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<span>Protocol</span> </div> </a> <ul id="toc-Protocol-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Experimental_results_and_records" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Experimental_results_and_records"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Experimental results and records</span> </div> </a> <button aria-controls="toc-Experimental_results_and_records-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Experimental results and records subsection</span> </button> <ul id="toc-Experimental_results_and_records-sublist" class="vector-toc-list"> <li id="toc-Quantum_teleportation_over_143_km" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_teleportation_over_143_km"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Quantum teleportation over 143 km</span> </div> </a> <ul id="toc-Quantum_teleportation_over_143_km-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_teleportation_across_the_Danube_River" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_teleportation_across_the_Danube_River"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Quantum teleportation across the Danube River</span> </div> </a> <ul id="toc-Quantum_teleportation_across_the_Danube_River-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Deterministic_quantum_teleportation_with_atoms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Deterministic_quantum_teleportation_with_atoms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Deterministic quantum teleportation with atoms</span> </div> </a> <ul id="toc-Deterministic_quantum_teleportation_with_atoms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ground-to-satellite_quantum_teleportation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ground-to-satellite_quantum_teleportation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Ground-to-satellite quantum teleportation</span> </div> </a> <ul id="toc-Ground-to-satellite_quantum_teleportation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_teleportation_over_internet_cables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_teleportation_over_internet_cables"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Quantum teleportation over internet cables</span> </div> </a> <ul id="toc-Quantum_teleportation_over_internet_cables-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Formal_presentation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Formal_presentation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Formal presentation</span> </div> </a> <ul id="toc-Formal_presentation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Certifying_quantum_teleportation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Certifying_quantum_teleportation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Certifying quantum teleportation</span> </div> </a> <ul id="toc-Certifying_quantum_teleportation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alternative_notations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Alternative_notations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Alternative notations</span> </div> </a> <ul id="toc-Alternative_notations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Entanglement_swapping" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Entanglement_swapping"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Entanglement swapping</span> </div> </a> <button aria-controls="toc-Entanglement_swapping-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Entanglement swapping subsection</span> </button> <ul id="toc-Entanglement_swapping-sublist" class="vector-toc-list"> <li id="toc-Algorithm_for_swapping_Bell_pairs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algorithm_for_swapping_Bell_pairs"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Algorithm for swapping Bell pairs</span> </div> </a> <ul id="toc-Algorithm_for_swapping_Bell_pairs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations_of_the_teleportation_protocol" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations_of_the_teleportation_protocol"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Generalizations of the teleportation protocol</span> </div> </a> <button aria-controls="toc-Generalizations_of_the_teleportation_protocol-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Generalizations of the teleportation protocol subsection</span> </button> <ul id="toc-Generalizations_of_the_teleportation_protocol-sublist" class="vector-toc-list"> <li id="toc-d-dimensional_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#d-dimensional_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span><i>d</i>-dimensional systems</span> </div> </a> <ul id="toc-d-dimensional_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multipartite_versions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multipartite_versions"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Multipartite versions</span> </div> </a> <ul id="toc-Multipartite_versions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Logic_gate_teleportation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Logic_gate_teleportation"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Logic gate teleportation</span> </div> </a> <button aria-controls="toc-Logic_gate_teleportation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Logic gate teleportation subsection</span> </button> <ul id="toc-Logic_gate_teleportation-sublist" class="vector-toc-list"> <li id="toc-General_description" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_description"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>General description</span> </div> </a> <ul id="toc-General_description-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_details" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Further_details"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Further details</span> </div> </a> <ul id="toc-Further_details-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Local_explanation_of_the_phenomenon" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Local_explanation_of_the_phenomenon"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Local explanation of the phenomenon</span> </div> </a> <ul id="toc-Local_explanation_of_the_phenomenon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Recent_developments" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Recent_developments"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Recent developments</span> </div> </a> <button aria-controls="toc-Recent_developments-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Recent developments subsection</span> </button> <ul id="toc-Recent_developments-sublist" class="vector-toc-list"> <li id="toc-Higher_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Higher_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>Higher dimensions</span> </div> </a> <ul id="toc-Higher_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Information_quality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Information_quality"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.2</span> <span>Information quality</span> </div> </a> <ul id="toc-Information_quality-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"> <span class="vector-toc-numb">12</span> <span>See also</span> </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"> <span class="vector-toc-numb">13</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Specific" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Specific"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.1</span> <span>Specific</span> </div> </a> <ul id="toc-Specific-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.2</span> <span>General</span> </div> </a> <ul id="toc-General-sublist" class="vector-toc-list"> </ul> </li> </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"> <span class="vector-toc-numb">14</span> <span>External links</span> </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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </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"><span class="mw-page-title-main">Quantum teleportation</span></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 41 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-41" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">41 languages</span> </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/Quantenteleportation" title="Quantenteleportation – Alemannic" lang="gsw" hreflang="gsw" data-title="Quantenteleportation" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%86%D8%AA%D9%82%D8%A7%D9%84_%D8%A2%D9%86%D9%8A_%D9%83%D9%85%D9%8A" title="انتقال آني كمي – Arabic" lang="ar" hreflang="ar" data-title="انتقال آني كمي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></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%9F%E0%A7%87%E0%A6%B2%E0%A6%BF%E0%A6%AA%E0%A7%8B%E0%A6%B0%E0%A7%8D%E0%A6%9F%E0%A7%87%E0%A6%B6%E0%A6%A8" title="কোয়ান্টাম টেলিপোর্টেশন – Bangla" lang="bn" hreflang="bn" data-title="কোয়ান্টাম টেলিপোর্টেশন" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%B0%D0%B2%D0%B0%D1%8F_%D1%82%D1%8D%D0%BB%D0%B5%D0%BF%D0%B0%D1%80%D1%82%D0%B0%D1%86%D1%8B%D1%8F" title="Квантавая тэлепартацыя – Belarusian" lang="be" hreflang="be" data-title="Квантавая тэлепартацыя" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></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%B0_%D1%82%D0%B5%D0%BB%D0%B5%D0%BF%D0%BE%D1%80%D1%82%D0%B0%D1%86%D0%B8%D1%8F" title="Квантова телепортация – Bulgarian" lang="bg" hreflang="bg" data-title="Квантова телепортация" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kvantna_teleportacija" title="Kvantna teleportacija – Bosnian" lang="bs" hreflang="bs" data-title="Kvantna teleportacija" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teleportaci%C3%B3_qu%C3%A0ntica" title="Teleportació quàntica – Catalan" lang="ca" hreflang="ca" data-title="Teleportació quàntica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kvantov%C3%A1_teleportace" title="Kvantová teleportace – Czech" lang="cs" hreflang="cs" data-title="Kvantová teleportace" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kvanteteleportation" title="Kvanteteleportation – Danish" lang="da" hreflang="da" data-title="Kvanteteleportation" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D8%AA%D9%86%D9%82%D8%A7%D9%84%D9%81%D9%84%D8%AD%D9%8A%D9%86_%D9%83%D9%88%D8%A7%D9%86%D8%AA%D9%8A" title="تنقالفلحين كوانتي – Moroccan Arabic" lang="ary" hreflang="ary" data-title="تنقالفلحين كوانتي" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Quantenteleportation" title="Quantenteleportation – German" lang="de" hreflang="de" data-title="Quantenteleportation" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kvantteleportatsioon" title="Kvantteleportatsioon – Estonian" lang="et" hreflang="et" data-title="Kvantteleportatsioon" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teleportaci%C3%B3n_cu%C3%A1ntica" title="Teleportación cuántica – Spanish" lang="es" hreflang="es" data-title="Teleportación cuántica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D9%88%D8%B1%D9%86%D9%88%D8%B1%D8%AF%DB%8C_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C" title="دورنوردی کوانتومی – Persian" lang="fa" hreflang="fa" data-title="دورنوردی کوانتومی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/T%C3%A9l%C3%A9portation_quantique" title="Téléportation quantique – French" lang="fr" hreflang="fr" data-title="Téléportation quantique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Teileap%C3%B3rt%C3%A1il_chandamach" title="Teileapórtáil chandamach – Irish" lang="ga" hreflang="ga" data-title="Teileapórtáil chandamach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teleportaci%C3%B3n_cu%C3%A1ntica" title="Teleportación cuántica – Galician" lang="gl" hreflang="gl" data-title="Teleportación cuántica" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></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%88%9C%EA%B0%84_%EC%9D%B4%EB%8F%99" title="양자 순간 이동 – Korean" lang="ko" hreflang="ko" data-title="양자 순간 이동" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></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%BF%D5%A5%D5%AC%D5%A5%D5%BA%D5%B8%D6%80%D5%BF%D5%A1%D6%81%D5%AB%D5%A1" title="Քվանտային տելեպորտացիա – Armenian" lang="hy" hreflang="hy" data-title="Քվանտային տելեպորտացիա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teleportasi_kuantum" title="Teleportasi kuantum – Indonesian" lang="id" hreflang="id" data-title="Teleportasi kuantum" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teletrasporto_quantistico" title="Teletrasporto quantistico – Italian" lang="it" hreflang="it" data-title="Teletrasporto quantistico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%98%D7%9C%D7%A4%D7%95%D7%A8%D7%98%D7%A6%D7%99%D7%94_%D7%A7%D7%95%D7%95%D7%A0%D7%98%D7%99%D7%AA" title="טלפורטציה קוונטית – Hebrew" lang="he" hreflang="he" data-title="טלפורטציה קוונטית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Teleportatio_quantica" title="Teleportatio quantica – Latin" lang="la" hreflang="la" data-title="Teleportatio quantica" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></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%9F%E0%B5%86%E0%B4%B2%E0%B4%BF%E0%B4%AA%E0%B5%8D%E0%B4%AA%E0%B5%8B%E0%B5%BC%E0%B4%9F%E0%B5%8D%E0%B4%9F%E0%B5%87%E0%B4%B7%E0%B5%BB" title="ക്വാണ്ടം ടെലിപ്പോർട്ടേഷൻ – Malayalam" lang="ml" hreflang="ml" data-title="ക്വാണ്ടം ടെലിപ്പോർട്ടേഷൻ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kwantumteleportatie" title="Kwantumteleportatie – Dutch" lang="nl" hreflang="nl" data-title="Kwantumteleportatie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></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%83%86%E3%83%AC%E3%83%9D%E3%83%BC%E3%83%86%E3%83%BC%E3%82%B7%E3%83%A7%E3%83%B3" title="量子テレポーテーション – Japanese" lang="ja" hreflang="ja" data-title="量子テレポーテーション" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></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%9F%E0%A9%88%E0%A8%B2%E0%A9%80%E0%A8%AA%E0%A9%8B%E0%A8%B0%E0%A8%9F%E0%A9%87%E0%A8%B8%E0%A8%BC%E0%A8%A8" title="ਕੁਆਂਟਮ ਟੈਲੀਪੋਰਟੇਸ਼ਨ – Punjabi" lang="pa" hreflang="pa" data-title="ਕੁਆਂਟਮ ਟੈਲੀਪੋਰਟੇਸ਼ਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%DA%A9%D9%88%D8%A7%D9%86%D9%BC%D9%85%D9%8A_%D9%BC%DB%8C%D9%84%D9%8A_%D9%BE%D9%88%D8%B1%D9%BC%DB%90%D8%B4%D9%86" title="کوانټمي ټیلي پورټېشن – Pashto" lang="ps" hreflang="ps" data-title="کوانټمي ټیلي پورټېشن" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Teleportacja_kwantowa" title="Teleportacja kwantowa – Polish" lang="pl" hreflang="pl" data-title="Teleportacja kwantowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teletransporte_qu%C3%A2ntico" title="Teletransporte quântico – Portuguese" lang="pt" hreflang="pt" data-title="Teletransporte quântico" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teleportare_cuantic%C4%83" title="Teleportare cuantică – Romanian" lang="ro" hreflang="ro" data-title="Teleportare cuantică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></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_%D1%82%D0%B5%D0%BB%D0%B5%D0%BF%D0%BE%D1%80%D1%82%D0%B0%D1%86%D0%B8%D1%8F" title="Квантовая телепортация – Russian" lang="ru" hreflang="ru" data-title="Квантовая телепортация" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Quantum_teleportation" title="Quantum teleportation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Quantum teleportation" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BD%D0%B0_%D1%82%D0%B5%D0%BB%D0%B5%D0%BF%D0%BE%D1%80%D1%82%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Квантна телепортација – Serbian" lang="sr" hreflang="sr" data-title="Квантна телепортација" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kvanttiteleportaatio" title="Kvanttiteleportaatio – Finnish" lang="fi" hreflang="fi" data-title="Kvanttiteleportaatio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kvantteleportering" title="Kvantteleportering – Swedish" lang="sv" hreflang="sv" data-title="Kvantteleportering" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></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_%D1%82%D0%B5%D0%BB%D0%B5%D0%BF%D0%BE%D1%80%D1%82%D0%B0%D1%86%D0%B8%D1%8F%D1%81%D0%B5" title="Квант телепортациясе – Tatar" lang="tt" hreflang="tt" data-title="Квант телепортациясе" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kuantum_%C4%B1%C5%9F%C4%B1nlama" title="Kuantum ışınlama – Turkish" lang="tr" hreflang="tr" data-title="Kuantum ışınlama" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a 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<div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Physical phenomenon</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Quantum_teleportation_video.ogg/330px--Quantum_teleportation_video.ogg.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="330" height="165" data-durationhint="26" data-mwtitle="Quantum_teleportation_video.ogg" data-mwprovider="wikimediacommons" resource="/wiki/File:Quantum_teleportation_video.ogg"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d9/Quantum_teleportation_video.ogg/Quantum_teleportation_video.ogg.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="428" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d9/Quantum_teleportation_video.ogg/Quantum_teleportation_video.ogg.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="640" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d9/Quantum_teleportation_video.ogg/Quantum_teleportation_video.ogg.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="960" /><source src="//upload.wikimedia.org/wikipedia/commons/d/d9/Quantum_teleportation_video.ogg" type="video/ogg; codecs="theora, vorbis"" data-width="2500" data-height="1250" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d9/Quantum_teleportation_video.ogg/Quantum_teleportation_video.ogg.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="128" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d9/Quantum_teleportation_video.ogg/Quantum_teleportation_video.ogg.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="214" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d9/Quantum_teleportation_video.ogg/Quantum_teleportation_video.ogg.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="320" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d9/Quantum_teleportation_video.ogg/Quantum_teleportation_video.ogg.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="320" /></video></span><figcaption>Schematic video demonstrating individual steps of quantum teleportation. A quantum state Q is sent from station A to station B using a pair of entangled particles created by source S. Station A measures its two particles and communicates the result to station B, which chooses an appropriate device based on the received message. Due to the action of the device, the state of the particle of station B turns into Q.</figcaption></figure> <p><b>Quantum teleportation</b> is a technique for transferring <a href="/wiki/Quantum_information" title="Quantum information">quantum information</a> from a sender at one location to a receiver some distance away. While <a href="/wiki/Teleportation" title="Teleportation">teleportation</a> is commonly portrayed in science fiction as a means to transfer physical objects from one location to the next, quantum teleportation only transfers quantum information. The sender does not have to know the particular quantum state being transferred. Moreover, the location of the recipient can be unknown, but to complete the quantum teleportation, classical information needs to be sent from sender to receiver. Because classical information needs to be sent, quantum teleportation cannot occur faster than the speed of light. </p><p>One of the first scientific articles to investigate quantum teleportation is "Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels"<sup id="cite_ref-BBCJPW93_1-0" class="reference"><a href="#cite_note-BBCJPW93-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> published by <a href="/wiki/Charles_H._Bennett_(physicist)" title="Charles H. Bennett (physicist)">C. H. Bennett</a>, <a href="/wiki/Gilles_Brassard" title="Gilles Brassard">G. Brassard</a>, <a href="/wiki/Claude_Cr%C3%A9peau" title="Claude Crépeau">C. Crépeau</a>, <a href="/wiki/Richard_Jozsa" title="Richard Jozsa">R. Jozsa</a>, <a href="/wiki/Asher_Peres" title="Asher Peres">A. Peres</a>, and <a href="/wiki/William_Wootters" title="William Wootters">W. K. Wootters</a> in 1993, in which they proposed using dual communication methods to send/receive quantum information. It was experimentally realized in 1997 by two research groups, led by <a href="/wiki/Sandu_Popescu" title="Sandu Popescu">Sandu Popescu</a> and <a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Anton Zeilinger</a>, respectively.