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quantum teleportation in nLab
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content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="quantum_systems">Quantum systems</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>, <a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">dependent</a> <a class="existingWikiWord" href="/nlab/show/linear+type+theory">linear type theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a> in <a class="existingWikiWord" href="/nlab/show/quantum+information+theory+via+dagger-compact+categories">†-compact categories</a></p> <p><a class="existingWikiWord" href="/nlab/show/tensor+networks">tensor networks</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a></p> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a></p> </li> </ul> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+systems">quantum systems</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/parameterized+quantum+systems">parameterized</a>, <a class="existingWikiWord" href="/nlab/show/open+quantum+system">open</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+state+collapse">quantum state collapse</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+decoherence">quantum decoherence</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">quantum adiabatic theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/Berry+phases">Berry phases</a></p> <p><a class="existingWikiWord" href="/nlab/show/Dyson+formula">Dyson formula</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+many-body+physics">quantum many-body physics</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/functorial+quantum+field+theory">functorial quantum field theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-</a>)<a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+material">quantum material</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/topological+phases+of+matter">topological</a>) <a class="existingWikiWord" href="/nlab/show/phases+of+matter">phases of matter</a></p> </li> </ul> <p><br /></p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a> – <a class="existingWikiWord" href="/nlab/show/observables">observables</a> and <a class="existingWikiWord" href="/nlab/show/states">states</a></strong></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/states">states</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/quasi-state">quasi-state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a></p> <p><a class="existingWikiWord" href="/nlab/show/dimer">dimer</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">quantum state preparation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a>, <a class="existingWikiWord" href="/nlab/show/quantum+fluctuation">quantum fluctuation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+superposition">quantum superposition</a>, <a class="existingWikiWord" href="/nlab/show/quantum+interference">quantum interference</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a></p> <p><a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a></p> <p><a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superselection+sector">superselection sector</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> <p><a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coherent+quantum+state">coherent quantum state</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>, <a class="existingWikiWord" href="/nlab/show/excited+state">excited state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>, <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+amplitude">vacuum amplitude</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+fluctuation">vacuum fluctuation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+polarization">vacuum polarization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/thermal+vacuum">thermal vacuum</a>, <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/false+vacuum">false vacuum</a>, <a class="existingWikiWord" href="/nlab/show/tachyon">tachyon</a>, <a class="existingWikiWord" href="/nlab/show/Coleman-De+Luccia+instanton">Coleman-De Luccia instanton</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theta+vacuum">theta vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/non-geometric+string+theory+vacuum">non-geometric string theory vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entangled+state">entangled state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix+product+state">matrix product state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tree+tensor+network+state">tree tensor network state</a></p> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/observables">observables</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>, <a class="existingWikiWord" href="/nlab/show/beable">beable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator (in geometric quantization)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+effect">quantum effect</a>, <a class="existingWikiWord" href="/nlab/show/effect+algebra">effect algebra</a></p> </li> <li> <p>in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+observable">linear observable</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+observable">regular observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>, <a class="existingWikiWord" href="/nlab/show/retarded+product">retarded product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra+of+observables">interacting field algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorems">theorems</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nuiten%27s+lemma">Nuiten's lemma</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner%27s+theorem">Wigner's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> </ul> </li> </ul> </div> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information+via+dagger-compact+categories">quantum information via dagger-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+channel">quantum