantics></math></div> <p>and thus, in particular:</p> <div class="maruku-equation" id="eq:BoundOnCorrelations"><span class="maruku-eq-number">(14)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">|</mo><mo stretchy="false">⟨</mo><mi>P</mi><mspace width="thinmathspace"></mspace><msub><mi>S</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><msub><mi>S</mi> <mi>j</mi></msub><mo stretchy="false">⟩</mo><mo maxsize="1.2em" minsize="1.2em">|</mo><mspace width="thickmathspace"></mspace><mo>≤</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">|</mo><mo stretchy="false">⟨</mo><mi>P</mi><mo stretchy="false">⟩</mo><mo maxsize="1.2em" minsize="1.2em">|</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \big\vert \langle P \, S_i \, S_j \rangle \big\vert \;\leq\; \big\vert \langle P \rangle \big\vert \,, </annotation></semantics></math></div> <p>for any <a class="existingWikiWord" href="/nlab/show/random+variable">random variable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>Λ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">P \,\colon\, \Lambda \to \mathbb{R}</annotation></semantics></math>.</p> <p>Using these (evident) ingredients, we directly compute as follows</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left left"><mtr><mtd><mo maxsize="1.2em" minsize="1.2em">|</mo><mo stretchy="false">⟨</mo><msub><mi>S</mi> <mn>1</mn></msub><msub><mi>S</mi> <mn>2</mn></msub><mo stretchy="false">⟩</mo><mo>−</mo><mo stretchy="false">⟨</mo><msub><mi>S</mi> <mn>3</mn></msub><msub><mi>S</mi> <mn>2</mn></msub><mo stretchy="false">⟩</mo><mo maxsize="1.2em" minsize="1.2em">|</mo></mtd> <mtd></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.8em" minsize="1.8em">|</mo><msub><mo>∫</mo> <mi>Λ</mi></msub><msub><mi>S</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>S</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>d</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mo>∫</mo> <mi>Λ</mi></msub><msub><mi>S</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>S</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>d</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo maxsize="1.8em" minsize="1.8em">|</mo></mtd> <mtd><mtext>by</mtext><mspace width="thickmathspace"></mspace><mtext><a class="maruku-eqref" href="#eq:TheCorrelations">(10)</a></mtext></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.8em" minsize="1.8em">|</mo><msub><mo>∫</mo> <mi>Λ</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>S</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>S</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><msub><mi>S</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>d</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo maxsize="1.8em" minsize="1.8em">|</mo></mtd> <mtd><mtext>by linearity of the integral</mtext></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.8em" minsize="1.8em">|</mo><msub><mo>∫</mo> <mi>Λ</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mn>1</mn><mo>−</mo><msub><mi>S</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>S</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><msub><mi>S</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>S</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mi>d</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo maxsize="1.8em" minsize="1.8em">|</mo></mtd> <mtd><mtext>by</mtext><mspace width="thickmathspace"></mspace><mtext><a class="maruku-eqref" href="#eq:IdempotencyOfTheRandomVariables">(13)</a></mtext></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>≤</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.8em" minsize="1.8em">|</mo><msub><mo>∫</mo> <mi>Λ</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mn>1</mn><mo>−</mo><msub><mi>S</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>S</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mi>d</mi><mi>ρ</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo><mo