<sup id="cite_ref-Rome1998_2-0" class="reference"><a href="#cite_note-Rome1998-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Bouwmeester-1997_3-0" class="reference"><a href="#cite_note-Bouwmeester-1997-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Experimental determinations<sup id="cite_ref-Feng,_Tianfeng_2020_4-0" class="reference"><a href="#cite_note-Feng,_Tianfeng_2020-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> of quantum teleportation have been made in information content – including photons, atoms, electrons, and <a href="/wiki/Superconducting_quantum_computing" title="Superconducting quantum computing">superconducting circuits</a> – as well as distance, with 1,400 km (870 mi) being the longest distance of successful teleportation by <a href="/wiki/Jian-Wei_Pan" class="mw-redirect" title="Jian-Wei Pan">Jian-Wei Pan</a>'s team using the <a href="/wiki/Quantum_Experiments_at_Space_Scale" title="Quantum Experiments at Space Scale">Micius satellite</a> for space-based quantum teleportation.<sup id="cite_ref-sat1400_6-0" class="reference"><a href="#cite_note-sat1400-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Non-technical_summary">Non-technical summary</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=1" title="Edit section: Non-technical summary"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In matters relating to <a href="/wiki/Quantum_information" title="Quantum information">quantum</a> <a href="/wiki/Information_theory" title="Information theory">information theory</a>, it is convenient to work with the simplest possible unit of information: the two-state system of the <a href="/wiki/Qubit" title="Qubit">qubit</a>. The qubit functions as the quantum analog of the classic computational part, the <a href="/wiki/Bit" title="Bit">bit</a>, as it can have a measurement value of <i>both</i> a 0 <i>and</i> a 1, whereas the classical bit can only be measured as a 0 <i>or</i> a 1. The quantum two-state system seeks to transfer quantum information from one location to another location without losing the information and preserving the quality of this information. This process involves moving the information <i>between carriers</i> and not movement of the <i>actual carriers</i>, similar to the traditional process of communications, as two parties remain stationary while the information (digital media, voice, text, etc.) is being transferred, contrary to the implications of the word "teleport". The main components needed for teleportation include a sender, the information (a qubit), a traditional channel, a quantum channel, and a receiver. The sender does not need to know the exact contents of the information being sent. The measurement postulate of quantum mechanics – when a measurement is made upon a quantum state, any subsequent measurements will "collapse" or that the observed state will be lost – creates an imposition within teleportation: if a sender measures their information, the state could collapse when the receiver obtains the data since the state had changed from when the sender made the initial measurement and in so making it different. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Quantum_teleportation_-_interactive_simulation_in_Virtual_Lab_by_Quantum_Flytrap.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Quantum_teleportation_-_interactive_simulation_in_Virtual_Lab_by_Quantum_Flytrap.png/440px-Quantum_teleportation_-_interactive_simulation_in_Virtual_Lab_by_Quantum_Flytrap.png" decoding="async" width="440" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Quantum_teleportation_-_interactive_simulation_in_Virtual_Lab_by_Quantum_Flytrap.png/660px-Quantum_teleportation_-_interactive_simulation_in_Virtual_Lab_by_Quantum_Flytrap.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Quantum_teleportation_-_interactive_simulation_in_Virtual_Lab_by_Quantum_Flytrap.png/880px-Quantum_teleportation_-_interactive_simulation_in_Virtual_Lab_by_Quantum_Flytrap.png 2x" data-file-width="940" data-file-height="534" /></a><figcaption>An interactive simulation of quantum teleportation in the Virtual Lab by Quantum Flytrap,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a rel="nofollow" class="external text" href="https://lab.quantumflytrap.com/lab/quantum-teleportation">available online</a>. In this optical setup, qubits are encoded using the <a href="/wiki/Polarization_(waves)" title="Polarization (waves)">polarization of light</a>. Teleportation occurs between the source photon (set to an arbitrary state) and one photon from an entangled pair. A Bell pair measurement is performed on the source photon and one entangled photon using a <a href="/wiki/Controlled_NOT_gate" title="Controlled NOT gate">quantum CNOT gate</a>, yielding two bits of classical information. The target photon is then rotated with two controllable <a href="/wiki/Waveplate" title="Waveplate">waveplates</a> based on this information.</figcaption></figure> <p>For actual teleportation, it is required that an <a href="/wiki/Entangled_state" class="mw-redirect" title="Entangled state">entangled quantum state</a> be created for the qubit to be transferred. Entanglement imposes statistical correlations between otherwise distinct physical systems by creating or placing two or more separate particles into a single, shared quantum state. This intermediate state contains two particles whose quantum states are related to each other: measuring one particle's state provides information about the measurement of the other particle's state. These correlations hold even when measurements are chosen and performed independently, out of causal contact from one another, as verified in <a href="/wiki/Bell_test_experiments" class="mw-redirect" title="Bell test experiments">Bell test experiments</a>. Thus, an observation resulting from a measurement choice made at one point in spacetime seems to instantaneously affect outcomes in another region, even though light hasn't yet had time to travel the distance, a conclusion seemingly at odds with <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>. This is known as the <a href="/wiki/EPR_paradox" class="mw-redirect" title="EPR paradox">EPR paradox</a>. However, such correlations can never be used to transmit any information faster than the speed of light, a statement encapsulated in the <a href="/wiki/No-communication_theorem" title="No-communication theorem">no-communication theorem</a>. Thus, teleportation as a whole can never be <a href="/wiki/Superluminal" class="mw-redirect" title="Superluminal">superluminal</a>, as a qubit cannot be reconstructed until the accompanying classical information arrives. </p><p>The sender will combine the particle, whose information is teleported, with one of the entangled particles, causing a change of the overall entangled quantum state. Of this changed state, the particles in the receiver's possession are then sent to an analyzer that will measure the change of the entangled state. The "change" measurement will allow the receiver to recreate the original information that the sender had, resulting in the information being teleported or carried between two people that have different locations. Since the initial quantum information is "destroyed" as it becomes part of the entangled state, the no-cloning theorem is maintained as the information is recreated from the entangled state and not copied during teleportation. </p><p>The <a href="/wiki/Quantum_channel" title="Quantum channel">quantum channel</a> is the communication mechanism that is used for all quantum information transmission and is the channel used for teleportation (relationship of quantum channel to traditional communication channel is akin to the qubit being the quantum analog of the classical bit). However, in addition to the quantum channel, a traditional channel must also be used to accompany a qubit to "preserve" the quantum information. When the change measurement between the original qubit and the entangled particle is made, the measurement result must be carried by a traditional channel so that the quantum information can be reconstructed and the receiver can get the original information. Because of this need for the traditional channel, the speed of teleportation can be no faster than the speed of light (hence the <a href="/wiki/No-communication_theorem" title="No-communication theorem">no-communication theorem</a> is not violated). The main advantage with this is that Bell states can be shared using <a href="/wiki/Photon" title="Photon">photons</a> from <a href="/wiki/Laser" title="Laser">lasers</a>, making teleportation achievable through open space, as there is no need to send information through physical cables or optical fibers. </p><p>Quantum states can be encoded in various degrees of freedom of atoms. For example, qubits can be encoded in the degrees of freedom of electrons surrounding the <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">atomic nucleus</a> or in the degrees of freedom of the nucleus itself. Thus, performing this kind of teleportation requires a stock of atoms at the receiving site, available for having qubits imprinted on them.<sup id="cite_ref-barrett_8-0" class="reference"><a href="#cite_note-barrett-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>As of 2015,<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Quantum_teleportation&action=edit">[update]</a></sup> the quantum states of single photons, photon modes, single atoms, atomic ensembles, defect centers in solids, single electrons, and superconducting circuits have been employed as information bearers.<sup id="cite_ref-Pirandola-2015_9-0" class="reference"><a href="#cite_note-Pirandola-2015-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Understanding quantum teleportation requires a good grounding in finite-dimensional <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a> and <a href="/wiki/Projection_matrix" title="Projection matrix">projection matrices</a>. A qubit is described using a two-dimensional <a href="/wiki/Complex_number" title="Complex number">complex number</a>-valued <a href="/wiki/Vector_space" title="Vector space">vector space</a> (a Hilbert space), which are the primary basis for the formal manipulations given below. A working knowledge of quantum mechanics is not absolutely required to understand the mathematics of quantum teleportation, although without such acquaintance, the deeper meaning of the equations may remain quite mysterious. </p> <div class="mw-heading mw-heading2"><h2 id="Protocol">Protocol</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=2" title="Edit section: Protocol"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Quantum_teleportation_diagram.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Quantum_teleportation_diagram.PNG/300px-Quantum_teleportation_diagram.PNG" decoding="async" width="300" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Quantum_teleportation_diagram.PNG/450px-Quantum_teleportation_diagram.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Quantum_teleportation_diagram.PNG/600px-Quantum_teleportation_diagram.PNG 2x" data-file-width="1019" data-file-height="896" /></a><figcaption> Diagram for quantum teleportation of a photon</figcaption></figure> <p>The resources required for quantum teleportation are a <a href="/wiki/Communication_channel" title="Communication channel">communication channel</a> capable of transmitting two classical bits, a means of generating an entangled <a href="/wiki/Bell_state" title="Bell state">Bell state</a> of qubits and distributing to two different locations, performing a <a href="/wiki/Bell_measurement" class="mw-redirect" title="Bell measurement">Bell measurement</a> on one of the Bell state qubits, and manipulating the quantum state of the other qubit from the pair. Of course, there must also be some input qubit (in the quantum state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312d43de853a9e6ca74888e63394fc8081f56a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle }"></span>) to be teleported. The <a href="/wiki/Protocol_(computing)" class="mw-redirect" title="Protocol (computing)">protocol</a> is then as follows: </p> <ol><li>A Bell state is generated with one qubit sent to location A and the other sent to location B.</li> <li>A Bell measurement of the Bell state qubit and the qubit to be teleported (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312d43de853a9e6ca74888e63394fc8081f56a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle }"></span> ) is performed at location A. This yields one of four measurement outcomes which can be encoded in two classical bits of information. Both qubits at location A are then discarded.</li> <li>Using the classical channel, the two bits are sent from A to B. (This is the only potentially time-consuming step after step 1 since information transfer is limited by the speed of light.)</li> <li>As a result of the measurement performed at location A, the Bell state qubit at location B is in one of four possible states. Of these four possible states, one is identical to the original quantum state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312d43de853a9e6ca74888e63394fc8081f56a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle }"></span> , and the other three are closely related. The identity of the state actually obtained is encoded in two classical bits and sent to location B. The Bell state qubit at location B is then modified in one of three ways, or not at all, which results in a qubit identical to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312d43de853a9e6ca74888e63394fc8081f56a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle }"></span>, the state of the qubit that was chosen for teleportation.</li></ol> <p>It is worth noticing that the above protocol assumes that the qubits are individually addressable, meaning that the qubits are distinguishable and physically labeled. However, there can be situations where two identical qubits are indistinguishable due to the spatial overlap of their wave functions. Under this condition, the qubits cannot be individually controlled or measured. Nevertheless, a teleportation protocol analogous to that described above can still be (conditionally) implemented by exploiting two independently prepared qubits, with no need of an initial Bell state. This can be made by addressing the internal degrees of freedom of the qubits (e.g., spins or polarisations) by spatially localized measurements performed in separated regions A and B where the two spatially overlapping, indistinguishable qubits can be found.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This theoretical prediction has been then verified experimentally via polarized photons in a quantum optical setup.<sup id="cite_ref-Sunetal_Teleportation_11-0" class="reference"><a href="#cite_note-Sunetal_Teleportation-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Experimental_results_and_records">Experimental results and records</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=3" title="Edit section: Experimental results and records"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Work in 1998 verified the initial predictions,<sup id="cite_ref-Rome1998_2-1" class="reference"><a href="#cite_note-Rome1998-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and the distance of teleportation was increased in August 2004 to 600 meters, using <a href="/wiki/Optical_fiber" title="Optical fiber">optical fiber</a>.<sup id="cite_ref-Danube2004_12-0" class="reference"><a href="#cite_note-Danube2004-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Subsequently, the record distance for quantum teleportation has been gradually increased to 16 kilometres (9.9 mi),<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> then to 97 km (60 mi),<sup id="cite_ref-Ma-2012_14-0" class="reference"><a href="#cite_note-Ma-2012-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> and is now 143 km (89 mi), set in open air experiments in the <a href="/wiki/Canary_Islands" title="Canary Islands">Canary Islands</a>, done between the two <a href="/wiki/Astronomical_observatory" class="mw-redirect" title="Astronomical observatory">astronomical observatories</a> of the <a href="/wiki/Instituto_de_Astrof%C3%ADsica_de_Canarias" title="Instituto de Astrofísica de Canarias">Instituto de Astrofísica de Canarias</a>.<sup id="cite_ref-Ma-2012_14-1" class="reference"><a href="#cite_note-Ma-2012-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> There has been a recent record set (as of September 2015<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Quantum_teleportation&action=edit">[update]</a></sup>) using superconducting nanowire detectors that reached the distance of 102 km (63 mi) over optical fiber.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> For material systems, the record distance is 21 metres (69 ft).<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>A variant of teleportation called "open-destination" teleportation, with receivers located at multiple locations, was demonstrated in 2004 using five-photon entanglement.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Teleportation of a composite state of two single qubits has also been realized.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> In April 2011, experimenters reported that they had demonstrated teleportation of wave packets of light up to a bandwidth of 10 MHz while preserving strongly nonclassical superposition states.<sup id="cite_ref-Lee2011_19-0" class="reference"><a href="#cite_note-Lee2011-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> In August 2013, the achievement of "fully deterministic" quantum teleportation, using a hybrid technique, was reported.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> On 29 May 2014, scientists announced a reliable way of transferring data by quantum teleportation. Quantum teleportation of data had been done before but with highly unreliable methods.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> On 26 February 2015, scientists at the <a href="/wiki/University_of_Science_and_Technology_of_China" title="University of Science and Technology of China">University of Science and Technology of China</a> in Hefei, led by <a href="/w/index.php?title=Chao-yang_Lu&action=edit&redlink=1" class="new" title="Chao-yang Lu (page does not exist)">Chao-yang Lu</a> and <a href="/wiki/Jian-Wei_Pan" class="mw-redirect" title="Jian-Wei Pan">Jian-Wei Pan</a> carried out the first experiment teleporting multiple degrees of freedom of a quantum particle. They managed to teleport the quantum information from ensemble of rubidium atoms to another ensemble of rubidium atoms over a distance of 150 metres (490 ft) using entangled photons.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-nature-20150226_25-0" class="reference"><a href="#cite_note-nature-20150226-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Quantum_Science_and_Technology_26-0" class="reference"><a href="#cite_note-Quantum_Science_and_Technology-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> In 2016, researchers demonstrated quantum teleportation with two independent sources which are separated by 6.5 km (4.0 mi) in Hefei optical fiber network.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> In September 2016, researchers at the University of Calgary demonstrated quantum teleportation over the Calgary metropolitan fiber network over a distance of 6.2 km (3.9 mi).<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> In December 2020, as part of the INQNET collaboration, researchers achieved quantum teleportation over a total distance of 44 km (27.3 mi) with fidelities exceeding 90%.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p><p>Researchers have also successfully used quantum teleportation to transmit information between clouds of gas atoms, notable because the clouds of gas are macroscopic atomic ensembles.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p><p>It is also possible to teleport <i>logical operations</i>, see <a href="/wiki/Quantum_gate_teleportation" title="Quantum gate teleportation">quantum gate teleportation</a>. In 2018, physicists at Yale demonstrated a deterministic teleported <a href="/wiki/Controlled_NOT_gate" title="Controlled NOT gate">CNOT</a> operation between <a href="/wiki/Quantum_error_correction#Bosonic_codes" title="Quantum error correction">logically encoded</a> qubits.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Conditional_quantum_teleportation_of_photon_polarization.