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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Meaning_of_entanglement"> <div class="vector-toc-text"> 2.1 Meaning of entanglement </div> </a> <ul id="toc-Meaning_of_entanglement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Paradox" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Paradox"> <div class="vector-toc-text"> 2.2 Paradox </div> </a> <ul id="toc-Paradox-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Failure_of_local_hidden-variable_theories" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Failure_of_local_hidden-variable_theories"> <div class="vector-toc-text"> 2.3 Failure of local hidden-variable theories </div> </a> <ul id="toc-Failure_of_local_hidden-variable_theories-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Nonlocality_and_entanglement" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Nonlocality_and_entanglement"> <div class="vector-toc-text"> 3 Nonlocality and entanglement </div> </a> <ul id="toc-Nonlocality_and_entanglement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematical_details" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mathematical_details"> <div class="vector-toc-text"> 4 Mathematical details </div> </a> <button aria-controls="toc-Mathematical_details-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Mathematical details subsection </button> <ul id="toc-Mathematical_details-sublist" class="vector-toc-list"> <li id="toc-Pure_states" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pure_states"> <div class="vector-toc-text"> 4.1 Pure states </div> </a> <ul id="toc-Pure_states-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ensembles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ensembles"> <div class="vector-toc-text"> 4.2 Ensembles </div> </a> <ul id="toc-Ensembles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reduced_density_matrices" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reduced_density_matrices"> <div class="vector-toc-text"> 4.3 Reduced density matrices </div> </a> <ul id="toc-Reduced_density_matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Entanglement_as_a_resource" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Entanglement_as_a_resource"> <div class="vector-toc-text"> 4.4 Entanglement as a resource </div> </a> <ul id="toc-Entanglement_as_a_resource-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multipartite_entanglement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multipartite_entanglement"> <div class="vector-toc-text"> 4.5 Multipartite entanglement </div> </a> <ul id="toc-Multipartite_entanglement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classification_of_entanglement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classification_of_entanglement"> <div class="vector-toc-text"> 4.6 Classification of entanglement </div> </a> <ul id="toc-Classification_of_entanglement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Entropy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Entropy"> <div class="vector-toc-text"> 4.7 Entropy </div> </a> <ul id="toc-Entropy-sublist" class="vector-toc-list"> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> 4.7.1 Definition </div> </a> <ul id="toc-Definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_a_measure_of_entanglement" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#As_a_measure_of_entanglement"> <div class="vector-toc-text"> 4.7.2 As a measure of entanglement </div> </a> <ul id="toc-As_a_measure_of_entanglement-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Entanglement_measures" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Entanglement_measures"> <div class="vector-toc-text"> 4.8 Entanglement measures </div> </a> <ul id="toc-Entanglement_measures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_field_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_field_theory"> <div class="vector-toc-text"> 4.9 Quantum field theory </div> </a> <ul id="toc-Quantum_field_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 5 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Entangled_states" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Entangled_states"> <div class="vector-toc-text"> 5.1 Entangled states </div> </a> <ul id="toc-Entangled_states-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Methods_of_creating_entanglement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Methods_of_creating_entanglement"> <div class="vector-toc-text"> 5.2 Methods of creating entanglement </div> </a> <ul id="toc-Methods_of_creating_entanglement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Testing_a_system_for_entanglement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Testing_a_system_for_entanglement"> <div class="vector-toc-text"> 5.3 Testing a system for entanglement </div> </a> <ul id="toc-Testing_a_system_for_entanglement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_quantum_gravity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_quantum_gravity"> <div class="vector-toc-text"> 5.4 In quantum gravity </div> </a> <ul id="toc-In_quantum_gravity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Experiments_demonstrating_and_using_entanglement" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Experiments_demonstrating_and_using_entanglement"> <div class="vector-toc-text"> 6 Experiments demonstrating and using entanglement </div> </a> <button aria-controls="toc-Experiments_demonstrating_and_using_entanglement-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Experiments demonstrating and using entanglement subsection </button> <ul id="toc-Experiments_demonstrating_and_using_entanglement-sublist" class="vector-toc-list"> <li id="toc-Bell_tests" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bell_tests"> <div class="vector-toc-text"> 6.1 Bell tests </div> </a> <ul id="toc-Bell_tests-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Entanglement_of_top_quarks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Entanglement_of_top_quarks"> <div class="vector-toc-text"> 6.2 Entanglement of top quarks </div> </a> <ul id="toc-Entanglement_of_top_quarks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Entanglement_of_macroscopic_objects" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Entanglement_of_macroscopic_objects"> <div class="vector-toc-text"> 6.3 Entanglement of macroscopic objects </div> </a> <ul id="toc-Entanglement_of_macroscopic_objects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Entanglement_of_elements_of_living_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Entanglement_of_elements_of_living_systems"> <div class="vector-toc-text"> 6.4 Entanglement of elements of living systems </div> </a> <ul id="toc-Entanglement_of_elements_of_living_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Entanglement_of_quarks_and_gluons_in_protons" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Entanglement_of_quarks_and_gluons_in_protons"> <div class="vector-toc-text"> 6.5 Entanglement of quarks and gluons in protons </div> </a> <ul id="toc-Entanglement_of_quarks_and_gluons_in_protons-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 7 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 8 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 9 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 10 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Quantum entanglement</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 65 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-65" aria-hidden="true" > 65 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Quantenverschr%C3%A4nkung" title="Quantenverschränkung – Alemannic" lang="gsw" hreflang="gsw" data-title="Quantenverschränkung" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%B4%D8%A7%D8%A8%D9%83_%D9%83%D9%85%D9%8A" title="تشابك كمي – Arabic" lang="ar" hreflang="ar" data-title="تشابك كمي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Entellazamientu_cu%C3%A1nticu" title="Entellazamientu cuánticu – Asturian" lang="ast" hreflang="ast" data-title="Entellazamientu cuánticu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8B%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%A8%E0%A7%8D%E0%A6%9F%E0%A6%BE%E0%A6%AE_%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A6%A1%E0%A6%BC%E0%A6%A8" title="কোয়ান্টাম বিজড়ন – Bangla" lang="bn" hreflang="bn" data-title="কোয়ান্টাম বিজড়ন" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Li%C5%8Dng-ch%C3%AD_tak-t%C3%AE%E2%81%BF" title="Liōng-chí tak-tîⁿ – Minnan" lang="nan" hreflang="nan" data-title="Liōng-chí tak-tîⁿ" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%BE_%D0%B7%D0%B0%D0%BF%D0%BB%D0%B8%D1%82%D0%B0%D0%BD%D0%B5" title="Квантово заплитане – Bulgarian" lang="bg" hreflang="bg" data-title="Квантово заплитане" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kvantna_zapletenost" title="Kvantna zapletenost – Bosnian" lang="bs" hreflang="bs" data-title="Kvantna zapletenost" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Entrella%C3%A7ament_qu%C3%A0ntic" title="Entrellaçament quàntic – Catalan" lang="ca" hreflang="ca" data-title="Entrellaçament quàntic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kvantov%C3%A9_prov%C3%A1z%C3%A1n%C3%AD" title="Kvantové provázání – Czech" lang="cs" hreflang="cs" data-title="Kvantové provázání" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kvantemekanisk_sammenfiltring" title="Kvantemekanisk sammenfiltring – Danish" lang="da" hreflang="da" data-title="Kvantemekanisk sammenfiltring" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D8%AA%D8%AE%D8%A8%D8%A7%D9%84_%D9%83%D9%88%D8%A7%D9%86%D8%AA%D9%8A" title="تخبال كوانتي – Moroccan Arabic" lang="ary" hreflang="ary" data-title="تخبال كوانتي" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target">الدارجة</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Quantenverschr%C3%A4nkung" title="Quantenverschränkung – German" lang="de" hreflang="de" data-title="Quantenverschränkung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kvantp%C3%B5imumine" title="Kvantpõimumine – Estonian" lang="et" hreflang="et" data-title="Kvantpõimumine" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%B2%CE%B1%CE%BD%CF%84%CE%B9%CE%BA%CE%AE_%CE%B4%CE%B9%CE%B5%CE%BC%CF%80%CE%BB%CE%BF%CE%BA%CE%AE" title="Κβαντική διεμπλοκή – Greek" lang="el" hreflang="el" data-title="Κβαντική διεμπλοκή" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Entrelazamiento_cu%C3%A1ntico" title="Entrelazamiento cuántico – Spanish" lang="es" hreflang="es" data-title="Entrelazamiento cuántico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Korapilatze_kuantiko" title="Korapilatze kuantiko – Basque" lang="eu" hreflang="eu" data-title="Korapilatze kuantiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%B1%D9%87%D9%85%E2%80%8C%D8%AA%D9%86%DB%8C%D8%AF%DA%AF%DB%8C_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C" title="درهم‌تنیدگی کوانتومی – Persian" lang="fa" hreflang="fa" data-title="درهم‌تنیدگی کوانتومی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Intrication_quantique" title="Intrication quantique – French" lang="fr" hreflang="fr" data-title="Intrication quantique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Fost%C3%BA_candamach" title="Fostú candamach – Irish" lang="ga" hreflang="ga" data-title="Fostú candamach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Entrelazamento_cu%C3%A1ntico" title="Entrelazamento cuántico – Galician" lang="gl" hreflang="gl" data-title="Entrelazamento cuántico" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%96%91%EC%9E%90_%EC%96%BD%ED%9E%98" title="양자 얽힘 – Korean" lang="ko" hreflang="ko" data-title="양자 얽힘" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%94%D5%BE%D5%A1%D5%B6%D5%BF%D5%A1%D5%B5%D5%AB%D5%B6_%D5%AD%D5%B3%D5%B3%D5%BE%D5%A1%D5%AE%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Քվանտային խճճվածություն – Armenian" lang="hy" hreflang="hy" data-title="Քվանտային խճճվածություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%A3%E0%A5%8D%E0%A4%9F%E0%A4%AE_%E0%A4%89%E0%A4%B2%E0%A4%9D%E0%A4%BE%E0%A4%B5" title="क्वाण्टम उलझाव – Hindi" lang="hi" hreflang="hi" data-title="क्वाण्टम उलझाव" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kvantno_sprezanje" title="Kvantno sprezanje – Croatian" lang="hr" hreflang="hr" data-title="Kvantno sprezanje" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Keterkaitan_kuantum" title="Keterkaitan kuantum – Indonesian" lang="id" hreflang="id" data-title="Keterkaitan kuantum" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Ukuphixana_kohoyana" title="Ukuphixana kohoyana – Zulu" lang="zu" hreflang="zu" data-title="Ukuphixana kohoyana" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target">IsiZulu</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Entanglement_quantistico" title="Entanglement quantistico – Italian" lang="it" hreflang="it" data-title="Entanglement quantistico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%96%D7%99%D7%A8%D7%94_%D7%A7%D7%95%D7%95%D7%A0%D7%98%D7%99%D7%AA" title="שזירה קוונטית – Hebrew" lang="he" hreflang="he" data-title="שזירה קוונטית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%95%E1%83%90%E1%83%9C%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%92%E1%83%90%E1%83%93%E1%83%90%E1%83%AE%E1%83%9A%E1%83%90%E1%83%A0%E1%83%97%E1%83%A3%E1%83%9A%E1%83%9D%E1%83%91%E1%83%90" title="კვანტური გადახლართულობა – Georgian" lang="ka" hreflang="ka" data-title="კვანტური გადახლართულობა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Mfungamano_wa_kwanta" title="Mfungamano wa kwanta – Swahili" lang="sw" hreflang="sw" data-title="Mfungamano wa kwanta" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Kvantin%C4%97_sietis" title="Kvantinė sietis – Lithuanian" lang="lt" hreflang="lt" data-title="Kvantinė sietis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Entanglement" title="Entanglement – Lombard" lang="lmo" hreflang="lmo" data-title="Entanglement" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Kvantum-%C3%B6sszefon%C3%B3d%C3%A1s" title="Kvantum-összefonódás – Hungarian" lang="hu" hreflang="hu" data-title="Kvantum-összefonódás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B5%8D%E0%B4%B5%E0%B4%BE%E0%B4%A3%E0%B5%8D%E0%B4%9F%E0%B4%82_%E0%B4%8E%E0%B5%BB%E2%80%8C%E0%B4%9F%E0%B4%BE%E0%B4%82%E0%B4%97%E0%B4%BF%E0%B5%BE%E0%B4%AE%E0%B5%86%E0%B4%A8%E0%B5%8D%E0%B4%B1%E0%B5%8D" title="ക്വാണ്ടം എൻ‌ടാംഗിൾമെന്റ് – Malayalam" lang="ml" hreflang="ml" data-title="ക്വാണ്ടം എൻ‌ടാംഗിൾമെന്റ്" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Keterlibatan_kuantum" title="Keterlibatan kuantum – Malay" lang="ms" hreflang="ms" data-title="Keterlibatan kuantum" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kwantumverstrengeling" title="Kwantumverstrengeling – Dutch" lang="nl" hreflang="nl" data-title="Kwantumverstrengeling" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BE_%E0%A4%B5%E0%A5%8D%E0%A4%AF%E0%A4%A4%E0%A4%BF%E0%A4%B7%E0%A4%99%E0%A5%8D%E0%A4%97" title="प्रमात्रा व्यतिषङ्ग – Nepali" lang="ne" hreflang="ne" data-title="प्रमात्रा व्यतिषङ्ग" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E3%82%82%E3%81%A4%E3%82%8C" title="量子もつれ – Japanese" lang="ja" hreflang="ja" data-title="量子もつれ" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kvantesammenfiltring" title="Kvantesammenfiltring – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kvantesammenfiltring" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kvantesamanfiltring" title="Kvantesamanfiltring – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kvantesamanfiltring" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%81%E0%A8%86%E0%A8%82%E0%A8%9F%E0%A8%AE_%E0%A8%87%E0%A9%B0%E0%A8%9F%E0%A9%88%E0%A8%82%E0%A8%97%E0%A8%B2%E0%A8%AE%E0%A9%88%E0%A8%82%E0%A8%9F" title="ਕੁਆਂਟਮ ਇੰਟੈਂਗਲਮੈਂਟ – Punjabi" lang="pa" hreflang="pa" data-title="ਕੁਆਂਟਮ ਇੰਟੈਂਗਲਮੈਂਟ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pwn mw-list-item"><a href="https://pwn.wikipedia.org/wiki/mapareumalj_a_quantum" title="mapareumalj a quantum – Paiwan" lang="pwn" hreflang="pwn" data-title="mapareumalj a quantum" data-language-autonym="Pinayuanan" data-language-local-name="Paiwan" class="interlanguage-link-target">Pinayuanan</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Stan_spl%C4%85tany" title="Stan splątany – Polish" lang="pl" hreflang="pl" data-title="Stan splątany" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Entrela%C3%A7amento_qu%C3%A2ntico" title="Entrelaçamento quântico – Portuguese" lang="pt" hreflang="pt" data-title="Entrelaçamento quântico" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Inseparabilitate_cuantic%C4%83" title="Inseparabilitate cuantică – Romanian" lang="ro" hreflang="ro" data-title="Inseparabilitate cuantică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0%D1%8F_%D0%B7%D0%B0%D0%BF%D1%83%D1%82%D0%B0%D0%BD%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Квантовая запутанность – Russian" lang="ru" hreflang="ru" data-title="Квантовая запутанность" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Currilazzioni_qu%C3%A0ntica" title="Currilazzioni quàntica – Sicilian" lang="scn" hreflang="scn" data-title="Currilazzioni quàntica" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Quantum_entanglement" title="Quantum entanglement – Simple English" lang="en-simple" hreflang="en-simple" data-title="Quantum entanglement" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kvantov%C3%A9_previazanie" title="Kvantové previazanie – Slovak" lang="sk" hreflang="sk" data-title="Kvantové previazanie" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kvantna_prepletenost" title="Kvantna prepletenost – Slovenian" lang="sl" hreflang="sl" data-title="Kvantna