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class="vector-toc-link" href="#Development_from_atomic_physics_and_relativity"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.2</span> <span>Development from atomic physics and relativity</span> </div> </a> <ul id="toc-Development_from_atomic_physics_and_relativity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Development_from_cryptography" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Development_from_cryptography"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.3</span> <span>Development from cryptography</span> </div> </a> <ul id="toc-Development_from_cryptography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Development_from_computer_science_and_mathematics" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Development_from_computer_science_and_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.4</span> <span>Development from computer science and mathematics</span> </div> </a> <ul id="toc-Development_from_computer_science_and_mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Development_from_information_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Development_from_information_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.5</span> <span>Development from information theory</span> </div> </a> <ul id="toc-Development_from_information_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Qubits_and_information_theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Qubits_and_information_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Qubits and information theory</span> </div> </a> <ul id="toc-Qubits_and_information_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_information_processing" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Quantum_information_processing"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Quantum information processing</span> </div> </a> <ul id="toc-Quantum_information_processing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_quantum_mechanics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relation_to_quantum_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Relation to quantum mechanics</span> </div> </a> <ul id="toc-Relation_to_quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Entropy_and_information" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Entropy_and_information"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Entropy and information</span> </div> </a> <button aria-controls="toc-Entropy_and_information-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 Entropy and information subsection</span> </button> <ul id="toc-Entropy_and_information-sublist" class="vector-toc-list"> <li id="toc-Classical_information_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classical_information_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Classical information theory</span> </div> </a> <ul id="toc-Classical_information_theory-sublist" class="vector-toc-list"> <li id="toc-Shannon_entropy" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Shannon_entropy"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.1</span> <span>Shannon entropy</span> </div> </a> <ul id="toc-Shannon_entropy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rényi_entropy" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Rényi_entropy"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.2</span> <span>Rényi entropy</span> </div> </a> <ul id="toc-Rényi_entropy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Quantum_information_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_information_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Quantum information theory</span> </div> </a> <ul id="toc-Quantum_information_theory-sublist" class="vector-toc-list"> <li id="toc-Von_Neumann_entropy" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Von_Neumann_entropy"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2.1</span> <span>Von Neumann entropy</span> </div> </a> <ul id="toc-Von_Neumann_entropy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-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 Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Quantum_communication" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_communication"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Quantum communication</span> </div> </a> <ul id="toc-Quantum_communication-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_key_distribution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_key_distribution"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Quantum key distribution</span> </div> </a> <ul id="toc-Quantum_key_distribution-sublist" class="vector-toc-list"> <li id="toc-BB84" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#BB84"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.1</span> <span>BB84</span> </div> </a> <ul id="toc-BB84-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-E91" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#E91"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.2</span> <span>E91</span> </div> </a> <ul id="toc-E91-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-B92" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#B92"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.3</span> <span>B92</span> </div> </a> <ul id="toc-B92-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Quantum_computation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_computation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Quantum computation</span> </div> </a> <ul id="toc-Quantum_computation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_decoherence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_decoherence"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Quantum decoherence</span> </div> </a> <ul id="toc-Quantum_decoherence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_error_correction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_error_correction"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Quantum error correction</span> </div> </a> <ul id="toc-Quantum_error_correction-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Journals" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Journals"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Journals</span> </div> </a> <ul id="toc-Journals-sublist" class="vector-toc-list"> </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">8</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">9</span> <span>References</span> </div> </a> <ul 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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%A4%E0%A6%A5%E0%A7%8D%E0%A6%AF" title="কোয়ান্টাম তথ্য – Bangla" lang="bn" hreflang="bn" data-title="কোয়ান্টাম তথ্য" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</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_%D0%B8%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D0%BE%D0%BD%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F" title="Квантова информационна теория – Bulgarian" lang="bg" hreflang="bg" data-title="Квантова информационна теория" 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_informacija" title="Kvantna informacija – Bosnian" lang="bs" hreflang="bs" data-title="Kvantna informacija" 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/Informaci%C3%B3_qu%C3%A0ntica" title="Informació quàntica – Catalan" lang="ca" hreflang="ca" data-title="Informació quàntica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Quanteninformation" title="Quanteninformation – German" lang="de" hreflang="de" data-title="Quanteninformation" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B7%D9%84%D8%A7%D8%B9%D8%A7%D8%AA_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C" title="اطلاعات کوانتومی – Persian" lang="fa" hreflang="fa" data-title="اطلاعات 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/Information_quantique" title="Information quantique – French" lang="fr" hreflang="fr" data-title="Information quantique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</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%A0%95%EB%B3%B4" title="양자정보 – Korean" lang="ko" hreflang="ko" data-title="양자정보" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%A3%E0%A5%8D%E0%A4%9F%E0%A4%AE_%E0%A4%B8%E0%A5%82%E0%A4%9A%E0%A4%A8%E0%A4%BE" title="क्वाण्टम सूचना – Hindi" lang="hi" hreflang="hi" data-title="क्वाण्टम सूचना" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Informasi_kuantum" title="Informasi kuantum – Indonesian" lang="id" hreflang="id" data-title="Informasi 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/Informazione_quantistica" title="Informazione quantistica – Italian" lang="it" hreflang="it" data-title="Informazione quantistica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</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%E6%83%85%E5%A0%B1" title="量子情報 – Japanese" lang="ja" hreflang="ja" data-title="量子情報" 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%B8%E0%A9%82%E0%A8%9A%E0%A8%A8%E0%A8%BE" title="ਕੁਆਂਟਮ ਸੂਚਨਾ – Punjabi" lang="pa" hreflang="pa" data-title="ਕੁਆਂਟਮ ਸੂਚਨਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Informacja_kwantowa" title="Informacja kwantowa – Polish" lang="pl" hreflang="pl" data-title="Informacja 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/Teoria_da_informa%C3%A7%C3%A3o_qu%C3%A2ntica" title="Teoria da informação quântica – Portuguese" lang="pt" hreflang="pt" data-title="Teoria da informação quântica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</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_%D0%B8%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%8F" title="Квантовая информация – Russian" lang="ru" hreflang="ru" data-title="Квантовая информация" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kvantti-informaatio" title="Kvantti-informaatio – Finnish" lang="fi" hreflang="fi" data-title="Kvantti-informaatio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0_%D1%96%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D1%96%D1%8F" title="Квантова інформація – Ukrainian" lang="uk" hreflang="uk" data-title="Квантова інформація" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E4%BF%A1%E6%81%AF" title="量子信息 – Chinese" lang="zh" hreflang="zh" data-title="量子信息" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2122243#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Quantum_information" title="View the 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<button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <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">Information held in the state of a quantum system</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">For the journal, see <a href="/wiki/Npj_Quantum_Information" title="Npj Quantum Information">npj Quantum Information</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Qubits_(5940500587).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Qubits_%285940500587%29.jpg/220px-Qubits_%285940500587%29.jpg" decoding="async" width="220" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Qubits_%285940500587%29.jpg/330px-Qubits_%285940500587%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Qubits_%285940500587%29.jpg/440px-Qubits_%285940500587%29.jpg 2x" data-file-width="948" data-file-height="658" /></a><figcaption>Optical lattices use lasers to separate rubidium atoms (red) for use as information bits in neutral-atom quantum processors—prototype devices which designers are trying to develop into full-fledged quantum computers.