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Conditional_quantum_teleportation_of_photon_polarization.png/220px-Conditional_quantum_teleportation_of_photon_polarization.png" decoding="async" width="220" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Conditional_quantum_teleportation_of_photon_polarization.png/330px-Conditional_quantum_teleportation_of_photon_polarization.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Conditional_quantum_teleportation_of_photon_polarization.png/440px-Conditional_quantum_teleportation_of_photon_polarization.png 2x" data-file-width="2459" data-file-height="1392" /></a><figcaption>Schematic of the quantum teleportation experiment performed by Zeilinger's group in 1997. For details, see the text.</figcaption></figure> <p>First proposed theoretically in 1993, quantum teleportation has since been demonstrated in many different guises. It has been carried out using two-level states of a single photon, a single atom and a trapped ion – among other quantum objects – and also using two photons. In 1997, two groups experimentally achieved quantum teleportation. The first group, led by <a href="/wiki/Sandu_Popescu" title="Sandu Popescu">Sandu Popescu</a>, was based in Italy. An experimental group led by <a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Anton Zeilinger</a> followed a few months later. </p><p>The results obtained from experiments done by Popescu's group concluded that classical channels alone could not replicate the teleportation of linearly polarized state and an elliptically polarized state. The Bell state measurement distinguished between the four Bell states, which can allow for a 100% success rate of teleportation, in an ideal representation.<sup id="cite_ref-Rome1998_2-2" class="reference"><a href="#cite_note-Rome1998-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Zeilinger's group produced a pair of entangled photons by implementing the process of parametric down-conversion. In order to ensure that the two photons cannot be distinguished by their arrival times, the photons were generated using a pulsed pump beam. The photons were then sent through narrow-bandwidth filters to produce a coherence time that is much longer than the length of the pump pulse. They then used a two-photon interferometry for analyzing the entanglement so that the quantum property could be recognized when it is transferred from one photon to the other.<sup id="cite_ref-Bouwmeester-1997_3-1" class="reference"><a href="#cite_note-Bouwmeester-1997-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Photon 1 was polarized at 45° in the first experiment conducted by Zeilinger's group. Quantum teleportation is verified when both photons are detected in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi ^{-}\rangle _{12}}"> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi ^{-}\rangle _{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e304e12f4bf2ef2fde98a643e52a92ba48ead663" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.747ex; height:3.009ex;" alt="{\displaystyle |\Psi ^{-}\rangle _{12}}"></span> state, which has a probability of 25%. Two detectors, f1 and f2, are placed behind the beam splitter, and recording the coincidence will identify the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi ^{-}\rangle _{12}}"> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi ^{-}\rangle _{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e304e12f4bf2ef2fde98a643e52a92ba48ead663" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.747ex; height:3.009ex;" alt="{\displaystyle |\Psi ^{-}\rangle _{12}}"></span> state. If there is a coincidence between detectors f1 and f2, then photon 3 is predicted to be polarized at a 45° angle. Photon 3 is passed through a polarizing beam splitter that selects +45° and −45° polarization. If quantum teleportation has happened, only detector d2, which is at the +45° output, will register a detection. Detector d1, located at the −45° output, will not detect a photon. If there is a coincidence between d2f1f2, with the 45° analysis, and a lack of a d1f1f2 coincidence, with −45° analysis, it is proof that the information from the polarized photon 1 has been teleported to photon 3 using quantum teleportation.<sup id="cite_ref-Bouwmeester-1997_3-2" class="reference"><a href="#cite_note-Bouwmeester-1997-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Quantum_teleportation_over_143_km">Quantum teleportation over 143 km</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=4" title="Edit section: Quantum teleportation over 143 km"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zeilinger's group developed an experiment using active feed-forward in real time and two free-space optical links, quantum and classical, between the Canary Islands of La Palma and Tenerife, a distance of over 143 kilometers. The results were published in 2012. In order to achieve teleportation, a frequency-uncorrelated polarization-entangled photon pair source, ultra-low-noise single-photon detectors and entanglement assisted clock synchronization were implemented. The two locations were entangled to share the auxiliary state:<sup id="cite_ref-Ma-2012_14-2" class="reference"><a href="#cite_note-Ma-2012-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi ^{-}\rangle _{23}={\frac {1}{\surd 2}}((|H\rangle _{2}|V\rangle _{3})-(|V\rangle _{2}|H\rangle _{3}))}"> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">√</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi ^{-}\rangle _{23}={\frac {1}{\surd 2}}((|H\rangle _{2}|V\rangle _{3})-(|V\rangle _{2}|H\rangle _{3}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d1e458ee6ecf8bac4c961d4089cd3287879f4d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.172ex; height:6.009ex;" alt="{\displaystyle |\Psi ^{-}\rangle _{23}={\frac {1}{\surd 2}}((|H\rangle _{2}|V\rangle _{3})-(|V\rangle _{2}|H\rangle _{3}))}"></span></dd></dl> <p>La Palma and Tenerife can be compared to the quantum characters Alice and Bob. Alice and Bob share the entangled state above, with photon 2 being with Alice and photon 3 being with Bob. A third party, Charlie, provides photon 1 (the input photon) which will be teleported to Alice in the generalized polarization state: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi \rangle _{1}=\alpha |H\rangle _{1}+\beta |V\rangle _{1}}"> <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"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle _{1}=\alpha |H\rangle _{1}+\beta |V\rangle _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef1f821d4342bbf30d8aa48eb818c55d44b53f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.812ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle _{1}=\alpha |H\rangle _{1}+\beta |V\rangle _{1}}"></span></dd></dl> <p>where the complex numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> are unknown to Alice or Bob. </p><p>Alice will perform a Bell-state measurement (BSM) that randomly projects the two photons onto one of the four Bell states with each one having a probability of 25%. Photon 3 will be projected onto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312d43de853a9e6ca74888e63394fc8081f56a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle }"></span>, the input state. Alice transmits the outcome of the BSM to Bob, via the classical channel, where Bob is able to apply the corresponding unitary operation to obtain photon 3 in the initial state of photon 1. Bob will not have to do anything if he detects the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi ^{-}\rangle _{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi ^{-}\rangle _{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d27e1ed968fc64e8774b9f1698205d136ff65cf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.452ex; height:3.009ex;" alt="{\displaystyle |\psi ^{-}\rangle _{12}}"></span> state. Bob will need to apply a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> phase shift to photon 3 between the horizontal and vertical component if the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi ^{+}\rangle _{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi ^{+}\rangle _{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b370c66c1d8f761f97012d159d6499041f1fbee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.452ex; height:3.009ex;" alt="{\displaystyle |\psi ^{+}\rangle _{12}}"></span> state is detected.<sup id="cite_ref-Ma-2012_14-3" class="reference"><a href="#cite_note-Ma-2012-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>The results of Zeilinger's group concluded that the average fidelity (overlap of the ideal teleported state with the measured density matrix) was 0.863 with a standard deviation of 0.038. The link attenuation during their experiments varied between 28.1 dB and 39.0 dB, which was a result of strong winds and rapid temperature changes. Despite the high loss in the quantum free-space channel, the average fidelity surpassed the classical limit of 2/3. Therefore, Zeilinger's group successfully demonstrated quantum teleportation over a distance of 143 km.<sup id="cite_ref-Ma-2012_14-4" class="reference"><a href="#cite_note-Ma-2012-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Quantum_teleportation_across_the_Danube_River">Quantum teleportation across the Danube River</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=5" title="Edit section: Quantum teleportation across the Danube River"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 2004, a quantum teleportation experiment was conducted across the Danube River in Vienna, a total of 600 meters. An 800-meter-long optical fiber wire was installed in a public sewer system underneath the Danube River, and it was exposed to temperature changes and other environmental influences. Alice must perform a joint Bell state measurement (BSM) on photon b, the input photon, and photon c, her part of the entangled photon pair (photons c and d). Photon d, Bob's receiver photon, will contain all of the information on the input photon b, except for a phase rotation that depends on the state that Alice observed. This experiment implemented an active feed-forward system that sends Alice's measurement results via a classical microwave channel with a fast electro-optical modulator in order to exactly replicate Alice's input photon. The teleportation fidelity obtained from the linear polarization state at 45° varied between 0.84 and 0.90, which is well above the classical fidelity limit of 0.66.<sup id="cite_ref-Danube2004_12-1" class="reference"><a href="#cite_note-Danube2004-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Deterministic_quantum_teleportation_with_atoms">Deterministic quantum teleportation with atoms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=6" title="Edit section: Deterministic quantum teleportation with atoms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Three qubits are required for this process: the source qubit from the sender, the ancillary qubit, and the receiver's target qubit, which is maximally entangled with the ancillary qubit. For this experiment, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ce {^{40}Ca+}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mmultiscripts> <mtext>Ca</mtext> <none /> <mo>+</mo> <mprescripts /> <none /> <mn>40</mn> </mmultiscripts> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ce {^{40}Ca+}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bff316376cdb7e12547ca667394f7a7d5168c3c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.286ex; height:2.676ex;" alt="{\displaystyle {\ce {^{40}Ca+}}}"></span> ions were used as the qubits. Ions 2 and 3 are prepared in the Bell state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi ^{+}\rangle _{23}={\frac {1}{\sqrt {2}}}(|0\rangle _{2}|1\rangle _{3}+|1\rangle _{2}|0\rangle _{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <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"> <mn>2</mn> </mrow> </msub> <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"> <mn>3</mn> </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"> <mn>2</mn> </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"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi ^{+}\rangle _{23}={\frac {1}{\sqrt {2}}}(|0\rangle _{2}|1\rangle _{3}+|1\rangle _{2}|0\rangle _{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b099b21189d47d0f16f41787fd63efbc34c509b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:33.207ex; height:6.176ex;" alt="{\displaystyle |\psi ^{+}\rangle _{23}={\frac {1}{\sqrt {2}}}(|0\rangle _{2}|1\rangle _{3}+|1\rangle _{2}|0\rangle _{3})}"></span>. The state of ion 1 is prepared arbitrarily. The quantum states of ions 1 and 2 are measured by illuminating them with light at a specific wavelength. The obtained fidelities for this experiment ranged between 73% and 76%. This is larger than the maximum possible average fidelity of 66.7% that can be obtained using completely classical resources.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Ground-to-satellite_quantum_teleportation">Ground-to-satellite quantum teleportation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=7" title="Edit section: Ground-to-satellite quantum teleportation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The quantum state being teleported in this experiment is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\chi \rangle _{1}=\alpha |H\rangle _{1}+\beta |V\rangle _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>χ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\chi \rangle _{1}=\alpha |H\rangle _{1}+\beta |V\rangle _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0d8084a10af2cdc452c789b81071089b5f1a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.882ex; height:2.843ex;" alt="{\displaystyle |\chi \rangle _{1}=\alpha |H\rangle _{1}+\beta |V\rangle _{1}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> are unknown complex numbers, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |H\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |H\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f010e5c1fd1410fa37d26b3ae7c769db1df7ab57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.615ex; height:2.843ex;" alt="{\displaystyle |H\rangle }"></span> represents the horizontal polarization state, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |V\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |V\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89463c7bdd8be86eae354b0fd9fc5ca09d423499" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.339ex; height:2.843ex;" alt="{\displaystyle |V\rangle }"></span> represents the vertical polarization state. The qubit prepared in this state is generated in a laboratory in Ngari, Tibet. The goal was to teleport the quantum information of the qubit to the Micius satellite that was launched on August 16, 2016, at an altitude of around 500 km. When a Bell state measurement is conducted on photons 1 and 2 and the resulting state is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi ^{+}\rangle _{12}={\frac {1}{\sqrt {2}}}(|H\rangle _{1}|H\rangle _{2}+|V\rangle _{1}|V\rangle _{2}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <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> <mi>H</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi ^{+}\rangle _{12}={\frac {1}{\sqrt {2}}}(|H\rangle _{1}|H\rangle _{2}+|V\rangle _{1}|V\rangle _{2}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693eb64956acbb0c4bce55882ca098f2f0af5919" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:37.036ex; height:6.176ex;" alt="{\displaystyle |\phi ^{+}\rangle _{12}={\frac {1}{\sqrt {2}}}(|H\rangle _{1}|H\rangle _{2}+|V\rangle _{1}|V\rangle _{2}))}"></span>, photon 3 carries this desired state. If the Bell state detected is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi ^{-}\rangle _{12}={\frac {1}{\sqrt {2}}}(|H\rangle _{1}|H\rangle _{2}-|V\rangle _{1}|V\rangle _{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <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> <mi>H</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi ^{-}\rangle _{12}={\frac {1}{\sqrt {2}}}(|H\rangle _{1}|H\rangle _{2}-|V\rangle _{1}|V\rangle _{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae11e734694ef0b731cef5d451d7627944a3cc83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:36.131ex; height:6.176ex;" alt="{\displaystyle |\phi ^{-}\rangle _{12}={\frac {1}{\sqrt {2}}}(|H\rangle _{1}|H\rangle _{2}-|V\rangle _{1}|V\rangle _{2})}"></span>, then a phase shift of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> is applied to the state to get the desired quantum state. The distance between the ground station and the satellite changes from as little as 500 km to as large as 1,400 km. Because of the changing distance, the channel loss of the uplink varies between 41 dB and 52 dB. The average fidelity obtained from this experiment was 0.80 with a standard deviation of 0.01. Therefore, this experiment successfully established a ground-to-satellite uplink over a distance of 500–1,400 km using quantum teleportation. This is an essential step towards creating a global-scale quantum internet.<sup id="cite_ref-sat1400_6-1" class="reference"><a href="#cite_note-sat1400-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Quantum_teleportation_over_internet_cables">Quantum teleportation over internet cables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=8" title="Edit section: Quantum teleportation over internet cables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Quantum teleportation has been demonstrated over fiber optic cables simultaneously carrying regular telecommunications traffic. This eliminates the need for separate, dedicated infrastructure for quantum networking and shows that quantum teleportation and classical communications can coexist on the same fiber optic cables. A less crowded wavelength of light was used for the quantum signal and special filters were required to reduce noise from other traffic.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Reliable_sources" title="Wikipedia:Reliable sources"><span title="The material near this tag may rely on an unreliable source. primary source has no citations (December 2024)">unreliable source?</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Formal_presentation">Formal presentation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=9" title="Edit section: Formal presentation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are a variety of ways in which the teleportation protocol can be written mathematically. Some are very compact but abstract, and some are verbose but straightforward and concrete. The presentation below is of the latter form: verbose, but has the benefit of showing each quantum state simply and directly. Later sections review more compact notations. </p><p>The teleportation protocol begins with a quantum state or qubit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></span>, in Alice's possession, that she wants to convey to Bob. This qubit can be written generally, in <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a>, as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle _{C}=\alpha |0\rangle _{C}+\beta |1\rangle _{C}.}"> <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>C</mi> </mrow> </msub> <mo>=</mo> <mi>α</mi> <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>C</mi> </mrow> </msub> <mo>+</mo> <mi>β</mi> <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>C</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle _{C}=\alpha |0\rangle _{C}+\beta |1\rangle _{C}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd62131f9cfed8422cdbaff9cdde3d4ad54e1920" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.342ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle _{C}=\alpha |0\rangle _{C}+\beta |1\rangle _{C}.}"></span></dd></dl> <p>The subscript <i>C</i> above is used only to distinguish this state from <i>A</i> and <i>B</i>, below. </p><p>Next, the protocol requires that Alice and Bob share a maximally entangled state. This state is fixed in advance, by mutual agreement between Alice and Bob, and can be any one of the four <a href="/wiki/Bell_state" title="Bell state">Bell states</a> shown. It does not matter which one. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Phi ^{+}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}+|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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <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 ^{+}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}+|1\rangle _{A}\otimes |1\rangle _{B})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d49769c421335dbee8ca608eaa4e68b90a3363d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:41.561ex; height:6.176ex;" alt="{\displaystyle |\Phi ^{+}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}+|1\rangle _{A}\otimes |1\rangle _{B})}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi ^{+}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}+|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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi ^{+}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}+|1\rangle _{A}\otimes |0\rangle _{B})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f1578792e7fc1f24f9ffdbd6e96a08fe8ece707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:41.691ex; height:6.176ex;" alt="{\displaystyle |\Psi ^{+}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}+|1\rangle _{A}\otimes |0\rangle _{B})}"></span>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi ^{-}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}-|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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi ^{-}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27052cbff5559f32b7cf2d2e2c64b18e2484bda9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:41.691ex; height:6.176ex;" alt="{\displaystyle |\Psi ^{-}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B})}"></span>.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Phi ^{-}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}-|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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <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 ^{-}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}-|1\rangle _{A}\otimes |1\rangle _{B})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37b9a92fa6ce4858f1172e6c5e5dd785bba67d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:41.561ex; height:6.176ex;" alt="{\displaystyle |\Phi ^{-}\rangle _{AB}={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}-|1\rangle _{A}\otimes |1\rangle _{B})}"></span>,</dd></dl> <p>In the following, assume that Alice and Bob share the state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Phi ^{+}\rangle _{AB}.}"> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Phi ^{+}\rangle _{AB}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dedd339505efa5c29850bf7cd9dc6c4c5ff7707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.099ex; height:3.009ex;" alt="{\displaystyle |\Phi ^{+}\rangle _{AB}.}"></span> Alice obtains one of the particles in the pair, with the other going to Bob. (This is implemented by preparing the particles together and shooting them to Alice and Bob from a common source.) The subscripts <i>A</i> and <i>B</i> in the entangled state refer to Alice's or Bob's particle. </p><p>At this point, Alice has two particles (<i>C</i>, the one she wants to teleport, and <i>A</i>, one of the entangled pair), and Bob has one particle, <i>B</i>. In the total system, the state of these three particles is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle _{C}\otimes |\Phi ^{+}\rangle _{AB}=(\alpha |0\rangle _{C}+\beta |1\rangle _{C})\otimes {\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}+|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> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>⊗</mo> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>α</mi> <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>C</mi> </mrow> </msub> <mo>+</mo> <mi>β</mi> <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>C</mi> </mrow> </msub> <mo 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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle _{C}\otimes |\Phi ^{+}\rangle _{AB}=(\alpha |0\rangle _{C}+\beta |1\rangle _{C})\otimes {\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}+|1\rangle _{A}\otimes |1\rangle _{B}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53d877a902f418b47fa7eba31973ab03f6078def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:68.294ex; height:6.176ex;" alt="{\displaystyle |\psi \rangle _{C}\otimes |\Phi ^{+}\rangle _{AB}=(\alpha |0\rangle _{C}+\beta |1\rangle _{C})\otimes {\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}+|1\rangle _{A}\otimes |1\rangle _{B}).}"></span></dd></dl> <p>Alice will then make a local measurement in the Bell basis (i.e. the four Bell states) on the two particles in her possession. To make the result of her measurement clear, it is best to write the state of Alice's two qubits as superpositions of the Bell basis. This is done by using the following general identities, which are easily verified: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle \otimes |0\rangle ={\frac {1}{\sqrt {2}}}(|\Phi ^{+}\rangle +|\Phi ^{-}\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> <mo>⊗</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"> <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> <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"> <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 stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle \otimes |0\rangle ={\frac {1}{\sqrt {2}}}(|\Phi ^{+}\rangle +|\Phi ^{-}\rangle ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2148698507b2e38dd180657eb11c9423fab1e624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:30.078ex; height:6.176ex;" alt="{\displaystyle |0\rangle \otimes |0\rangle ={\frac {1}{\sqrt {2}}}(|\Phi ^{+}\rangle +|\Phi ^{-}\rangle ),}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle \otimes |1\rangle ={\frac {1}{\sqrt {2}}}(|\Psi ^{+}\rangle +|\Psi ^{-}\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> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <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> <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"> <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 stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle \otimes |1\rangle ={\frac {1}{\sqrt {2}}}(|\Psi ^{+}\rangle +|\Psi ^{-}\rangle ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6dee9ba0cfcb92ab6aafa88726b2f93251e6545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:30.339ex; height:6.176ex;" alt="{\displaystyle |0\rangle \otimes |1\rangle ={\frac {1}{\sqrt {2}}}(|\Psi ^{+}\rangle +|\Psi ^{-}\rangle ),}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle \otimes |0\rangle ={\frac {1}{\sqrt {2}}}(|\Psi ^{+}\rangle -|\Psi ^{-}\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> <mo>⊗</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"> <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> <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"> <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 stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |1\rangle \otimes |0\rangle ={\frac {1}{\sqrt {2}}}(|\Psi ^{+}\rangle -|\Psi ^{-}\rangle ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65c9250c7893c1cdbbe1b67d60a736e37dc9c556" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:30.339ex; height:6.176ex;" alt="{\displaystyle |1\rangle \otimes |0\rangle ={\frac {1}{\sqrt {2}}}(|\Psi ^{+}\rangle -|\Psi ^{-}\rangle ),}"></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle \otimes |1\rangle ={\frac {1}{\sqrt {2}}}(|\Phi ^{+}\rangle -|\Phi ^{-}\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> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <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> <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"> <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 stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |1\rangle \otimes |1\rangle ={\frac {1}{\sqrt {2}}}(|\Phi ^{+}\rangle -|\Phi ^{-}\rangle ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ab134d9b6a99c12dc53488360ea914a783593b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:30.078ex; height:6.176ex;" alt="{\displaystyle |1\rangle \otimes |1\rangle ={\frac {1}{\sqrt {2}}}(|\Phi ^{+}\rangle -|\Phi ^{-}\rangle ).}"></span></dd></dl> <p>After expanding the expression for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\begin{aligned}|&\psi \rangle _{C}\otimes \ |\Phi ^{+}\rangle _{AB}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>⊗</mo> <mtext> </mtext> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\begin{aligned}|&\psi \rangle _{C}\otimes \ |\Phi ^{+}\rangle _{AB}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88cdd1f777232277635f291661cc236c24370a96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.171ex; height:3.009ex;" alt="{\textstyle {\begin{aligned}|&\psi \rangle _{C}\otimes \ |\Phi ^{+}\rangle _{AB}\end{aligned}}}"></span>, one applies these identities to the qubits with <i>A</i> and <i>C</i> subscripts. In particular,<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha {\frac {1}{\sqrt {2}}}|0\rangle _{C}\otimes |0\rangle _{A}\otimes |0\rangle _{B}=\alpha {\frac {1}{2}}(|\Phi ^{+}\rangle _{CA}+|\Phi ^{-}\rangle _{CA})\otimes |0\rangle _{B},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </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>C</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>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> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> <mo>+</mo> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> <mo stretchy="false">)</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha {\frac {1}{\sqrt {2}}}|0\rangle _{C}\otimes |0\rangle _{A}\otimes |0\rangle _{B}=\alpha {\frac {1}{2}}(|\Phi ^{+}\rangle _{CA}+|\Phi ^{-}\rangle _{CA})\otimes |0\rangle _{B},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83964540bb39ac34caac15e6db0633dcbf7ea104" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:57.494ex; height:6.176ex;" alt="{\displaystyle \alpha {\frac {1}{\sqrt {2}}}|0\rangle _{C}\otimes |0\rangle _{A}\otimes |0\rangle _{B}=\alpha {\frac {1}{2}}(|\Phi ^{+}\rangle _{CA}+|\Phi ^{-}\rangle _{CA})\otimes |0\rangle _{B},}"></span>and the other terms follow similarly. Combining similar terms, the total three particle state of <i>A</i>, <i>B</i> and <i>C</i> together becomes the following four-term superposition:<sup id="cite_ref-Nielsen-2010_37-0" class="reference"><a href="#cite_note-Nielsen-2010-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}|&\psi \rangle _{C}\otimes \ |\Phi ^{+}\rangle _{AB}\ =\\{\frac {1}{2}}{\Big \lbrack }\ &|\Phi ^{+}\rangle _{CA}\otimes (\alpha |0\rangle _{B}+\beta |1\rangle _{B})\ +\ |\Phi ^{-}\rangle _{CA}\otimes (\alpha |0\rangle _{B}-\beta |1\rangle _{B})\\\ +\ &|\Psi ^{+}\rangle _{CA}\otimes (\alpha |1\rangle _{B}+\beta |0\rangle _{B})\ +\ |\Psi ^{-}\rangle _{CA}\otimes (\alpha |1\rangle _{B}-\beta |0\rangle _{B}){\Big \rbrack }.\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>⊗</mo> <mtext> </mtext> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mtext> </mtext> <mo>=</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mtext> </mtext> </mtd> <mtd> <mi></mi> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mo stretchy="false">(</mo> <mi>α</mi> <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> <mi>β</mi> <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> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mo stretchy="false">(</mo> <mi>α</mi> <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> <mi>β</mi> <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> </mtd> </mtr> <mtr> <mtd> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mo stretchy="false">(</mo> <mi>α</mi> <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> <mi>β</mi> <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> <mtext> </mtext> <mo>+</mo> <mtext> </mtext> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mo stretchy="false">(</mo> <mi>α</mi> <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> <mi>β</mi> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}|&\psi \rangle _{C}\otimes \ |\Phi ^{+}\rangle _{AB}\ =\\{\frac {1}{2}}{\Big \lbrack }\ &|\Phi ^{+}\rangle _{CA}\otimes (\alpha |0\rangle _{B}+\beta |1\rangle _{B})\ +\ |\Phi ^{-}\rangle _{CA}\otimes (\alpha |0\rangle _{B}-\beta |1\rangle _{B})\\\ +\ &|\Psi ^{+}\rangle _{CA}\otimes (\alpha |1\rangle _{B}+\beta |0\rangle _{B})\ +\ |\Psi ^{-}\rangle _{CA}\otimes (\alpha |1\rangle _{B}-\beta |0\rangle _{B}){\Big \rbrack }.\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc25e489c84f5fdb07776ea4497042ada175ead" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:62.736ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}|&\psi \rangle _{C}\otimes \ |\Phi ^{+}\rangle _{AB}\ =\\{\frac {1}{2}}{\Big \lbrack }\ &|\Phi ^{+}\rangle _{CA}\otimes (\alpha |0\rangle _{B}+\beta |1\rangle _{B})\ +\ |\Phi ^{-}\rangle _{CA}\otimes (\alpha |0\rangle _{B}-\beta |1\rangle _{B})\\\ +\ &|\Psi ^{+}\rangle _{CA}\otimes (\alpha |1\rangle _{B}+\beta |0\rangle _{B})\ +\ |\Psi ^{-}\rangle _{CA}\otimes (\alpha |1\rangle _{B}-\beta |0\rangle _{B}){\Big \rbrack }.\\\end{aligned}}}"></span></dd></dl> <p>Note that all three particles are still in the same total state since no operations have been performed. Rather, the above is just a change of basis on Alice's part of the system. This change has moved the entanglement from particles A and B to particles C and A. The actual teleportation occurs when Alice measures her two qubits (C and A) in the Bell basis </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bell_State_Decoder.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Bell_State_Decoder.jpg/250px-Bell_State_Decoder.jpg" decoding="async" width="250" height="87" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Bell_State_Decoder.jpg/375px-Bell_State_Decoder.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Bell_State_Decoder.jpg/500px-Bell_State_Decoder.jpg 2x" data-file-width="765" data-file-height="266" /></a><figcaption>A simple quantum circuit that maps one of the four Bell states (the EPR pair in the picture) into one of the four two-qubit computational basis states. The circuit consists of a <a href="/wiki/Controlled_NOT_gate" title="Controlled NOT gate">CNOT</a> gate followed by a <a href="/wiki/Quantum_logic_gate#Hadamard_gate" title="Quantum logic gate">Hadamard</a> operation. In the outputs, a and b take on values of 0 or 1.</figcaption></figure> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Phi ^{+}\rangle _{CA},|\Phi ^{-}\rangle _{CA},|\Psi ^{+}\rangle _{CA},|\Psi ^{-}\rangle _{CA}.}"> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> <mo>,</mo> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> <mo>,</mo> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> <mo>,</mo> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Phi ^{+}\rangle _{CA},|\Phi ^{-}\rangle _{CA},|\Psi ^{+}\rangle _{CA},|\Psi ^{-}\rangle _{CA}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4175b517506283ba9936d2bf18a6b56b307d27a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.825ex; height:3.009ex;" alt="{\displaystyle |\Phi ^{+}\rangle _{CA},|\Phi ^{-}\rangle _{CA},|\Psi ^{+}\rangle _{CA},|\Psi ^{-}\rangle _{CA}.}"></span></dd></dl> <p>Equivalently, the measurement may be done in the computational basis, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 \}}"></span>, by mapping each Bell state uniquely to one of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|0\rangle \otimes |0\rangle ,|0\rangle \otimes |1\rangle ,|1\rangle \otimes |0\rangle ,|1\rangle \otimes |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>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{|0\rangle \otimes |0\rangle ,|0\rangle \otimes |1\rangle ,|1\rangle \otimes |0\rangle ,|1\rangle \otimes |1\rangle \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3021a9498f5069f9d6e9272d24133ef1243d57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.5ex; height:2.843ex;" alt="{\displaystyle \{|0\rangle \otimes |0\rangle ,|0\rangle \otimes |1\rangle ,|1\rangle \otimes |0\rangle ,|1\rangle \otimes |1\rangle \}}"></span> with the quantum circuit in the figure to the right. </p><p>The result of Alice's (local) measurement is a collection of two classical bits (00, 01, 10 or 11) related to one of the following four states (with equal probability of 1/4), after the three-particle state has <a href="/wiki/Wave_function_collapse" title="Wave function collapse">collapsed</a> into one of the states: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Phi ^{+}\rangle _{CA}\otimes (\alpha |0\rangle _{B}+\beta |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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mo stretchy="false">(</mo> <mi>α</mi> <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> <mi>β</mi> <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 ^{+}\rangle _{CA}\otimes (\alpha |0\rangle _{B}+\beta |1\rangle _{B})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/436acb01237ad71393db0079af7b39f29e3e7934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.151ex; height:3.009ex;" alt="{\displaystyle |\Phi ^{+}\rangle _{CA}\otimes (\alpha |0\rangle _{B}+\beta |1\rangle _{B})}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Phi ^{-}\rangle _{CA}\otimes (\alpha |0\rangle _{B}-\beta |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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mo stretchy="false">(</mo> <mi>α</mi> <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> <mi>β</mi> <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 ^{-}\rangle _{CA}\otimes (\alpha |0\rangle _{B}-\beta |1\rangle _{B})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7918dadfa10dfc1db95a9ecb5dbbf98c70408c9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.151ex; height:3.009ex;" alt="{\displaystyle |\Phi ^{-}\rangle _{CA}\otimes (\alpha |0\rangle _{B}-\beta |1\rangle _{B})}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi ^{+}\rangle _{CA}\otimes (\alpha |1\rangle _{B}+\beta |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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mo stretchy="false">(</mo> <mi>α</mi> <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> <mi>β</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi ^{+}\rangle _{CA}\otimes (\alpha |1\rangle _{B}+\beta |0\rangle _{B})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af9db9b6e4820268fc00e492be9737a00e9f75ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.281ex; height:3.009ex;" alt="{\displaystyle |\Psi ^{+}\rangle _{CA}\otimes (\alpha |1\rangle _{B}+\beta |0\rangle _{B})}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi ^{-}\rangle _{CA}\otimes (\alpha |1\rangle _{B}-\beta |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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <mo stretchy="false">(</mo> <mi>α</mi> <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> <mi>β</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi ^{-}\rangle _{CA}\otimes (\alpha |1\rangle _{B}-\beta |0\rangle _{B})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61351719ba1f3a7c92cbdd34ce2781855f5ef999" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.281ex; height:3.009ex;" alt="{\displaystyle |\Psi ^{-}\rangle _{CA}\otimes (\alpha |1\rangle _{B}-\beta |0\rangle _{B})}"></span></li></ul> <p>Alice's two particles are now entangled to each other, in one of the four <a href="/wiki/Bell_states" class="mw-redirect" title="Bell states">Bell states</a>, and the entanglement originally shared between Alice's and Bob's particles is now broken. Bob's particle takes on one of the four superposition states shown above. Note how Bob's qubit is now in a state that resembles the state to be teleported. The four possible states for Bob's qubit are unitary images of the state to be teleported. </p><p>The result of Alice's Bell measurement tells her which of the above four states the system is in. She can now send her result to Bob through a classical channel. Two classical bits can communicate which of the four results she obtained. After Bob receives the message from Alice, he will know which of the four states his particle is in. Using this information, he performs a unitary operation on his particle to transform it to the desired state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha |0\rangle _{B}+\beta |1\rangle _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <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> <mi>β</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha |0\rangle _{B}+\beta |1\rangle _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82f4ec0632b4c48aecf261b4734aafe566e994d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.047ex; height:2.843ex;" alt="{\displaystyle \alpha |0\rangle _{B}+\beta |1\rangle _{B}}"></span>: </p> <ul><li>If Alice indicates her result is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Phi ^{+}\rangle _{CA}}"> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Phi ^{+}\rangle _{CA}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d71756139737c7898d99332944128f73c1fd91d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.454ex; height:3.009ex;" alt="{\displaystyle |\Phi ^{+}\rangle _{CA}}"></span>, Bob knows his qubit is already in the desired state and does nothing. This amounts to the trivial unitary operation, the identity operator.