prepletenost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Kvantno_sprezanje" title="Kvantno sprezanje – Serbian" lang="sr" hreflang="sr" data-title="Kvantno sprezanje" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lomittuminen" title="Lomittuminen – Finnish" lang="fi" hreflang="fi" data-title="Lomittuminen" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kvantsammanfl%C3%A4tning" title="Kvantsammanflätning – Swedish" lang="sv" hreflang="sv" data-title="Kvantsammanflätning" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Pagkakabuhol_na_quantum" title="Pagkakabuhol na quantum – Tagalog" lang="tl" hreflang="tl" data-title="Pagkakabuhol na quantum" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%81%E0%AE%B5%E0%AE%BE%E0%AE%A3%E0%AF%8D%E0%AE%9F%E0%AE%AE%E0%AF%8D_%E0%AE%AA%E0%AE%BF%E0%AE%A9%E0%AF%8D%E0%AE%A9%E0%AE%B2%E0%AF%8D" title="குவாண்டம் பின்னல் – Tamil" lang="ta" hreflang="ta" data-title="குவாண்டம் பின்னல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Amcubbek_Anectan" title="Amcubbek Anectan – Kabyle" lang="kab" hreflang="kab" data-title="Amcubbek Anectan" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target">Taqbaylit</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82_%D0%B1%D1%83%D1%82%D0%B0%D0%BB%D1%87%D1%8B%D0%BB%D1%8B%D0%B3%D1%8B" title="Квант буталчылыгы – Tatar" lang="tt" hreflang="tt" data-title="Квант буталчылыгы" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%9E%E0%B8%B1%E0%B8%A7%E0%B8%9E%E0%B8%B1%E0%B8%99%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%84%E0%B8%A7%E0%B8%AD%E0%B8%99%E0%B8%95%E0%B8%B1%E0%B8%A1" title="การพัวพันเชิงควอนตัม – Thai" lang="th" hreflang="th" data-title="การพัวพันเชิงควอนตัม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%94%D0%B0%D1%80%D2%B3%D0%B0%D0%BC%D1%82%D0%B0%D0%BD%D0%B8%D0%B4%D0%B0%D0%B3%D0%B8%D0%B8_%D0%BA%D0%B2%D0%BE%D0%BD%D1%82%D1%83%D0%BC%D3%A3" title="Дарҳамтанидагии квонтумӣ – Tajik" lang="tg" hreflang="tg" data-title="Дарҳамтанидагии квонтумӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kuantum_dolan%C4%B1kl%C4%B1k" title="Kuantum dolanıklık – Turkish" lang="tr" hreflang="tr" data-title="Kuantum dolanıklık" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%BF%D0%BB%D1%83%D1%82%D0%B0%D0%BD%D1%96_%D0%BA%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D1%96_%D1%81%D1%82%D0%B0%D0%BD%D0%B8" title="Сплутані квантові стани – Ukrainian" lang="uk" hreflang="uk" data-title="Сплутані квантові стани" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/V%C6%B0%E1%BB%9Bng_m%E1%BA%AFc_l%C6%B0%E1%BB%A3ng_t%E1%BB%AD" title="Vướng mắc lượng tử – Vietnamese" lang="vi" hreflang="vi" data-title="Vướng mắc lượng tử" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E7%BA%A0%E7%BC%A0" title="量子纠缠 – Wu" lang="wuu" hreflang="wuu" data-title="量子纠缠" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E7%B3%BE%E7%BA%8F" title="量子糾纏 – Cantonese" lang="yue" hreflang="yue" data-title="量子糾纏" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://zh.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E7%BA%8F%E7%B5%90" title="量子纏結 – Chinese" lang="zh" hreflang="zh" data-title="量子纏結" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q215675#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> </div> </div> </div> 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href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist nowraplinks" style="width:19.0em;"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></th></tr><tr><td class="sidebar-image"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1799e4a910c7d26396922a20ef5ceec25ca1871c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.882ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"><div class="sidebar-caption" style="font-size:90%;padding-top:0.4em;font-style:italic;"><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a></div></td></tr><tr><td class="sidebar-above hlist nowrap" style="display:block;margin-bottom:0.4em;"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a></li></ul></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)">Complementarity</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a class="mw-selflink selflink">Entanglement</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_number" title="Quantum number">Quantum number</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">State</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li></ul></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Experiments</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell's inequality</a></li> <li><a href="/wiki/CHSH_inequality" title="CHSH inequality">CHSH inequality</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Leggett_inequality" title="Leggett inequality">Leggett inequality</a></li> <li><a href="/wiki/Leggett%E2%80%93Garg_inequality" title="Leggett–Garg inequality">Leggett–Garg inequality</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li></ul> </div> <ul><li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a> <ul><li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice</a></li></ul></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed-choice</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Overview</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase-space</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Sum-over-histories (path integral)</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective-collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Advanced topics</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Yakir_Aharonov" title="Yakir Aharonov">Aharonov</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Hans_Bethe" title="Hans Bethe">Bethe</a></li> <li><a href="/wiki/Patrick_Blackett" title="Patrick Blackett">Blackett</a></li> <li><a href="/wiki/Felix_Bloch" title="Felix Bloch">Bloch</a></li> <li><a href="/wiki/David_Bohm" title="David Bohm">Bohm</a></li> <li><a href="/wiki/Niels_Bohr" title="Niels Bohr">Bohr</a></li> <li><a href="/wiki/Max_Born" title="Max Born">Born</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a></li> <li><a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">de Broglie</a></li> <li><a href="/wiki/Arthur_Compton" title="Arthur Compton">Compton</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Clinton_Davisson" title="Clinton Davisson">Davisson</a></li> <li><a href="/wiki/Peter_Debye" title="Peter Debye">Debye</a></li> <li><a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hugh_Everett_III" title="Hugh Everett III">Everett</a></li> <li><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Roy_J._Glauber" title="Roy J. Glauber">Glauber</a></li> <li><a href="/wiki/Martin_Gutzwiller" title="Martin Gutzwiller">Gutzwiller</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Hans_Kramers" title="Hans Kramers">Kramers</a></li> <li><a href="/wiki/Willis_Lamb" title="Willis Lamb">Lamb</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li> <li><a href="/wiki/Max_von_Laue" title="Max von Laue">Laue</a></li> <li><a href="/wiki/Henry_Moseley" title="Henry Moseley">Moseley</a></li> <li><a href="/wiki/Robert_Andrews_Millikan" title="Robert Andrews Millikan">Millikan</a></li> <li><a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/Isidor_Isaac_Rabi" title="Isidor Isaac Rabi">Rabi</a></li> <li><a href="/wiki/C._V._Raman" title="C. V. Raman">Raman</a></li> <li><a href="/wiki/Johannes_Rydberg" title="Johannes Rydberg">Rydberg</a></li> <li><a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a></li> <li><a href="/wiki/Michelle_Simmons" title="Michelle Simmons">Simmons</a></li> <li><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Wilhelm_Wien" title="Wilhelm Wien">Wien</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Pieter_Zeeman" title="Pieter Zeeman">Zeeman</a></li> <li><a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Zeilinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;padding-top:0.1em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics" title="Template:Quantum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics" title="Template talk:Quantum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics" title="Special:EditPage/Template:Quantum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> Quantum entanglement is the phenomenon of a group of <a href="/wiki/Particle" title="Particle">particles</a> being generated, interacting, or sharing spatial proximity in a manner such that the <a href="/wiki/Quantum_state" title="Quantum state">quantum state</a> of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between <a href="/wiki/Classical_physics" title="Classical physics">classical physics</a> and <a href="/wiki/Quantum_physics" class="mw-redirect" title="Quantum physics">quantum physics</a>: entanglement is a primary feature of quantum mechanics not present in classical mechanics.<a href="#cite_note-horodecki2007-1">[1]</a>: <span title="Page / location: 867
Quotation: "In this way entanglement is that feature of quantum formalism which makes it impossible to simulate quantum correlations within any classical formalism."" class="tooltip tooltip-dashed" style="border-bottom: 1px dashed;">867  <a href="/wiki/Measurement#Quantum_mechanics" title="Measurement">Measurements</a> of <a href="/wiki/Physical_properties" class="mw-redirect" title="Physical properties">physical properties</a> such as <a href="/wiki/Position_(vector)" class="mw-redirect" title="Position (vector)">position</a>, <a href="/wiki/Momentum" title="Momentum">momentum</a>, <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a>, and <a href="/wiki/Polarization_(waves)" title="Polarization (waves)">polarization</a> performed on entangled particles can, in some cases, be found to be perfectly <a href="/wiki/Correlated" class="mw-redirect" title="Correlated">correlated</a>. For example, if a pair of entangled particles is generated such that their total spin is known to be zero, and one particle is found to have clockwise spin on a first axis, then the spin of the other particle, measured on the same axis, is found to be anticlockwise. However, this behavior gives rise to seemingly <a href="/wiki/Paradox" title="Paradox">paradoxical</a> effects: any measurement of a particle's properties results in an apparent and irreversible <a href="/wiki/Wave_function_collapse" title="Wave function collapse">wave function collapse</a> of that particle and changes the original quantum state. With entangled particles, such measurements affect the entangled system as a whole. Such phenomena were the subject of a 1935 paper by <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>, <a href="/wiki/Boris_Podolsky" title="Boris Podolsky">Boris Podolsky</a>, and <a href="/wiki/Nathan_Rosen" title="Nathan Rosen">Nathan Rosen</a>,<a href="#cite_note-Einstein1935-2">[2]</a> and several papers by <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a> shortly thereafter,<a href="#cite_note-Schrödinger1935-3">[3]</a><a href="#cite_note-Schrödinger1936-4">[4]</a> describing what came to be known as the <a href="/wiki/EPR_paradox" class="mw-redirect" title="EPR paradox">EPR paradox</a>. Einstein and others considered such behavior impossible, as it violated the <a href="/wiki/Local_realism" class="mw-redirect" title="Local realism">local realism</a> view of <a href="/wiki/Causality" title="Causality">causality</a> (Einstein referring to it as "spooky <a href="/wiki/Action_at_a_distance" title="Action at a distance">action at a distance</a>")<a href="#cite_note-5">[5]</a> and argued that the accepted formulation of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> must therefore be incomplete. Later, however, the counterintuitive predictions of quantum mechanics were verified in tests where polarization or spin of entangled particles were measured at separate locations, statistically violating <a href="/wiki/Bell%27s_theorem" title="Bell's theorem">Bell's inequality</a>.<a href="#cite_note-Clauser-6">[6]</a><a href="#cite_note-:0-7">[7]</a><a href="#cite_note-:1-8">[8]</a><a href="#cite_note-:2-9">[9]</a> This established that the correlations produced from quantum entanglement cannot be explained in terms of <a href="/wiki/Local_hidden_variable_theory" class="mw-redirect" title="Local hidden variable theory">local hidden variables</a>, i.e., properties contained within the individual particles themselves. However, despite the fact that entanglement can produce statistical <a href="/wiki/Correlation" title="Correlation">correlations</a> between events in widely separated places, it cannot be used for <a href="/wiki/Faster-than-light_communication" class="mw-redirect" title="Faster-than-light communication">faster-than-light communication</a>.<a href="#cite_note-10">[10]</a><a href="#cite_note-11">[11]</a><a href="#cite_note-Griffiths-12">[12]</a>: 453  Quantum entanglement has been demonstrated experimentally with <a href="/wiki/Photon" title="Photon">photons</a>,<a href="#cite_note-Kocher1-13">[13]</a><a href="#cite_note-Kocherphd-14">[14]</a> <a href="/wiki/Electron" title="Electron">electrons</a>,<a href="#cite_note-NTR-20151021-15">[15]</a><a href="#cite_note-NYT-20151021-16">[16]</a> <a href="/wiki/Top_quark" title="Top quark">top quarks</a>,<a href="#cite_note-17">[17]</a> molecules<a href="#cite_note-18">[18]</a> and even small diamonds.<a href="#cite_note-19">[19]</a> The use of quantum entanglement in <a href="/wiki/Quantum_communication" class="mw-redirect" title="Quantum communication">communication</a> and <a href="/wiki/Quantum_computing" title="Quantum computing">computation</a> is an active area of research and development. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=1" title="Edit section: History">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Background: <a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History of quantum mechanics</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:NYT_May_4,_1935.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NYT_May_4%2C_1935.jpg/224px-NYT_May_4%2C_1935.jpg" decoding="async" width="224" height="268" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NYT_May_4%2C_1935.jpg/336px-NYT_May_4%2C_1935.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NYT_May_4%2C_1935.jpg/448px-NYT_May_4%2C_1935.jpg 2x" data-file-width="569" data-file-height="680" /></a><figcaption>Article headline regarding the <a href="/wiki/EPR_paradox" class="mw-redirect" title="EPR paradox">Einstein–Podolsky–Rosen (EPR) paradox</a> paper, in the 4 May 1935 issue of <a href="/wiki/The_New_York_Times" title="The New York Times">The New York Times</a></figcaption></figure> Albert Einstein and Niels Bohr engaged in a long-running collegial dispute about the meaning of quantum mechanics, now known as the <a href="/wiki/Bohr%E2%80%93Einstein_debates" title="Bohr–Einstein debates">Bohr–Einstein debates</a>. During these debates, Einstein introduced a <a href="/wiki/Thought_experiment" title="Thought experiment">thought experiment</a> about a box that emits a photon. He noted that the experimenter's choice of what measurement to make upon the box will change what can be predicted about the photon, even if the photon is very far away. This argument, which Einstein had formulated by 1931, was an early recognition of the phenomenon that would later be called entanglement.<a href="#cite_note-20">[20]</a> That same year, <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> observed in his textbook on <a href="/wiki/Group_theory" title="Group theory">group theory</a> and quantum mechanics that quantum systems made of multiple interacting pieces exhibit a kind of <a href="/wiki/Gestalt_psychology" title="Gestalt psychology">Gestalt</a>, in which "the whole is greater than the sum of its parts".<a href="#cite_note-21">[21]</a><a href="#cite_note-22">[22]</a> In 1932, <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a> wrote down the defining equations of quantum entanglement but set them aside, unpublished.<a href="#cite_note-23">[23]</a> In 1935, <a href="/wiki/Grete_Hermann" title="Grete Hermann">Grete Hermann</a> studied the mathematics of an electron interacting with a photon and noted the phenomenon that would come to be called entanglement.<a href="#cite_note-24">[24]</a> Later that same year, Einstein, <a href="/wiki/Boris_Podolsky" title="Boris Podolsky">Boris Podolsky</a> and <a href="/wiki/Nathan_Rosen" title="Nathan Rosen">Nathan Rosen</a> published a paper on what is now known as the <a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">Einstein–Podolsky–Rosen (EPR) paradox</a>, a thought experiment that attempted to show that "the <a href="/wiki/Quantum-mechanical" class="mw-redirect" title="Quantum-mechanical">quantum-mechanical</a> description of physical reality given by wave functions is not complete".<a href="#cite_note-Einstein1935-2">[2]</a> Their thought experiment had two systems interact, then separate, and they showed that afterwards quantum mechanics cannot describe the two systems individually. Shortly after this paper appeared, <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a> wrote a letter to Einstein in <a href="/wiki/German_language" title="German language">German</a> in which he used the word Verschränkung (translated by himself as entanglement) to describe situations like that of the EPR scenario.<a href="#cite_note-MK-25">[25]</a> Schrödinger followed up with a full paper defining and discussing the notion of entanglement,<a href="#cite_note-Schroeder-2017-26">[26]</a> saying "I would not call [entanglement] one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical</a> lines of thought."