</figcaption></figure> <p><b>Quantum information</b> is the information of the <a href="/wiki/Quantum_state" title="Quantum state">state</a> of a <a href="/wiki/Quantum_system" class="mw-redirect" title="Quantum system">quantum system</a>. It is the basic entity of study in <b>quantum information theory</b>,<sup id="cite_ref-Vedral2006_1-0" class="reference"><a href="#cite_note-Vedral2006-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Nielsen2010_2-0" class="reference"><a href="#cite_note-Nielsen2010-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Hayashi2006_3-0" class="reference"><a href="#cite_note-Hayashi2006-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and can be manipulated using <a href="/wiki/Quantum_information_processing" class="mw-redirect" title="Quantum information processing">quantum information processing</a> techniques. Quantum information refers to both the technical definition in terms of <a href="/wiki/Von_Neumann_entropy" title="Von Neumann entropy">Von Neumann entropy</a> and the general computational term. </p><p>It is an interdisciplinary field that involves <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, <a href="/wiki/Computer_science" title="Computer science">computer science</a>, <a href="/wiki/Information_theory" title="Information theory">information theory</a>, <a href="/wiki/Philosophy" title="Philosophy">philosophy</a> and <a href="/wiki/Cryptography" title="Cryptography">cryptography</a> among other fields.<sup id="cite_ref-Bokulich2010_4-0" class="reference"><a href="#cite_note-Bokulich2010-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Benatti2010_5-0" class="reference"><a href="#cite_note-Benatti2010-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Benatti2009_6-0" class="reference"><a href="#cite_note-Benatti2009-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Its study is also relevant to disciplines such as <a href="/wiki/Cognitive_science" title="Cognitive science">cognitive science</a>, <a href="/wiki/Psychology" title="Psychology">psychology</a> and <a href="/wiki/Neuroscience" title="Neuroscience">neuroscience</a>.<sup id="cite_ref-Hayashi2015_7-0" class="reference"><a href="#cite_note-Hayashi2015-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Hayashi2017_8-0" class="reference"><a href="#cite_note-Hayashi2017-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Georgiev2017_9-0" class="reference"><a href="#cite_note-Georgiev2017-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Georgiev2020_10-0" class="reference"><a href="#cite_note-Georgiev2020-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Its main focus is in extracting information from matter at the microscopic scale. Observation in science is one of the most important ways of acquiring information and measurement is required in order to quantify the observation, making this crucial to the <a href="/wiki/Scientific_method" title="Scientific method">scientific method</a>. In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, due to the <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a>, <a href="/wiki/Commutative_property" title="Commutative property">non-commuting</a> <a href="/wiki/Observable" title="Observable">observables</a> cannot be precisely measured simultaneously, as an <a href="/wiki/Quantum_state" title="Quantum state">eigenstate</a> in one basis is not an eigenstate in the other basis. According to the eigenstate–eigenvalue link, an observable is well-defined (definite) when the state of the system is an eigenstate of the observable.<sup id="cite_ref-Gilton2016_11-0" class="reference"><a href="#cite_note-Gilton2016-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Since any two non-commuting observables are not simultaneously well-defined, a quantum state can never contain definitive information about both non-commuting observables.<sup id="cite_ref-Hayashi2017_8-1" class="reference"><a href="#cite_note-Hayashi2017-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Data can be encoded into the <a href="/wiki/State_(computer_science)" title="State (computer science)">quantum state</a> of a quantum system as <a class="mw-selflink selflink">quantum information</a>.<sup id="cite_ref-Preskill2018_12-0" class="reference"><a href="#cite_note-Preskill2018-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> While quantum mechanics deals with examining properties of matter at the microscopic level,<sup id="cite_ref-Feynman2013_13-0" class="reference"><a href="#cite_note-Feynman2013-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Hayashi2017_8-2" class="reference"><a href="#cite_note-Hayashi2017-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Quantum_information_science" title="Quantum information science">quantum information science</a> focuses on extracting information from those properties,<sup id="cite_ref-Hayashi2017_8-3" class="reference"><a href="#cite_note-Hayashi2017-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Quantum_computing" title="Quantum computing">quantum computation</a> manipulates and processes information – performs logical operations – using <a href="/wiki/Quantum_information_processing" class="mw-redirect" title="Quantum information processing">quantum information processing</a> techniques.<sup id="cite_ref-Lo1998_14-0" class="reference"><a href="#cite_note-Lo1998-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Quantum information, like classical information, can be processed using <a href="/wiki/Digital_computer" class="mw-redirect" title="Digital computer">digital computers</a>, <a href="/wiki/Communications_channel" class="mw-redirect" title="Communications channel">transmitted</a> from one location to another, manipulated with <a href="/wiki/Algorithm" title="Algorithm">algorithms</a>, and analyzed with computer science and <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>. Just like the basic unit of classical information is the bit, quantum information deals with <a href="/wiki/Qubit" title="Qubit">qubits</a>.<sup id="cite_ref-Bennett1998_15-0" class="reference"><a href="#cite_note-Bennett1998-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Quantum information can be measured using Von Neumann entropy. </p><p>Recently, the field of <a href="/wiki/Quantum_computing" title="Quantum computing">quantum computing</a> has become an active research area because of the possibility to disrupt modern computation, communication, and <a href="/wiki/Cryptography" title="Cryptography">cryptography</a>.<sup id="cite_ref-Lo1998_14-1" class="reference"><a href="#cite_note-Lo1998-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Garlinghouse2020_16-0" class="reference"><a href="#cite_note-Garlinghouse2020-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History_and_development">History and development</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=1" title="Edit section: History and development"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Development_from_fundamental_quantum_mechanics">Development from fundamental quantum mechanics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=2" title="Edit section: Development from fundamental quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The history of quantum information theory began at the turn of the 20th century when <a href="/wiki/Classical_physics" title="Classical physics">classical physics</a> was revolutionized into <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum physics</a>. The theories of classical physics were predicting absurdities such as the <a href="/wiki/Ultraviolet_catastrophe" title="Ultraviolet catastrophe">ultraviolet catastrophe</a>, or electrons spiraling into the nucleus. At first these problems were brushed aside by adding ad hoc hypotheses to classical physics. Soon, it became apparent that a new theory must be created in order to make sense of these absurdities, and the theory of quantum mechanics was born.<sup id="cite_ref-Nielsen2010_2-1" class="reference"><a href="#cite_note-Nielsen2010-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a> was formulated by <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a> using wave mechanics and <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a> using <a href="/wiki/Matrix_mechanics" title="Matrix mechanics">matrix mechanics</a>.<sup id="cite_ref-Mahan2009_17-0" class="reference"><a href="#cite_note-Mahan2009-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The equivalence of these methods was proven later.<sup id="cite_ref-Perlman1964_18-0" class="reference"><a href="#cite_note-Perlman1964-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Their formulations described the dynamics of microscopic systems but had several unsatisfactory aspects in describing measurement processes. Von Neumann formulated quantum theory using <a href="/wiki/Operator_algebra" title="Operator algebra">operator algebra</a> in a way that it described measurement as well as dynamics.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> These studies emphasized the philosophical aspects of measurement rather than a quantitative approach to extracting information via measurements. </p><p>See: <a href="/wiki/Dynamical_pictures" title="Dynamical pictures">Dynamical Pictures</a> </p> <table class="wikitable" style="padding:0.3em; clear:right; margin:1em auto; text-align:center;"> <tbody><tr> <td rowspan="2" style="background-color:#E0FFEE;">Evolution of: </td> <td colspan="3" style="background-color:#E0FFEE;"><a href="/wiki/Dynamical_pictures" title="Dynamical pictures">Picture</a> (<style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist 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a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Pictures_in_quantum_mechanics" title="Template:Pictures in quantum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Pictures_in_quantum_mechanics" title="Template talk:Pictures in quantum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Pictures_in_quantum_mechanics" title="Special:EditPage/Template:Pictures in quantum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div>) </td></tr> <tr> <td style="background-color:#E0F0FF;"><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a> (S) </td> <td style="background-color:#E0F0FF;"><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a> (H) </td> <td style="background-color:#E0F0FF;"><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a> (I) </td></tr> <tr> <td style="background-color:#D0FFDD;"><a href="/wiki/Bra-ket_notation" class="mw-redirect" title="Bra-ket notation">Ket state</a> </td> <td><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 _{\rm {S}}(t)\rangle =e^{-iH_{\rm {S}}~t/\hbar }|\psi _{\rm {S}}(0)\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi _{\rm {S}}(t)\rangle =e^{-iH_{\rm {S}}~t/\hbar }|\psi _{\rm {S}}(0)\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f7e8476598d735d3fa5be6979777f685fb8ea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.495ex; height:3.343ex;" alt="{\displaystyle |\psi _{\rm {S}}(t)\rangle =e^{-iH_{\rm {S}}~t/\hbar }|\psi _{\rm {S}}(0)\rangle }"></span> </td> <td>constant </td> <td><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 _{\rm {I}}(t)\rangle =e^{iH_{0,\mathrm {S} }~t/\hbar }|\psi _{\rm {S}}(t)\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi _{\rm {I}}(t)\rangle =e^{iH_{0,\mathrm {S} }~t/\hbar }|\psi _{\rm {S}}(t)\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6239a430d0eac4e07abc9f7926d65b12b4755c8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.612ex; height:3.343ex;" alt="{\displaystyle |\psi _{\rm {I}}(t)\rangle =e^{iH_{0,\mathrm {S} }~t/\hbar }|\psi _{\rm {S}}(t)\rangle }"></span> </td></tr> <tr> <td style="background-color:#D0FFDD;"><a href="/wiki/Observable" title="Observable">Observable</a> </td> <td>constant </td> <td><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_{\rm {H}}(t)=e^{iH_{\rm {S}}~t/\hbar }A_{\rm {S}}e^{-iH_{\rm {S}}~t/\hbar }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\rm {H}}(t)=e^{iH_{\rm {S}}~t/\hbar }A_{\rm {S}}e^{-iH_{\rm {S}}~t/\hbar }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fed2a30b52c2506bf999ba176813e78a5cd6f28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.273ex; height:3.343ex;" alt="{\displaystyle A_{\rm {H}}(t)=e^{iH_{\rm {S}}~t/\hbar }A_{\rm {S}}e^{-iH_{\rm {S}}~t/\hbar }}"></span> </td> <td><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_{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }A_{\rm {S}}e^{-iH_{0,\mathrm {S} }~t/\hbar }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }A_{\rm {S}}e^{-iH_{0,\mathrm {S} }~t/\hbar }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e17359cb3e0e75ad9db7c54218cb5ddedcb7605e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.712ex; height:3.343ex;" alt="{\displaystyle A_{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }A_{\rm {S}}e^{-iH_{0,\mathrm {S} }~t/\hbar }}"></span> </td></tr> <tr> <td style="background-color:#D0FFDD;"><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a> </td> <td><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 _{\rm {S}}(t)=e^{-iH_{\rm {S}}~t/\hbar }\rho _{\rm {S}}(0)e^{iH_{\rm {S}}~t/\hbar }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\rm {S}}(t)=e^{-iH_{\rm {S}}~t/\hbar }\rho _{\rm {S}}(0)e^{iH_{\rm {S}}~t/\hbar }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/238d1dec8256c54d7316d3656ca666eb32784d6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.844ex; height:3.343ex;" alt="{\displaystyle \rho _{\rm {S}}(t)=e^{-iH_{\rm {S}}~t/\hbar }\rho _{\rm {S}}(0)e^{iH_{\rm {S}}~t/\hbar }}"></span> </td> <td>constant </td> <td><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 _{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }\rho _{\rm {S}}(t)e^{-iH_{0,\mathrm {S} }~t/\hbar }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }\rho _{\rm {S}}(t)e^{-iH_{0,\mathrm {S} }~t/\hbar }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f578fe89295524333c363ccf0b9781275894c7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.278ex; height:3.343ex;" alt="{\displaystyle \rho _{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }\rho _{\rm {S}}(t)e^{-iH_{0,\mathrm {S} }~t/\hbar }}"></span> </td></tr> </tbody></table> <div class="mw-heading mw-heading4"><h4 id="Development_from_communication">Development from communication</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=3" title="Edit section: Development from communication"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the 1960s, <a href="/wiki/Ruslan_Stratonovich" title="Ruslan Stratonovich">Ruslan Stratonovich</a>, <a href="/wiki/Carl_W._Helstrom" title="Carl W. Helstrom">Carl Helstrom</a> and Gordon<sup id="cite_ref-Gordon1962_20-0" class="reference"><a href="#cite_note-Gordon1962-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> proposed a formulation of optical communications using quantum mechanics. This was the first historical appearance of quantum information theory. They mainly studied error probabilities and channel capacities for communication.<sup id="cite_ref-Gordon1962_20-1" class="reference"><a href="#cite_note-Gordon1962-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Helstrom1969_21-0" class="reference"><a href="#cite_note-Helstrom1969-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Helstrom1976_22-0" class="reference"><a href="#cite_note-Helstrom1976-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Later, <a href="/wiki/Alexander_Holevo" title="Alexander Holevo">Alexander Holevo</a> obtained an upper bound of communication speed in the transmission of a classical message via a <a href="/wiki/Quantum_channel" title="Quantum channel">quantum channel</a>.<sup id="cite_ref-Holevo1973_23-0" class="reference"><a href="#cite_note-Holevo1973-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Holevo1979_24-0" class="reference"><a href="#cite_note-Holevo1979-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Development_from_atomic_physics_and_relativity">Development from atomic physics and relativity</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=4" title="Edit section: Development from atomic physics and relativity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the 1970s, techniques for manipulating single-atom quantum states, such as the <a href="/wiki/Atom_trap" class="mw-redirect" title="Atom trap">atom trap</a> and the <a href="/wiki/Scanning_tunneling_microscope" title="Scanning tunneling microscope">scanning tunneling microscope</a>, began to be developed, making it possible to isolate single atoms and arrange them in arrays. Prior to these developments, precise control over single quantum systems was not possible, and experiments used coarser, simultaneous control over a large number of quantum systems.<sup id="cite_ref-Nielsen2010_2-2" class="reference"><a href="#cite_note-Nielsen2010-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The development of viable single-state manipulation techniques led to increased interest in the field of quantum information and computation. </p><p>In the 1980s, interest arose in whether it might be possible to use quantum effects to disprove <a href="/wiki/Theory_of_relativity" title="Theory of relativity">Einstein's theory of relativity</a>. If it were possible to clone an unknown quantum state, it would be possible to use <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">entangled</a> quantum states to transmit information faster than the speed of light, disproving Einstein's theory. However, the <a href="/wiki/No-cloning_theorem" title="No-cloning theorem">no-cloning theorem</a> showed that such cloning is impossible. The theorem was one of the earliest results of quantum information theory.<sup id="cite_ref-Nielsen2010_2-3" class="reference"><a href="#cite_note-Nielsen2010-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Development_from_cryptography">Development from cryptography</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=5" title="Edit section: Development from cryptography"><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">See also: <a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></div> <p>Despite all the excitement and interest over studying isolated quantum systems and trying to find a way to circumvent the theory of relativity, research in quantum information theory became stagnant in the 1980s. However, around the same time another avenue started dabbling into quantum information and computation: <a href="/wiki/Cryptography" title="Cryptography">Cryptography</a>. In a general sense, <i>cryptography is the problem of doing communication or computation involving two or more parties who may not trust one another.</i><sup id="cite_ref-Nielsen2010_2-4" class="reference"><a href="#cite_note-Nielsen2010-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Bennett and Brassard developed a communication channel on which it is impossible to eavesdrop without being detected, a way of communicating secretly at long distances using the <a href="/wiki/BB84" title="BB84">BB84</a> quantum cryptographic protocol.<sup id="cite_ref-Bennett2014_25-0" class="reference"><a href="#cite_note-Bennett2014-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> The key idea was the use of the fundamental principle of quantum mechanics that observation disturbs the observed, and the introduction of an eavesdropper in a secure communication line will immediately let the two parties trying to communicate know of the presence of the eavesdropper. </p> <div class="mw-heading mw-heading4"><h4 id="Development_from_computer_science_and_mathematics">Development from computer science and mathematics</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=6" title="Edit section: Development from computer science and mathematics"><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">See also: <a href="/wiki/Quantum_supremacy" title="Quantum supremacy">Quantum supremacy</a> and <a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithm</a></div> <p>With the advent of <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a>'s revolutionary ideas of a programmable computer, or <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a>, he showed that any real-world computation can be translated into an equivalent computation involving a Turing machine.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Deutsch1985_27-0" class="reference"><a href="#cite_note-Deutsch1985-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> This is known as the <a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a>. </p><p>Soon enough, the first computers were made, and computer hardware grew at such a fast pace that the growth, through experience in production, was codified into an empirical relationship called <a href="/wiki/Moore%27s_law" title="Moore's law">Moore's law</a>. This 'law' is a projective trend that states that the number of transistors in an <a href="/wiki/Integrated_circuit" title="Integrated circuit">integrated circuit</a> doubles every two years.<sup id="cite_ref-Moore1998_28-0" class="reference"><a href="#cite_note-Moore1998-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> As transistors began to become smaller and smaller in order to pack more power per surface area, quantum effects started to show up in the electronics resulting in inadvertent interference. This led to the advent of quantum computing, which uses quantum mechanics to design algorithms. </p><p>At this point, quantum computers showed promise of being much faster than classical computers for certain specific problems. One such example problem was developed by <a href="/wiki/David_Deutsch" title="David Deutsch">David Deutsch</a> and <a href="/wiki/Richard_Jozsa" title="Richard Jozsa">Richard Jozsa</a>, known as the <a href="/wiki/Deutsch_algorithm" class="mw-redirect" title="Deutsch algorithm">Deutsch–Jozsa algorithm</a>. This problem however held little to no practical applications.<sup id="cite_ref-Nielsen2010_2-5" class="reference"><a href="#cite_note-Nielsen2010-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Peter_Shor" title="Peter Shor">Peter Shor</a> in 1994 came up with a very important and practical <a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">problem</a>, one of finding the prime factors of an integer. The <a href="/wiki/Discrete_logarithm" title="Discrete logarithm">discrete logarithm</a> problem as it was called, could theoretically be solved efficiently on a quantum computer but not on a classical computer hence showing that quantum computers should be more powerful than Turing machines. </p> <div class="mw-heading mw-heading4"><h4 id="Development_from_information_theory">Development from information theory</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=7" title="Edit section: Development from information theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Around the time computer science was making a revolution, so was information theory and communication, through <a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a>.<sup id="cite_ref-Shannon1948a_29-0" class="reference"><a href="#cite_note-Shannon1948a-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Shannon1948b_30-0" class="reference"><a href="#cite_note-Shannon1948b-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Shannon1964_31-0" class="reference"><a href="#cite_note-Shannon1964-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Shannon developed two fundamental theorems of information theory: noiseless channel coding theorem and <a href="/wiki/Noisy-channel_coding_theorem" title="Noisy-channel coding theorem">noisy channel coding theorem</a>. He also showed that <a href="/wiki/Error_correction_code" title="Error correction code">error correcting codes</a> could be used to protect information being sent. </p><p>Quantum information theory also followed a similar trajectory, Ben Schumacher in 1995 made an analogue to Shannon's <a href="/wiki/Shannon%27s_source_coding_theorem" title="Shannon's source coding theorem">noiseless coding theorem</a> using the <a href="/wiki/Qubit" title="Qubit">qubit</a>. A theory of error-correction also developed, which allows quantum computers to make efficient computations regardless of noise and make reliable communication over noisy quantum channels.