</li> <li>If the message indicates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Phi ^{-}\rangle _{CA}}"> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Phi ^{-}\rangle _{CA}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b43c821a0102e9efeadbd117d0934b1fe0fdca9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.454ex; height:3.009ex;" alt="{\displaystyle |\Phi ^{-}\rangle _{CA}}"></span>, Bob would send his qubit through the unitary <a href="/wiki/Quantum_gate" class="mw-redirect" title="Quantum gate">quantum gate</a> given by the <a href="/wiki/Pauli_matrix" class="mw-redirect" title="Pauli matrix">Pauli matrix</a></li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{3}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{3}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06c79964899ad3256d5fcc25a485c2b80f47e467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.142ex; height:6.176ex;" alt="{\displaystyle \sigma _{3}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}"></span></dd></dl> <p>to recover the state. </p> <ul><li>If Alice's message corresponds to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi ^{+}\rangle _{CA}}"> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi ^{+}\rangle _{CA}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e362330ee6286bcff1d6f6231f9481a38ec887b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.584ex; height:3.009ex;" alt="{\displaystyle |\Psi ^{+}\rangle _{CA}}"></span>, Bob applies the gate</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1}={\begin{bmatrix}0&1\\1&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{1}={\begin{bmatrix}0&1\\1&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9ad3f14668aaf3248a797965f9515ed868e6399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.334ex; height:6.176ex;" alt="{\displaystyle \sigma _{1}={\begin{bmatrix}0&1\\1&0\end{bmatrix}}}"></span></dd></dl> <p>to his qubit. </p> <ul><li>Finally, for the remaining case, the appropriate gate is given by</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{3}\sigma _{1}=-\sigma _{1}\sigma _{3}=i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mi>i</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{3}\sigma _{1}=-\sigma _{1}\sigma _{3}=i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b46d154628042901cff7a07089a947b52edd7aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.123ex; height:6.176ex;" alt="{\displaystyle \sigma _{3}\sigma _{1}=-\sigma _{1}\sigma _{3}=i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}.}"></span></dd></dl> <p>Teleportation is thus achieved. The above-mentioned three gates correspond to rotations of π radians (180°) about appropriate axes (X, Y and Z) in the <a href="/wiki/Bloch_sphere" title="Bloch sphere">Bloch sphere</a> picture of a qubit. </p><p>Some remarks: </p> <ul><li>After this operation, Bob's qubit will take on the state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle _{B}=\alpha |0\rangle _{B}+\beta |1\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>B</mi> </mrow> </msub> <mo>=</mo> <mi>α</mi> <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> <mi>β</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle _{B}=\alpha |0\rangle _{B}+\beta |1\rangle _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa3ebbb57cc31cb3eeda01198753f0f12f2e893" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.69ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle _{B}=\alpha |0\rangle _{B}+\beta |1\rangle _{B}}"></span>, and Alice's qubit becomes an (undefined) part of an entangled state. Teleportation does not result in the copying of qubits, and hence is consistent with the <a href="/wiki/No-cloning_theorem" title="No-cloning theorem">no-cloning theorem</a>.</li> <li>There is no transfer of matter or energy involved. Alice's particle has not been physically moved to Bob; only its state has been transferred. The term "teleportation", coined by Bennett, Brassard, Crépeau, Jozsa, Peres and Wootters, reflects the indistinguishability of quantum mechanical particles.</li> <li>For every qubit teleported, Alice needs to send Bob two classical bits of information. These two classical bits do not carry complete information about the qubit being teleported. If an eavesdropper intercepts the two bits, she may know exactly what Bob needs to do in order to recover the desired state. However, this information is useless if she cannot interact with the entangled particle in Bob's possession.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Certifying_quantum_teleportation">Certifying quantum teleportation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=10" title="Edit section: Certifying quantum teleportation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When implementing the quantum teleportation protocol, different experimental noises may arise affecting the state transference.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> The usual way to benchmark a particular teleportation procedure is by using the average <a href="/wiki/Fidelity_of_quantum_states" title="Fidelity of quantum states">fidelity</a>: Given an arbitrary teleportation protocol producing output states <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f9e51d2ad9cab8f8a5d962502ddf4189b713f7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.002ex; height:2.176ex;" alt="{\displaystyle \rho _{i}}"></span> with probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span> for an initial state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =|\psi \rangle \langle \psi |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =|\psi \rangle \langle \psi |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caef57c47c042ade8573808c1d00117880088e4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.43ex; height:2.843ex;" alt="{\displaystyle \rho =|\psi \rangle \langle \psi |}"></span>, the average fidelity is defined as:<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle {\overline {F}}\rangle =\int \sum _{i}p_{i}F(\rho ,\rho _{i})d\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo fence="false" 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> <mi>F</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>,</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>d</mi> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\overline {F}}\rangle =\int \sum _{i}p_{i}F(\rho ,\rho _{i})d\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/550da4cdd828569a649ec0f62cee333edd583f29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.732ex; height:6.343ex;" alt="{\displaystyle \langle {\overline {F}}\rangle =\int \sum _{i}p_{i}F(\rho ,\rho _{i})d\psi }"></span> </p><p>where the integration is performed over the <a href="/wiki/Haar_measure" title="Haar measure">Haar measure</a> defined by assuming maximal uncertainty over the initial quantum states <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\rho ,\rho _{i})=\left({\text{Tr}}{\sqrt {{\sqrt {\rho }}\rho _{i}{\sqrt {\rho }}}}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>,</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Tr</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ρ</mi> </msqrt> </mrow> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ρ</mi> </msqrt> </mrow> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(\rho ,\rho _{i})=\left({\text{Tr}}{\sqrt {{\sqrt {\rho }}\rho _{i}{\sqrt {\rho }}}}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/122e42a3639b21a6bb0ef8fcfdb5eedc7e8ab523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.552ex; height:6.509ex;" alt="{\displaystyle F(\rho ,\rho _{i})=\left({\text{Tr}}{\sqrt {{\sqrt {\rho }}\rho _{i}{\sqrt {\rho }}}}\right)^{2}}"></span> is the Uhlmann-Jozsa <a href="/wiki/Fidelity_of_quantum_states" title="Fidelity of quantum states">fidelity</a>. </p><p>The widely known classical threshold is obtained by optimizing the average fidelity over all classical protocols (i.e. when the sender Alice and the receiver Bob can use just a classical channel to communicate with each other). When teleportation involves qubit states, the maximal classical average fidelity is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2/3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2/3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f73d0fc4f429ca39387dffad817c8147cf73d0ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle 2/3}"></span>.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> In this way, a particular protocol with average fidelity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle {\overline {F}}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\overline {F}}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97eea8e59aa67192a10cad99ab7d775000f46b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.825ex; height:3.509ex;" alt="{\displaystyle \langle {\overline {F}}\rangle }"></span> is certified as <i>useful</i> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle {\overline {F}}\rangle \geq 2/3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>≥</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\overline {F}}\rangle \geq 2/3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/787d903987aa545c0303c33ccc5f0069945294b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.411ex; height:3.509ex;" alt="{\displaystyle \langle {\overline {F}}\rangle \geq 2/3}"></span>.<sup id="cite_ref-sat1400_6-2" class="reference"><a href="#cite_note-sat1400-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Danube2004_12-2" class="reference"><a href="#cite_note-Danube2004-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Ma-2012_14-5" class="reference"><a href="#cite_note-Ma-2012-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>However, using the Uhlmann-Jozsa <a href="/wiki/Fidelity_of_quantum_states" title="Fidelity of quantum states">fidelity</a> as the <i>unique</i> distance measure for benchmarking teleportation is not justified, and one may choose different distinguishability measures.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> For example, there may exist reasons depending on the context in which other measures might be more suitable than fidelity.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> In this way, the average distance of teleportation is defined as:<sup id="cite_ref-Bussandri-2024_44-0" class="reference"><a href="#cite_note-Bussandri-2024-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle {\overline {D}}\rangle =\int \sum _{i}p_{i}D(\rho ,\rho _{i})d\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo fence="false" 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> <mi>D</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>,</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>d</mi> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\overline {D}}\rangle =\int \sum _{i}p_{i}D(\rho ,\rho _{i})d\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76bb9d7e86f1efef22eaf8716d05c9b8945ca441" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.939ex; height:6.343ex;" alt="{\displaystyle \langle {\overline {D}}\rangle =\int \sum _{i}p_{i}D(\rho ,\rho _{i})d\psi }"></span> </p><p>being <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(\rho ,\sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(\rho ,\sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92fd26266b9fe008fdfe549cdc8db3befb8ce505" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.299ex; height:2.843ex;" alt="{\displaystyle D(\rho ,\sigma )}"></span> a well-behaved (i.e. satisfying identity of indiscernibles and unitary invariance) distinguishability measure between quantum states. Consequently, different classical thresholds exist, depending on the considered distance measure (classical thresholds for <a href="/wiki/Trace_distance" title="Trace distance">Trace distance</a>, quantum <a href="/wiki/Jensen%E2%80%93Shannon_divergence" title="Jensen–Shannon divergence">Jensen–Shannon divergence</a>, transmission distance, <a href="/wiki/Bures_distance" class="mw-redirect" title="Bures distance">Bures distance</a>, wootters distance, and quantum Hellinger distance, among others, were obtained in Ref. <sup id="cite_ref-Bussandri-2024_44-1" class="reference"><a href="#cite_note-Bussandri-2024-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup>). This points out a particular issue when certifying quantum teleportation: Given a teleportation protocol, its certification is not a universal fact in the sense that depends on the distance used. Then, a particular protocol might be certified as useful for a set of distance quantifiers, and non-useful for other distinguishability measures.<sup id="cite_ref-Bussandri-2024_44-2" class="reference"><a href="#cite_note-Bussandri-2024-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Alternative_notations">Alternative notations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=11" title="Edit section: Alternative notations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="thumb tnone" style="margin-left:auto;margin-right:auto;overflow:hidden;width:auto;max-width:788px"><div class="thumbinner"><div class="noresize" style="overflow:auto"><span typeof="mw:File"><a href="/wiki/File:Teleport.png" class="mw-file-description" title="Quantum teleportation in its diagrammatic form. employing Penrose graphical notation. Formally, such a computation takes place in a dagger compact category. This results in the abstract description of quantum teleportation as employed in categorical quantum mechanics."><img alt="" src="//upload.wikimedia.org/wikipedia/commons/e/e8/Teleport.png" decoding="async" width="780" height="169" class="mw-file-element" data-file-width="780" data-file-height="169" /></a></span></div><div class="thumbcaption"><div class="magnify"><a href="/wiki/File:Teleport.png" title="File:Teleport.png"> </a></div>Quantum teleportation in its diagrammatic form.<sup id="cite_ref-coecke_45-0" class="reference"><a href="#cite_note-coecke-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> employing <a href="/wiki/Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a>.<sup id="cite_ref-Penrose_46-0" class="reference"><a href="#cite_note-Penrose-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> Formally, such a computation takes place in a <a href="/wiki/Dagger_compact_category" title="Dagger compact category">dagger compact category</a>. This results in the abstract description of quantum teleportation as employed in <a href="/wiki/Categorical_quantum_mechanics" title="Categorical quantum mechanics">categorical quantum mechanics</a>.</div></div></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Quantum_teleportation_circuit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Quantum_teleportation_circuit.svg/330px-Quantum_teleportation_circuit.svg.png" decoding="async" width="330" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Quantum_teleportation_circuit.svg/495px-Quantum_teleportation_circuit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dc/Quantum_teleportation_circuit.svg/660px-Quantum_teleportation_circuit.svg.png 2x" data-file-width="200" data-file-height="75" /></a><figcaption> <a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a> representation for teleportation of a quantum state,<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> as <a href="#Formal_presentation">described above</a>. The circuit consumes the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span> <a href="/wiki/Bell_state" title="Bell state">Bell state</a> and the qubit to teleport as input, and consists of <a href="/wiki/Quantum_logic_gate#CNOT" title="Quantum logic gate">CNOT</a>, <a href="/wiki/Quantum_logic_gate#Hadamard" title="Quantum logic gate">Hadamard</a>, two <a href="/wiki/Measurement_in_quantum_mechanics#Quantum_circuits" title="Measurement in quantum mechanics">measurements</a> of two qubits, and finally, two gates with <a href="/wiki/Quantum_logic_gate#Classical_control" title="Quantum logic gate">classical control</a>: a <a href="/wiki/Quantum_logic_gate#X" title="Quantum logic gate">Pauli X</a>, and a <a href="/wiki/Quantum_logic_gate#Z" title="Quantum logic gate">Pauli Z</a>, meaning that if the result from the measurement was <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span>, then the classically controlled Pauli gate is executed. After the circuit has run to completion, the value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle _{C}}"> <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>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle _{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/241b6fa2bbf4d7c342076e84de2433854a740139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.546ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle _{C}}"></span> will have moved to, or <i>teleported</i> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \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>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/064d77ee677368f8b4df49ef172fc1ac80d55875" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.544ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle _{B}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle _{C}}"> <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>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle _{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/241b6fa2bbf4d7c342076e84de2433854a740139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.546ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle _{C}}"></span> will have its value set to either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span>, depending on the result from the measurement on that qubit.<br />This circuit can also be used for <i>entanglement swapping</i>, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle _{C}}"> <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>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle _{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/241b6fa2bbf4d7c342076e84de2433854a740139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.546ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle _{C}}"></span> is one of the qubits that make up an entangled state, as described in <a href="#Entanglement_swapping">the text</a>.