maxsize="1.8em" minsize="1.8em">|</mo></mtd> <mtd><mtext>by</mtext><mspace width="thickmathspace"></mspace><mtext><a class="maruku-eqref" href="#eq:BoundOnCorrelations">(14)</a></mtext></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>1</mn><mo>−</mo><mo stretchy="false">⟨</mo><msub><mi>S</mi> <mn>1</mn></msub><msub><mi>S</mi> <mn>3</mn></msub><mo stretchy="false">⟩</mo></mtd> <mtd><mtext>by</mtext><mspace width="thickmathspace"></mspace><mtext><a class="maruku-eqref" href="#eq:NormalizationOfProbability">(12)</a></mtext><mspace width="thickmathspace"></mspace><mtext>and</mtext><mspace width="thickmathspace"></mspace><mtext><a class="maruku-eqref" href="#eq:TheCorrelations">(10)</a></mtext></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{ll} \big\vert \langle S_1 S_2\rangle - \langle S_3 S_2\rangle \big\vert & \\ \;=\; \Big\vert \int_{\Lambda} S_1(\lambda) \, S_2(\lambda) \, d\rho(\lambda) - \int_{\Lambda} S_3(\lambda) \, S_2(\lambda) \, d\rho(\lambda) \Big\vert & \text{by}\;\text{<a class="maruku-eqref" href="#eq:TheCorrelations">(10)</a>} \\ \;=\; \Big\vert \int_{\Lambda} \big( S_1(\lambda) - S_3(\lambda) \big) \, S_2(\lambda) \, d\rho(\lambda) \Big\vert & \text{by linearity of the integral} \\ \;=\; \Big\vert \int_{\Lambda} \big( 1 - S_1(\lambda) \, S_3(\lambda) \big) S_1(\lambda) \, S_2(\lambda) \, d\rho(\lambda) \Big\vert & \text{by}\;\text{<a class="maruku-eqref" href="#eq:IdempotencyOfTheRandomVariables">(13)</a>} \\ \;\leq\; \Big\vert \int_{\Lambda} \big( 1 - S_1(\lambda) \, S_3(\lambda) \big) \, d\rho(\lambda) \Big\vert & \text{by}\;\text{<a class="maruku-eqref" href="#eq:BoundOnCorrelations">(14)</a>} \\ \;=\; 1 - \langle S_1 S_3\rangle & \text{by}\;\text{<a class="maruku-eqref" href="#eq:NormalizationOfProbability">(12)</a>}\;\text{and}\; \text{<a class="maruku-eqref" href="#eq:TheCorrelations">(10)</a>} \end{array} </annotation></semantics></math></div> <p>This is the inequality <a class="maruku-eqref" href="#eq:TheInequality">(11)</a>.</p> </div> </p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Einstein-Podolsky-Rosen+paradox">Einstein-Podolsky-Rosen paradox</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+inequality">Grothendieck inequality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interpretation+of+quantum+mechanics">interpretation of quantum mechanics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a></p> </li> </ul> <p>Other theorems about the foundations and <a class="existingWikiWord" href="/nlab/show/interpretation+of+quantum+mechanics">interpretation of quantum mechanics</a> include:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner+theorem">Wigner theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bub-Clifton+theorem">Bub-Clifton theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kadison-Singer+problem">Kadison-Singer problem</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The original article:</p> <ul> <li id="Bell64"><a class="existingWikiWord" href="/nlab/show/John+Bell">John Bell</a>, <em>On the Einstein Podolsky Rosen paradox</em>, Physics <strong>1</strong> 195 (1964) [<a href="https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195">doi:10.1103/PhysicsPhysiqueFizika.1.195</a>, <a href="http://www.drchinese.com/David/Bell_Compact.pdf">pdf</a>]</li> </ul> <p>Relation to the <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/N.+David+Mermin">N. David Mermin</a>, <em>Hidden Variables and the Two Theorems of John Bell</em>, Reviews of Modern Physics <strong>65</strong> (1993) 803-815 [<a href="https://doi.org/10.1103/RevModPhys.65.803">doi:10.1103/RevModPhys.65.803</a>, <a href="https://arxiv.org/abs/1802.10119">arXiv:1802.10119</a>]</li> </ul> <p>Introduction and review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+F.