<a href="#cite_note-Schrödinger1935-3">[3]</a> Like Einstein, Schrödinger was dissatisfied with the concept of entanglement, because it seemed to violate the speed limit on the transmission of information implicit in the <a href="/wiki/Theory_of_relativity" title="Theory of relativity">theory of relativity</a>.<a href="#cite_note-27">[27]</a> Einstein later referred to the effects of entanglement as "spukhafte Fernwirkung"<a href="#cite_note-spukhafte-28">[28]</a> or "<a href="/wiki/Spooky_action_at_a_distance" class="mw-redirect" title="Spooky action at a distance">spooky action at a distance</a>", meaning the acquisition of a value of a property at one location resulting from a measurement at a distant location.<a href="#cite_note-MerminMoon-1985-29">[29]</a> In 1946, <a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">John Archibald Wheeler</a> suggested studying the <a href="/wiki/Polarization_(physics)" class="mw-redirect" title="Polarization (physics)">polarization</a> of pairs of <a href="/wiki/Gamma-ray" class="mw-redirect" title="Gamma-ray">gamma-ray</a> photons produced by electron–<a href="/wiki/Positron" title="Positron">positron</a> annihilation.<a href="#cite_note-30">[30]</a> <a href="/wiki/Chien-Shiung_Wu" title="Chien-Shiung Wu">Chien-Shiung Wu</a> and I. Shaknov carried out this experiment in 1949,<a href="#cite_note-:3-31">[31]</a> thereby demonstrating that the entangled particle pairs considered by EPR could be created in the laboratory.<a href="#cite_note-32">[32]</a> Despite Schrödinger's claim of its importance, little work on entanglement was published for decades after his paper was published.<a href="#cite_note-Schroeder-2017-26">[26]</a> In 1964 <a href="/wiki/John_S._Bell" class="mw-redirect" title="John S. Bell">John S. Bell</a> demonstrated an upper limit, seen in <a href="/wiki/Bell%27s_inequality" class="mw-redirect" title="Bell's inequality">Bell's inequality</a>, regarding the strength of correlations that can be produced in any theory obeying <a href="/wiki/Local_realism" class="mw-redirect" title="Local realism">local realism</a>, and showed that quantum theory predicts violations of this limit for certain entangled systems.<a href="#cite_note-:4-33">[33]</a><a href="#cite_note-34">[34]</a>: 405  His inequality is experimentally testable, and there have been numerous <a href="/wiki/Bell_test_experiments" class="mw-redirect" title="Bell test experiments">relevant experiments</a>, starting with the pioneering work of <a href="/wiki/Stuart_Freedman" title="Stuart Freedman">Stuart Freedman</a> and <a href="/wiki/John_Clauser" title="John Clauser">John Clauser</a> in 1972<a href="#cite_note-Clauser-6">[6]</a> and <a href="/wiki/Alain_Aspect" title="Alain Aspect">Alain Aspect</a>'s experiments in 1982.<a href="#cite_note-Aspect1982-35">[35]</a><a href="#cite_note-hanson-36">[36]</a><a href="#cite_note-37">[37]</a> While Bell actively discouraged students from pursuing work like his as too esoteric, after a talk at Oxford a student named <a href="/wiki/Artur_Ekert" title="Artur Ekert">Artur Ekert</a> suggested that the violation of a Bell inequality could be used as a resource for communication.<a href="#cite_note-Gilder2009-38">[38]</a>: 315  Ekert followed up by publishing a <a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">quantum key distribution</a> protocol called <a href="/wiki/E91_protocol" class="mw-redirect" title="E91 protocol">E91</a> based on it.<a href="#cite_note-ekert91-39">[39]</a><a href="#cite_note-horodecki2007-1">[1]</a>: <span title="Page / location: 874
Quotation: "The first discovery within quantum information theory, which involves entanglement, is due to Ekert 1991."" class="tooltip tooltip-dashed" style="border-bottom: 1px dashed;">874  In 1992, the entanglement concept was leveraged to propose <a href="/wiki/Quantum_teleportation" title="Quantum teleportation">quantum teleportation</a>,<a href="#cite_note-BBCJPW93-40">[40]</a> an effect that was realized experimentally in 1997.<a href="#cite_note-41">[41]</a><a href="#cite_note-Bouwmeester-1997-42">[42]</a><a href="#cite_note-Rome1998-43">[43]</a> Beginning in the mid-1990s, <a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Anton Zeilinger</a> used the generation of entanglement via <a href="/wiki/Spontaneous_parametric_down-conversion" title="Spontaneous parametric down-conversion"> parametric down-conversion</a> to develop <a href="/wiki/Entanglement_swapping" title="Entanglement swapping">entanglement swapping</a><a href="#cite_note-Gilder2009-38">[38]</a>: 317  and demonstrate <a href="/wiki/Quantum_cryptography" title="Quantum cryptography">quantum cryptography</a> with entangled photons.<a href="#cite_note-44">[44]</a><a href="#cite_note-45">[45]</a> In 2022, the <a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a> was awarded to Aspect, Clauser, and Zeilinger "for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science".<a href="#cite_note-NobelPrize-46">[46]</a> <div class="mw-heading mw-heading2"><h2 id="Concept">Concept</h2>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=2" title="Edit section: Concept">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Meaning_of_entanglement">Meaning of entanglement</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=3" title="Edit section: Meaning of entanglement">edit</a>]</div> Just as <a href="/wiki/Energy" title="Energy">energy</a> is a resource that facilitates mechanical operations, entanglement is a resource that facilitates performing tasks that involve communication and computation.<a href="#cite_note-Nielsen-2010-47">[47]</a>: 106 <a href="#cite_note-Rieffel2011-48">[48]</a>: 218 <a href="#cite_note-Bengtsson2017-49">[49]</a>: 435 <a href="#cite_note-Bub2023-50">[50]</a> The mathematical definition of entanglement can be paraphrased as saying that maximal knowledge about the whole of a system does not imply maximal knowledge about the individual parts of that system.<a href="#cite_note-Rau2021-51">[51]</a> If the quantum state that describes a pair of particles is entangled, then the results of measurements upon one half of the pair can be strongly correlated with the results of measurements upon the other. However, entanglement is not the same as "correlation" as understood in classical probability theory and in daily life. Instead, entanglement can be thought of as potential correlation that can be used to generate actual correlation in an appropriate experiment.<a href="#cite_note-Fuchs2011-52">[52]</a>: 130  The correlations generated from an entangled quantum state cannot in general be replicated by classical probability.<a href="#cite_note-Holevo2001-53">[53]</a>: 33  An example of entanglement is a <a href="/wiki/Subatomic_particle" title="Subatomic particle">subatomic particle</a> that <a href="/wiki/Particle_decay" title="Particle decay">decays</a> into an entangled pair of other particles. The decay events obey the various <a href="/wiki/Conservation_laws" class="mw-redirect" title="Conservation laws">conservation laws</a>, and as a result, the measurement outcomes of one daughter particle must be highly correlated with the measurement outcomes of the other daughter particle (so that the total momenta, angular momenta, energy, and so forth remains roughly the same before and after this process). For instance, a <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a>-zero particle could decay into a pair of spin-1/2 particles. If there is no orbital angular momentum, the total spin angular momentum after this decay must be zero (by the <a href="/wiki/Conservation_of_angular_momentum" class="mw-redirect" title="Conservation of angular momentum">conservation of angular momentum</a>). Whenever the first particle is measured to be <a href="/wiki/Spin_(physics)#Direction" title="Spin (physics)">spin up</a> on some axis, the other, when measured on the same axis, is always found to be <a href="/wiki/Spin_(physics)#Direction" title="Spin (physics)">spin down</a>. This is called the spin anti-correlated case and the pair is said to be in the <a href="/wiki/Singlet_state" title="Singlet state">singlet state</a>. Perfect anti-correlations like this could be explained by "hidden variables" within the particles. For example, we could hypothesize that the particles are made in pairs such that one carries a value of "up" while the other carries a value of "down". Then, knowing the result of the spin measurement upon one particle, we could predict that the other will have the opposite value. Bell illustrated this with a story about a colleague, Bertlmann, who always wore socks with mismatching colors. "Which colour he will have on a given foot on a given day is quite unpredictable," Bell wrote, but upon observing "that the first sock is pink you can be already sure that the second sock will not be pink."<a href="#cite_note-54">[54]</a> Revealing the remarkable features of quantum entanglement requires considering multiple distinct experiments, such as spin measurements along different axes, and comparing the correlations obtained in these different configurations.<a href="#cite_note-Zwiebach2022-55">[55]</a>: §18.8  Quantum <a href="/wiki/Physical_system" title="Physical system">systems</a> can become entangled through various types of interactions. For some ways in which entanglement may be achieved for experimental purposes, see the section below on <a href="#Methods_of_creating_entanglement">methods</a>. Entanglement is broken when the entangled particles <a href="/wiki/Quantum_decoherence" title="Quantum decoherence">decohere</a> through interaction with the environment; for example, when a measurement is made. In more detail, this process involves the particles becoming entangled with the environment, as a consequence of which, the quantum state describing the particles themselves is no longer entangled.<a href="#cite_note-Peres1993-56">[56]</a>: 369 <a href="#cite_note-57">[57]</a> Mathematically, an entangled system can be defined to be one whose quantum state cannot be factored as a product of states of its local constituents; that is to say, they are not individual particles but are an inseparable whole. When entanglement is present, one constituent cannot be fully described without considering the other(s).<a href="#cite_note-Mermin2007-58">[58]</a>: 18–19 <a href="#cite_note-Zwiebach2022-55">[55]</a>: §1.5  The state of a composite system is always expressible as a sum, or <a href="/wiki/Quantum_superposition" title="Quantum superposition">superposition</a>, of products of states of local constituents; it is entangled if this sum cannot be written as a single product term.<a href="#cite_note-Rieffel2011-48">[48]</a>: 39  <div class="mw-heading mw-heading3"><h3 id="Paradox">Paradox</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=4" title="Edit section: Paradox">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/EPR_paradox" class="mw-redirect" title="EPR paradox">EPR paradox</a></div> The singlet state described above is the basis for one version of the EPR paradox. In this variant, introduced by <a href="/wiki/David_Bohm" title="David Bohm">David Bohm</a>, a source emits particles and sends them in opposite directions. The state describing each pair is entangled.<a href="#cite_note-59">[59]</a> In the standard textbook presentation of quantum mechanics, performing a spin measurement on one of the particles causes the wave function for the whole pair to <a href="/wiki/Wave_function_collapse" title="Wave function collapse">collapse</a> into a state in which each particle has a definite spin (either up or down) along the axis of measurement. The outcome is random, with each possibility having a probability of 50%. However, if both spins are measured along the same axis, they are found to be anti-correlated. This means that the random outcome of the measurement made on one particle seems to have been transmitted to the other, so that it can make the "right choice" when it too is measured.<a href="#cite_note-Zwiebach2022-55">[55]</a>: §18.8 <a href="#cite_note-Griffiths-12">[12]</a>: 447–448  The distance and timing of the measurements can be chosen so as to make the interval between the two measurements <a href="/wiki/Spacelike" class="mw-redirect" title="Spacelike">spacelike</a>, hence, any causal effect connecting the events would have to travel faster than light. According to the principles of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>, it is not possible for any information to travel between two such measuring events. It is not even possible to say which of the measurements came first. For two spacelike separated events x1 and x2 there are <a href="/wiki/Inertial_frame" class="mw-redirect" title="Inertial frame">inertial frames</a> in which x1 is first and others in which x2 is first. Therefore, the correlation between the two measurements cannot be explained as one measurement determining the other: different observers would disagree about the role of cause and effect.<a href="#cite_note-60">[60]</a> <div class="mw-heading mw-heading3"><h3 id="Failure_of_local_hidden-variable_theories">Failure of local hidden-variable theories</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=5" title="Edit section: Failure of local hidden-variable theories">edit</a>]</div> A possible resolution to the paradox is to assume that quantum theory is incomplete, and the result of measurements depends on predetermined "<a href="/wiki/Hidden-variables_theory" class="mw-redirect" title="Hidden-variables theory">hidden variables</a>".<a href="#cite_note-Gibney2017-61">[61]</a> The state of the particles being measured contains some hidden variables, whose values effectively determine, right from the moment of separation, what the outcomes of the spin measurements are going to be. This would mean that each particle carries all the required information with it, and nothing needs to be transmitted from one particle to the other at the time of measurement. Einstein and others (see the previous section) originally believed this was the only way out of the paradox, and the accepted quantum mechanical description (with a random measurement outcome) must be incomplete. <a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local hidden variable theories</a> fail, however, when measurements of the spin of entangled particles along different axes are considered. If a large number of pairs of such measurements are made (on a large number of pairs of entangled particles), then statistically, if the local realist or hidden variables view were correct, the results would always satisfy <a href="/wiki/Bell%27s_inequality" class="mw-redirect" title="Bell's inequality">Bell's inequality</a>. A <a href="/wiki/Bell_test" title="Bell test">number of experiments</a> have shown in practice that Bell's inequality is not satisfied.<a href="#cite_note-Clauser-6">[6]</a><a href="#cite_note-62">[62]</a><a href="#cite_note-63">[63]</a><a href="#cite_note-64">[64]</a> Moreover, when measurements of the entangled particles are made in moving <a href="/wiki/Special_relativity" title="Special relativity">relativistic</a> reference frames, in which each measurement (in its own relativistic time frame) occurs before the other, the measurement results remain correlated.<a href="#cite_note-65">[65]</a><a href="#cite_note-Gilder2009-38">[38]</a>: 321–324  The fundamental issue about measuring spin along different axes is that these measurements cannot have definite values at the same time―they are <a href="/wiki/Incompatible_observables" class="mw-redirect" title="Incompatible observables">incompatible</a> in the sense that these measurements' maximum simultaneous precision is constrained by the <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a>. This is contrary to what is found in classical physics, where any number of properties can be measured simultaneously with arbitrary accuracy. It has been proven mathematically that compatible measurements cannot show Bell-inequality-violating correlations,<a href="#cite_note-66">[66]</a> and thus entanglement is a fundamentally non-classical phenomenon. <div class="mw-heading mw-heading2"><h2 id="Nonlocality_and_entanglement">Nonlocality and entanglement</h2>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=6" title="Edit section: Nonlocality and entanglement">edit</a>]</div> As discussed above, entanglement is necessary to produce a violation of a <a href="/wiki/Bell%27s_theorem" title="Bell's theorem">Bell inequality</a>. However, the mere presence of entanglement alone is insufficient,<a href="#cite_note-Brunner-RMP2014-67">[67]</a> as Bell himself noted in his 1964 paper.<a href="#cite_note-:4-33">[33]</a> This is demonstrated, for example, by <a href="/wiki/Werner_state" title="Werner state">Werner states</a>, which are a family of states describing pairs of particles. For appropriate choices of the key parameter that identifies a given Werner state within the full set thereof, the Werner states exhibit entanglement. Yet pairs of particles described by Werner states always admit a local hidden variable model. In other words, these states cannot power the violation of a Bell inequality, despite possessing entanglement.<a href="#cite_note-werner1989-68">[68]</a> This can be generalized from pairs of particles to larger collections as well.<a href="#cite_note-Augusiak2015-69">[69]</a> The violation of Bell inequalities is often called <a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">quantum nonlocality</a>. This term is not without controversy.<a href="#cite_note-70">[70]</a> It is sometimes argued that using the term nonlocality carries the unwarranted implication that the violation of Bell inequalities must be explained by physical, faster-than-light signals.