channel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+teleportation">quantum teleportation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+entanglement+entropy">topological entanglement entropy</a></p> </li> </ul> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adiabatic+quantum+computation">adiabatic quantum computation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measurement-based+quantum+computation">measurement-based quantum computation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+gate">quantum gate</a>, <a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+programming+language">quantum programming language</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+error+correction">quantum error correction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/HaPPY+code">HaPPY code</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Majorana+dimer+code">Majorana dimer code</a></p> </li> </ul> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/qbit">qbit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/spin+resonance+qbit">spin resonance qbit</a></li> </ul> <p>quantum algorithms:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grover%27s+algorithm">Grover's algorithm</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Shor%27s+algorithm">Shor's algorithm</a></p> </li> </ul> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+sensing">quantum sensing</a></strong></p> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+communication">quantum communication</a></strong></p> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#HistoryAndPerspective'>History and perspective</a></li> <li><a href='#Statement'>Statement</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>What has been called quantum “teleportation” (<a href="#BBCJPW93">BBCJPW 1993</a>) is a procedure (a “protocol”) in <a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a> by which the exact transmission of a <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> (such as that of some <a class="existingWikiWord" href="/nlab/show/qbits">qbits</a>) is equivalently established by the transmission of <a class="existingWikiWord" href="/nlab/show/classical+physics">classical</a> information, <em>provided</em> that sender and receiver “share a <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a>”, hence that each have access to one half of a pair of <a class="existingWikiWord" href="/nlab/show/subsystems">subsystems</a> which have been <a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">prepared</a> in a maximally <a class="existingWikiWord" href="/nlab/show/entanglement">entangled</a> <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> with each other. The protocol involves a <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> and its ensuing <a class="existingWikiWord" href="/nlab/show/wavefunction+collapse">wavefunction collapse</a> destroys (“uses up”) this entangled state in order to transmit the remaining quantum state – which somewhat relativizes the imagery of “teleportation”.</p> <p>While this prose may appear somewhat exotic, the actual mathematics underlying the “quantum teleportation protocol” is an elementary and direct consequence of the most basic <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a> of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> (in fact it is all but a self-evident consequence when formulated in modern notation, see <a href="#HistoryAndPerspective">below</a>). What makes quantum teleportation interesting is the (<a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum</a>) <a class="existingWikiWord" href="/nlab/show/information+theory">information theoretic</a> perspective on it.</p> <h2 id="HistoryAndPerspective">History and perspective</h2> <p>Given that the basic axioms of quantum physics were established in the 1920s and widely appreciated by the 1930s, the observation of the possibility of “quantum teleportation” only <a href="#BBCJPW93">in 1993</a>, using tools and notation available to the founding fathers, may make one wonder (e.g <a href="#Coecke10">Coecke 2010, p. 1</a>):</p> <blockquote> <p>why did it take us 60 years to discover the conceptually intriguing and easily derivable physical phenomenon of ‘quantum teleportation’?</p> </blockquote> <p>Traditional reviews of the matter maintain that the phenomenon just happens to be intrinsically tricky to see through – e.g. <a href="#Aaronson18">Aaronson 2018</a>, <a href="https://www.scottaaronson.com/qclec.pdf#page=71">p. 71</a>, apparently in reply to his student’s puzzlement “How do people come up with this stuff?”, asserts that:</p> <blockquote> <p>These sorts of protocols can be hard to find.</p> </blockquote> <p>On the other hand, it was observed around <a href="#Coecke04">Coecke 2004</a>, <a href="#AbramskyCoecke04">Abramsky & Coecke 2004</a> – and pronouncedly brought out in <a href="#Coecke05">Coecke 2005</a>, <a href="#Coecke10">2010</a> – that the logical structure of <a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a> in general and that of quantum teleportation in particular is transparently captured by <a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>-calculus for the ambient <a class="existingWikiWord" href="/nlab/show/compact+closed+category">compact closed</a> <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a> of <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite dimensional</a> <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a> (see <em><a class="existingWikiWord" href="/nlab/show/finite+quantum+mechanics+in+terms+of+dagger-compact+categories">finite quantum mechanics in terms of dagger-compact categories</a></em>):</p> <p>In this <a class="existingWikiWord" href="/nlab/show/category+theory">category-theoretic</a> language, the quantum teleportation protocol is essentially just the <a class="existingWikiWord" href="/nlab/show/zig-zag+identity">zig-zag identity</a> for <a class="existingWikiWord" href="/nlab/show/dualizable+objects">dualizable objects</a> (here: <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite-dimensional</a> <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a>) in any <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, see <a href="#StatementInStringDiagramLanguage">below</a>.