<sup id="cite_ref-Nielsen2010_2-6" class="reference"><a href="#cite_note-Nielsen2010-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Qubits_and_information_theory">Qubits and information theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=8" title="Edit section: Qubits and information theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Quantum information differs strongly from classical information, epitomized by the <a href="/wiki/Bit" title="Bit">bit</a>, in many striking and unfamiliar ways. While the fundamental unit of classical information is the <a href="/wiki/Bit" title="Bit">bit</a>, the most basic unit of quantum information is the <a href="/wiki/Qubit" title="Qubit">qubit</a>. Classical information is measured using <a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">Shannon entropy</a>, while the quantum mechanical analogue is <a href="/wiki/Von_Neumann_entropy" title="Von Neumann entropy">Von Neumann entropy</a>. Given a <a href="/wiki/Statistical_ensemble" class="mw-redirect" title="Statistical ensemble">statistical ensemble</a> of quantum mechanical systems with the <a href="/wiki/Density_matrix" title="Density matrix">density matrix</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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>, it is given by <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 S(\rho )=-\operatorname {Tr} (\rho \ln \rho ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>Tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mi>ln</mi> <mo>⁡</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\rho )=-\operatorname {Tr} (\rho \ln \rho ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f35089bd22e6bff5a6c28d3bf92c9c6251eb7d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.967ex; height:2.843ex;" alt="{\displaystyle S(\rho )=-\operatorname {Tr} (\rho \ln \rho ).}"></span><sup id="cite_ref-Nielsen2010_2-7" class="reference"><a href="#cite_note-Nielsen2010-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Many of the same entropy measures in classical <a href="/wiki/Information_theory" title="Information theory">information theory</a> can also be generalized to the quantum case, such as Holevo entropy<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> and the <a href="/wiki/Conditional_quantum_entropy" title="Conditional quantum entropy">conditional quantum entropy</a>. </p><p>Unlike classical digital states (which are discrete), a qubit is continuous-valued, describable by a direction on the <a href="/wiki/Bloch_sphere" title="Bloch sphere">Bloch sphere</a>. Despite being continuously valued in this way, a qubit is the <i>smallest</i> possible unit of quantum information, and despite the qubit state being continuous-valued, it is <a href="/wiki/EPR_paradox" class="mw-redirect" title="EPR paradox">impossible</a> to <a href="/wiki/Quantum_measurement" class="mw-redirect" title="Quantum measurement">measure</a> the value precisely. Five famous theorems describe the limits on manipulation of quantum information.<sup id="cite_ref-Nielsen2010_2-8" class="reference"><a href="#cite_note-Nielsen2010-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <ol><li><a href="/wiki/No-teleportation_theorem" title="No-teleportation theorem">no-teleportation theorem</a>, which states that a qubit cannot be (wholly) converted into classical bits; that is, it cannot be fully "read".</li> <li><a href="/wiki/No-cloning_theorem" title="No-cloning theorem">no-cloning theorem</a>, which prevents an arbitrary qubit from being copied.</li> <li><a href="/wiki/No-deleting_theorem" title="No-deleting theorem">no-deleting theorem</a>, which prevents an arbitrary qubit from being deleted.</li> <li><a href="/wiki/No-broadcast_theorem" class="mw-redirect" title="No-broadcast theorem">no-broadcast theorem</a>, which prevents an arbitrary qubit from being delivered to multiple recipients, although it can be transported from place to place (<i>e.g.</i> via <a href="/wiki/Quantum_teleportation" title="Quantum teleportation">quantum teleportation</a>).</li> <li><a href="/wiki/No-hiding_theorem" title="No-hiding theorem">no-hiding theorem</a>, which demonstrates the conservation of quantum information.</li></ol> <p>These theorems are proven from <a href="/wiki/Unitarity_(physics)" title="Unitarity (physics)">unitarity</a>, which according to <a href="/wiki/Leonard_Susskind" title="Leonard Susskind">Leonard Susskind</a> is the technical term for the statement that quantum information within the universe is conserved.<sup id="cite_ref-Susskind2014_33-0" class="reference"><a href="#cite_note-Susskind2014-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap">: <span title="Page: 94
Quotation: "The minus first law says that information is never lost. If two identical isolated systems start out in different states, they stay in different states. Moreover, in the past they were also in different states. On the other hand, if two identical systems are in the same state at some point in time, then their histories and their future evolutions must also be identical. Distinctions are conserved. The quantum version of the minus first law has a name — unitarity."" class="tooltip tooltip-dashed" style="border-bottom: 1px dashed;">94</span> </sup> The five theorems open possibilities in quantum information processing. </p> <div class="mw-heading mw-heading2"><h2 id="Quantum_information_processing">Quantum information processing</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=9" title="Edit section: Quantum information processing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The state of a qubit contains all of its information. This state is frequently expressed as a vector on the Bloch sphere. This state can be changed by applying <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformations</a> or <a href="/wiki/Quantum_gate" class="mw-redirect" title="Quantum gate">quantum gates</a> to them. These <a href="/wiki/Unitary_transformation_(quantum_mechanics)" title="Unitary transformation (quantum mechanics)">unitary transformations</a> are described as rotations on the Bloch sphere. While classical gates correspond to the familiar operations of <a href="/wiki/Boolean_Logic" class="mw-redirect" title="Boolean Logic">Boolean logic</a>, quantum gates are physical <a href="/wiki/Unitary_operator" title="Unitary operator">unitary operators</a>. </p> <ul><li>Due to the volatility of quantum systems and the impossibility of copying states, the storing of quantum information is much more difficult than storing classical information. Nevertheless, with the use of <a href="/wiki/Quantum_error_correction" title="Quantum error correction">quantum error correction</a> quantum information can still be reliably stored in principle. The existence of quantum error correcting codes has also led to the possibility of <a href="/wiki/Fault_tolerance" title="Fault tolerance">fault-tolerant</a> <a href="/wiki/Quantum_computation" class="mw-redirect" title="Quantum computation">quantum computation</a>.</li> <li>Classical bits can be encoded into and subsequently retrieved from configurations of qubits, through the use of quantum gates. By itself, a single qubit can convey no more than one bit of accessible classical information about its preparation. This is <a href="/wiki/Holevo%27s_theorem" title="Holevo's theorem">Holevo's theorem</a>. However, in <a href="/wiki/Superdense_coding" title="Superdense coding">superdense coding</a> a sender, by acting on one of two <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">entangled</a> qubits, can convey two bits of accessible information about their joint state to a receiver.</li> <li>Quantum information can be moved about, in a <a href="/wiki/Quantum_channel" title="Quantum channel">quantum channel</a>, analogous to the concept of a classical <a href="/wiki/Communications_channel" class="mw-redirect" title="Communications channel">communications channel</a>. Quantum messages have a finite size, measured in qubits; quantum channels have a finite <a href="/wiki/Channel_capacity" title="Channel capacity">channel capacity</a>, measured in qubits per second.</li> <li>Quantum information, and changes in quantum information, can be quantitatively measured by using an analogue of <a href="/wiki/Claude_Shannon" title="Claude Shannon">Shannon</a> <a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">entropy</a>, called the von Neumann entropy.</li> <li>In some cases, <a href="/wiki/Quantum_algorithm" title="Quantum algorithm">quantum algorithms</a> can be used to perform computations faster than in any known classical algorithm. The most famous example of this is <a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's algorithm</a> that can factor numbers in polynomial time, compared to the best classical algorithms that take sub-exponential time. As factorization is an important part of the safety of <a href="/wiki/RSA_(cryptosystem)" title="RSA (cryptosystem)">RSA encryption</a>, Shor's algorithm sparked the new field of <a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">post-quantum cryptography</a> that tries to find encryption schemes that remain safe even when quantum computers are in play. Other examples of algorithms that demonstrate <a href="/wiki/Quantum_supremacy" title="Quantum supremacy">quantum supremacy</a> include <a href="/wiki/Grover%27s_algorithm" title="Grover's algorithm">Grover's search algorithm</a>, where the quantum algorithm gives a quadratic speed-up over the best possible classical algorithm. The <a href="/wiki/Complexity_class" title="Complexity class">complexity class</a> of problems efficiently solvable by a <a href="/wiki/Quantum_Computer" class="mw-redirect" title="Quantum Computer">quantum computer</a> is known as <a href="/wiki/BQP" title="BQP">BQP</a>.</li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a> (QKD) allows unconditionally secure transmission of classical information, unlike classical encryption, which can always be broken in principle, if not in practice. Note that certain subtle points regarding the safety of QKD are debated.</li></ul> <p>The study of the above topics and differences comprises quantum information theory. </p> <div class="mw-heading mw-heading2"><h2 id="Relation_to_quantum_mechanics">Relation to quantum mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=10" title="Edit section: Relation to quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a> is the study of how microscopic physical systems change dynamically in nature. In the field of quantum information theory, the quantum systems studied are abstracted away from any real world counterpart. A qubit might for instance physically be a <a href="/wiki/Photon" title="Photon">photon</a> in a <a href="/wiki/Linear_optical_quantum_computing" title="Linear optical quantum computing">linear optical quantum computer</a>, an ion in a <a href="/wiki/Trapped_ion_quantum_computer" class="mw-redirect" title="Trapped ion quantum computer">trapped ion quantum computer</a>, or it might be a large collection of atoms as in a <a href="/wiki/Superconducting_quantum_computing" title="Superconducting quantum computing">superconducting quantum computer</a>. Regardless of the physical implementation, the limits and features of qubits implied by quantum information theory hold as all these systems are mathematically described by the same apparatus of <a href="/wiki/Density_matrix" title="Density matrix">density matrices</a> over the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. Another important difference with quantum mechanics is that while quantum mechanics often studies <a href="/wiki/Infinite-dimensional" class="mw-redirect" title="Infinite-dimensional">infinite-dimensional</a> systems such as a <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">harmonic oscillator</a>, quantum information theory is concerned with both continuous-variable systems<sup id="cite_ref-Weedbrook2012_34-0" class="reference"><a href="#cite_note-Weedbrook2012-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> and finite-dimensional systems.