<a href="#cite_note-Scarani-71">[71]</a> In other words, the failure of local hidden-variable models to reproduce quantum mechanics is not necessarily a sign of true nonlocality in quantum mechanics itself.<a href="#cite_note-72">[72]</a><a href="#cite_note-73">[73]</a><a href="#cite_note-74">[74]</a> Despite these reservations, the term nonlocality has become a widespread convention.<a href="#cite_note-Scarani-71">[71]</a> The term nonlocality is also sometimes applied to other concepts besides the nonexistence of a local hidden-variable model, such as <a href="/wiki/Unextendible_product_basis" title="Unextendible product basis">whether states can be distinguished by local measurements</a>.<a href="#cite_note-75">[75]</a> Moreover, <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> is often said to be local because <a href="/wiki/Observable" title="Observable">observables</a> defined within spacetime regions that are <a href="/wiki/Spacelike" class="mw-redirect" title="Spacelike">spacelike</a> separated must commute.<a href="#cite_note-Brunner-RMP2014-67">[67]</a><a href="#cite_note-76">[76]</a> These other uses of local and nonlocal are not discussed further here. <div class="mw-heading mw-heading2"><h2 id="Mathematical_details">Mathematical details</h2>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=7" title="Edit section: Mathematical details">edit</a>]</div> The following subsections use the formalism and theoretical framework developed in the articles <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a> and <a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">mathematical formulation of quantum mechanics</a>. <div class="mw-heading mw-heading3"><h3 id="Pure_states">Pure states</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=8" title="Edit section: Pure states">edit</a>]</div> Consider two arbitrary quantum systems A and B, with respective <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a> HA and HB. The Hilbert space of the composite system is the <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{A}\otimes H_{B}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{A}\otimes H_{B}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19fca01f34c847473055f02b00ae2c3e51409901" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.294ex; height:2.509ex;" alt="{\displaystyle H_{A}\otimes H_{B}.}"></dd></dl> If the first system is in state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle _{A}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/435c58e1961f1e68f74f8734418e7c386efd5ef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.529ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle _{A}}"> and the second in state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi \rangle _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle _{B}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2be1ace3f2710536cfa89e755db7a9f4f9f5136e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.417ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle _{B}}">, the state of the composite system is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle _{A}\otimes |\phi \rangle _{B}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle _{A}\otimes |\phi \rangle _{B}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c828da659b247951f5792107fc723ebe7c5577e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.433ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle _{A}\otimes |\phi \rangle _{B}.}"></dd></dl> States of the composite system that can be represented in this form are called separable states, or <a href="/wiki/Product_state" class="mw-redirect" title="Product state">product states</a>. However, not all states of the composite system are separable. Fix a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|i\rangle _{A}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>i</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{|i\rangle _{A}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3744b6c619227428bd614c13bb9aac750b0e82fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.144ex; height:2.843ex;" alt="{\displaystyle \{|i\rangle _{A}\}}"> for HA and a basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|j\rangle _{B}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>j</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{|j\rangle _{B}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e8a89a80bddaf7a469e3ecc3ec555eede209a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.314ex; height:2.843ex;" alt="{\displaystyle \{|j\rangle _{B}\}}"> for HB. The most general state in HA ⊗ HB is of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle _{AB}=\sum _{i,j}c_{ij}|i\rangle _{A}\otimes |j\rangle _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>i</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>j</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle _{AB}=\sum _{i,j}c_{ij}|i\rangle _{A}\otimes |j\rangle _{B}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d211d5745f4b52151d8489776dfbdf1fee13dc1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.75ex; height:5.843ex;" alt="{\displaystyle |\psi \rangle _{AB}=\sum _{i,j}c_{ij}|i\rangle _{A}\otimes |j\rangle _{B}}">.</dd></dl> This state is separable if there exist vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [c_{i}^{A}],[c_{j}^{B}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo>,</mo> <mo stretchy="false">[</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [c_{i}^{A}],[c_{j}^{B}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2b47f980dac5bc92f73609aef7a6280f2f2be7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.579ex; height:3.509ex;" alt="{\displaystyle [c_{i}^{A}],[c_{j}^{B}]}"> so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{ij}=c_{i}^{A}c_{j}^{B},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{ij}=c_{i}^{A}c_{j}^{B},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/febfdcca5d8a1c72ab47af48880d828bb7d60aab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.187ex; height:3.509ex;" alt="{\displaystyle c_{ij}=c_{i}^{A}c_{j}^{B},}"> yielding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |\psi \rangle _{A}=\sum _{i}c_{i}^{A}|i\rangle _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>i</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle |\psi \rangle _{A}=\sum _{i}c_{i}^{A}|i\rangle _{A}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f0bbef833d99d88a2228377927c3b0c01229d78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.559ex; height:3.176ex;" alt="{\textstyle |\psi \rangle _{A}=\sum _{i}c_{i}^{A}|i\rangle _{A}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |\phi \rangle _{B}=\sum _{j}c_{j}^{B}|j\rangle _{B}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>j</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle |\phi \rangle _{B}=\sum _{j}c_{j}^{B}|j\rangle _{B}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d82f19111902fc3a2c36f633c6925a9314f48d72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:18.388ex; height:3.509ex;" alt="{\textstyle |\phi \rangle _{B}=\sum _{j}c_{j}^{B}|j\rangle _{B}.}"> It is inseparable if for any vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [c_{i}^{A}],[c_{j}^{B}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo>,</mo> <mo stretchy="false">[</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [c_{i}^{A}],[c_{j}^{B}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2b47f980dac5bc92f73609aef7a6280f2f2be7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.579ex; height:3.509ex;" alt="{\displaystyle [c_{i}^{A}],[c_{j}^{B}]}"> at least for one pair of coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}^{A},c_{j}^{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}^{A},c_{j}^{B}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79bd35827efb95e0fcfebd87e1b5071ecc50709f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.992ex; height:3.509ex;" alt="{\displaystyle c_{i}^{A},c_{j}^{B}}"> we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{ij}\neq c_{i}^{A}c_{j}^{B}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>≠</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{ij}\neq c_{i}^{A}c_{j}^{B}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edc74f42a3e75058769623c147eb649b8e284d1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.187ex; height:3.509ex;" alt="{\displaystyle c_{ij}\neq c_{i}^{A}c_{j}^{B}.}"> If a state is inseparable, it is called an 'entangled state'.<a href="#cite_note-Rieffel2011-48">[48]</a>: 218 <a href="#cite_note-Zwiebach2022-55">[55]</a>: §1.5  For example, given two basis vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|0\rangle _{A},|1\rangle _{A}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{|0\rangle _{A},|1\rangle _{A}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29ccb462de8b00e179e027c5d0834ac3691bc45b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.716ex; height:2.843ex;" alt="{\displaystyle \{|0\rangle _{A},|1\rangle _{A}\}}"> of HA and two basis vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|0\rangle _{B},|1\rangle _{B}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{|0\rangle _{B},|1\rangle _{B}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/357e2a7e30db6187849c070e3398ff8d031d1178" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.746ex; height:2.843ex;" alt="{\displaystyle \{|0\rangle _{B},|1\rangle _{B}\}}"> of HB, the following is an entangled state: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\sqrt {2}}}\left(|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\sqrt {2}}}\left(|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4053aa0fcea6250c60262618348a170fc18ac03b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.523ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{\sqrt {2}}}\left(|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B}\right).}"></dd></dl> If the composite system is in this state, it is impossible to attribute to either system A or system B a definite <a href="/wiki/Pure_state" class="mw-redirect" title="Pure state">pure state</a>. Another way to say this is that while the <a href="/wiki/Von_Neumann_entropy" title="Von Neumann entropy">von Neumann entropy</a> of the whole state is zero (as it is for any pure state), the entropy of the subsystems is greater than zero. In this sense, the systems are "entangled". The above example is one of four <a href="/wiki/Bell_states" class="mw-redirect" title="Bell states">Bell states</a>, which are (maximally) entangled pure states (pure states of the HA ⊗ HB space, but which cannot be separated into pure states of each HA and HB).<a href="#cite_note-Zwiebach2022-55">[55]</a>: §18.6  Now suppose Alice is an observer for system A, and Bob is an observer for system B. If in the entangled state given above Alice makes a measurement in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|0\rangle ,|1\rangle \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{|0\rangle ,|1\rangle \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a72099dd8347d05e92c98f19db855b1651f8402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.787ex; height:2.843ex;" alt="{\displaystyle \{|0\rangle ,|1\rangle \}}"> eigenbasis of A, there are two possible outcomes, occurring with equal probability: Alice can obtain the outcome 0, or she can obtain the outcome 1. If she obtains the outcome 0, then she can predict with certainty that Bob's result will be 1. Likewise, if she obtains the outcome 1, then she can predict with certainty that Bob's result will be 0. In other words, the results of measurements on the two qubits will be perfectly anti-correlated. This remains true even if the systems A and B are spatially separated. This is the foundation of the EPR paradox.<a href="#cite_note-Nielsen-2010-47">[47]</a>: 113–114  The outcome of Alice's measurement is random. Alice cannot decide which state to collapse the composite system into, and therefore cannot transmit information to Bob by acting on her system. Causality is thus preserved, in this particular scheme. For the general argument, see <a href="/wiki/No-communication_theorem" title="No-communication theorem">no-communication theorem</a>. <div class="mw-heading mw-heading3"><h3 id="Ensembles">Ensembles</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=9" title="Edit section: Ensembles">edit</a>]</div> As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space. More generally, if one has less information about the system, then one calls it an 'ensemble' and describes it by a <a href="/wiki/Density_matrix" title="Density matrix">density matrix</a>, which is a <a href="/wiki/Positive-semidefinite_matrix" class="mw-redirect" title="Positive-semidefinite matrix">positive-semidefinite matrix</a>, or a <a href="/wiki/Trace_class" title="Trace class">trace class</a> when the state space is infinite-dimensional, and which has trace 1. By the <a href="/wiki/Spectral_theorem" title="Spectral theorem">spectral theorem</a>, such a matrix takes the general form: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =\sum _{i}w_{i}|\alpha _{i}\rangle \langle \alpha _{i}|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\sum _{i}w_{i}|\alpha _{i}\rangle \langle \alpha _{i}|,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b4708b78f8713fd851ab3f5d8e97d8f85b873c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.831ex; height:5.509ex;" alt="{\displaystyle \rho =\sum _{i}w_{i}|\alpha _{i}\rangle \langle \alpha _{i}|,}"></dd></dl> where the wi are positive-valued probabilities (they sum up to 1), the vectors αi are unit vectors, and in the infinite-dimensional case, we would take the closure of such states in the trace norm. We can interpret ρ as representing an ensemble where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe22f0329d3ecb2e1880d44d191aba0e5475db68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\displaystyle w_{i}}"> is the proportion of the ensemble whose states are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha _{i}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\alpha _{i}\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0894b0cce03f69124a81e6f240e13af70918a098" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.839ex; height:2.843ex;" alt="{\displaystyle |\alpha _{i}\rangle }">. When a mixed state has rank 1, it therefore describes a 'pure ensemble'. When there is less than total information about the state of a quantum system we need <a href="#Reduced_density_matrices">density matrices</a> to represent the state.<a href="#cite_note-Peres1993-56">[56]</a>: 73–74 <a href="#cite_note-Holevo2001-53">[53]</a>: 13–15 <a href="#cite_note-Zwiebach2022-55">[55]</a>: §22.2  Experimentally, a mixed ensemble might be realized as follows. Consider a "black box" apparatus that spits <a href="/wiki/Electron" title="Electron">electrons</a> towards an observer. The electrons' Hilbert spaces are <a href="/wiki/Identical_particles" class="mw-redirect" title="Identical particles">identical</a>. The apparatus might produce electrons that are all in the same state; in this case, the electrons received by the observer are then a pure ensemble. However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {z} +\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo>+</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {z} +\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/567f15dd6efff02e103e9637ee7cf4954ff92662" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.548ex; height:2.843ex;" alt="{\displaystyle |\mathbf {z} +\rangle }"> with spins aligned in the positive z direction, and the other with state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {y} -\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>−</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {y} -\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2032e156f9a7256cfb8095258bce23a33e0345" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.771ex; height:2.843ex;" alt="{\displaystyle |\mathbf {y} -\rangle }"> with spins aligned in the negative y direction. Generally, this is a mixed ensemble, as there can be any number of populations, each corresponding to a different state. Following the definition above, for a bipartite composite system, mixed states are just density matrices on HA ⊗ HB. That is, it has the general form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =\sum _{i}w_{i}\left[\sum _{j}{\bar {c}}_{ij}(|\alpha _{ij}\rangle \otimes |\beta _{ij}\rangle )\right]\left[\sum _{k}c_{ik}(\langle \alpha _{ik}|\otimes \langle \beta _{ik}|)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>[</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>c</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⊗</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\sum _{i}w_{i}\left[\sum _{j}{\bar {c}}_{ij}(|\alpha _{ij}\rangle \otimes |\beta _{ij}\rangle )\right]\left[\sum _{k}c_{ik}(\langle \alpha _{ik}|\otimes \langle \beta _{ik}|)\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbf2f7c56adc4e273d3e3c73ee6a66e190c3f35c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:56.996ex; height:7.676ex;" alt="{\displaystyle \rho =\sum _{i}w_{i}\left[\sum _{j}{\bar {c}}_{ij}(|\alpha _{ij}\rangle \otimes |\beta _{ij}\rangle )\right]\left[\sum _{k}c_{ik}(\langle \alpha _{ik}|\otimes \langle \beta _{ik}|)\right]}"></dd></dl> where the wi are positively valued probabilities, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{j}|c_{ij}|^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{j}|c_{ij}|^{2}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/407c8b79636e8aafbad96aa6aaaeeefd52d73008" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.843ex; height:3.843ex;" alt="{\textstyle \sum _{j}|c_{ij}|^{2}=1}">, and the vectors are unit vectors. This is self-adjoint and positive and has trace 1. Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as<a href="#cite_note-Laloe-77">[77]</a>: 131–132  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =\sum _{i}w_{i}\rho _{i}^{A}\otimes \rho _{i}^{B},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mo>⊗</mo> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\sum _{i}w_{i}\rho _{i}^{A}\otimes \rho _{i}^{B},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6597264612d7732f42fd586b145e28e9b04aa329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.342ex; height:5.509ex;" alt="{\displaystyle \rho =\sum _{i}w_{i}\rho _{i}^{A}\otimes \rho _{i}^{B},}"></dd></dl> where the wi are positively valued probabilities and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{i}^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{i}^{A}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60b5bc5b7612e68b45a4ce589c0e4c289df55b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.667ex; height:3.176ex;" alt="{\displaystyle \rho _{i}^{A}}">s and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{i}^{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{i}^{B}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/475221b5b58c88f20750357c28e1c45b9087f9bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.682ex; height:3.176ex;" alt="{\displaystyle \rho _{i}^{B}}">s are themselves mixed states (density operators) on the subsystems A and B respectively. In other words, a state is separable if it is a probability distribution over uncorrelated states, or product states. By writing the density matrices as sums of pure ensembles and expanding, we may assume without loss of generality that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{i}^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{i}^{A}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60b5bc5b7612e68b45a4ce589c0e4c289df55b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.667ex; height:3.176ex;" alt="{\displaystyle \rho _{i}^{A}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{i}^{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{i}^{B}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/475221b5b58c88f20750357c28e1c45b9087f9bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.682ex; height:3.176ex;" alt="{\displaystyle \rho _{i}^{B}}"> are themselves pure ensembles. A state is then said to be entangled if it is not separable. In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be <a href="/wiki/NP-hard" class="mw-redirect" title="NP-hard">NP-hard</a>.<a href="#cite_note-78">[78]</a> For the 2 × 2 and 2 × 3 cases, a necessary and sufficient criterion for separability is given by the famous <a href="/wiki/Peres-Horodecki_criterion" class="mw-redirect" title="Peres-Horodecki criterion">Positive Partial Transpose (PPT)</a> condition.<a href="#cite_note-79">[79]</a> <div class="mw-heading mw-heading3"><h3 id="Reduced_density_matrices">Reduced density matrices</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=10" title="Edit section: Reduced density matrices">edit</a>]</div> The idea of a reduced density matrix was introduced by <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a> in 1930.<a href="#cite_note-Dirac1930-80">[80]</a> Consider as above systems A and B each with a Hilbert space HA, HB. Let the state of the composite system be <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi \rangle \in H_{A}\otimes H_{B}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∈</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi \rangle \in H_{A}\otimes H_{B}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b222fc006a0cca79472f2fbb1313f03c2b512490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.494ex; height:2.843ex;" alt="{\displaystyle |\Psi \rangle \in H_{A}\otimes H_{B}.}"></dd></dl> As indicated above, in general there is no way to associate a pure state to the component system A. However, it still is possible to associate a density matrix. Let <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{T}=|\Psi \rangle \;\langle \Psi |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thickmathspace" /> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{T}=|\Psi \rangle \;\langle \Psi |}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/354e47cf72ebfb19333d551ac631ca85d9fc2c0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.054ex; height:2.843ex;" alt="{\displaystyle \rho _{T}=|\Psi \rangle \;\langle \Psi |}">.</dd></dl> which is the <a href="/wiki/Projection_operator" class="mw-redirect" title="Projection operator">projection operator</a> onto this state. The state of A is the <a href="/wiki/Partial_trace" title="Partial trace">partial trace</a> of ρT over the basis of system B: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{A}\ {\stackrel {\mathrm {def} }{=}}\ \sum _{j}^{N_{B}}\left(I_{A}\otimes \langle j|_{B}\right)\left(|\Psi \rangle \langle \Psi |\right)\left(I_{A}\otimes |j\rangle _{B}\right)={\hbox{Tr}}_{B}\;\rho _{T}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>j</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>j</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Tr</mtext> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mspace width="thickmathspace" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{A}\ {\stackrel {\mathrm {def} }{=}}\ \sum _{j}^{N_{B}}\left(I_{A}\otimes \langle j|_{B}\right)\left(|\Psi \rangle \langle \Psi |\right)\left(I_{A}\otimes |j\rangle _{B}\right)={\hbox{Tr}}_{B}\;\rho _{T}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e603e36b10b46642f2ab810508663f112c64b376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:52.82ex; height:7.843ex;" alt="{\displaystyle \rho _{A}\ {\stackrel {\mathrm {def} }{=}}\ \sum _{j}^{N_{B}}\left(I_{A}\otimes \langle j|_{B}\right)\left(|\Psi \rangle \langle \Psi |\right)\left(I_{A}\otimes |j\rangle _{B}\right)={\hbox{Tr}}_{B}\;\rho _{T}.}"></dd></dl> The sum occurs over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{B}:=\dim(H_{B})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>:=</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{B}:=\dim(H_{B})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc45ee38d11c298d2c78941259ec8477b0aee5cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.186ex; height:2.843ex;" alt="{\displaystyle N_{B}:=\dim(H_{B})}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{A}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fad11afeef762471a89c8a258282a40af27d6516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.488ex; height:2.509ex;" alt="{\displaystyle I_{A}}"> the identity operator in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{A}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f561a051435337d6fd20d1b3a2cb945fe97f3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.396ex; height:2.509ex;" alt="{\displaystyle H_{A}}">. ρA is sometimes called the reduced density matrix of ρ on subsystem A. Colloquially, we "trace out" or "trace over" system B to obtain the reduced density matrix on A.<a href="#cite_note-Rieffel2011-48">[48]</a>: 207–212 <a href="#cite_note-Rau2021-51">[51]</a>: 133 <a href="#cite_note-Zwiebach2022-55">[55]</a>: §22.4  For example, the reduced density matrix of A for the entangled state <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\sqrt {2}}}\left(|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\sqrt {2}}}\left(|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a819774559f3e64569ce18b2c7de993d2066d24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.523ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{\sqrt {2}}}\left(|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B}\right),}"></dd></dl> discussed above is<a href="#cite_note-Zwiebach2022-55">[55]</a>: §22.4  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{A}={\tfrac {1}{2}}\left(|0\rangle _{A}\langle 0|_{A}+|1\rangle _{A}\langle 1|_{A}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <mn>1</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{A}={\tfrac {1}{2}}\left(|0\rangle _{A}\langle 0|_{A}+|1\rangle _{A}\langle 1|_{A}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96d96b1e42a9c740ebe571acad39d0844a8473c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:30.209ex; height:3.509ex;" alt="{\displaystyle \rho _{A}={\tfrac {1}{2}}\left(|0\rangle _{A}\langle 0|_{A}+|1\rangle _{A}\langle 1|_{A}\right).}"></dd></dl> This demonstrates that the reduced density matrix for an entangled pure ensemble is a mixed ensemble. In contrast, the density matrix of A for the pure product state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle _{A}\otimes |\phi \rangle _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle _{A}\otimes |\phi \rangle _{B}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ff74a24657c12842e203b89a4427f0cfabd067d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.786ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle _{A}\otimes |\phi \rangle _{B}}"> discussed above is<a href="#cite_note-Nielsen-2010-47">[47]</a>: 106  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{A}=|\psi \rangle _{A}\langle \psi |_{A},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{A}=|\psi \rangle _{A}\langle \psi |_{A},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee961e82ef88b7cf94c6214e0eda56eed6f61c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.471ex; height:2.843ex;" alt="{\displaystyle \rho _{A}=|\psi \rangle _{A}\langle \psi |_{A},}"></dd></dl> the projection operator onto <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle _{A}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/435c58e1961f1e68f74f8734418e7c386efd5ef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.529ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle _{A}}">. In general, a bipartite pure state ρ is entangled if and only if its reduced states are mixed rather than pure.<a href="#cite_note-Rau2021-51">[51]</a>: 131  <div class="mw-heading mw-heading3"><h3 id="Entanglement_as_a_resource">Entanglement as a resource</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=11" title="Edit section: Entanglement as a resource">edit</a>]</div> In quantum information theory, entangled states are considered a 'resource', i.e., something costly to produce and that allows implementing valuable transformations.<a href="#cite_note-Chitambar2019-81">[81]</a><a href="#cite_note-GG-2022-82">[82]</a> The setting in which this perspective is most evident is that of "distant labs", i.e., two quantum systems labelled "A" and "B" on each of which arbitrary <a href="/wiki/Quantum_operation" title="Quantum operation">quantum operations</a> can be performed, but which do not interact with each other quantum mechanically. The only interaction allowed is the exchange of classical information, which combined with the most general local quantum operations gives rise to the class of operations called <a href="/wiki/LOCC" title="LOCC">LOCC</a> (local operations and classical communication). These operations do not allow the production of entangled states between systems A and B. But if A and B are provided with a supply of entangled states, then these, together with LOCC operations can enable a larger class of transformations. If Alice and Bob share an entangled state, Alice can tell Bob over a telephone call how to reproduce a quantum state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e77f6b1e903837c5765c9683da41dd93199621c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |\Psi \rangle }"> she has in her lab. Alice performs a joint measurement on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e77f6b1e903837c5765c9683da41dd93199621c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |\Psi \rangle }"> together with her half of the entangled state and tells Bob the results. Using Alice's results Bob operates on his half of the entangled state to make it equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e77f6b1e903837c5765c9683da41dd93199621c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |\Psi \rangle }">. Since Alice's measurement necessarily erases the quantum state of the system in her lab, the state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e77f6b1e903837c5765c9683da41dd93199621c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |\Psi \rangle }"> is not copied, but transferred: it is said to be "<a href="/wiki/Quantum_teleportation" title="Quantum teleportation">teleported</a>" to Bob's laboratory through this protocol.<a href="#cite_note-Nielsen-2010-47">[47]</a>: 27 <a href="#cite_note-horodecki2007-1">[1]</a>: 875 <a href="#cite_note-83">[83]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Entanglement_swapping.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Entanglement_swapping.svg/220px-Entanglement_swapping.svg.png" decoding="async" width="220" height="193" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Entanglement_swapping.svg/330px-Entanglement_swapping.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Entanglement_swapping.svg/440px-Entanglement_swapping.svg.png 2x" data-file-width="512" data-file-height="450" /></a><figcaption>Entanglement of states from independent sources can be swapped through Bell state measurement.<a href="#cite_note-GuoReview2023-84">[84]</a>: 341 </figcaption></figure> <a href="/wiki/Entanglement_swapping" title="Entanglement swapping">Entanglement swapping</a> is variant of teleportation that allows two parties that have never interacted to share an entangled state. The swapping protocol begins with two EPR sources. One source emits an entangled pair of particles A and B, while the other emits a second entangled pair of particles C and D. Particles B and C are subjected to a measurement in the basis of Bell states. The state of the remaining particles, C and D, collapses to a Bell state, leaving them entangled despite never having interacted with each other.<a href="#cite_note-horodecki2007-1">[1]</a><a href="#cite_note-Pan1998-85">[85]</a> An interaction between a qubit of A and a qubit of B can be realized by first teleporting A's qubit to B, then letting it interact with B's qubit (which is now a LOCC operation, since both qubits are in B's lab) and then teleporting the qubit back to A. Two maximally entangled states of two qubits are used up in this process. Thus entangled states are a resource that enables the realization of quantum interactions (or of quantum channels) in a setting where only LOCC are available, but they are consumed in the process. There are other applications where entanglement can be seen as a resource, e.g., private communication or distinguishing quantum states.<a href="#cite_note-horodecki2007-1">[1]</a> <div class="mw-heading mw-heading3"><h3 id="Multipartite_entanglement">Multipartite entanglement</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=12" title="Edit section: Multipartite entanglement">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Multipartite_entanglement" title="Multipartite entanglement">Multipartite entanglement</a></div> Quantum states describing systems made of more than two pieces can also be entangled. An example for a three-qubit system is the <a href="/wiki/Greenberger%E2%80%93Horne%E2%80%93Zeilinger_state" title="Greenberger–Horne–Zeilinger state">Greenberger–Horne–Zeilinger (GHZ) state</a>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathrm {GHZ} \rangle ={\frac {|000\rangle +|111\rangle }{\sqrt {2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">Z</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>000</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>111</mn> <mo fence="false" stretchy="false">⟩</mo> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathrm {GHZ} \rangle ={\frac {|000\rangle +|111\rangle }{\sqrt {2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/806963b685467e9f5252c8199a6946394a43f460" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.039ex; height:6.676ex;" alt="{\displaystyle |\mathrm {GHZ} \rangle ={\frac {|000\rangle +|111\rangle }{\sqrt {2}}}.}"> Another three-qubit example is the <a href="/wiki/W_state" title="W state">W state</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathrm {W} \rangle ={\frac {|001\rangle +|010\rangle +|100\rangle }{\sqrt {3}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">W</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>001</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>010</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>100</mn> <mo fence="false" stretchy="false">⟩</mo> </mrow> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathrm {W} \rangle ={\frac {|001\rangle +|010\rangle +|100\rangle }{\sqrt {3}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/532e85aa6d0ab1895ef5e81c257ed75f9c7d8d34" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.319ex; height:6.676ex;" alt="{\displaystyle |\mathrm {W} \rangle ={\frac {|001\rangle +|010\rangle +|100\rangle }{\sqrt {3}}}.}"> Tracing out any one of the three qubits turns the GHZ state into a separable state, whereas the result of tracing over any of the three qubits in the W state is still entangled. This illustrates how multipartite entanglement is a more complicated topic than bipartite entanglement: systems composed of three or more parts can exhibit multiple qualitatively different types of entanglement.