+Clauser">John F. Clauser</a>, <a class="existingWikiWord" href="/nlab/show/Abner+Shimony">Abner Shimony</a>, <em>Bell’s theorem. Experimental tests and implications</em>, Rep. Prog. Phys. <strong>41</strong> (1978) 1881 [<a href="https://iopscience.iop.org/article/10.1088/0034-4885/41/12/002">doi:10.1088/0034-4885/41/12/002</a>]</p> </li> <li id="Kuperberg05"> <p><a class="existingWikiWord" href="/nlab/show/Greg+Kuperberg">Greg Kuperberg</a>, section 1.6.2 of: <em>A concise introduction to quantum probability, quantum mechanics, and quantum computation</em> (2005) [<a href="http://www.math.ucdavis.edu/~greg/intro-2005.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Kuperberg-ConciseQuantum.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Valter+Moretti">Valter Moretti</a>, Thm. 4.49 of: <em>Fundamental Mathematical Structures of Quantum Theory</em>, Springer (2019) [<a href="https://doi.org/10.1007/978-3-030-18346-2">doi:10.1007/978-3-030-18346-2</a>]</p> </li> <li> <p>M.S.Guimaraes, I. Roditi, S.P. Sorella: <em>Introduction to Bell’s inequality in Quantum Mechanics</em> [<a href="https://arxiv.org/abs/2409.07597">arXiv:2409.07597</a>]</p> </li> </ul> <p>and on a background of <a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a>:</p> <ul> <li>Laura Molenaar, <em>Quantum logic and the EPR paradox</em>, Delft (2014) [<a href="http://resolver.tudelft.nl/uuid:cfee567c-425a-4b2d-9550-f7d7eea41b8b">uuid:cfee567c-425a-4b2d-9550-f7d7eea41b8b</a>]</li> </ul> <p>Further on experimental verification:</p> <ul> <li id="Aspect15"><a class="existingWikiWord" href="/nlab/show/Alain+Aspect">Alain Aspect</a>, <em><a href="https://physics.aps.org/articles/v8/123">Closing the Door on Einstein and Bohr’s Quantum Debate</a></em>, Physics <strong>8</strong> 123 (2015)</li> </ul> <p>Relation to the <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Leandro+Aolita">Leandro Aolita</a>, Rodrigo Gallego, Antonio Acín, Andrea Chiuri, Giuseppe Vallone, Paolo Mataloni, Adán Cabello, <em>Fully nonlocal quantum correlations</em>, Phys. Rev. A <strong>85</strong> 032107 (2012) [<a href="https://arxiv.org/abs/1105.3598">arXiv:1105.3598</a>, <a href="https://doi.org/10.1103/PhysRevA.85.032107">doi:10.1103/PhysRevA.85.032107</a>]</li> </ul> <p>See also:</p> <ul> <li> <p>Wikipedia, <em><a href="http://en.wikipedia.org/wiki/Bell_inequality">Bell’s theorem</a></em></p> </li> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Bell_test">Bell test</a></em></p> </li> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Leggett%E2%80%93Garg_inequality">Leggett-Garg inequality</a></em></p> </li> <li> <p>Stanford Encyclopedia of Philosophy, <em>Bell’s theorem</em> (<a href="http://plato.stanford.edu/entries/bell-theorem/">url</a>)</p> </li> </ul> <p>In relation to the <a class="existingWikiWord" href="/nlab/show/Grothendieck+inequality">Grothendieck inequality</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Boris+S.+Tsirelson">Boris S. Tsirelson</a>, <em>Quantum analogues of the Bell inequalities. The case of two spatially separated domains</em>, Journal of Soviet Mathematics <strong>36</strong> (1987) 557–570 [<a href="https://doi.org/10.1007/BF01663472">doi:10.1007/BF01663472</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boris+S.+Tsirelson">Boris S. Tsirelson</a>, <em>Some results and problems on quantum Bell-type inequalities</em> Hadronic Journal Supplement <strong>8</strong> 4 (1993) 329-345 [<a href="https://www.tau.ac.il/~tsirel/download/hadron.