<sup id="cite_ref-Hayashi2017_8-4" class="reference"><a href="#cite_note-Hayashi2017-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Watrous2018_35-0" class="reference"><a href="#cite_note-Watrous2018-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Wilde2017_36-0" class="reference"><a href="#cite_note-Wilde2017-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Entropy_and_information">Entropy and information</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=11" title="Edit section: Entropy and information"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Entropy measures the uncertainty in the state of a physical system.<sup id="cite_ref-Nielsen2010_2-9" class="reference"><a href="#cite_note-Nielsen2010-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Entropy can be studied from the point of view of both the classical and quantum information theories. </p> <div class="mw-heading mw-heading3"><h3 id="Classical_information_theory">Classical information theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=12" title="Edit section: Classical information theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Classical information is based on the concepts of information laid out by <a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a>. Classical information, in principle, can be stored in a bit of binary strings. Any system having two states is a capable bit.<sup id="cite_ref-Jaeger2007_37-0" class="reference"><a href="#cite_note-Jaeger2007-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Shannon_entropy">Shannon entropy</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=13" title="Edit section: Shannon entropy"><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/Entropy_(information_theory)" title="Entropy (information theory)">Entropy (information theory)</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Shannon%27s_source_coding_theorem" title="Shannon's source coding theorem">Shannon's source coding theorem</a></div> <p>Shannon entropy is the quantification of the information gained by measuring the value of a random variable. Another way of thinking about it is by looking at the uncertainty of a system prior to measurement. As a result, entropy, as pictured by Shannon, can be seen either as a measure of the uncertainty prior to making a measurement or as a measure of information gained after making said measurement.<sup id="cite_ref-Nielsen2010_2-10" class="reference"><a href="#cite_note-Nielsen2010-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Shannon entropy, written as a functional of a discrete probability distribution, <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(x_{1}),P(x_{2}),...,P(x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x_{1}),P(x_{2}),...,P(x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d59d7535c0cefdcf5b05152b4fc40b14e9aa0eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.184ex; height:2.843ex;" alt="{\displaystyle P(x_{1}),P(x_{2}),...,P(x_{n})}"></span> associated with events <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 x_{1},...,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},...,x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f979c14353ba9d99b39d68265ad6db58c5faaae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.102ex; height:2.009ex;" alt="{\displaystyle x_{1},...,x_{n}}"></span>, can be seen as the average information associated with this set of events, in units of bits: </p><p><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 H(X)=H[P(x_{1}),P(x_{2}),...,P(x_{n})]=-\sum _{i=1}^{n}P(x_{i})\log _{2}P(x_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo stretchy="false">[</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(X)=H[P(x_{1}),P(x_{2}),...,P(x_{n})]=-\sum _{i=1}^{n}P(x_{i})\log _{2}P(x_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6fddb19d2d1948497eb53261dc6b6fed63e9ec9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:61.697ex; height:6.843ex;" alt="{\displaystyle H(X)=H[P(x_{1}),P(x_{2}),...,P(x_{n})]=-\sum _{i=1}^{n}P(x_{i})\log _{2}P(x_{i})}"></span> </p><p>This definition of entropy can be used to quantify the physical resources required to store the output of an information source. The ways of interpreting Shannon entropy discussed above are usually only meaningful when the number of samples of an experiment is large.<sup id="cite_ref-Watrous2018_35-1" class="reference"><a href="#cite_note-Watrous2018-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Rényi_entropy"><span id="R.C3.A9nyi_entropy"></span>Rényi entropy</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=14" title="Edit section: Rényi entropy"><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/R%C3%A9nyi_entropy" title="Rényi entropy">Rényi entropy</a></div> <p>The <a href="/wiki/R%C3%A9nyi_entropy" title="Rényi entropy">Rényi entropy</a> is a generalization of Shannon entropy defined above. The Rényi entropy of order r, written as a function of a discrete probability distribution, <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(a_{1}),P(a_{2}),...,P(a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(a_{1}),P(a_{2}),...,P(a_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d622ec0db3a26e8b5bacf6e48aec6147692c5ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.884ex; height:2.843ex;" alt="{\displaystyle P(a_{1}),P(a_{2}),...,P(a_{n})}"></span>, associated with events <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},...,a_{n}}"> <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> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},...,a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9bcc0e499b63bdf3e61e2769f2df6d70b2aff65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.902ex; height:2.009ex;" alt="{\displaystyle a_{1},...,a_{n}}"></span>, is defined as:<sup id="cite_ref-Jaeger2007_37-1" class="reference"><a href="#cite_note-Jaeger2007-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p><p><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 H_{r}(A)={1 \over 1-r}\log _{2}\sum _{i=1}^{n}P^{r}(a_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>r</mi> </mrow> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{r}(A)={1 \over 1-r}\log _{2}\sum _{i=1}^{n}P^{r}(a_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef99187ac395715bb29ec3c3b9dfdb64b5f9e605" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.62ex; height:6.843ex;" alt="{\displaystyle H_{r}(A)={1 \over 1-r}\log _{2}\sum _{i=1}^{n}P^{r}(a_{i})}"></span> </p><p>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="{\displaystyle 0<r<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>r</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<r<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f02c920222e9b2ac9b9d79c6e6909ee015218f63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.732ex; height:2.176ex;" alt="{\displaystyle 0<r<\infty }"></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 r\neq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>≠</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\neq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e99cedb8e04f4d403afffce3aba8a7559741a36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.31ex; height:2.676ex;" alt="{\displaystyle r\neq 1}"></span>. </p><p>We arrive at the definition of Shannon entropy from Rényi when <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 r\rightarrow 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">→</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\rightarrow 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40071d59f67deed2a19d24c9354cbe0505b0d5da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.825ex; height:2.176ex;" alt="{\displaystyle r\rightarrow 1}"></span>, of <a href="/wiki/Hartley_entropy" class="mw-redirect" title="Hartley entropy">Hartley entropy</a> (or max-entropy) when <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 r\rightarrow 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\rightarrow 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47fe923b35572a4c03c177f7cd857e4ebc9b77b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.825ex; height:2.176ex;" alt="{\displaystyle r\rightarrow 0}"></span>, and <a href="/wiki/Min-entropy" title="Min-entropy">min-entropy</a> when <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 r\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/170e3065b668161cd003c9fd57119e51d3424ce3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.986ex; height:1.843ex;" alt="{\displaystyle r\rightarrow \infty }"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Quantum_information_theory">Quantum information theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=15" title="Edit section: Quantum information theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Quantum information theory is largely an extension of classical information theory to quantum systems. Classical information is produced when measurements of quantum systems are made.<sup id="cite_ref-Jaeger2007_37-2" class="reference"><a href="#cite_note-Jaeger2007-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Von_Neumann_entropy">Von Neumann entropy</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=16" title="Edit section: Von Neumann entropy"><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/Von_Neumann_entropy" title="Von Neumann entropy">Von Neumann entropy</a></div> <p>One interpretation of Shannon entropy was the uncertainty associated with a probability distribution. When we want to describe the information or the uncertainty of a quantum state, the probability distributions are simply replaced by <a href="/wiki/Density_operator" class="mw-redirect" title="Density operator">density operators</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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>: </p><p><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 S(\rho )\equiv -\mathrm {tr} (\rho \ \log _{2}\ \rho )=-\sum _{i}\lambda _{i}\ \log _{2}\ \lambda _{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mtext> </mtext> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mtext> </mtext> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mtext> </mtext> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mtext> </mtext> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\rho )\equiv -\mathrm {tr} (\rho \ \log _{2}\ \rho )=-\sum _{i}\lambda _{i}\ \log _{2}\ \lambda _{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60d8001c795f3d61be849af02a1a0f4dadb40aa7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.362ex; height:5.509ex;" alt="{\displaystyle S(\rho )\equiv -\mathrm {tr} (\rho \ \log _{2}\ \rho )=-\sum _{i}\lambda _{i}\ \log _{2}\ \lambda _{i},}"></span> </p><p>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 \lambda _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde940918edf84caf3d406cc7d31949166820f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}"></span> are the eigenvalues 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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>. </p><p>Von Neumann entropy plays a role in quantum information similar to the role Shannon entropy plays in classical information. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=17" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Quantum_communication">Quantum communication</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=18" title="Edit section: Quantum communication"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum communication</a> is one of the applications of quantum physics and quantum information. There are some famous theorems such as the no-cloning theorem that illustrate some important properties in quantum communication. <a href="/wiki/Superdense_coding" title="Superdense coding">Dense coding</a> and <a href="/wiki/Quantum_teleportation" title="Quantum teleportation">quantum teleportation</a> are also applications of quantum communication. They are two opposite ways to communicate using qubits. While teleportation transfers one qubit from Alice and Bob by communicating two classical bits under the assumption that Alice and Bob have a pre-shared <a href="/wiki/Bell_state" title="Bell state">Bell state</a>, dense coding transfers two classical bits from Alice to Bob by using one qubit, again under the same assumption, that Alice and Bob have a pre-shared Bell state. </p> <div class="mw-heading mw-heading3"><h3 id="Quantum_key_distribution">Quantum key distribution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=19" title="Edit section: Quantum key distribution"><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_key_distribution" title="Quantum key distribution">Quantum key distribution</a></div> <p>One of the best known applications of quantum cryptography is <a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">quantum key distribution</a> which provide a theoretical solution to the security issue of a classical key. The advantage of quantum key distribution is that it is impossible to copy a quantum key because of the <a href="/wiki/No-cloning_theorem" title="No-cloning theorem">no-cloning theorem</a>. If someone tries to read encoded data, the quantum state being transmitted will change. This could be used to detect eavesdropping. </p> <div class="mw-heading mw-heading4"><h4 id="BB84">BB84</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=20" title="Edit section: BB84"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first quantum key distribution scheme, <a href="/wiki/BB84" title="BB84">BB84</a>, was developed by Charles Bennett and <a href="/wiki/Gilles_Brassard" title="Gilles Brassard">Gilles Brassard</a> in 1984. It is usually explained as a method of securely communicating a private key from a third party to another for use in one-time pad encryption.<sup id="cite_ref-Nielsen2010_2-11" class="reference"><a href="#cite_note-Nielsen2010-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="E91">E91</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=21" title="Edit section: E91"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/E91_protocol" class="mw-redirect" title="E91 protocol">E91</a> was made by <a href="/wiki/Artur_Ekert" title="Artur Ekert">Artur Ekert</a> in 1991. His scheme uses entangled pairs of photons. These two photons can be created by Alice, Bob, or by a third party including eavesdropper Eve. One of the photons is distributed to Alice and the other to Bob so that each one ends up with one photon from the pair. </p><p>This scheme relies on two properties of quantum entanglement: </p> <ol><li>The entangled states are perfectly correlated which means that if Alice and Bob both measure their particles having either a vertical or horizontal polarization, they always get the same answer with 100% probability. The same is true if they both measure any other pair of complementary (orthogonal) polarizations. This necessitates that the two distant parties have exact directionality synchronization. However, from quantum mechanics theory the quantum state is completely random so that it is impossible for Alice to predict if she will get vertical polarization or horizontal polarization results.</li> <li>Any attempt at eavesdropping by Eve destroys this quantum entanglement such that Alice and Bob can detect.</li></ol> <div class="mw-heading mw-heading4"><h4 id="B92">B92</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=22" title="Edit section: B92"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>B92 is a simpler version of BB84.<sup id="cite_ref-Bennett1992_38-0" class="reference"><a href="#cite_note-Bennett1992-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p><p>The main difference between B92 and BB84: </p> <ul><li>B92 only needs two states</li> <li>BB84 needs 4 polarization states</li></ul> <p>Like the BB84, Alice transmits to Bob a string of photons encoded with randomly chosen bits but this time the bits Alice chooses the bases she must use. Bob still randomly chooses a basis by which to measure but if he chooses the wrong basis, he will not measure anything which is guaranteed by quantum mechanics theories. Bob can simply tell Alice after each bit she sends whether he measured it correctly.<sup id="cite_ref-Haitjema2007_39-0" class="reference"><a href="#cite_note-Haitjema2007-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Quantum_computation">Quantum computation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=23" title="Edit section: Quantum computation"><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_computing" title="Quantum computing">Quantum computing</a></div> <p>The most widely used model in quantum computation is the <a href="/wiki/Quantum_circuit" title="Quantum circuit">quantum circuit</a>, which are based on the quantum bit "<a href="/wiki/Qubit" title="Qubit">qubit</a>". Qubit is somewhat analogous to the <a href="/wiki/Bit" title="Bit">bit</a> in classical computation. Qubits can be in a 1 or 0 <a href="/wiki/Quantum_state" title="Quantum state">quantum state</a>, or they can be in a <a href="/wiki/Quantum_superposition" title="Quantum superposition">superposition</a> of the 1 and 0 states. However, when qubits are measured, the result of the measurement is always either a 0 or a 1; the <a href="/wiki/Probability" title="Probability">probabilities</a> of these two outcomes depend on the <a href="/wiki/Quantum_state" title="Quantum state">quantum state</a> that the qubits were in immediately prior to the measurement. </p><p>Any quantum computation algorithm can be represented as a network of <a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">quantum logic gates</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Quantum_decoherence">Quantum decoherence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=24" title="Edit section: Quantum decoherence"><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_decoherence" title="Quantum decoherence">Quantum decoherence</a></div> <p>If a quantum system were perfectly isolated, it would maintain coherence perfectly, but it would be impossible to test the entire system. If it is not perfectly isolated, for example during a measurement, coherence is shared with the environment and appears to be lost with time; this process is called quantum decoherence. As a result of this process, quantum behavior is apparently lost, just as energy appears to be lost by friction in classical mechanics. </p> <div class="mw-heading mw-heading3"><h3 id="Quantum_error_correction">Quantum error correction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=25" title="Edit section: Quantum error correction"><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_error_correction" title="Quantum error correction">Quantum error correction</a></div> <p><b>QEC</b> is used in <a href="/wiki/Quantum_computer" class="mw-redirect" title="Quantum computer">quantum computing</a> to protect quantum information from errors due to <a href="/wiki/Decoherence" class="mw-redirect" title="Decoherence">decoherence</a> and other <a href="/wiki/Quantum_noise" title="Quantum noise">quantum noise</a>. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum information, but also with faulty quantum gates, faulty quantum preparation, and faulty measurements. </p><p><a href="/wiki/Peter_Shor" title="Peter Shor">Peter Shor</a> first discovered this method of formulating a <i>quantum error correcting code</i> by storing the information of one qubit onto a highly entangled state of <a href="/wiki/Ancilla_bit" title="Ancilla bit">ancilla qubits</a>. A quantum error correcting code protects quantum information against errors. </p> <div class="mw-heading mw-heading2"><h2 id="Journals">Journals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_information&action=edit&section=26" title="Edit section: Journals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many journals publish research in <a href="/wiki/Quantum_information_science" title="Quantum information science">quantum information science</a>, although only a few are dedicated to this area. Among these are: </p> <ul><li><i><a href="/wiki/International_Journal_of_Quantum_Information" title="International Journal of Quantum Information">International Journal of Quantum Information</a></i></li> <li><i><a href="/wiki/Npj_Quantum_Information" title="Npj Quantum Information">npj Quantum Information</a></i></li> <li><i><a href="/wiki/Quantum_(journal)" title="Quantum (journal)">Quantum</a></i></li> <li><i><a rel="nofollow" class="external text" href="https://www.rintonpress.com/journals/qic/">Quantum Information & Computation</a></i></li> <li><i><a rel="nofollow" class="external text" href="https://www.springer.com/journal/11128">Quantum Information Processing</a></i></li> <li><i><a rel="nofollow" class="external text" href="https://iopscience.iop.org/journal/2058-9565">Quantum Science and Technology</a></i></li></ul> <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_information&action=edit&section=27" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col div-col-small"> <ul><li><a href="/wiki/Categorical_quantum_mechanics" title="Categorical quantum mechanics">Categorical quantum mechanics</a></li> <li><a href="/wiki/Einstein%27s_thought_experiments" title="Einstein's thought experiments">Einstein's thought experiments</a></li> <li><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations of quantum mechanics</a></li> <li><a href="/wiki/POVM" title="POVM">Positive Operator Valued Measure (POVM)</a></li> <li><a href="/wiki/Quantum_clock" class="mw-redirect" title="Quantum clock">Quantum clock</a></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Quantum entanglement</a></li> <li><a href="/wiki/Quantum_foundations" title="Quantum foundations">Quantum foundations</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Qubit" title="Qubit">Qubit</a></li> <li><a href="/wiki/Qutrit" title="Qutrit">Qutrit</a></li> <li><a href="/wiki/Typical_subspace" title="Typical subspace">Typical subspace</a></li></ul> </div> <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_information&action=edit&section=28" title="Edit section: References"><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 reflist-columns-3"> <ol class="references"> <li id="cite_note-Vedral2006-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Vedral2006_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFVedral2006" class="citation book cs1"><a href="/wiki/Vlatko_Vedral" title="Vlatko Vedral">Vedral, Vlatko</a> (2006). <i>Introduction to Quantum Information Science</i>. Oxford: Oxford University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Facprof%3Aoso%2F9780199215706.001.0001">10.1093/acprof:oso/9780199215706.001.0001</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780199215706" title="Special:BookSources/9780199215706"><bdi>9780199215706</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/822959053">822959053</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Quantum+Information+Science&rft.place=Oxford&rft.pub=Oxford+University+Press&rft.date=2006&rft_id=info%3Aoclcnum%2F822959053&rft_id=info%3Adoi%2F10.1093%2Facprof%3Aoso%2F9780199215706.001.0001&rft.isbn=9780199215706&rft.aulast=Vedral&rft.aufirst=Vlatko&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Nielsen2010-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Nielsen2010_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Nielsen2010_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Nielsen2010_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Nielsen2010_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Nielsen2010_2-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Nielsen2010_2-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Nielsen2010_2-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-Nielsen2010_2-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-Nielsen2010_2-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-Nielsen2010_2-9"><sup><i><b>j</b></i></sup></a> <a href="#cite_ref-Nielsen2010_2-10"><sup><i><b>k</b></i></sup></a> <a href="#cite_ref-Nielsen2010_2-11"><sup><i><b>l</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNielsenChuang2010" class="citation book cs1">Nielsen, Michael A.; Chuang, Isaac L. (2010). <i>Quantum Computation and Quantum Information</i> (10th anniversary ed.). Cambridge: Cambridge University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fcbo9780511976667">10.1017/cbo9780511976667</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780511976667" title="Special:BookSources/9780511976667"><bdi>9780511976667</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/665137861">665137861</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:59717455">59717455</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Computation+and+Quantum+Information&rft.place=Cambridge&rft.edition=10th+anniversary&rft.pub=Cambridge+University+Press&rft.date=2010&rft_id=info%3Aoclcnum%2F665137861&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A59717455%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2Fcbo9780511976667&rft.isbn=9780511976667&rft.aulast=Nielsen&rft.aufirst=Michael+A.