<a href="#cite_note-Bengtsson2017-49">[49]</a>: 493–497  A single particle cannot be maximally entangled with more than a particle at a time, a property called <a href="/wiki/Monogamy_of_entanglement" title="Monogamy of entanglement">monogamy</a>.<a href="#cite_note-86">[86]</a> <div class="mw-heading mw-heading3"><h3 id="Classification_of_entanglement">Classification of entanglement</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=13" title="Edit section: Classification of entanglement">edit</a>]</div> Not all quantum states are equally valuable as a resource. One method to quantify this value is to use an <a href="#Entanglement_measures">entanglement measure</a> that assigns a numerical value to each quantum state. However, it is often interesting to settle for a coarser way to compare quantum states. This gives rise to different classification schemes. Most entanglement classes are defined based on whether states can be converted to other states using LOCC or a subclass of these operations. The smaller the set of allowed operations, the finer the classification. Important examples are: <ul><li>If two states can be transformed into each other by a local unitary operation, they are said to be in the same LU class. This is the finest of the usually considered classes. Two states in the same LU class have the same value for entanglement measures and the same value as a resource in the distant-labs setting. There is an infinite number of different LU classes (even in the simplest case of two qubits in a pure state).<a href="#cite_note-GRB1998-87">[87]</a><a href="#cite_note-Kraus2010-88">[88]</a></li> <li>If two states can be transformed into each other by local operations including measurements with probability larger than 0, they are said to be in the same 'SLOCC class' ("stochastic LOCC"). Qualitatively, two states <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e0f2d347f2a0ed7f7c9808c427a89813b957017" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:2.176ex;" alt="{\displaystyle \rho _{1}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/793b211571b3ffe34c4639654d567296d29d7f72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:2.176ex;" alt="{\displaystyle \rho _{2}}"> in the same SLOCC class are equally powerful, since one can transform each into the other, but since the transformations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{1}\to \rho _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{1}\to \rho _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/942457dace61f7de964e603f92a3f2f16138d687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.126ex; height:2.343ex;" alt="{\displaystyle \rho _{1}\to \rho _{2}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{2}\to \rho _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{2}\to \rho _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7190e9057b105b0847bbb547e18f29bb647b6996" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.126ex; height:2.343ex;" alt="{\displaystyle \rho _{2}\to \rho _{1}}"> may succeed with different probability, they are no longer equally valuable. E.g., for two pure qubits there are only two SLOCC classes: the entangled states (which contains both the (maximally entangled) Bell states and weakly entangled states like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |00\rangle +0.01|11\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>00</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mn>0.01</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>11</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |00\rangle +0.01|11\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/587cf78fb2f0efce5b5bc27af0fa8fe01d95bf25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.727ex; height:2.843ex;" alt="{\displaystyle |00\rangle +0.01|11\rangle }">) and the separable ones (i.e., product states like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |00\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>00</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |00\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79016644d7bb5d4282f69bf8af1befa3131445fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.876ex; height:2.843ex;" alt="{\displaystyle |00\rangle }">).<a href="#cite_note-89">[89]</a><a href="#cite_note-GoWa2010-90">[90]</a></li> <li>Instead of considering transformations of single copies of a state (like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{1}\to \rho _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{1}\to \rho _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/942457dace61f7de964e603f92a3f2f16138d687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.126ex; height:2.343ex;" alt="{\displaystyle \rho _{1}\to \rho _{2}}">) one can define classes based on the possibility of multi-copy transformations. E.g., there are examples when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{1}\to \rho _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{1}\to \rho _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/942457dace61f7de964e603f92a3f2f16138d687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.126ex; height:2.343ex;" alt="{\displaystyle \rho _{1}\to \rho _{2}}"> is impossible by LOCC, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{1}\otimes \rho _{1}\to \rho _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{1}\otimes \rho _{1}\to \rho _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18f007d3601506321810c4cea20dd31586650876" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.223ex; height:2.509ex;" alt="{\displaystyle \rho _{1}\otimes \rho _{1}\to \rho _{2}}"> is possible. A very important (and very coarse) classification is based on the property whether it is possible to transform an arbitrarily large number of copies of a state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"> into at least one pure entangled state. States that have this property are called <a href="/wiki/Entanglement_distillation" title="Entanglement distillation">distillable</a>. These states are the most useful quantum states since, given enough of them, they can be transformed (with local operations) into any entangled state and hence allow for all possible uses. It came initially as a surprise that not all entangled states are distillable; those that are not are called '<a href="/wiki/Bound_entanglement" title="Bound entanglement">bound entangled</a>'.<a href="#cite_note-HHH97-91">[91]</a><a href="#cite_note-horodecki2007-1">[1]</a></li></ul> A different entanglement classification is based on what the quantum correlations present in a state allow A and B to do: one distinguishes three subsets of entangled states: (1) the <a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">non-local</a> states, which produce correlations that cannot be explained by a local hidden variable model and thus violate a Bell inequality, (2) the <a href="/wiki/Quantum_steering" title="Quantum steering">steerable</a> states that contain sufficient correlations for A to modify ("steer") by local measurements the conditional reduced state of B in such a way, that A can prove to B that the state they possess is indeed entangled, and finally (3) those entangled states that are neither non-local nor steerable. All three sets are non-empty.<a href="#cite_note-WJD2007-92">[92]</a> <div class="mw-heading mw-heading3"><h3 id="Entropy">Entropy</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=14" title="Edit section: Entropy">edit</a>]</div> In this section, the entropy of a mixed state is discussed as well as how it can be viewed as a measure of quantum entanglement. <div class="mw-heading mw-heading4"><h4 id="Definition">Definition</h4>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=15" title="Edit section: Definition">edit</a>]</div> In classical <a href="/wiki/Information_theory" title="Information theory">information theory</a> H, the <a href="/wiki/Shannon_entropy" class="mw-redirect" title="Shannon entropy">Shannon entropy</a>, is associated to a probability distribution, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1},\cdots ,p_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1},\cdots ,p_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3711f3a8dc0f30eca8b99ff7fb660d911007ab7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.879ex; height:2.009ex;" alt="{\displaystyle p_{1},\cdots ,p_{n}}">, in the following way:<a href="#cite_note-SE-93">[93]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(p_{1},\cdots ,p_{n})=-\sum _{i}p_{i}\log _{2}p_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(p_{1},\cdots ,p_{n})=-\sum _{i}p_{i}\log _{2}p_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/875a53f0d12455b69f2b0ee8230fc98e151bc87a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.084ex; height:5.509ex;" alt="{\displaystyle H(p_{1},\cdots ,p_{n})=-\sum _{i}p_{i}\log _{2}p_{i}.}"></dd></dl> Since a mixed state ρ is a probability distribution over an ensemble, this leads naturally to the definition of the <a href="/wiki/Von_Neumann_entropy" title="Von Neumann entropy">von Neumann entropy</a>:<a href="#cite_note-Peres1993-56">[56]</a>: 264  <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\rho )=-{\hbox{Tr}}\left(\rho \log _{2}{\rho }\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Tr</mtext> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mi>ρ</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\rho )=-{\hbox{Tr}}\left(\rho \log _{2}{\rho }\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e68e4845f55bd5775f163c25624fb141d1d1330" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.441ex; height:2.843ex;" alt="{\displaystyle S(\rho )=-{\hbox{Tr}}\left(\rho \log _{2}{\rho }\right),}"></dd></dl> which can be expressed in terms of the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of ρ: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\rho )=-{\hbox{Tr}}\left(\rho \log _{2}{\rho }\right)=-\sum _{i}\lambda _{i}\log _{2}\lambda _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Tr</mtext> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mi>ρ</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\rho )=-{\hbox{Tr}}\left(\rho \log _{2}{\rho }\right)=-\sum _{i}\lambda _{i}\log _{2}\lambda _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/928c2d6ade5615fad5fe82e67e17ca9b07640bb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.553ex; height:5.509ex;" alt="{\displaystyle S(\rho )=-{\hbox{Tr}}\left(\rho \log _{2}{\rho }\right)=-\sum _{i}\lambda _{i}\log _{2}\lambda _{i}}">.</dd></dl> Since an event of probability 0 should not contribute to the entropy, and given that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{p\to 0}p\log p=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mi>p</mi> <mi>log</mi> <mo>⁡</mo> <mi>p</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{p\to 0}p\log p=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4050c3b4a3183e1f8af387f25474c7e83bd980" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.063ex; width:14.735ex; height:4.176ex;" alt="{\displaystyle \lim _{p\to 0}p\log p=0,}"></dd></dl> the convention 0 log(0) = 0 is adopted. When a pair of particles is described by the spin singlet state discussed above, the von Neumann entropy of either particle is log(2), which can be shown to be the maximum entropy for 2 × 2 mixed states.<a href="#cite_note-Holevo2001-53">[53]</a>: 15  <div class="mw-heading mw-heading4"><h4 id="As_a_measure_of_entanglement">As a measure of entanglement</h4>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=16" title="Edit section: As a measure of entanglement">edit</a>]</div> Entropy provides one tool that can be used to quantify entanglement, although other entanglement measures exist.<a href="#cite_note-Plenio-94">[94]</a><a href="#cite_note-Vedral2002-95">[95]</a> If the overall system is pure, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems. For bipartite pure states, the von Neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure.<a href="#cite_note-96">[96]</a> It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {1/n, ..., 1/n}.<a href="#cite_note-Nielsen-2010-47">[47]</a>: 505  Therefore, a bipartite pure state ρ ∈ HA ⊗ HB is said to be a maximally entangled state if the reduced state of each subsystem of ρ is the diagonal matrix<a href="#cite_note-97">[97]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\frac {1}{n}}&&\\&\ddots &\\&&{\frac {1}{n}}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mtd> <mtd /> <mtd /> </mtr> <mtr> <mtd /> <mtd> <mo>⋱</mo> </mtd> <mtd /> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\frac {1}{n}}&&\\&\ddots &\\&&{\frac {1}{n}}\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac2c4a2dbfa47eec57c86a3f4939f6aaf0adc23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:15.768ex; height:12.176ex;" alt="{\displaystyle {\begin{bmatrix}{\frac {1}{n}}&&\\&\ddots &\\&&{\frac {1}{n}}\end{bmatrix}}.}"></dd></dl> For mixed states, the reduced von Neumann entropy is not the only reasonable entanglement measure.<a href="#cite_note-Bengtsson2017-49">[49]</a>: 471  <a href="/wiki/R%C3%A9nyi_entropy" title="Rényi entropy">Rényi entropy</a> also can be used as a measure of entanglement.<a href="#cite_note-Bengtsson2017-49">[49]</a>: 447, 480 <a href="#cite_note-98">[98]</a> <div class="mw-heading mw-heading3"><h3 id="Entanglement_measures">Entanglement measures</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=17" title="Edit section: Entanglement measures">edit</a>]</div> Entanglement measures quantify the amount of entanglement in a (often viewed as a bipartite) quantum state. As aforementioned, <a href="/wiki/Entropy_of_entanglement" title="Entropy of entanglement">entanglement entropy</a> is the standard measure of entanglement for pure states (but no longer a measure of entanglement for mixed states). For mixed states, there are some entanglement measures in the literature<a href="#cite_note-Plenio-94">[94]</a> and no single one is standard. <ul><li>Entanglement cost</li> <li><a href="/wiki/Entanglement_distillation" title="Entanglement distillation">Distillable entanglement</a></li> <li><a href="/wiki/Entanglement_of_formation" title="Entanglement of formation">Entanglement of formation</a></li> <li><a href="/wiki/Concurrence_(quantum_computing)" title="Concurrence (quantum computing)">Concurrence</a></li> <li><a href="/wiki/Quantum_relative_entropy" title="Quantum relative entropy">Relative entropy of entanglement</a></li> <li><a href="/wiki/Squashed_entanglement" title="Squashed entanglement">Squashed entanglement</a></li> <li><a href="/wiki/Negativity_(quantum_mechanics)#Logarithmic_negativity" title="Negativity (quantum mechanics)">Logarithmic negativity</a></li></ul> Most (but not all) of these entanglement measures reduce for pure states to entanglement entropy, and are difficult (<a href="/wiki/NP-hard" class="mw-redirect" title="NP-hard">NP-hard</a>) to compute for mixed states as the dimension of the entangled system grows.<a href="#cite_note-99">[99]</a> <div class="mw-heading mw-heading3"><h3 id="Quantum_field_theory">Quantum field theory</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=18" title="Edit section: Quantum field theory">edit</a>]</div> The <a href="/wiki/Reeh%E2%80%93Schlieder_theorem" title="Reeh–Schlieder theorem">Reeh–Schlieder theorem</a> of <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> is sometimes interpreted as saying that entanglement is omnipresent in the <a href="/wiki/Quantum_vacuum" class="mw-redirect" title="Quantum vacuum">quantum vacuum</a>.<a href="#cite_note-100">[100]</a> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=19" title="Edit section: Applications">edit</a>]</div> Entanglement has many applications in <a href="/wiki/Quantum_information_theory" class="mw-redirect" title="Quantum information theory">quantum information theory</a>. With the aid of entanglement, otherwise impossible tasks may be achieved. Among the best-known applications of entanglement are <a href="/wiki/Superdense_coding" title="Superdense coding">superdense coding</a> and quantum teleportation.<a href="#cite_note-Bouwmeester-1997-42">[42]</a> Most researchers believe that entanglement is necessary to realize <a href="/wiki/Quantum_computer" class="mw-redirect" title="Quantum computer">quantum computing</a> (although this is disputed by some).<a href="#cite_note-jozsa02-101">[101]</a> Entanglement is used in some protocols of <a href="/wiki/Quantum_cryptography" title="Quantum cryptography">quantum cryptography</a>,<a href="#cite_note-ekert91-39">[39]</a><a href="#cite_note-horodecki10-102">[102]</a> but to prove the security of <a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">quantum key distribution</a> (QKD) under standard assumptions does not require entanglement.<a href="#cite_note-103">[103]</a> However, the <a href="/wiki/Device-independent_quantum_cryptography" title="Device-independent quantum cryptography">device independent</a> security of QKD is shown exploiting entanglement between the communication partners.<a href="#cite_note-104">[104]</a> In August 2014, Brazilian researcher Gabriela Barreto Lemos, from the University of Vienna, and team were able to "take pictures" of objects using photons that had not interacted with the subjects, but were entangled with photons that did interact with such objects.<a href="#cite_note-105">[105]</a> The idea has been adapted to make infrared images using only standard cameras that are insensitive to infrared.