</p> <p>This suggests the following alternative answer, from <a href="#Coecke05">Coecke 2005, p. 1</a>:</p> <blockquote> <p>Why did discovering quantum teleportation take 60 year?</p> <p>We claim that this is due to a ‘bad quantum formalism’ (bad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≠</mo></mrow><annotation encoding="application/x-tex">\neq</annotation></semantics></math> wrong) and this badness is in particular due to the fact that the formalism is ‘too low level’.</p> </blockquote> <p>Here “bad quantum formalism” refers to the traditional notation as in the original <a href="#BBCJPW93">BBCJPW93</a> and in traditional reviews (e.g. <a href="#Aaronson18">Aaronson 2018</a>, see also <a href="#Coecke05">Coecke 05</a> <a href="https://arxiv.org/pdf/quant-ph/0510032.pdf#page=8">ftn. 5 on p. 8</a>), while the suggested high-level improvement is <a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>-calculus for the ambient <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a> of finite-dimensional Hilbert spaces, further discussed at <em><a class="existingWikiWord" href="/nlab/show/finite+quantum+mechanics+in+terms+of+dagger-compact+categories">finite quantum mechanics in terms of dagger-compact categories</a></em>.</p> <h2 id="Statement">Statement</h2> <p id="StatementInStringDiagramLanguage">The quantum teleportation protocol is neatly expressed in <a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>-calculus as follows (essentially following <a href="#Coecke05">Coecke 2005</a>):</p> <p><img src="/nlab/files/QuantumTeleportationProtocol-220906b.jpg" width="850" /></p> <p>Here, by the general laws of <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a>: a solid line pointing to the right represents a given <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite dimensional</a> <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℋ</mi><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><msup><mi>ℂ</mi> <mi>D</mi></msup></mrow><annotation encoding="application/x-tex">\mathscr{H} \,\simeq\, \mathbb{C}^D</annotation></semantics></math> (of <a class="existingWikiWord" href="/nlab/show/dimension+of+a+vector+space">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>), the same line pointing backwards denotes its <a class="existingWikiWord" href="/nlab/show/dual+vector+space">dual vector space</a>, two lines running parallel reflects the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+Hilbert+spaces">tensor product</a> of the corresponding Hilbert spaces, and the curve “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊃</mo></mrow><annotation encoding="application/x-tex">\supset</annotation></semantics></math>” denotes <a class="existingWikiWord" href="/nlab/show/evaluation">evaluation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi class="mathscript">ℋ</mi> <mo>*</mo></msup><mo>⊗</mo><mi class="mathscript">ℋ</mi><mover><mo>→</mo><mi>ev</mi></mover><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathscr{H}^\ast \otimes \mathscr{H} \xrightarrow{ev} \mathbb{C}</annotation></semantics></math>.</p> <p>The wiggly line denotes a classically <a class="existingWikiWord" href="/nlab/show/controlled+quantum+gate">controlled quantum gate</a>, controlled by the <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a>-outcome which is reflected by the projection operator shown at the top.</p> <p>That – for any fixed measurement outcome at the top with the correspondingly controlled correction gate operation at the bottom – this <a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a> (represents a <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> which) takes the <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi></mrow><annotation encoding="application/x-tex">\Psi</annotation></semantics></math> from “Agent A” (“Alice”) to the same state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi></mrow><annotation encoding="application/x-tex">\Psi</annotation></semantics></math> for “Agent B” (“Bob”), as shown is <em>immediate</em> from the <a class="existingWikiWord" href="/nlab/show/zig-zag+identity">zig-zag identity</a>, hence from the fact that the curved line may be “yanked straight” to an <a class="existingWikiWord" href="/nlab/show/identity">identity</a> line, without changing the operational value of the diagram. This is the transparent <a class="existingWikiWord" href="/nlab/show/proof">proof</a> of the quantum teleporation protocol.</p> <p>Notice that this is a <a class="existingWikiWord" href="/nlab/show/rigorous+proof">rigorous proof</a>, and yet this <a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a> is close to informal flow-charts traditionally used to illustrate the idea of quantum teleportation (e.g <a href="#BouwmeesterEkertZeilinger20">Bouwmeester, Ekert & Zeilinger 2020, Fig. 3.1</a>).</p> <p>In more detail, here the top box shows the <a class="existingWikiWord" href="/nlab/show/projector">projector</a> onto a <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> with one variable transformed by a <a class="existingWikiWord" href="/nlab/show/unitary+operator">unitary operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>. This reflects the (random/unpredictable!) result of a <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> in a corresponding <a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℋ</mi><mo>⊗</mo><msup><mi class="mathscript">ℋ</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathscr{H} \otimes \mathscr{H}^\ast</annotation></semantics></math> (according to the <a class="existingWikiWord" href="/nlab/show/wavefunction+collapse">collapse postulate</a>).