&rft.au=Chuang%2C+Isaac+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Hayashi2006-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hayashi2006_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHayashi2006" class="citation book cs1">Hayashi, Masahito (2006). <i>Quantum Information: An Introduction</i>. Berlin: Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-30266-2">10.1007/3-540-30266-2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-30266-7" title="Special:BookSources/978-3-540-30266-7"><bdi>978-3-540-30266-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/68629072">68629072</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Information%3A+An+Introduction&rft.place=Berlin&rft.pub=Springer&rft.date=2006&rft_id=info%3Aoclcnum%2F68629072&rft_id=info%3Adoi%2F10.1007%2F3-540-30266-2&rft.isbn=978-3-540-30266-7&rft.aulast=Hayashi&rft.aufirst=Masahito&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Bokulich2010-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bokulich2010_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBokulichJaeger2010" class="citation book cs1">Bokulich, Alisa; Jaeger, Gregg (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZC_XB7MLcC8C"><i>Philosophy of Quantum Information and Entanglement</i></a>. Cambridge: Cambridge University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511676550">10.1017/CBO9780511676550</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780511676550" title="Special:BookSources/9780511676550"><bdi>9780511676550</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Philosophy+of+Quantum+Information+and+Entanglement&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2010&rft_id=info%3Adoi%2F10.1017%2FCBO9780511676550&rft.isbn=9780511676550&rft.aulast=Bokulich&rft.aufirst=Alisa&rft.au=Jaeger%2C+Gregg&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZC_XB7MLcC8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Benatti2010-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Benatti2010_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBenattiFannesFloreaniniPetritis2010" class="citation book cs1">Benatti, Fabio; Fannes, Mark; Floreanini, Roberto; Petritis, Dimitri (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_0A8Qf1DYIgC"><i>Quantum Information, Computation and Cryptography: An Introductory Survey of Theory, Technology and Experiments</i></a>. 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Berlin: Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-11914-9">10.1007/978-3-642-11914-9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-11914-9" title="Special:BookSources/978-3-642-11914-9"><bdi>978-3-642-11914-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Information%2C+Computation+and+Cryptography%3A+An+Introductory+Survey+of+Theory%2C+Technology+and+Experiments&rft.place=Berlin&rft.series=Lecture+Notes+in+Physics&rft.pub=Springer&rft.date=2010&rft_id=info%3Adoi%2F10.1007%2F978-3-642-11914-9&rft.isbn=978-3-642-11914-9&rft.aulast=Benatti&rft.aufirst=Fabio&rft.au=Fannes%2C+Mark&rft.au=Floreanini%2C+Roberto&rft.au=Petritis%2C+Dimitri&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_0A8Qf1DYIgC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Benatti2009-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Benatti2009_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBenatti2009" class="citation book cs1">Benatti, Fabio (2009). 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Berlin: Springer. <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/2015iqis.book.....H">2015iqis.book.....H</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%2F978-3-662-43502-1">10.1007/978-3-662-43502-1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-43502-1" title="Special:BookSources/978-3-662-43502-1"><bdi>978-3-662-43502-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Quantum+Information+Science&rft.place=Berlin&rft.pub=Springer&rft.date=2015&rft_id=info%3Adoi%2F10.1007%2F978-3-662-43502-1&rft_id=info%3Abibcode%2F2015iqis.book.....H&rft.isbn=978-3-662-43502-1&rft.aulast=Hayashi&rft.aufirst=Masahito&rft.au=Ishizaka%2C+Satoshi&rft.au=Kawachi%2C+Akinori&rft.au=Kimura%2C+Gen&rft.au=Ogawa%2C+Tomohiro&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Hayashi2017-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hayashi2017_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hayashi2017_8-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Hayashi2017_8-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Hayashi2017_8-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Hayashi2017_8-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHayashi2017" class="citation book cs1">Hayashi, Masahito (2017). <i>Quantum Information Theory: Mathematical Foundation</i>. 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Berlin: Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-662-49725-8">10.1007/978-3-662-49725-8</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-49725-8" title="Special:BookSources/978-3-662-49725-8"><bdi>978-3-662-49725-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Information+Theory%3A+Mathematical+Foundation&rft.place=Berlin&rft.series=Graduate+Texts+in+Physics&rft.pub=Springer&rft.date=2017&rft_id=info%3Adoi%2F10.1007%2F978-3-662-49725-8&rft.isbn=978-3-662-49725-8&rft.aulast=Hayashi&rft.aufirst=Masahito&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Georgiev2017-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Georgiev2017_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorgiev2017" class="citation book cs1">Georgiev, Danko D. 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Boca Raton: CRC Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1201%2F9780203732519">10.1201/9780203732519</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781138104488" title="Special:BookSources/9781138104488"><bdi>9781138104488</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1003273264">1003273264</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1390.81001">1390.81001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Information+and+Consciousness%3A+A+Gentle+Introduction&rft.place=Boca+Raton&rft.pub=CRC+Press&rft.date=2017-12-06&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1390.81001%23id-name%3DZbl&rft_id=info%3Adoi%2F10.1201%2F9780203732519&rft_id=info%3Aoclcnum%2F1003273264&rft.isbn=9781138104488&rft.aulast=Georgiev&rft.aufirst=Danko+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOtRBDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Georgiev2020-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Georgiev2020_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorgiev2020" class="citation journal cs1">Georgiev, Danko D. 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Singapore: World Scientific. <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/1998iqci.book.....S">1998iqci.book.....S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F3724">10.1142/3724</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4496-35-3" title="Special:BookSources/978-981-4496-35-3"><bdi>978-981-4496-35-3</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/52859247">52859247</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Quantum+Computation+and+Information&rft.place=Singapore&rft.pub=World+Scientific&rft.date=1998&rft_id=info%3Aoclcnum%2F52859247&rft_id=info%3Adoi%2F10.1142%2F3724&rft_id=info%3Abibcode%2F1998iqci.book.....S&rft.isbn=978-981-4496-35-3&rft.aulast=Lo&rft.aufirst=Hoi-Kwong&rft.au=Popescu%2C+Sandu&rft.au=Spiller%2C+Tim&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D18_-rgkdQGIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Bennett1998-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bennett1998_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBennettShor1998" class="citation journal cs1"><a href="/wiki/Charles_H._Bennett_(physicist)" title="Charles H. Bennett (physicist)">Bennett, Charles H.</a>; <a href="/wiki/Peter_Shor" title="Peter Shor">Shor, Peter Williston</a> (1998). "Quantum information theory". <i>IEEE Transactions on Information Theory</i>. <b>44</b> (6): 2724–2742. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.89.1572">10.1.1.89.1572</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F18.720553">10.1109/18.720553</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Information+Theory&rft.atitle=Quantum+information+theory&rft.volume=44&rft.issue=6&rft.pages=2724-2742&rft.date=1998&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.89.1572%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1109%2F18.720553&rft.aulast=Bennett&rft.aufirst=Charles+H.&rft.au=Shor%2C+Peter+Williston&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Garlinghouse2020-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Garlinghouse2020_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGarlinghouse2020" class="citation journal cs1">Garlinghouse, Tom (2020). <a rel="nofollow" class="external text" href="https://www.princeton.edu/news/2020/01/21/quantum-computing-opening-new-realms-possibilities">"Quantum computing: Opening new realms of possibilities"</a>. <i>Discovery: Research at Princeton</i>: 12–17.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discovery%3A+Research+at+Princeton&rft.atitle=Quantum+computing%3A+Opening+new+realms+of+possibilities&rft.pages=12-17&rft.date=2020&rft.aulast=Garlinghouse&rft.aufirst=Tom&rft_id=https%3A%2F%2Fwww.princeton.edu%2Fnews%2F2020%2F01%2F21%2Fquantum-computing-opening-new-realms-possibilities&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Mahan2009-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-Mahan2009_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMahan2009" class="citation book cs1">Mahan, Gerald D. (2009). <i>Quantum Mechanics in a Nutshell</i>. Princeton: Princeton University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2Fj.ctt7s8nw">10.2307/j.ctt7s8nw</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4008-3338-2" title="Special:BookSources/978-1-4008-3338-2"><bdi>978-1-4008-3338-2</bdi></a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/j.ctt7s8nw">j.ctt7s8nw</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics+in+a+Nutshell&rft.place=Princeton&rft.pub=Princeton+University+Press&rft.date=2009&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2Fj.ctt7s8nw%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2Fj.ctt7s8nw&rft.isbn=978-1-4008-3338-2&rft.aulast=Mahan&rft.aufirst=Gerald+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Perlman1964-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Perlman1964_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPerlman1964" class="citation journal cs1">Perlman, H. S. (1964). "Equivalence of the Schroedinger and Heisenberg pictures". <i>Nature</i>. <b>204</b> (4960): 771–772. <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/1964Natur.204..771P">1964Natur.204..771P</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%2F204771b0">10.1038/204771b0</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:4194913">4194913</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=Equivalence+of+the+Schroedinger+and+Heisenberg+pictures&rft.volume=204&rft.issue=4960&rft.pages=771-772&rft.date=1964&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4194913%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1038%2F204771b0&rft_id=info%3Abibcode%2F1964Natur.204..771P&rft.aulast=Perlman&rft.aufirst=H.