</figcaption></figure> <p>There are a variety of different notations in use that describe the teleportation protocol. One common one is by using the notation of <a href="/wiki/Quantum_gate" class="mw-redirect" title="Quantum gate">quantum gates</a>. </p><p>In the above derivation, the unitary transformation that is the change of basis (from the standard product basis into the Bell basis) can be written using quantum gates. Direct calculation shows that this gate is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=(H\otimes I)\operatorname {CNOT} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>H</mi> <mo>⊗</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mi>CNOT</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=(H\otimes I)\operatorname {CNOT} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ce7af8834d2bad23b8304c936d3376e34754ca5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.105ex; height:2.843ex;" alt="{\displaystyle G=(H\otimes I)\operatorname {CNOT} }"></span></dd></dl> <p>where <i>H</i> is the one qubit <a href="/wiki/Hadamard_gate" class="mw-redirect" title="Hadamard gate">Walsh-Hadamard gate</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {CNOT} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>CNOT</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {CNOT} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19c94e5f7d3e2a2b536438b4def0b15b83152056" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.907ex; height:2.176ex;" alt="{\displaystyle \operatorname {CNOT} }"></span> is the Controlled NOT gate. </p> <div class="mw-heading mw-heading2"><h2 id="Entanglement_swapping">Entanglement swapping</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=12" title="Edit section: Entanglement swapping"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Further information: <a href="/wiki/Quantum_entanglement_swapping" class="mw-redirect" title="Quantum entanglement swapping">Quantum entanglement swapping</a></div> <p>Teleportation can be applied not just to pure states, but also <a href="/wiki/Density_matrix" title="Density matrix">mixed states</a>, that can be regarded as the state of a single subsystem of an entangled pair. The so-called entanglement swapping is a simple and illustrative example. </p><p>If Alice and Bob share an entangled pair, and Bob teleports his particle to Carol, then Alice's particle is now entangled with Carol's particle. This situation can also be viewed symmetrically as follows: </p><p>Alice and Bob share an entangled pair, and Bob and Carol share a different entangled pair. Now let Bob perform a projective measurement on his two particles in the Bell basis and communicate the result to Carol. These actions are precisely the teleportation protocol described above with Bob's first particle, the one entangled with Alice's particle, as the state to be teleported. When Carol finishes the protocol she now has a particle with the teleported state, that is an entangled state with Alice's particle. Thus, although Alice and Carol never interacted with each other, their particles are now entangled. </p><p>A detailed diagrammatic derivation of entanglement swapping has been given by <a href="/wiki/Bob_Coecke" title="Bob Coecke">Bob Coecke</a>,<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> presented in terms of <a href="/wiki/Categorical_quantum_mechanics" title="Categorical quantum mechanics">categorical quantum mechanics</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Algorithm_for_swapping_Bell_pairs">Algorithm for swapping Bell pairs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=13" title="Edit section: Algorithm for swapping Bell pairs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An important application of entanglement swapping is distributing Bell states for use in entanglement distributed <a href="/wiki/Quantum_network" title="Quantum network">quantum networks</a>. A technical description of the entanglement swapping protocol is given here for pure Bell states. </p> <ol><li>Alice and Bob locally prepare known Bell pairs resulting in the initial state: <br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle _{\rm {in}}=|\Phi ^{+}\rangle _{A_{1},A_{2}}|\Phi ^{+}\rangle _{B_{1},B_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle _{\rm {in}}=|\Phi ^{+}\rangle _{A_{1},A_{2}}|\Phi ^{+}\rangle _{B_{1},B_{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d394f8be6994f9c30f624606838bd63eab6b829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.913ex; height:3.176ex;" alt="{\displaystyle |\psi \rangle _{\rm {in}}=|\Phi ^{+}\rangle _{A_{1},A_{2}}|\Phi ^{+}\rangle _{B_{1},B_{2}}}"></span></li> <li>Alice sends qubit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{1}}"></span> to a third party Carol</li> <li>Bob sends qubit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fa091eb428443c9c5c5fcf32a69d3665c89e00c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle B_{1}}"></span> to Carol</li> <li>Carol performs a Bell projection between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc2435b217c1a0f46f8a517ffa225c6f9440e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fa091eb428443c9c5c5fcf32a69d3665c89e00c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle B_{1}}"></span> that by chance (all four Bell states are possible and recognizable) results in the measurement outcome: <br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \Phi ^{+}|_{A_{1},B_{1}}|\psi \rangle _{\rm {in}}=|\Phi ^{+}\rangle _{A_{2},B_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \Phi ^{+}|_{A_{1},B_{1}}|\psi \rangle _{\rm {in}}=|\Phi ^{+}\rangle _{A_{2},B_{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dc9720aa5516fc7e8a9f499aaa84b6b529dc3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.913ex; height:3.343ex;" alt="{\displaystyle \langle \Phi ^{+}|_{A_{1},B_{1}}|\psi \rangle _{\rm {in}}=|\Phi ^{+}\rangle _{A_{2},B_{2}}}"></span></li> <li>In the case of the other three Bell projection outcomes, local corrections given by Pauli operators are made by Alice and or Bob after Carol has communicated the results of the measurement. <br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \Phi ^{-}|_{A_{1},B_{1}}|\psi \rangle _{\rm {in}}={\hat {Z}}_{B_{2}}|\Phi ^{+}\rangle _{A_{2},B_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \Phi ^{-}|_{A_{1},B_{1}}|\psi \rangle _{\rm {in}}={\hat {Z}}_{B_{2}}|\Phi ^{+}\rangle _{A_{2},B_{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f55b172ab8bb08cb233088ea7cb44a291feff5bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:30.904ex; height:3.676ex;" alt="{\displaystyle \langle \Phi ^{-}|_{A_{1},B_{1}}|\psi \rangle _{\rm {in}}={\hat {Z}}_{B_{2}}|\Phi ^{+}\rangle _{A_{2},B_{2}}}"></span> <br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \Psi ^{+}|_{A_{1},B_{1}}|\psi \rangle _{\rm {in}}={\hat {X}}_{B_{2}}|\Phi ^{+}\rangle _{A_{2},B_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \Psi ^{+}|_{A_{1},B_{1}}|\psi \rangle _{\rm {in}}={\hat {X}}_{B_{2}}|\Phi ^{+}\rangle _{A_{2},B_{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be0531b9147b43b4e91cdb26bbe849e0f6eca4d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:31.334ex; height:3.676ex;" alt="{\displaystyle \langle \Psi ^{+}|_{A_{1},B_{1}}|\psi \rangle _{\rm {in}}={\hat {X}}_{B_{2}}|\Phi ^{+}\rangle _{A_{2},B_{2}}}"></span> <br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \Psi ^{-}|_{A_{1},B_{1}}|\psi \rangle _{\rm {in}}={\hat {X}}_{B_{2}}{\hat {Z}}_{B_{2}}|\Phi ^{+}\rangle _{A_{2},B_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \Psi ^{-}|_{A_{1},B_{1}}|\psi \rangle _{\rm {in}}={\hat {X}}_{B_{2}}{\hat {Z}}_{B_{2}}|\Phi ^{+}\rangle _{A_{2},B_{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd001b446b241916ed6cd0b0cfeb3cbe510e372f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:35.326ex; height:3.676ex;" alt="{\displaystyle \langle \Psi ^{-}|_{A_{1},B_{1}}|\psi \rangle _{\rm {in}}={\hat {X}}_{B_{2}}{\hat {Z}}_{B_{2}}|\Phi ^{+}\rangle _{A_{2},B_{2}}}"></span></li> <li>Alice and Bob now have a Bell pair between qubits <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ec73b8bc9abc3efb934f5a6ec2803713771f4bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.509ex;" alt="{\displaystyle A_{2}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/199944d59dcc18842dfd1deab6000a1d1dadcbae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle B_{2}}"></span> <br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle _{\rm {out}}=|\Phi ^{+}\rangle _{A_{2},B_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>=</mo> <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> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle _{\rm {out}}=|\Phi ^{+}\rangle _{A_{2},B_{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/219fd29ae457eb21130d33b466c3ccb154dfd510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.344ex; height:3.176ex;" alt="{\displaystyle |\psi \rangle _{\rm {out}}=|\Phi ^{+}\rangle _{A_{2},B_{2}}}"></span></li></ol> <div class="mw-heading mw-heading2"><h2 id="Generalizations_of_the_teleportation_protocol">Generalizations of the teleportation protocol</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=14" title="Edit section: Generalizations of the teleportation protocol"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The basic teleportation protocol for a qubit described above has been generalized in several directions, in particular regarding the dimension of the system teleported and the number of parties involved (either as sender, controller, or receiver). </p> <div class="mw-heading mw-heading3"><h3 id="d-dimensional_systems"><i>d</i>-dimensional systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=15" title="Edit section: d-dimensional systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A generalization to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>-level systems (so-called <a href="/wiki/Qudit" class="mw-redirect" title="Qudit">qudits</a>) is straight forward and was already discussed in the original paper by Bennett <i>et al.</i>:<sup id="cite_ref-BBCJPW93_1-1" class="reference"><a href="#cite_note-BBCJPW93-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> the maximally entangled state of two qubits has to be replaced by a maximally entangled state of two qudits and the Bell measurement by a measurement defined by a maximally entangled orthonormal basis. All possible such generalizations were discussed by Werner in 2001.<sup id="cite_ref-Werner2001_50-0" class="reference"><a href="#cite_note-Werner2001-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> </p><p>The generalization to infinite-dimensional so-called <a href="/wiki/Continuous-variable_quantum_information" title="Continuous-variable quantum information">continuous-variable systems</a> was proposed by Braunstein and Kimble<sup id="cite_ref-BrKi98_51-0" class="reference"><a href="#cite_note-BrKi98-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> and led to the first teleportation experiment that worked unconditionally.<sup id="cite_ref-Furu98_52-0" class="reference"><a href="#cite_note-Furu98-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Multipartite_versions">Multipartite versions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=16" title="Edit section: Multipartite versions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The use of <a href="/wiki/Multipartite_entanglement" title="Multipartite entanglement">multipartite entangled states</a> instead of a bipartite maximally entangled state allows for several new features: either the sender can teleport information to several receivers either sending the same state to all of them (which allows to reduce the amount of entanglement needed for the process)<sup id="cite_ref-DuCi00_53-0" class="reference"><a href="#cite_note-DuCi00-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> or teleporting multipartite states<sup id="cite_ref-YeCh06_54-0" class="reference"><a href="#cite_note-YeCh06-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> or sending a single state in such a way that the receiving parties need to cooperate to extract the information.<sup id="cite_ref-KaBo98_55-0" class="reference"><a href="#cite_note-KaBo98-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> A different way of viewing the latter setting is that some of the parties can control whether the others can teleport. </p> <div class="mw-heading mw-heading2"><h2 id="Logic_gate_teleportation">Logic gate teleportation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=17" title="Edit section: Logic gate teleportation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Quantum_gate_teleportation" title="Quantum gate teleportation">Quantum gate teleportation</a></div> <p>In general, <a href="/wiki/Mixed_state_(physics)" class="mw-redirect" title="Mixed state (physics)">mixed states</a> ρ may be transported, and a linear transformation ω applied during teleportation, thus allowing data processing of quantum information. This is one of the foundational building blocks of quantum information processing. This is demonstrated below. </p> <div class="mw-heading mw-heading3"><h3 id="General_description">General description</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=18" title="Edit section: General description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A general teleportation scheme can be described as follows. Three quantum systems are involved. System 1 is the (unknown) state <i>ρ</i> to be teleported by Alice. Systems 2 and 3 are in a maximally entangled state <i>ω</i> that are distributed to Alice and Bob, respectively. The total system is then in the state </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \otimes \omega .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>⊗</mo> <mi>ω</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \otimes \omega .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2c656f4c9e4372e34073c7659ed87a1f4ab94a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.135ex; height:2.509ex;" alt="{\displaystyle \rho \otimes \omega .}"></span></dd></dl> <p>A successful teleportation process is a <a href="/wiki/LOCC" title="LOCC">LOCC</a> <a href="/wiki/Quantum_channel" title="Quantum channel">quantum channel</a> Φ that satisfies </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\operatorname {Tr} _{12}\circ \Phi )(\rho \otimes \omega )=\rho \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>Tr</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>∘</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>⊗</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ρ</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\operatorname {Tr} _{12}\circ \Phi )(\rho \otimes \omega )=\rho \,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4150886555c8b4467d231797c72c8de82be84ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.135ex; height:2.843ex;" alt="{\displaystyle (\operatorname {Tr} _{12}\circ \Phi )(\rho \otimes \omega )=\rho \,,}"></span></dd></dl> <p>where Tr<sub>12</sub> is the <a href="/wiki/Partial_trace" title="Partial trace">partial trace</a> operation with respect systems 1 and 2, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"></span> denotes the composition of maps. This describes the channel in the Schrödinger picture. </p><p>Taking adjoint maps in the Heisenberg picture, the success condition becomes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \Phi (\rho \otimes \omega )|I\otimes O\rangle =\langle \rho |O\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>⊗</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>I</mi> <mo>⊗</mo> <mi>O</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>O</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \Phi (\rho \otimes \omega )|I\otimes O\rangle =\langle \rho |O\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bfa7a374fecb9d3eb28c1dfc969350a1384aeb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.747ex; height:2.843ex;" alt="{\displaystyle \langle \Phi (\rho \otimes \omega )|I\otimes O\rangle =\langle \rho |O\rangle }"></span></dd></dl> <p>for all observable <i>O</i> on Bob's system. The tensor factor in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\otimes O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>⊗</mo> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I\otimes O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc185a40fb8237ce0318cd6f024def1976252af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.785ex; height:2.343ex;" alt="{\displaystyle I\otimes O}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12\otimes 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> <mo>⊗</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12\otimes 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2358dcd21445139204133c82663b67385502168a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.328ex; height:2.343ex;" alt="{\displaystyle 12\otimes 3}"></span> while that of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \otimes \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>⊗</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \otimes \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98e1f88c7e2c71974ff2d6cad918f2a823d611e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.488ex; height:2.509ex;" alt="{\displaystyle \rho \otimes \omega }"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\otimes 23}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⊗</mo> <mn>23</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\otimes 23}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/184a95032146a112916001d2652f29a49c2638a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.328ex; height:2.343ex;" alt="{\displaystyle 1\otimes 23}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Further_details">Further details</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=19" title="Edit section: Further details"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The proposed channel Φ can be described more explicitly. To begin teleportation, Alice performs a local measurement on the two subsystems (1 and 2) in her possession. Assume the local measurement have <i>effects</i> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {F_{i}}={M_{i}^{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {F_{i}}={M_{i}^{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a870f653e83cd1564310ea08c1efd9a0c0d698e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.593ex; height:3.176ex;" alt="{\displaystyle {F_{i}}={M_{i}^{2}}.}"></span></dd></dl> <p>If the measurement registers the <i>i</i>-th outcome, the overall state collapses to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊗</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>⊗</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊗</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39ba714f350945d9e6e8af2d124ec020180d7f4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.695ex; height:2.843ex;" alt="{\displaystyle (M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I).}"></span></dd></dl> <p>The tensor factor in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M_{i}\otimes I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊗</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M_{i}\otimes I)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03b534bfb5093cfc91c9b097a0aab727fcc4de9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.875ex; height:2.843ex;" alt="{\displaystyle (M_{i}\otimes I)}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12\otimes 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> <mo>⊗</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12\otimes 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2358dcd21445139204133c82663b67385502168a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.328ex; height:2.343ex;" alt="{\displaystyle 12\otimes 3}"></span> while that of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \otimes \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>⊗</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \otimes \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98e1f88c7e2c71974ff2d6cad918f2a823d611e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.488ex; height:2.509ex;" alt="{\displaystyle \rho \otimes \omega }"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\otimes 23}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⊗</mo> <mn>23</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\otimes 23}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/184a95032146a112916001d2652f29a49c2638a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.328ex; height:2.343ex;" alt="{\displaystyle 1\otimes 23}"></span>. Bob then applies a corresponding local operation Ψ<i><sub>i</sub></i> on system 3. On the combined system, this is described by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Id\otimes \Psi _{i})(M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>I</mi> <mi>d</mi> <mo>⊗</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊗</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>⊗</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊗</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Id\otimes \Psi _{i})(M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e18c8c3cb03a71ce2b50a69ee753564dd8a8a4f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.34ex; height:2.843ex;" alt="{\displaystyle (Id\otimes \Psi _{i})(M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I).