<a href="#cite_note-106">[106]</a> <div class="mw-heading mw-heading3"><h3 id="Entangled_states">Entangled states</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=20" title="Edit section: Entangled states">edit</a>]</div> There are several canonical entangled states that appear often in theory and experiments. For two <a href="/wiki/Qubits" class="mw-redirect" title="Qubits">qubits</a>, the <a href="/wiki/Bell_state" title="Bell state">Bell states</a> are <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Phi ^{\pm }\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}\pm |1\rangle _{A}\otimes |1\rangle _{B})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Phi ^{\pm }\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}\pm |1\rangle _{A}\otimes |1\rangle _{B})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c5e3e97f9337b2928160879b1039b554fa4efd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:38.848ex; height:6.176ex;" alt="{\displaystyle |\Phi ^{\pm }\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}\pm |1\rangle _{A}\otimes |1\rangle _{B})}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi ^{\pm }\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}\pm |1\rangle _{A}\otimes |0\rangle _{B}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi ^{\pm }\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}\pm |1\rangle _{A}\otimes |0\rangle _{B}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3ad61f595b438e703d3fdf2ff88bbe5946b7e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:39.625ex; height:6.176ex;" alt="{\displaystyle |\Psi ^{\pm }\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}\pm |1\rangle _{A}\otimes |0\rangle _{B}).}"></dd></dl> These four pure states are all maximally entangled and form an <a href="/wiki/Orthonormal" class="mw-redirect" title="Orthonormal">orthonormal</a> <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> of the Hilbert space of the two qubits.<a href="#cite_note-Rieffel2011-48">[48]</a>: 38–39 <a href="#cite_note-Nielsen-2010-47">[47]</a>: 98  They provide examples of how quantum mechanics can violate <a href="/wiki/Bell%27s_theorem" title="Bell's theorem">Bell-type inequalities</a>.<a href="#cite_note-Rieffel2011-48">[48]</a>: 62 <a href="#cite_note-Nielsen-2010-47">[47]</a>: 116  For M > 2 qubits, the <a href="/wiki/Greenberger%E2%80%93Horne%E2%80%93Zeilinger_state" title="Greenberger–Horne–Zeilinger state">GHZ state</a> is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathrm {GHZ} \rangle ={\frac {|0\rangle ^{\otimes M}+|1\rangle ^{\otimes M}}{\sqrt {2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">Z</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msup> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⊗</mo> <mi>M</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <msup> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⊗</mo> <mi>M</mi> </mrow> </msup> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathrm {GHZ} \rangle ={\frac {|0\rangle ^{\otimes M}+|1\rangle ^{\otimes M}}{\sqrt {2}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e51e220f2b023d20c1e3d4c7461928e4bbafc1bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.864ex; height:6.843ex;" alt="{\displaystyle |\mathrm {GHZ} \rangle ={\frac {|0\rangle ^{\otimes M}+|1\rangle ^{\otimes M}}{\sqrt {2}}},}"></dd></dl> which reduces to the Bell state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Phi ^{+}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Phi ^{+}\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4570afa1cbf91ef240b83f9ad1476558042f4361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.74ex; height:3.009ex;" alt="{\displaystyle |\Phi ^{+}\rangle }"> for M = 2. The traditional GHZ state was defined for M = 3. GHZ states are occasionally extended to <a href="/wiki/Qudit" class="mw-redirect" title="Qudit">qudits</a>, i.e., systems of d rather than 2 dimensions.<a href="#cite_note-107">[107]</a><a href="#cite_note-108">[108]</a> Also for M > 2 qubits, there are <a href="/wiki/Spin_squeezing" title="Spin squeezing">spin squeezed states</a>, a class of <a href="/wiki/Squeezed_coherent_states" class="mw-redirect" title="Squeezed coherent states">squeezed coherent states</a> satisfying certain restrictions on the uncertainty of spin measurements, which are necessarily entangled.<a href="#cite_note-109">[109]</a> Spin squeezed states are good candidates for enhancing precision measurements using quantum entanglement.<a href="#cite_note-110">[110]</a> For two <a href="/wiki/Boson" title="Boson">bosonic</a> modes, a <a href="/wiki/NOON_state" title="NOON state">NOON state</a> is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi _{\text{NOON}}\rangle ={\frac {|N\rangle _{a}|0\rangle _{b}+|{0}\rangle _{a}|{N}\rangle _{b}}{\sqrt {2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>NOON</mtext> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>N</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi _{\text{NOON}}\rangle ={\frac {|N\rangle _{a}|0\rangle _{b}+|{0}\rangle _{a}|{N}\rangle _{b}}{\sqrt {2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f8fd5adf9cfd79648f2cceb5f5534b61c91e72b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.478ex; height:6.676ex;" alt="{\displaystyle |\psi _{\text{NOON}}\rangle ={\frac {|N\rangle _{a}|0\rangle _{b}+|{0}\rangle _{a}|{N}\rangle _{b}}{\sqrt {2}}}.}"></dd></dl> This is like the Bell state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi ^{+}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi ^{+}\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb516a603ad0d72c91c8e3e736aaad102d073f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.87ex; height:3.009ex;" alt="{\displaystyle |\Psi ^{+}\rangle }"> except the basis states <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |0\rangle }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |1\rangle }"> have been replaced with "the N photons are in one mode" and "the N photons are in the other mode".<a href="#cite_note-Kishore2007-111">[111]</a> Finally, there also exist <a href="/w/index.php?title=Twin_Fock_states&action=edit&redlink=1" class="new" title="Twin Fock states (page does not exist)">twin Fock states</a> for bosonic modes, which can be created by feeding a <a href="/wiki/Fock_state" title="Fock state">Fock state</a> into two arms leading to a beam splitter. They are the sum of multiple NOON states, and can be used to achieve the <a href="/wiki/Heisenberg_limit" class="mw-redirect" title="Heisenberg limit">Heisenberg limit</a>.<a href="#cite_note-112">[112]</a> For the appropriately chosen measures of entanglement, Bell, GHZ, and NOON states are maximally entangled while spin squeezed and twin Fock states are only partially entangled.<a href="#cite_note-113">[113]</a><a href="#cite_note-Kishore2007-111">[111]</a><a href="#cite_note-114">[114]</a> <div class="mw-heading mw-heading3"><h3 id="Methods_of_creating_entanglement">Methods of creating entanglement</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=21" title="Edit section: Methods of creating entanglement">edit</a>]</div> Entanglement is usually created by direct interactions between subatomic particles. These interactions can take numerous forms. One of the most commonly used methods is <a href="/wiki/Spontaneous_parametric_down-conversion" title="Spontaneous parametric down-conversion">spontaneous parametric down-conversion</a> to generate a pair of photons entangled in polarization.<a href="#cite_note-horodecki2007-1">[1]</a><a href="#cite_note-Shadbolt2012-115">[115]</a> Other methods include the use of a <a href="/w/index.php?title=Fibre_coupler&action=edit&redlink=1" class="new" title="Fibre coupler (page does not exist)">fibre coupler</a> to confine and mix photons, photons emitted from decay cascade of the bi-exciton in a <a href="/wiki/Quantum_dot" title="Quantum dot">quantum dot</a>,<a href="#cite_note-116">[116]</a> or the use of the <a href="/wiki/Hong%E2%80%93Ou%E2%80%93Mandel_effect" title="Hong–Ou–Mandel effect">Hong–Ou–Mandel effect</a>.<a href="#cite_note-117">[117]</a> Quantum entanglement of a <a href="/wiki/Elementary_particle" title="Elementary particle">particle</a> and its <a href="/wiki/Antiparticle" title="Antiparticle">antiparticle</a>, such as an electron and a <a href="/wiki/Positron" title="Positron">positron</a>, can be created by partial overlap of the corresponding <a href="/wiki/Quantum_wave_function" class="mw-redirect" title="Quantum wave function">quantum wave functions</a> in <a href="/wiki/Hardy%27s_paradox" title="Hardy's paradox">Hardy's interferometer</a>.<a href="#cite_note-Hardy1992-118">[118]</a><a href="#cite_note-Georgiev2022-119">[119]</a> In the earliest tests of Bell's theorem, the entangled particles were generated using <a href="/wiki/Atomic_cascade" class="mw-redirect" title="Atomic cascade">atomic cascades</a>.<a href="#cite_note-Clauser-6">[6]</a> It is also possible to create entanglement between quantum systems that never directly interacted, through the use of <a href="/wiki/Quantum_teleportation#Entanglement_swapping" title="Quantum teleportation">entanglement swapping</a>. Two independently prepared, identical particles may also be entangled if their wave functions merely spatially overlap, at least partially.<a href="#cite_note-120">[120]</a> <div class="mw-heading mw-heading3"><h3 id="Testing_a_system_for_entanglement">Testing a system for entanglement</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=22" title="Edit section: Testing a system for entanglement">edit</a>]</div> A density matrix ρ is called separable if it can be written as a convex sum of product states, namely <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rho =\sum _{j}p_{j}\rho _{j}^{(A)}\otimes \rho _{j}^{(B)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>⊗</mo> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rho =\sum _{j}p_{j}\rho _{j}^{(A)}\otimes \rho _{j}^{(B)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/595cfaf3cbe5191d7b447eecd2003515a410ae2a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:20.869ex; height:6.009ex;" alt="{\displaystyle {\rho =\sum _{j}p_{j}\rho _{j}^{(A)}\otimes \rho _{j}^{(B)}}}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq p_{j}\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq p_{j}\leq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be98598a3b3c77ddc979cf9a2617effaa0f62c47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.601ex; height:2.843ex;" alt="{\displaystyle 0\leq p_{j}\leq 1}"> probabilities. By definition, a state is entangled if it is not separable. For 2-qubit and qubit-qutrit systems (2 × 2 and 2 × 3 respectively) the simple <a href="/wiki/Peres%E2%80%93Horodecki_criterion" title="Peres–Horodecki criterion">Peres–Horodecki criterion</a> provides both a necessary and a sufficient criterion for separability, and thus—inadvertently—for detecting entanglement. However, for the general case, the criterion is merely a necessary one for separability, as the problem becomes NP-hard when generalized.<a href="#cite_note-NP-hard1-121">[121]</a><a href="#cite_note-NP-hard2-122">[122]</a> Other separability criteria include (but not limited to) the <a href="/wiki/Range_criterion" title="Range criterion">range criterion</a>, <a href="/wiki/Reduction_criterion" title="Reduction criterion">reduction criterion</a>, and those based on uncertainty relations.<a href="#cite_note-123">[123]</a><a href="#cite_note-124">[124]</a><a href="#cite_note-125">[125]</a><a href="#cite_note-126">[126]</a> See Ref.<a href="#cite_note-127">[127]</a> for a review of separability criteria in discrete-variable systems and Ref.<a href="#cite_note-FriisEtAl2019entanglement-128">[128]</a> for a review on techniques and challenges in experimental entanglement certification in discrete-variable systems. A numerical approach to the problem is suggested by <a href="/wiki/Jon_Magne_Leinaas" title="Jon Magne Leinaas">Jon Magne Leinaas</a>, <a href="/wiki/Jan_Myrheim" title="Jan Myrheim">Jan Myrheim</a> and <a href="/w/index.php?title=Eirik_Ovrum&action=edit&redlink=1" class="new" title="Eirik Ovrum (page does not exist)">Eirik Ovrum</a> in their paper "Geometrical aspects of entanglement".<a href="#cite_note-129">[129]</a> Leinaas et al. offer a numerical approach, iteratively refining an estimated separable state towards the target state to be tested, and checking if the target state can indeed be reached. In continuous variable systems, the Peres–Horodecki criterion also applies. Specifically, Simon<a href="#cite_note-130">[130]</a> formulated a particular version of the Peres–Horodecki criterion in terms of the second-order moments of canonical operators and showed that it is necessary and sufficient for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\oplus 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⊕</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\oplus 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/910ec7d56dcdfb30a81400532fa3399299261342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.165ex; height:2.343ex;" alt="{\displaystyle 1\oplus 1}">-mode Gaussian states (see Ref.<a href="#cite_note-131">[131]</a> for a seemingly different but essentially equivalent approach). It was later found<a href="#cite_note-132">[132]</a> that Simon's condition is also necessary and sufficient for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\oplus n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⊕</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\oplus n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49fdecc4a94f6e2bf4ea50e1cebd24d9e346173c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle 1\oplus n}">-mode Gaussian states, but no longer sufficient for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\oplus 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>⊕</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\oplus 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60ea3c2d231414b3e14766f90606b747314d3fd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.165ex; height:2.343ex;" alt="{\displaystyle 2\oplus 2}">-mode Gaussian states. Simon's condition can be generalized by taking into account the higher order moments of canonical operators<a href="#cite_note-133">[133]</a><a href="#cite_note-134">[134]</a> or by using entropic measures.<a href="#cite_note-135">[135]</a><a href="#cite_note-136">[136]</a> <div class="mw-heading mw-heading3"><h3 id="In_quantum_gravity">In quantum gravity</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=23" title="Edit section: In quantum gravity">edit</a>]</div> There is a fundamental conflict, referred to as the <a href="/wiki/Problem_of_time" title="Problem of time">problem of time</a>, between the way the concept of time is used in quantum mechanics, and the role it plays in <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. In standard quantum theories time acts as an independent background through which states evolve, while general relativity treats time as a dynamical variable which relates directly with matter. Part of the effort to reconcile these approaches to time results in the <a href="/wiki/Wheeler%E2%80%93DeWitt_equation" title="Wheeler–DeWitt equation">Wheeler–DeWitt equation</a>, which predicts the state of the universe is timeless or static, contrary to ordinary experience.<a href="#cite_note-Moreva2014-137">[137]</a> Work started by <a href="/wiki/Don_Page_(physicist)" title="Don Page (physicist)">Don Page</a> and <a href="/wiki/William_Wootters" title="William Wootters">William Wootters</a><a href="#cite_note-138">[138]</a><a href="#cite_note-139">[139]</a><a href="#cite_note-140">[140]</a> suggests that the universe appears to evolve for observers on the inside because of energy entanglement between an evolving system and a clock system, both within the universe.<a href="#cite_note-Moreva2014-137">[137]</a> In this way the overall system can remain timeless while parts experience time via entanglement. The issue remains an open question closely related to attempts at theories of <a href="/wiki/Quantum_gravity" title="Quantum gravity">quantum gravity</a>.<a href="#cite_note-141">[141]</a><a href="#cite_note-142">[142]</a> In general relativity, gravity arises from the curvature of spacetime and that curvature derives from the distribution of matter. However, matter is governed by quantum mechanics. Integration of these two theories faces many problems. In an (unrealistic) model space called the <a href="/wiki/Anti-de_Sitter_space" title="Anti-de Sitter space">anti-de Sitter space</a>, the <a href="/wiki/AdS/CFT_correspondence" title="AdS/CFT correspondence">AdS/CFT correspondence</a> allows a quantum gravitational system to be related to a quantum field theory without gravity.<a href="#cite_note-Swingle2018-143">[143]</a> Using this correspondence, <a href="/wiki/Mark_Van_Raamsdonk" title="Mark Van Raamsdonk">Mark Van Raamsdonk</a> suggested that <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> arises as an emergent phenomenon of the quantum degrees of freedom that are entangled and live in the boundary of the spacetime.