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeumann2018" class="citation book cs1">Neumann, John von (2018-02-27). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=B3OYDwAAQBAJ&q=foundations+of+quantum+mechanics+von+neumann&pg=PR1"><i>Mathematical Foundations of Quantum Mechanics: New Edition</i></a>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-17856-1" title="Special:BookSources/978-0-691-17856-1"><bdi>978-0-691-17856-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Foundations+of+Quantum+Mechanics%3A+New+Edition&rft.pub=Princeton+University+Press&rft.date=2018-02-27&rft.isbn=978-0-691-17856-1&rft.aulast=Neumann&rft.aufirst=John+von&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DB3OYDwAAQBAJ%26q%3Dfoundations%2Bof%2Bquantum%2Bmechanics%2Bvon%2Bneumann%26pg%3DPR1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Gordon1962-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-Gordon1962_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Gordon1962_20-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGordon1962" class="citation journal cs1">Gordon, J. 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"Quantum effects in communications systems". <i>Proceedings of the IRE</i>. <b>50</b> (9): 1898–1908. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Fjrproc.1962.288169">10.1109/jrproc.1962.288169</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:51631629">51631629</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+IRE&rft.atitle=Quantum+effects+in+communications+systems&rft.volume=50&rft.issue=9&rft.pages=1898-1908&rft.date=1962&rft_id=info%3Adoi%2F10.1109%2Fjrproc.1962.288169&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A51631629%23id-name%3DS2CID&rft.aulast=Gordon&rft.aufirst=J.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Helstrom1969-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-Helstrom1969_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHelstrom1969" class="citation journal cs1">Helstrom, Carl W. (1969). "Quantum detection and estimation theory". <i>Journal of Statistical Physics</i>. <b>1</b> (2): 231–252. <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/1969JSP.....1..231H">1969JSP.....1..231H</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%2Fbf01007479">10.1007/bf01007479</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2060%2F19690016211">2060/19690016211</a></span>. <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:121571330">121571330</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+Statistical+Physics&rft.atitle=Quantum+detection+and+estimation+theory&rft.volume=1&rft.issue=2&rft.pages=231-252&rft.date=1969&rft_id=info%3Ahdl%2F2060%2F19690016211&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121571330%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fbf01007479&rft_id=info%3Abibcode%2F1969JSP.....1..231H&rft.aulast=Helstrom&rft.aufirst=Carl+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Helstrom1976-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-Helstrom1976_22-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHelstrom1976" class="citation book cs1">Helstrom, Carl W. 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New York: Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-36944-0">10.1007/978-0-387-36944-0</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-36944-0" title="Special:BookSources/978-0-387-36944-0"><bdi>978-0-387-36944-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/255569451">255569451</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Information%3A+An+Overview&rft.place=New+York&rft.pub=Springer&rft.date=2007&rft_id=info%3Aoclcnum%2F255569451&rft_id=info%3Adoi%2F10.1007%2F978-0-387-36944-0&rft.isbn=978-0-387-36944-0&rft.aulast=Jaeger&rft.aufirst=Gregg&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DE0ho97k7S4oC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Bennett1992-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bennett1992_38-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBennett1992" class="citation journal cs1"><a href="/wiki/Charles_H._Bennett_(physicist)" title="Charles H. Bennett (physicist)">Bennett, Charles H.</a> (1992). "Quantum cryptography using any two nonorthogonal states". <i>Physical Review Letters</i>. <b>68</b> (21): 3121–3124. <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/1992PhRvL..68.3121B">1992PhRvL..68.3121B</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.68.3121">10.1103/PhysRevLett.68.3121</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/10045619">10045619</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:19708593">19708593</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Quantum+cryptography+using+any+two+nonorthogonal+states&rft.volume=68&rft.issue=21&rft.pages=3121-3124&rft.date=1992&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.68.3121&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A19708593%23id-name%3DS2CID&rft_id=info%3Apmid%2F10045619&rft_id=info%3Abibcode%2F1992PhRvL..68.3121B&rft.aulast=Bennett&rft.aufirst=Charles+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> <li id="cite_note-Haitjema2007-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-Haitjema2007_39-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaitjema2007" class="citation book cs1">Haitjema, Mart (2007). <a rel="nofollow" class="external text" href="https://www.cse.wustl.edu/~jain/cse571-07/ftp/quantum/#b92"><i>A Survey of the Prominent Quantum Key Distribution Protocols</i></a>. Washington University in St. Louis. <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:18346434">18346434</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Survey+of+the+Prominent+Quantum+Key+Distribution+Protocols&rft.pub=Washington+University+in+St.+Louis&rft.date=2007&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18346434%23id-name%3DS2CID&rft.aulast=Haitjema&rft.aufirst=Mart&rft_id=https%3A%2F%2Fwww.cse.wustl.edu%2F~jain%2Fcse571-07%2Fftp%2Fquantum%2F%23b92&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+information" class="Z3988"></span></span> </li> </ol></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output 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.mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Quantum_information_science" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_information" title="Template:Quantum information"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_information" title="Template talk:Quantum information"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_information" title="Special:EditPage/Template:Quantum information"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_information_science" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/DiVincenzo%27s_criteria" title="DiVincenzo's criteria">DiVincenzo's criteria</a></li> <li><a href="/wiki/Noisy_intermediate-scale_quantum_era" title="Noisy intermediate-scale quantum era">NISQ era</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">timeline</a></li></ul></li> <li><a class="mw-selflink selflink">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/Monogamy_of_entanglement" title="Monogamy of entanglement">Monogamy of entanglement</a></li> <li><a href="/wiki/LOCC" title="LOCC">LOCC</a></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a> <ul><li><a href="/wiki/Quantum_network" title="Quantum network">quantum network</a></li></ul></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a> <ul><li><a href="/wiki/Quantum_gate_teleportation" title="Quantum gate teleportation">quantum gate teleportation</a></li></ul></li> <li><a href="/wiki/Superdense_coding" title="Superdense coding">Superdense coding</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Quantum_cryptography" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">Post-quantum cryptography</a></li> <li><a href="/wiki/Quantum_coin_flipping" title="Quantum coin flipping">Quantum coin flipping</a></li> <li><a href="/wiki/Quantum_money" title="Quantum money">Quantum money</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a> <ul><li><a href="/wiki/BB84" title="BB84">BB84</a></li> <li><a href="/wiki/SARG04" title="SARG04">SARG04</a></li> <li><a href="/wiki/List_of_quantum_key_distribution_protocols" title="List of quantum key distribution protocols">other protocols</a></li></ul></li> <li><a href="/wiki/Quantum_secret_sharing" title="Quantum secret sharing">Quantum secret sharing</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amplitude_amplification" title="Amplitude amplification">Amplitude amplification</a></li> <li><a href="/wiki/Bernstein%E2%80%93Vazirani_algorithm" title="Bernstein–Vazirani algorithm">Bernstein–Vazirani</a></li> <li><a href="/wiki/BHT_algorithm" title="BHT algorithm">BHT</a></li> <li><a href="/wiki/Boson_sampling" title="Boson sampling">Boson sampling</a></li> <li><a href="/wiki/Deutsch%E2%80%93Jozsa_algorithm" title="Deutsch–Jozsa algorithm">Deutsch–Jozsa</a></li> <li><a href="/wiki/Grover%27s_algorithm" title="Grover's algorithm">Grover's</a></li> <li><a href="/wiki/HHL_algorithm" title="HHL algorithm">HHL</a></li> <li><a href="/wiki/Hidden_subgroup_problem" title="Hidden subgroup problem">Hidden subgroup</a></li> <li><a href="/wiki/Quantum_annealing" title="Quantum annealing">Quantum annealing</a></li> <li><a href="/wiki/Quantum_counting_algorithm" title="Quantum counting algorithm">Quantum counting</a></li> <li><a href="/wiki/Quantum_Fourier_transform" title="Quantum Fourier transform">Quantum Fourier transform</a></li> <li><a href="/wiki/Quantum_optimization_algorithms" title="Quantum optimization algorithms">Quantum optimization</a></li> <li><a href="/wiki/Quantum_phase_estimation_algorithm" title="Quantum phase estimation algorithm">Quantum phase estimation</a></li> <li><a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's</a></li> <li><a href="/wiki/Simon%27s_problem" title="Simon's problem">Simon's</a></li> <li><a href="/wiki/Variational_quantum_eigensolver" title="Variational quantum eigensolver">VQE</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum<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></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_mechanics" 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_mechanics" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a> <ul><li><a href="/wiki/Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Born_rule" title="Born rule">Born rule</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a> <ul><li><a href="/wiki/Ground_state" title="Ground state">Ground state</a></li> <li><a href="/wiki/Excited_state" title="Excited state">Excited state</a></li> <li><a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a href="/wiki/Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a class="mw-selflink selflink">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a> <ul><li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat_in_popular_culture" title="Schrödinger's cat in popular culture">in popular culture</a></li></ul></li> <li><a href="/wiki/Wigner%27s_friend" title="Wigner's friend">Wigner's friend</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><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_technologies" 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 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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> 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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 href="/wiki/Quantum_teleportation" title="Quantum teleportation">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> 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