}"></span></dd></dl> <p>where <i>Id</i> is the identity map on the composite system <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\otimes 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⊗</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\otimes 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ad0467bed659214688c54f8a8768b62e5f91e0" 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\otimes 2}"></span>. </p><p>Therefore, the channel Φ is defined by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (\rho \otimes \omega )=\sum _{i}(Id\otimes \Psi _{i})(M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>⊗</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mi>I</mi> <mi>d</mi> <mo>⊗</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊗</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>⊗</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊗</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (\rho \otimes \omega )=\sum _{i}(Id\otimes \Psi _{i})(M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf2a4c153d2275372ce744b8abd6f60b31d6c71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.122ex; height:5.509ex;" alt="{\displaystyle \Phi (\rho \otimes \omega )=\sum _{i}(Id\otimes \Psi _{i})(M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I)}"></span></dd></dl> <p>Notice Φ satisfies the definition of <a href="/wiki/LOCC" title="LOCC">LOCC</a>. As stated above, the teleportation is said to be successful if, for all observable <i>O</i> on Bob's system, the equality </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \Phi (\rho \otimes \omega ),I\otimes O\rangle =\langle \rho ,O\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>⊗</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>I</mi> <mo>⊗</mo> <mi>O</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ρ</mi> <mo>,</mo> <mi>O</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \Phi (\rho \otimes \omega ),I\otimes O\rangle =\langle \rho ,O\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e02a76dc7a248465f46db7e2bd7b975b766222" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.521ex; height:2.843ex;" alt="{\displaystyle \langle \Phi (\rho \otimes \omega ),I\otimes O\rangle =\langle \rho ,O\rangle }"></span></dd></dl> <p>holds. The left hand side of the equation is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}\langle (Id\otimes \Psi _{i})(M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I),\;I\otimes O\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mi>d</mi> <mo>⊗</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊗</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>⊗</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊗</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mi>I</mi> <mo>⊗</mo> <mi>O</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}\langle (Id\otimes \Psi _{i})(M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I),\;I\otimes O\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e741dce0663aab1cad9dfaf46aed06128132faa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.322ex; height:5.509ex;" alt="{\displaystyle \sum _{i}\langle (Id\otimes \Psi _{i})(M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I),\;I\otimes O\rangle }"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\sum _{i}\langle (M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I),\;I\otimes \Psi _{i}^{*}(O)\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊗</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>⊗</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊗</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mi>I</mi> <mo>⊗</mo> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\sum _{i}\langle (M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I),\;I\otimes \Psi _{i}^{*}(O)\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20fa01d176ae63860cb40205025e564c4e3cb065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.802ex; height:5.509ex;" alt="{\displaystyle =\sum _{i}\langle (M_{i}\otimes I)(\rho \otimes \omega )(M_{i}\otimes I),\;I\otimes \Psi _{i}^{*}(O)\rangle }"></span></dd></dl> <p>where Ψ<i><sub>i</sub>*</i> is the adjoint of Ψ<i><sub>i</sub></i> in the Heisenberg picture. Assuming all objects are finite dimensional, this becomes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}\operatorname {Tr} \;(\rho \otimes \omega )(F_{i}\otimes \Psi _{i}^{*}(O)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>Tr</mi> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>⊗</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊗</mo> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}\operatorname {Tr} \;(\rho \otimes \omega )(F_{i}\otimes \Psi _{i}^{*}(O)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/867f7d9442cd1dae69ed6e5d13108d0d696f0f70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.31ex; height:5.509ex;" alt="{\displaystyle \sum _{i}\operatorname {Tr} \;(\rho \otimes \omega )(F_{i}\otimes \Psi _{i}^{*}(O)).}"></span></dd></dl> <p>The success criterion for teleportation has the expression </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}\operatorname {Tr} \;(\rho \otimes \omega )(F_{i}\otimes \Psi _{i}^{*}(O))=\operatorname {Tr} \;\rho \cdot O.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mi>Tr</mi> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>⊗</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊗</mo> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>O</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Tr</mi> <mspace width="thickmathspace" /> <mi>ρ</mi> <mo>⋅</mo> <mi>O</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}\operatorname {Tr} \;(\rho \otimes \omega )(F_{i}\otimes \Psi _{i}^{*}(O))=\operatorname {Tr} \;\rho \cdot O.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aed4be57ad81ccc1feeec507c69ea5b94408019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.685ex; height:5.509ex;" alt="{\displaystyle \sum _{i}\operatorname {Tr} \;(\rho \otimes \omega )(F_{i}\otimes \Psi _{i}^{*}(O))=\operatorname {Tr} \;\rho \cdot O.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Local_explanation_of_the_phenomenon">Local explanation of the phenomenon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=20" title="Edit section: Local explanation of the phenomenon"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Principle_of_locality#Quantum_mechanics" title="Principle of locality">local</a> explanation of quantum teleportation is put forward by <a href="/wiki/David_Deutsch" title="David Deutsch">David Deutsch</a> and <a href="/wiki/Patrick_Hayden_(scientist)" title="Patrick Hayden (scientist)">Patrick Hayden</a>, with respect to the <a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">many-worlds interpretation</a> of quantum mechanics. Their paper asserts that the two bits that Alice sends Bob contain "locally inaccessible information" resulting in the teleportation of the quantum state. "The ability of quantum information to flow through a classical channel [...], surviving decoherence, is [...] the basis of quantum teleportation."<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Recent_developments">Recent developments</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=21" title="Edit section: Recent developments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While quantum teleportation is in an infancy stage, there are many aspects pertaining to teleportation that scientists are working to better understand or improve the process that include: </p> <div class="mw-heading mw-heading3"><h3 id="Higher_dimensions">Higher dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=22" title="Edit section: Higher dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Quantum teleportation can improve the errors associated with <a href="/wiki/Quantum_threshold_theorem" class="mw-redirect" title="Quantum threshold theorem">fault tolerant quantum computation</a> via an arrangement of logic gates. Experiments by D. Gottesman and I. L. Chuang have determined that a "Clifford hierarchy"<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> gate arrangement which acts to enhance protection against environmental errors. Overall, a higher threshold of error is allowed with the Clifford hierarchy as the sequence of gates requires less resources that are needed for computation. While the more gates that are used in a quantum computer create more noise, the gates arrangement and use of teleportation in logic transfer can reduce this noise as it calls for less "traffic" that is compiled in these quantum networks.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> The more qubits used for a quantum computer, the more levels are added to a gate arrangement, with the diagonalization of gate arrangement varying in degree. Higher dimension analysis involves the higher level gate arrangement of the Clifford hierarchy.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Information_quality">Information quality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=23" title="Edit section: Information quality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Considering the previously mentioned requirement of an intermediate entangled state for quantum teleportation, there needs to be consideration of the purity of this state for information quality. A protection that has been developed involves the use of continuous variable information (rather than a typical discrete variable) creating a superimposed coherent intermediate state. This involves making a phase shift in the received information and then adding a mixing step upon reception using a preferred state, which could be an odd or even coherent state, that will be "conditioned to the classical information of the sender", creating a two mode state that contains the originally sent information.<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> </p><p>There have also been developments with teleporting information between systems that already have quantum information in them. Experiments done by Feng, Xu, Zhou et al. have demonstrated that teleportation of a qubit to a photon that already has a qubit's worth of information is possible due to using an optical qubit-ququart entangling gate.<sup id="cite_ref-Feng,_Tianfeng_2020_4-1" class="reference"><a href="#cite_note-Feng,_Tianfeng_2020-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This quality can increase computation possibilities as calculations can be done based on previously stored information, allowing for improvements on past calculations. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=24" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Superdense_coding" title="Superdense coding">Superdense coding</a></li> <li><a href="/wiki/Quantum_complex_network" title="Quantum complex network">Quantum complex network</a></li> <li><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction to quantum mechanics</a></li> <li><a href="/wiki/Quantum_computer" class="mw-redirect" title="Quantum computer">Quantum computer</a></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Quantum nonlocality</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Heisenberg uncertainty principle</a></li></ul></li> <li><a href="/wiki/Wheeler%E2%80%93Feynman_absorber_theory" title="Wheeler–Feynman absorber theory">Wheeler–Feynman absorber theory</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=25" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Specific">Specific</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=26" title="Edit section: Specific"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-BBCJPW93-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-BBCJPW93_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-BBCJPW93_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output 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"High success standard quantum teleportation using entangled coherent state and two-level atoms in cavities". <i>Quantum Inf Process</i>. <b>20</b> (10). 322. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2010.06829">2010.06829</a></span>. <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/2021QuIP...20..322P">2021QuIP...20..322P</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%2Fs11128-021-03264-0">10.1007/s11128-021-03264-0</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:222341312">222341312</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quantum+Inf+Process&rft.atitle=High+success+standard+quantum+teleportation+using+entangled+coherent+state+and+two-level+atoms+in+cavities&rft.volume=20&rft.issue=10&rft.pages=322&rft.date=2021-09-27&rft_id=info%3Aarxiv%2F2010.06829&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A222341312%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs11128-021-03264-0&rft_id=info%3Abibcode%2F2021QuIP...20..322P&rft.aulast=Pandey&rft.aufirst=Ravi&rft.au=Prakash%2C+Ranjana&rft.au=Prakash%2C+Hari&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+teleportation" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="General">General</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_teleportation&action=edit&section=27" title="Edit section: General"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li>Review on theory and experiments: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPirandolaEisertWeedbrookFurusawa2015" class="citation journal cs1">Pirandola, S.; Eisert, J.; Weedbrook, C.; Furusawa, A.; Braunstein, S. L. (2015). "Advances in Quantum Teleportation". <i><a href="/wiki/Nature_Photonics" title="Nature Photonics">Nature Photonics</a></i>. <b>9</b> (10): <span class="nowrap">641–</span>652. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1505.07831">1505.07831</a></span>. <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/2015NaPho...9..641P">2015NaPho...9..641P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnphoton.2015.154">10.1038/nphoton.2015.154</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:15074330">15074330</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Photonics&rft.atitle=Advances+in+Quantum+Teleportation&rft.volume=9&rft.issue=10&rft.pages=%3Cspan+class%3D%22nowrap%22%3E641-%3C%2Fspan%3E652&rft.date=2015&rft_id=info%3Aarxiv%2F1505.07831&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15074330%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2Fnphoton.2015.154&rft_id=info%3Abibcode%2F2015NaPho...9..641P&rft.aulast=Pirandola&rft.aufirst=S.&rft.au=Eisert%2C+J.&rft.au=Weedbrook%2C+C.&rft.au=Furusawa%2C+A.&rft.au=Braunstein%2C+S.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+teleportation" class="Z3988"></span></li></ul></li> <li>Theoretical proposal: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBennettBrassardCrépeauJozsa1993" class="citation journal cs1">Bennett, Charles H.; <a href="/wiki/Gilles_Brassard" title="Gilles Brassard">Brassard, Gilles</a>; Crépeau, Claude; Jozsa, Richard; Peres, Asher; Wootters, William K. (29 March 1993). "Teleporting an Unknown Quantum State via Dual Classical and Einstein–Podolsky–Rosen Channels". <i>Phys. Rev. Lett</i>. <b>70</b> (13): <span class="nowrap">1895–</span>1899. <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/1993PhRvL..70.1895B">1993PhRvL..70.1895B</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%2FPhysRevLett.70.1895">10.1103/PhysRevLett.70.1895</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10053414">10053414</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:6599630">6599630</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.+Lett.&rft.atitle=Teleporting+an+Unknown+Quantum+State+via+Dual+Classical+and+Einstein%E2%80%93Podolsky%E2%80%93Rosen+Channels&rft.volume=70&rft.issue=13&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1895-%3C%2Fspan%3E1899&rft.date=1993-03-29&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.70.1895&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6599630%23id-name%3DS2CID&rft_id=info%3Apmid%2F10053414&rft_id=info%3Abibcode%2F1993PhRvL..70.1895B&rft.aulast=Bennett&rft.aufirst=Charles+H.&rft.au=Brassard%2C+Gilles&rft.au=Cr%C3%A9peau%2C+Claude&rft.au=Jozsa%2C+Richard&rft.au=Peres%2C+Asher&rft.au=Wootters%2C+William+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+teleportation" class="Z3988"></span> This is the seminal paper that laid out the entanglement protocol.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVaidman1994" class="citation journal cs1"><a href="/wiki/Lev_Vaidman" title="Lev Vaidman">Vaidman, L.</a> (February 1994). 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Date: 2006-02-16</li> <li><a rel="nofollow" class="external text" href="http://www.research.ibm.com/quantuminfo/teleportation/">Quantum Teleportation at IBM</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110107152340/http://www.research.ibm.com/quantuminfo/teleportation/">Archived</a> 7 January 2011 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www.spacedaily.com/news/physics-04zi.html">Physicists Succeed In Transferring Information Between Matter And Light</a></li> <li><a rel="nofollow" class="external text" href="http://www.physorg.com/news10924.html">Quantum telecloning: Captain Kirk's clone and the eavesdropper</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20081218105859/http://www.math.uwaterloo.ca/~amchilds/talks/pi03.ppt">"Teleportation-based approaches to universal quantum computation"</a>. 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"Deterministic quantum teleportation with atoms". <i>Nature</i>. <b>429</b> (6993): <span class="nowrap">734–</span>737. <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/2004Natur.429..734R">2004Natur.429..734R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature02570">10.1038/nature02570</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/15201903">15201903</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Deterministic+quantum+teleportation+with+atoms&rft.volume=429&rft.issue=6993&rft.pages=%3Cspan+class%3D%22nowrap%22%3E734-%3C%2Fspan%3E737&rft.date=2004-06&rft_id=info%3Apmid%2F15201903&rft_id=info%3Adoi%2F10.1038%2Fnature02570&rft_id=info%3Abibcode%2F2004Natur.429..734R&rft.aulast=Riebe&rft.aufirst=M.&rft.au=H%C3%A4ffner%2C+H.&rft.au=Roos%2C+C.+F.&rft.au=H%C3%A4nsel%2C+W.&rft.au=Benhelm%2C+J.&rft.au=Lancaster%2C+G.+P.+T.&rft.au=K%C3%B6rber%2C+T.+W.&rft.au=Becher%2C+C.&rft.au=Schmidt-Kaler%2C+F.&rft.au=James%2C+D.+F.+V.&rft.au=Blatt%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+teleportation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKimKulikShih2002" class="citation journal cs1">Kim, Yoon-Ho; Kulik, Sergei; Shih, Yanhua (January 2002). "Quantum teleportation with a complete Bell state measurement". <i>Journal of Modern Optics</i>. <b>49</b> (<span class="nowrap">1–</span>2): <span class="nowrap">221–</span>236. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0010046">quant-ph/0010046</a></span>. <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/2002JMOp...49..221K">2002JMOp...49..221K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F09500340110087633">10.1080/09500340110087633</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Modern+Optics&rft.atitle=Quantum+teleportation+with+a+complete+Bell+state+measurement&rft.volume=49&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E221-%3C%2Fspan%3E236&rft.date=2002-01&rft_id=info%3Aarxiv%2Fquant-ph%2F0010046&rft_id=info%3Adoi%2F10.1080%2F09500340110087633&rft_id=info%3Abibcode%2F2002JMOp...49..221K&rft.aulast=Kim&rft.aufirst=Yoon-Ho&rft.au=Kulik%2C+Sergei&rft.au=Shih%2C+Yanhua&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+teleportation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20121130091556/http://www.sciencenews.org/view/feature/id/34762/title/Welcome_to_the_Quantum_Internet">"Welcome to the quantum Internet"</a>. <i>Science News</i>. 16 August 2008. Archived from <a rel="nofollow" class="external text" href="http://sciencenews.org/view/feature/id/34762/title/Welcome_to_the_Quantum_Internet">the original</a> on 2012-11-30<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-08-14</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science+News&rft.atitle=Welcome+to+the+quantum+Internet&rft.date=2008-08-16&rft_id=http%3A%2F%2Fsciencenews.org%2Fview%2Ffeature%2Fid%2F34762%2Ftitle%2FWelcome_to_the_Quantum_Internet&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+teleportation" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://www.QuantumLab.de">Quantum experiments – interactive.