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Tsirelson-QuantumBellType.pdf" title="pdf">pdf</a> <a href="https://www.tau.ac.il/~tsirel/download/hadron.html">web</a>]</p> <blockquote> <p>(but see the erratum <a href="https://www.tau.ac.il/~tsirel/Research/bellopalg/main.html">here</a>)</p> </blockquote> </li> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Tsirelson%27s_bound">Tsirelson’s bound</a></em></p> </li> </ul> <h3 id="in_quantum_field_theory">In quantum field theory</h3> <p>In the generality of <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>:</p> <p>On Bell inequalities in <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> and possible relation to the <a class="existingWikiWord" href="/nlab/show/weak+gravity+conjecture">weak gravity conjecture</a>:</p> <ul> <li>Aninda Sinha, Ahmadullah Zahed, <em>Bell inequalities in 2-2 scattering</em> [<a href="https://arxiv.org/abs/2212.10213">arXiv:2212.10213</a>]</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST invariant</a> Bell inequality in <a class="existingWikiWord" href="/nlab/show/gauge+field+theory">gauge field theory</a>:</p> <ul> <li>David Dudal, Philipe De Fabritiis, Marcelo S. Guimaraes, Giovani Peruzzo, Silvio P. Sorella: <em>BRST invariant formulation of the Bell-CHSH inequality in gauge field theories</em> [<a href="https://arxiv.org/abs/2304.01028">arXiv:2304.01028</a>]</li> </ul> <h3 id="ReferencesProbabilisticOpposition">Probabilistic opposition</h3> <p>Identification of Bell’s inequalities with much older inequalities in classical <a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a>, due to <a class="existingWikiWord" href="/nlab/show/George+Boole">George Boole</a>‘s <em><a class="existingWikiWord" href="/nlab/show/Boole+--+The+Laws+of+Thought">The Laws of Thought</a></em>, was pointed out by (among others, called the “probabilistic opposition” in <a href="#Khrennikov07">Khrennikov 2007, p. 3</a>) by:</p> <ul> <li id="Pitowsky89a"> <p><a class="existingWikiWord" href="/nlab/show/Itamar+Pitowsky">Itamar Pitowsky</a>, <em>From George Boole To John Bell — The Origins of Bell’s Inequality</em>, in: <em>Bell’s Theorem, Quantum Theory and Conceptions of the Universe</em>, Fundamental Theories of Physics <strong>37</strong> Springer (1989) [<a href="https://doi.org/10.1007/978-94-017-0849-4_6">doi:10.1007/978-94-017-0849-4_6</a>]</p> </li> <li id="Pitowsky89b"> <p><a class="existingWikiWord" href="/nlab/show/Itamar+Pitowsky">Itamar Pitowsky</a>, <em>Quantum Probability – Quantum Logic</em>, Lecture Notes in Physics <strong>321</strong>, Springer (1989) [<a href="https://doi.org/10.1007/BFb0021186">doi:10.1007/BFb0021186</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Luigi+Accardi">Luigi Accardi</a>, <em>The Probabilistic Roots of the Quantum Mechanical Paradoxes</em>, in: <em>The Wave-Particle Dualism</em>, Fundamental Theories of Physics <strong>3</strong> Springer (1984) [<a href="https://doi.org/10.1007/978-94-009-6286-6_16">doi:10.1007/978-94-009-6286-6_16</a>]</p> </li> </ul> <p>reviewed in:</p> <ul> <li> <p>Elemer E Rosinger, <em>George Boole and the Bell inequalities</em> [<a href="https://arxiv.org/abs/quant-ph/0406004">arXiv:quant-ph/0406004</a>]</p> </li> <li id="Khrennikov07"> <p><a class="existingWikiWord" href="/nlab/show/Andrei+Khrennikov">Andrei Khrennikov</a>, <em>Bell’s inequality: Physics meets Probability</em> [<a href="https://arxiv.org/abs/0709.3909">arXiv:0709.3909</a>]</p> </li> <li id="Khrennikov08"> <p><a class="existingWikiWord" href="/nlab/show/Andrei+Khrennikov">Andrei Khrennikov</a>, <em>Bell-Boole Inequality: Nonlocality or Probabilistic Incompatibility of Random Variables?</em>, Entropy <strong>10</strong> 2 (2008) 19-32 [<a href="https://doi.org/10.3390/entropy-e10020019">doi:10.3390/entropy-e10020019</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 13, 2024 at 05:44:14. 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