<a href="#cite_note-144">[144]</a> <div class="mw-heading mw-heading2"><h2 id="Experiments_demonstrating_and_using_entanglement">Experiments demonstrating and using entanglement</h2>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=24" title="Edit section: Experiments demonstrating and using entanglement">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Bell_tests">Bell tests</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=25" title="Edit section: Bell tests">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Bell_test" title="Bell test">Bell test</a></div> A <a href="/wiki/Bell_test" title="Bell test">Bell test</a>, also known as Bell inequality test or Bell experiment, is a real-world physics experiment designed to test the theory of quantum mechanics against the hypothesis of local hidden variables. These tests empirically evaluate the implications of <a href="/wiki/Bell%27s_theorem" title="Bell's theorem">Bell's theorem</a>. To date, all Bell tests have found that the hypothesis of local hidden variables is inconsistent with the way that physical systems behave. Many types of Bell tests have been performed in physics laboratories, often with the goal of ameliorating problems of experimental design or set-up that could in principle affect the validity of the findings of earlier Bell tests. This is known as "closing loopholes in Bell tests". In earlier tests, it could not be ruled out that the result at one point could have been subtly transmitted to the remote point, affecting the outcome at the second location.<a href="#cite_note-:2-9">[9]</a> However, so-called "loophole-free" Bell tests have since been performed where the locations were sufficiently separated that communications at the speed of light would have taken longer—in one case, 10,000 times longer—than the interval between the measurements.<a href="#cite_note-:1-8">[8]</a><a href="#cite_note-:0-7">[7]</a><a href="#cite_note-NTR-20151021-15">[15]</a><a href="#cite_note-hanson-36">[36]</a> In 2017, Yin et al. reported setting a new quantum entanglement distance record of 1,203 km, demonstrating the survival of a two-photon pair and a violation of a Bell inequality, reaching a <a href="/wiki/CHSH_inequality" title="CHSH inequality">CHSH valuation</a> of 2.37±0.09, under strict Einstein locality conditions, from the <a href="/wiki/Quantum_Experiments_at_Space_Scale" title="Quantum Experiments at Space Scale">Micius satellite</a> to bases in Lijian, Yunnan and Delingha, Qinghai, increasing the efficiency of transmission over prior fiberoptic experiments by an order of magnitude.<a href="#cite_note-145">[145]</a><a href="#cite_note-146">[146]</a> <div class="mw-heading mw-heading3"><h3 id="Entanglement_of_top_quarks">Entanglement of top quarks</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=26" title="Edit section: Entanglement of top quarks">edit</a>]</div> In 2023 the <a href="/wiki/Large_Hadron_Collider" title="Large Hadron Collider">LHC</a> using techniques from <a href="/wiki/Quantum_tomography" title="Quantum tomography">quantum tomography</a> measured entanglement at the highest energy so far,<a href="#cite_note-147">[147]</a><a href="#cite_note-148">[148]</a><a href="#cite_note-149">[149]</a> a rare intersection between quantum information and high energy physics based on theoretical work first proposed in 2021.<a href="#cite_note-150">[150]</a> The experiment was carried by the <a href="/wiki/ATLAS_experiment" title="ATLAS experiment">ATLAS</a> detector measuring the spin of top-quark pair production and the effect was observed with a more than 5<a href="/wiki/Standard_deviation" title="Standard deviation">σ</a> level of significance, the top quark is the heaviest known particle and therefore has a very short lifetime (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"> ≈ 10−25 s) being the only quark that decays before undergoing <a href="/wiki/Hadronization" title="Hadronization">hadronization</a> (~ 10−23 s) and spin decorrelation (~ 10−21 s), so the spin information is transferred without much loss to the leptonic decays products that will be caught by the detector.<a href="#cite_note-151">[151]</a> The <a href="/wiki/Spin_polarization" title="Spin polarization">spin polarization</a> and correlation of the particles was measured and tested for entanglement with <a href="/wiki/Concurrence_(quantum_computing)" title="Concurrence (quantum computing)">concurrence</a> as well as the <a href="/wiki/Peres%E2%80%93Horodecki_criterion" title="Peres–Horodecki criterion">Peres–Horodecki criterion</a> and subsequently the effect has been confirmed too in the <a href="/wiki/CMS_experiment" class="mw-redirect" title="CMS experiment">CMS</a> detector.<a href="#cite_note-152">[152]</a><a href="#cite_note-153">[153]</a> <div class="mw-heading mw-heading3"><h3 id="Entanglement_of_macroscopic_objects">Entanglement of macroscopic objects</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=27" title="Edit section: Entanglement of macroscopic objects">edit</a>]</div> In 2020, researchers reported the quantum entanglement between the <a href="/wiki/Vibrations_of_a_circular_membrane" title="Vibrations of a circular membrane">motion of a millimetre-sized mechanical oscillator</a> and a disparate distant spin system of a cloud of atoms.<a href="#cite_note-154">[154]</a> Later work complemented this work by quantum-entangling two mechanical oscillators.<a href="#cite_note-155">[155]</a><a href="#cite_note-156">[156]</a><a href="#cite_note-157">[157]</a> <div class="mw-heading mw-heading3"><h3 id="Entanglement_of_elements_of_living_systems">Entanglement of elements of living systems</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=28" title="Edit section: Entanglement of elements of living systems">edit</a>]</div> In October 2018, physicists reported producing quantum entanglement using <a href="/wiki/Living_organism" class="mw-redirect" title="Living organism">living organisms</a>, particularly between photosynthetic molecules within living <a href="/wiki/Bacteria" title="Bacteria">bacteria</a> and <a href="/wiki/Photon" title="Photon">quantized light</a>.<a href="#cite_note-JPC-20181010-158">[158]</a><a href="#cite_note-SA-20181029-159">[159]</a> Living organisms (green sulphur bacteria) have been studied as mediators to create quantum entanglement between otherwise non-interacting light modes, showing high entanglement between light and bacterial modes, and to some extent, even entanglement within the bacteria.<a href="#cite_note-160">[160]</a> <div class="mw-heading mw-heading3"><h3 id="Entanglement_of_quarks_and_gluons_in_protons">Entanglement of quarks and gluons in protons</h3>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=29" title="Edit section: Entanglement of quarks and gluons in protons">edit</a>]</div> Physicists at <a href="/wiki/Brookhaven_National_Laboratory" title="Brookhaven National Laboratory">Brookhaven National Laboratory</a> demonstrated quantum entanglement within <a href="/wiki/Protons" class="mw-redirect" title="Protons">protons</a>, showing <a href="/wiki/Quarks" class="mw-redirect" title="Quarks">quarks</a> and <a href="/wiki/Gluons" class="mw-redirect" title="Gluons">gluons</a> are interdependent rather than isolated particles.<a href="#cite_note-161">[161]</a> Using high-energy electron-proton collisions, they revealed maximal entanglement, reshaping our understanding of proton structure.<a href="#cite_note-162">[162]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=30" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 16em;"> <ul><li><a href="/wiki/Concurrence_(quantum_computing)" title="Concurrence (quantum computing)">Concurrence</a></li> <li><a href="/wiki/Controlled_NOT_gate" title="Controlled NOT gate">CNOT gate</a></li> <li><a href="/wiki/Einstein%27s_thought_experiments" title="Einstein's thought experiments">Einstein's thought experiments</a></li> <li><a href="/wiki/Entanglement_witness" title="Entanglement witness">Entanglement witness</a></li> <li><a href="/wiki/ER_%3D_EPR" title="ER = EPR">ER = EPR</a></li> <li><a href="/wiki/Multipartite_entanglement" title="Multipartite entanglement">Multipartite entanglement</a></li> <li><a href="/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent" class="mw-redirect" title="Normally distributed and uncorrelated does not imply independent">Normally distributed and uncorrelated does not imply independent</a></li> <li><a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a></li> <li><a href="/wiki/Quantum_coherence" class="mw-redirect" title="Quantum coherence">Quantum coherence</a></li> <li><a href="/wiki/Quantum_discord" title="Quantum discord">Quantum discord</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_phase_transition" title="Quantum phase transition">Quantum phase transition</a></li> <li><a href="/wiki/Quantum_pseudo-telepathy" title="Quantum pseudo-telepathy">Quantum pseudo-telepathy</a></li> <li><a href="/wiki/Retrocausality" title="Retrocausality">Retrocausality</a></li> <li><a href="/wiki/Squashed_entanglement" title="Squashed entanglement">Squashed entanglement</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach experiment</a></li> <li><a href="/wiki/John_Clive_Ward" title="John Clive Ward">Ward's probability amplitude</a></li></ul> </div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output 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.reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-horodecki2007-1">^ <a href="#cite_ref-horodecki2007_1-0">a</a> <a href="#cite_ref-horodecki2007_1-1">b</a> <a href="#cite_ref-horodecki2007_1-2">c</a> <a href="#cite_ref-horodecki2007_1-3">d</a> <a href="#cite_ref-horodecki2007_1-4">e</a> <a href="#cite_ref-horodecki2007_1-5">f</a> <a href="#cite_ref-horodecki2007_1-6">g</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHorodeckiHorodeckiHorodeckiHorodecki2009" class="citation journal cs1">Horodecki, Ryszard; Horodecki, Pawel; Horodecki, Michal; Horodecki, Karol (2009). 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To avoid such action at a distance they have to attribute, to the space-time regions in question, real properties in advance of observation, correlated properties, which predetermine the outcomes of these particular observations. Since these real properties, fixed in advance of observation, are not contained in quantum formalism, that formalism for EPR is incomplete. It may be correct, as far as it goes, but the usual quantum formalism cannot be the whole story." And again on p. 144 Bell says: "Einstein had no difficulty accepting that affairs in different places could be correlated. What he could not accept was that an intervention at one place could influence, immediately, affairs at the other." Downloaded 5 July 2011 from <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBell1987" class="citation book cs1">Bell, J. S. 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Quantum Information Processing. 16 (5): 118. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1607.00164">1607.00164</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017QuIP...16..118B">2017QuIP...16..118B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11128-017-1568-0">10.1007/s11128-017-1568-0</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:43754114">43754114</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quantum+Information+Processing&rft.atitle=Generalized+concurrence+measure+for+faithful+quantification+of+multiparticle+pure+state+entanglement+using+Lagrange%27s+identity+and+wedge+product&rft.volume=16&rft.issue=5&rft.pages=118&rft.date=2017&rft_id=info%3Aarxiv%2F1607.00164&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A43754114%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs11128-017-1568-0&rft_id=info%3Abibcode%2F2017QuIP...16..118B&rft.aulast=Bhaskara&rft.aufirst=VS&rft.au=Panigrahi%2C+PK&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+entanglement" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSwainBhaskaraPanigrahi2022" class="citation journal cs1">Swain SN, Bhaskara VS, Panigrahi PK (2022). "Generalized entanglement measure for continuous-variable systems". Physical Review A. 105 (5): 052441. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1706.01448">1706.01448</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2022PhRvA.105e2441S">2022PhRvA.105e2441S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevA.105.052441">10.1103/PhysRevA.105.052441</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:239885759">239885759</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+A&rft.atitle=Generalized+entanglement+measure+for+continuous-variable+systems&rft.volume=105&rft.issue=5&rft.pages=052441&rft.date=2022&rft_id=info%3Aarxiv%2F1706.01448&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A239885759%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevA.105.052441&rft_id=info%3Abibcode%2F2022PhRvA.105e2441S&rft.aulast=Swain&rft.aufirst=SN&rft.au=Bhaskara%2C+VS&rft.au=Panigrahi%2C+PK&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+entanglement" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJaeger2009" class="citation book cs1">Jaeger, G. (2009). Entanglement, Information, and the Interpretation of Quantum Mechanics. Heildelberg, Germany: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-92127-1" title="Special:BookSources/978-3-540-92127-1"><bdi>978-3-540-92127-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Entanglement%2C+Information%2C+and+the+Interpretation+of+Quantum+Mechanics&rft.place=Heildelberg%2C+Germany&rft.pub=Springer&rft.date=2009&rft.isbn=978-3-540-92127-1&rft.aulast=Jaeger&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+entanglement" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteward2008" class="citation book cs1">Steward, E. G. (2008). Quantum Mechanics: Its Early Development and the Road to Entanglement. Imperial College Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-86094-978-4" title="Special:BookSources/978-1-86094-978-4"><bdi>978-1-86094-978-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics%3A+Its+Early+Development+and+the+Road+to+Entanglement&rft.pub=Imperial+College+Press&rft.date=2008&rft.isbn=978-1-86094-978-4&rft.aulast=Steward&rft.aufirst=E.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+entanglement" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilde2017" class="citation book cs1"><a href="/wiki/Mark_Wilde" title="Mark Wilde">Wilde, Mark M.</a> (2017). Quantum Information Theory (2nd ed.). Cambridge University Press. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1106.1445">1106.1445</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2F9781316809976">10.1017/9781316809976</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781316809976" title="Special:BookSources/9781316809976"><bdi>9781316809976</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Information+Theory&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2017&rft_id=info%3Aarxiv%2F1106.1445&rft_id=info%3Adoi%2F10.1017%2F9781316809976&rft.isbn=9781316809976&rft.aulast=Wilde&rft.aufirst=Mark+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+entanglement" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Quantum_entanglement&action=edit&section=33" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><a href="/wiki/File:Wikiquote-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/34px-Wikiquote-logo.svg.png" decoding="async" width="34" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/51px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/68px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></a></div> <div class="side-box-text plainlist">Wikiquote has quotations related to <a href="https://en.wikiquote.org/wiki/Special:Search/Quantum_entanglement" class="extiw" title="q:Special:Search/Quantum entanglement">Quantum entanglement</a>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=xM3GOXaci7w">Explanatory video by Scientific American magazine</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20121025073450/http://www.didaktik.physik.uni-erlangen.de/quantumlab/english/index.html">Entanglement experiment with photon pairs – interactive</a></li> <li>Audio – Cain/Gay (2009) <a rel="nofollow" class="external text" href="http://www.astronomycast.com/physics/ep-140-entanglement/">Astronomy Cast</a> Entanglement</li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=ta09WXiUqcQ">"Spooky Actions at a Distance?": Oppenheimer Lecture, Prof. David Mermin (Cornell University) Univ. California, Berkeley, 2008.</a> Non-mathematical popular lecture on YouTube, posted Mar 2008</li> <li><a rel="nofollow" class="external text" href="https://demonstrations.wolfram.com/QuantumEntanglementVersusClassicalCorrelation/">"Quantum Entanglement versus Classical Correlation" (Interactive demonstration)</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 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.navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Quantum_mechanics332" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics_topics" title="Template:Quantum mechanics topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics_topics" title="Template talk:Quantum mechanics topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics_topics" title="Special:EditPage/Template:Quantum mechanics topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_mechanics332" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a> <ul><li><a href="/wiki/Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Born_rule" title="Born rule">Born rule</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a> <ul><li><a href="/wiki/Ground_state" title="Ground state">Ground state</a></li> <li><a href="/wiki/Excited_state" title="Excited state">Excited state</a></li> <li><a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li> <li><a class="mw-selflink selflink">Entanglement</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a href="/wiki/Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a> <ul><li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat_in_popular_culture" title="Schrödinger's cat in popular culture">in popular culture</a></li></ul></li> <li><a href="/wiki/Wigner%27s_friend" title="Wigner's friend">Wigner's friend</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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