</a></li> <li><a rel="nofollow" class="external text" href="http://lightlike.com/teleport/">A (mostly serious) introduction to quantum teleportation for non-physicists</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output 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href="/wiki/Template:Quantum_information" title="Template:Quantum information"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_information" title="Template talk:Quantum information"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_information" title="Special:EditPage/Template:Quantum information"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_information_science667" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/DiVincenzo%27s_criteria" title="DiVincenzo's criteria">DiVincenzo's criteria</a></li> <li><a href="/wiki/Noisy_intermediate-scale_quantum_era" title="Noisy intermediate-scale quantum era">NISQ era</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">timeline</a></li></ul></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulation</a></li> <li><a href="/wiki/Qubit" title="Qubit">Qubit</a> <ul><li><a href="/wiki/Physical_and_logical_qubits" title="Physical and logical qubits">physical vs. logical</a></li></ul></li> <li><a href="/wiki/List_of_quantum_processors" title="List of quantum processors">Quantum processors</a> <ul><li><a href="/wiki/Cloud-based_quantum_computing" title="Cloud-based quantum computing">cloud-based</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell%27s_theorem" title="Bell's theorem">Bell's</a></li> <li><a href="/wiki/Eastin%E2%80%93Knill_theorem" title="Eastin–Knill theorem">Eastin–Knill</a></li> <li><a href="/wiki/Gleason%27s_theorem" title="Gleason's theorem">Gleason's</a></li> <li><a href="/wiki/Gottesman%E2%80%93Knill_theorem" title="Gottesman–Knill theorem">Gottesman–Knill</a></li> <li><a href="/wiki/Holevo%27s_theorem" title="Holevo's theorem">Holevo's</a></li> <li><a href="/wiki/No-broadcasting_theorem" title="No-broadcasting theorem">No-broadcasting</a></li> <li><a href="/wiki/No-cloning_theorem" title="No-cloning theorem">No-cloning</a></li> <li><a href="/wiki/No-communication_theorem" title="No-communication theorem">No-communication</a></li> <li><a href="/wiki/No-deleting_theorem" title="No-deleting theorem">No-deleting</a></li> <li><a href="/wiki/No-hiding_theorem" title="No-hiding theorem">No-hiding</a></li> <li><a href="/wiki/No-teleportation_theorem" title="No-teleportation theorem">No-teleportation</a></li> <li><a href="/wiki/PBR_theorem" class="mw-redirect" title="PBR theorem">PBR</a></li> <li><a href="/wiki/Quantum_speed_limit_theorems" class="mw-redirect" title="Quantum speed limit theorems">Quantum speed limit</a></li> <li><a href="/wiki/Threshold_theorem" title="Threshold theorem">Threshold</a></li> <li><a href="/wiki/Solovay%E2%80%93Kitaev_theorem" title="Solovay–Kitaev theorem">Solovay–Kitaev</a></li> <li><a href="/wiki/Schr%C3%B6dinger%E2%80%93HJW_theorem" title="Schrödinger–HJW theorem">Purification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum<br />communication</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_capacity" title="Classical capacity">Classical capacity</a> <ul><li><a href="/wiki/Entanglement-assisted_classical_capacity" title="Entanglement-assisted classical capacity">entanglement-assisted</a></li> <li><a href="/wiki/Quantum_capacity" title="Quantum capacity">quantum capacity</a></li></ul></li> <li><a href="/wiki/Entanglement_distillation" title="Entanglement distillation">Entanglement distillation</a></li> <li><a href="/wiki/Entanglement_swapping" title="Entanglement swapping">Entanglement swapping</a></li> <li><a href="/wiki/Monogamy_of_entanglement" title="Monogamy of entanglement">Monogamy of entanglement</a></li> <li><a href="/wiki/LOCC" title="LOCC">LOCC</a></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a> <ul><li><a href="/wiki/Quantum_network" title="Quantum network">quantum network</a></li></ul></li> <li><a class="mw-selflink selflink">Quantum teleportation</a> <ul><li><a href="/wiki/Quantum_gate_teleportation" title="Quantum gate teleportation">quantum gate teleportation</a></li></ul></li> <li><a href="/wiki/Superdense_coding" title="Superdense coding">Superdense coding</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Quantum_cryptography24" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">Post-quantum cryptography</a></li> <li><a href="/wiki/Quantum_coin_flipping" title="Quantum coin flipping">Quantum coin flipping</a></li> <li><a href="/wiki/Quantum_money" title="Quantum money">Quantum money</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a> <ul><li><a href="/wiki/BB84" title="BB84">BB84</a></li> <li><a href="/wiki/SARG04" title="SARG04">SARG04</a></li> <li><a href="/wiki/List_of_quantum_key_distribution_protocols" title="List of quantum key distribution protocols">other protocols</a></li></ul></li> <li><a href="/wiki/Quantum_secret_sharing" title="Quantum secret sharing">Quantum secret sharing</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amplitude_amplification" title="Amplitude amplification">Amplitude amplification</a></li> <li><a href="/wiki/Bernstein%E2%80%93Vazirani_algorithm" title="Bernstein–Vazirani algorithm">Bernstein–Vazirani</a></li> <li><a href="/wiki/BHT_algorithm" title="BHT algorithm">BHT</a></li> <li><a href="/wiki/Boson_sampling" title="Boson sampling">Boson sampling</a></li> <li><a href="/wiki/Deutsch%E2%80%93Jozsa_algorithm" title="Deutsch–Jozsa algorithm">Deutsch–Jozsa</a></li> <li><a href="/wiki/Grover%27s_algorithm" title="Grover's algorithm">Grover's</a></li> <li><a href="/wiki/HHL_algorithm" title="HHL algorithm">HHL</a></li> <li><a href="/wiki/Hidden_subgroup_problem" title="Hidden subgroup problem">Hidden subgroup</a></li> <li><a href="/wiki/Quantum_annealing" title="Quantum annealing">Quantum annealing</a></li> <li><a href="/wiki/Quantum_counting_algorithm" title="Quantum counting algorithm">Quantum counting</a></li> <li><a href="/wiki/Quantum_Fourier_transform" title="Quantum Fourier transform">Quantum Fourier transform</a></li> <li><a href="/wiki/Quantum_optimization_algorithms" title="Quantum optimization algorithms">Quantum optimization</a></li> <li><a href="/wiki/Quantum_phase_estimation_algorithm" title="Quantum phase estimation algorithm">Quantum phase estimation</a></li> <li><a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's</a></li> <li><a href="/wiki/Simon%27s_problem" title="Simon's problem">Simon's</a></li> <li><a href="/wiki/Variational_quantum_eigensolver" title="Variational quantum eigensolver">VQE</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum<br />complexity theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/BQP" title="BQP">BQP</a></li> <li><a href="/wiki/Exact_quantum_polynomial_time" title="Exact quantum polynomial time">EQP</a></li> <li><a href="/wiki/QIP_(complexity)" title="QIP (complexity)">QIP</a></li> <li><a href="/wiki/QMA" title="QMA">QMA</a></li> <li><a href="/wiki/PostBQP" title="PostBQP">PostBQP</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum <br /> processor benchmarks</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_supremacy" title="Quantum supremacy">Quantum supremacy</a></li> <li><a href="/wiki/Quantum_volume" title="Quantum volume">Quantum volume</a></li> <li><a href="/wiki/Randomized_benchmarking" title="Randomized benchmarking">Randomized benchmarking</a> <ul><li><a href="/wiki/Cross-entropy_benchmarking" title="Cross-entropy benchmarking">XEB</a></li></ul></li> <li><a href="/wiki/Relaxation_(NMR)" title="Relaxation (NMR)">Relaxation times</a> <ul><li><a href="/wiki/Spin%E2%80%93lattice_relaxation" title="Spin–lattice relaxation"><i>T</i><sub>1</sub></a></li> <li><a href="/wiki/Spin%E2%80%93spin_relaxation" title="Spin–spin relaxation"><i>T</i><sub>2</sub></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum<br /><a href="/wiki/Model_of_computation" title="Model of computation">computing models</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adiabatic_quantum_computation" title="Adiabatic quantum computation">Adiabatic quantum computation</a></li> <li><a href="/wiki/Continuous-variable_quantum_information" title="Continuous-variable quantum information">Continuous-variable quantum information</a></li> <li><a href="/wiki/One-way_quantum_computer" title="One-way quantum computer">One-way quantum computer</a> <ul><li><a href="/wiki/Cluster_state" title="Cluster state">cluster state</a></li></ul></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a> <ul><li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">quantum logic gate</a></li></ul></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a> <ul><li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">quantum neural network</a></li></ul></li> <li><a href="/wiki/Quantum_Turing_machine" title="Quantum Turing machine">Quantum Turing machine</a></li> <li><a href="/wiki/Topological_quantum_computer" title="Topological quantum computer">Topological quantum computer</a></li> <li><a href="/wiki/Hamiltonian_quantum_computation" title="Hamiltonian quantum computation">Hamiltonian quantum computation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum<br />error correction</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Codes <ul><li><a href="/wiki/CSS_code" title="CSS code">CSS</a></li> <li><a href="/wiki/Quantum_convolutional_code" title="Quantum convolutional code">quantum convolutional</a></li> <li><a href="/wiki/Stabilizer_code" title="Stabilizer code">stabilizer</a></li> <li><a href="/wiki/Shor_code" class="mw-redirect" title="Shor code">Shor</a></li> <li><a href="/wiki/Bacon%E2%80%93Shor_code" title="Bacon–Shor code">Bacon–Shor</a></li> <li><a href="/wiki/Steane_code" title="Steane code">Steane</a></li> <li><a href="/wiki/Toric_code" title="Toric code">Toric</a></li> <li><a href="/wiki/Gnu_code" title="Gnu code"><i>gnu</i></a></li></ul></li> <li><a href="/wiki/Entanglement-assisted_stabilizer_formalism" title="Entanglement-assisted stabilizer formalism">Entanglement-assisted</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Physical<br />implementations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cavity_quantum_electrodynamics" title="Cavity quantum electrodynamics">Cavity QED</a></li> <li><a href="/wiki/Circuit_quantum_electrodynamics" title="Circuit quantum electrodynamics">Circuit QED</a></li> <li><a href="/wiki/Linear_optical_quantum_computing" title="Linear optical quantum computing">Linear optical QC</a></li> <li><a href="/wiki/KLM_protocol" title="KLM protocol">KLM protocol</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Ultracold_atom" title="Ultracold atom">Ultracold atoms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Neutral_atom_quantum_computer" title="Neutral atom quantum computer">Neutral atom QC</a></li> <li><a href="/wiki/Trapped-ion_quantum_computer" title="Trapped-ion quantum computer">Trapped-ion QC</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Spin_(physics)" title="Spin (physics)">Spin</a>-based</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kane_quantum_computer" title="Kane quantum computer">Kane QC</a></li> <li><a href="/wiki/Spin_qubit_quantum_computer" title="Spin qubit quantum computer">Spin qubit QC</a></li> <li><a href="/wiki/Nitrogen-vacancy_center" title="Nitrogen-vacancy center">NV center</a></li> <li><a href="/wiki/Nuclear_magnetic_resonance_quantum_computer" title="Nuclear magnetic resonance quantum computer">NMR QC</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Superconducting_quantum_computing" title="Superconducting quantum computing">Superconducting</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Charge_qubit" title="Charge qubit">Charge qubit</a></li> <li><a href="/wiki/Flux_qubit" title="Flux qubit">Flux qubit</a></li> <li><a href="/wiki/Phase_qubit" title="Phase qubit">Phase qubit</a></li> <li><a href="/wiki/Transmon" title="Transmon">Transmon</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum<br />programming</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/OpenQASM" title="OpenQASM">OpenQASM</a>–<a href="/wiki/Qiskit" title="Qiskit">Qiskit</a>–<a href="/wiki/IBM_Quantum_Experience" class="mw-redirect" title="IBM Quantum Experience">IBM QX</a></li> <li><a href="/wiki/Quil_(instruction_set_architecture)" title="Quil (instruction set architecture)">Quil</a>–<a href="/wiki/Rigetti_Computing" title="Rigetti Computing">Forest/Rigetti QCS</a></li> <li><a href="/wiki/Cirq" title="Cirq">Cirq</a></li> <li><a href="/wiki/Q_Sharp" title="Q Sharp">Q#</a></li> <li><a href="/wiki/Libquantum" title="Libquantum">libquantum</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">many others...</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Quantum_information_science" title="Category:Quantum information science">Quantum information science</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Template"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/16px-Symbol_template_class_pink.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/23px-Symbol_template_class_pink.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/31px-Symbol_template_class_pink.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Template:Quantum_mechanics_topics" title="Template:Quantum mechanics topics">Quantum mechanics topics</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Quantum_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 href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a href="/wiki/Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a class="mw-selflink selflink">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><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Quantum_mechanics" title="Category:Quantum mechanics">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Emerging_technologies167" 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" style="text-align: center;"><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:Emerging_technologies" title="Template:Emerging technologies"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Emerging_technologies" title="Template talk:Emerging technologies"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Emerging_technologies" title="Special:EditPage/Template:Emerging technologies"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Emerging_technologies167" style="font-size:114%;margin:0 4em"><a href="/wiki/Emerging_technologies" title="Emerging technologies">Emerging technologies</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%">Fields</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Quantum</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_algorithm" title="Quantum algorithm">algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">cellular automata</a></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">computing</a></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">cryptography</a> <ul><li><a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">post-quantum</a></li></ul></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">dynamics</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">error correction</a></li> <li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">finite automata</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">image processing</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">imaging</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">logic</a></li> <li><a href="/wiki/Quantum_logic_clock" title="Quantum logic clock">logic clock</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">logic gate</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">metamaterial</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">simulator</a></li> <li><a class="mw-selflink selflink">teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Other</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/Acoustic_levitation" title="Acoustic levitation">Acoustic levitation</a></li> <li><a href="/wiki/Anti-gravity" title="Anti-gravity">Anti-gravity</a></li> <li><a href="/wiki/Cloak_of_invisibility" title="Cloak of invisibility">Cloak of invisibility</a></li> <li><a href="/wiki/Digital_scent_technology" title="Digital scent technology">Digital scent technology</a></li> <li><a href="/wiki/Force_field_(technology)" title="Force field (technology)">Force field</a> <ul><li><a href="/wiki/Plasma_window" title="Plasma window">Plasma window</a></li></ul></li> <li><a href="/wiki/Immersion_(virtual_reality)" title="Immersion (virtual reality)">Immersive virtual reality</a></li> <li><a href="/wiki/Magnetic_refrigeration" title="Magnetic refrigeration">Magnetic refrigeration</a></li> <li><a href="/wiki/Phased-array_optics" title="Phased-array optics">Phased-array optics</a></li> <li><a href="/wiki/Thermoacoustic_heat_engine" title="Thermoacoustic heat engine">Thermoacoustic heat engine</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="text-align: center;"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <b><a href="/wiki/List_of_emerging_technologies" title="List of emerging technologies">List</a></b></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q1135332#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata943" 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"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q1135332#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata943" style="font-size:114%;margin:0 4em"><a 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558.945 1 Template:Reflist"," 8.98% 111.881 6 Template:Navbox"," 7.73% 96.399 1 Template:Short_description"," 7.55% 94.115 1 Template:Quantum_computing"," 7.17% 89.403 2 Template:Harvnb"," 5.44% 67.796 2 Template:Pagetype"," 2.86% 35.700 1 Template:Unreliable_source?"," 2.66% 33.189 1 Template:Fix"]},"scribunto":{"limitreport-timeusage":{"value":"0.886","limit":"10.000"},"limitreport-memusage":{"value":10082230,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFBarrettChiaveriniSchaetzBritton2004\"] = 2,\n [\"CITEREFBennettBrassardCrépeauJozsa1993\"] = 2,\n [\"CITEREFBoschiBrancaDe_MartiniHardy1998\"] = 1,\n [\"CITEREFBouwmeesterPanMattleEibl1997\"] = 1,\n [\"CITEREFBrassardBraunsteinCleve1998\"] = 1,\n [\"CITEREFBraunsteinKimble1998\"] = 1,\n [\"CITEREFBussandriBosykToscano2024\"] = 1,\n [\"CITEREFCardosoAvelarBaseiaDe_Almeida2005\"] = 1,\n [\"CITEREFChang2004\"] = 1,\n [\"CITEREFChildress1999\"] = 1,\n [\"CITEREFChouBlumoffWangReinhold2018\"] = 1,\n [\"CITEREFCoecke2004\"] = 1,\n [\"CITEREFCoecke2009\"] = 1,\n [\"CITEREFCommissariat2015\"] = 1,\n [\"CITEREFDeutschHayden1999\"] = 1,\n [\"CITEREFDürCirac2000\"] = 1,\n [\"CITEREFFurusawaSørensenBraunsteinFuchs1998\"] = 1,\n [\"CITEREFGottesmanChuang1999\"] = 1,\n [\"CITEREFJinRenYangYi2010\"] = 1,\n [\"CITEREFKarlssonBourennane1998\"] = 1,\n [\"CITEREFKimKulikShih2001\"] = 1,\n [\"CITEREFKimKulikShih2002\"] = 1,\n [\"CITEREFKnollSchmiegelowLarotonda2014\"] = 1,\n [\"CITEREFKrauterSalartMuschikPetersen2013\"] = 1,\n [\"CITEREFLeeHugo_BenichiYuishi_TakenoShuntaro_Takeda2011\"] = 1,\n [\"CITEREFLeuenbergerFlattéAwschalom2005\"] = 1,\n [\"CITEREFLo_FrancoCompagno2018\"] = 1,\n [\"CITEREFLuoChenErhardZhong2020\"] = 1,\n [\"CITEREFMaHerbstScheidlWang2012\"] = 1,\n [\"CITEREFMarcikicTittelZbindenGisin2003\"] = 1,\n [\"CITEREFMarkoff2014\"] = 1,\n [\"CITEREFMassarPopescu1995\"] = 1,\n [\"CITEREFMigdałJankiewiczGrabarzDecaroli2022\"] = 1,\n [\"CITEREFNielsenChuang2010\"] = 2,\n [\"CITEREFNöllekeNeuznerReisererHahn2013\"] = 1,\n [\"CITEREFOhLeeLee2002\"] = 1,\n [\"CITEREFOlmschenkMatsukevichMaunzHayes2009\"] = 1,\n [\"CITEREFPandeyPrakashPrakash2021\"] = 1,\n [\"CITEREFPeres2003\"] = 1,\n [\"CITEREFPfaffHensenBernienvan_Dam2014\"] = 1,\n [\"CITEREFPirandolaEisertWeedbrookFurusawa2015\"] = 3,\n [\"CITEREFPopescu1994\"] = 1,\n [\"CITEREFPyrkovByrnes2014\"] = 1,\n [\"CITEREFRenXuYongZhang2017\"] = 1,\n [\"CITEREFRibeiroRigolin2024\"] = 1,\n [\"CITEREFRiebeChwallaBenhelmHäffner2007\"] = 1,\n [\"CITEREFRiebeHäffnerRoosHänsel2004\"] = 3,\n [\"CITEREFRigolin2005\"] = 1,\n [\"CITEREFShahriarPradhanGopalMorzinski2006\"] = 1,\n [\"CITEREFSunMaoChenZhang2016\"] = 1,\n [\"CITEREFSunWangLiuXu2020\"] = 1,\n [\"CITEREFTakedaMizutaFuwavan_Loock2013\"] = 1,\n [\"CITEREFTakesueDyerStevensVerma2015\"] = 1,\n [\"CITEREFTangermann2020\"] = 1,\n [\"CITEREFThomasYehChenMambretti2024\"] = 1,\n [\"CITEREFTianfeng_FengQiao_XuLinxiang_ZhouMaolin_Luo2022\"] = 1,\n [\"CITEREFTrute\"] = 1,\n [\"CITEREFUniversity_of_Copenhagen2013\"] = 1,\n [\"CITEREFUrsinJenneweinAspelmeyerKaltenbaek2004\"] = 2,\n [\"CITEREFVaidman1994\"] = 1,\n [\"CITEREFValivarthiDavisPeñaXie2020\"] = 1,\n [\"CITEREFValivarthiPuigibertZhouAguilar2016\"] = 1,\n [\"CITEREFVidalLatorrePascualTarrach1999\"] = 1,\n [\"CITEREFWei2016\"] = 1,\n [\"CITEREFWerner2001\"] = 1,\n [\"CITEREFWilliams2010\"] = 1,\n [\"CITEREFXi-Lin_WangXin-Dong_CaiZu-En_SuMing-Cheng_Chen2015\"] = 1,\n [\"CITEREFXiaQi-Chao_SunQiang_ZhangJian-Wei_Pan2018\"] = 1,\n [\"CITEREFYeoChua2006\"] = 1,\n [\"CITEREFZhangGoebelWagenknechtChen2006\"] = 1,\n [\"CITEREFZhaoChenZhangYang2004\"] = 1,\n [\"CITEREFde_Silva2021\"] = 1,\n}\ntemplate_list = table#1 {\n [\"As of\"] = 2,\n [\"Authority control\"] = 1,\n [\"Cite arXiv\"] = 1,\n [\"Cite book\"] = 4,\n [\"Cite journal\"] = 68,\n [\"Cite news\"] = 2,\n [\"Cite report\"] = 1,\n [\"Cite web\"] = 7,\n [\"Convert\"] = 3,\n [\"Cvt\"] = 5,\n [\"DEFAULTSORT:Quantum Teleportation\"] = 1,\n [\"Emerging technologies\"] = 1,\n [\"Harvnb\"] = 2,\n [\"Main\"] = 1,\n [\"More\"] = 1,\n [\"Quantum computing\"] = 1,\n [\"Quantum mechanics topics\"] = 1,\n [\"Refbegin\"] = 1,\n [\"Refend\"] = 1,\n [\"Reflist\"] = 1,\n [\"Short description\"] = 1,\n [\"Snd\"] = 4,\n [\"Unreliable source?\"] = 1,\n [\"Use dmy dates\"] = 1,\n [\"Webarchive\"] = 1,\n [\"Wide image\"] = 1,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n"},"cachereport":{"origin":"mw-api-int.codfw.main-5b65fffc7d-bwqs9","timestamp":"20250214225727","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Quantum teleportation","url":"https:\/\/en.wikipedia.org\/wiki\/Quantum_teleportation","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1135332","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1135332","author":{"@type":"Organization","name":"Contributors to Wikimedia 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