</p> <p>Concretely, if the <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> represented by a single solid line is that of a single <a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, i.e. if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℋ</mi><mo>=</mo><msup><mi>ℂ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathscr{H} = \mathbb{C}^2</annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">D = 2</annotation></semantics></math>), then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> may be taken to range over the <a class="existingWikiWord" href="/nlab/show/unitary+operator">unitary</a> <a class="existingWikiWord" href="/nlab/show/Pauli+matrices">Pauli matrices</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>0</mn></msub><mo>≔</mo><mrow><mo>(</mo><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center center"><mtr><mtd><mn>1</mn></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mn>1</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>σ</mi> <mn>1</mn></msub><mo>≔</mo><mrow><mo>(</mo><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center center"><mtr><mtd><mn>0</mn></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd><mn>0</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>σ</mi> <mn>2</mn></msub><mo>≔</mo><mrow><mo>(</mo><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center center"><mtr><mtd><mn>0</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mi mathvariant="normal">i</mi></mtd></mtr> <mtr><mtd><mi mathvariant="normal">i</mi></mtd> <mtd><mn>0</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>σ</mi> <mn>3</mn></msub><mo>≔</mo><mrow><mo>(</mo><mtable displaystyle="false" rowspacing="0.5ex" columnalign="center center"><mtr><mtd><mn>1</mn></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd></mtr></mtable><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \sigma_0 \coloneqq \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) ,\;\;\;\; \sigma_1 \coloneqq \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) ,\;\;\;\; \sigma_2 \coloneqq \left( \begin{array}{cc} 0 & -\mathrm{i} \\ \mathrm{i} & 0 \end{array} \right) ,\;\;\;\; \sigma_3 \coloneqq \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \,. </annotation></semantics></math></div> <p id="TraditionalCircuit"> In this case, the component-evaluation of the above diagram yields the quantum teleportation protocol in its traditional form (due to <a href="#BBCJPW93">BBCJPW93</a>) which is the following <a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a> (cf. <a href="Quipper#GLRSV13">GLRSV13, p. 5</a>) on <a class="existingWikiWord" href="/nlab/show/qbits">qbits</a> (where “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>” denotes the <a class="existingWikiWord" href="/nlab/show/Hadamard+gate">Hadamard gate</a>, the circles denote the <a class="existingWikiWord" href="/nlab/show/quantum+CNOT+gates">quantum CNOT gates</a> and the boxed pointers denote <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> in the <a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>-basis):</p> <center> <img src="/nlab/files/QBitQuantumTeleportationProtocol-221109.jpg" width="800" /> </center> <p><br /></p> <p>The concrete implementation of the above protocol on given quantum hardware will look more complicated.</p> <p id="AsABraidGateCircuit"> For instance, the implementation of the quantum teleportation protocol as a circuit of <a class="existingWikiWord" href="/nlab/show/braid+representation">braid</a> gates of <a class="existingWikiWord" href="/nlab/show/Ising+anyons">Ising anyons</a> (<a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a>) is given in <a href="#XuZhou22">Xu & Zhou 2022</a>:</p> <center> <img src="https://ncatlab.org/nlab/files/BraidCircuitQuantumTeleProtocol-220915.jpg" width="450" /> </center> <p>See also <a href="#Valera23">Valera (2023)</a>.</p> <p><br /></p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superdense+coding">superdense coding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measurement-based+quantum+computation">measurement-based quantum computation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a></p> </li> </ul> <h2 id="references">References</h2> <p>The original article:</p> <ul> <li id="BBCJPW93"><a class="existingWikiWord" href="/nlab/show/Charles+H.+Bennett">Charles H. Bennett</a>, <a class="existingWikiWord" href="/nlab/show/Gilles+Brassard">Gilles Brassard</a>, <a class="existingWikiWord" href="/nlab/show/Claude+Cr%C3%A9peau">Claude Crépeau</a>, <a class="existingWikiWord" href="/nlab/show/Richard+Jozsa">Richard Jozsa</a>, <a class="existingWikiWord" href="/nlab/show/Asher+Peres">Asher Peres</a>, <a class="existingWikiWord" href="/nlab/show/William+K.+Wootters">William K. Wootters</a>: <em>Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels</em>, Phys. Rev. Lett. <strong>70</strong> 1895 (1993) [<a href="https://doi.org/10.1103/PhysRevLett.70.1895">doi:10.1103/PhysRevLett.70.1895</a>]</li> </ul> <p>Traditional review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+A.+Nielsen">Michael A. Nielsen</a>, <a class="existingWikiWord" href="/nlab/show/Isaac+L.+Chuang">Isaac L. Chuang</a>, §1.3.7 in: <em>Quantum computation and quantum information</em>, Cambridge University Press (2000) [<a href="https://doi.org/10.1017/CBO9780511976667">doi:10.1017/CBO9780511976667</a>, <a href="http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/NielsenChuangQuantumComputation.pdf" title="pdf">pdf</a>]</p> </li> <li id="Aaronson18"> <p><a class="existingWikiWord" href="/nlab/show/Scott+Aaronson">Scott Aaronson</a>, §10.1 in <em>Introduction to Quantum Information Science</em> (2018) [<a href="https://www.scottaaronson.com/qclec.pdf">pdf</a>, <a href="https://www.scottaaronson.com/cs378/">webpage</a>]</p> </li> </ul> <p>and phrased in the <a class="existingWikiWord" href="/nlab/show/quantum+programming+language">quantum programming language</a> <a class="existingWikiWord" href="/nlab/show/Quipper">Quipper</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alexander+Green">Alexander Green</a>, <a class="existingWikiWord" href="/nlab/show/Peter+LeFanu+Lumsdaine">Peter LeFanu Lumsdaine</a>, <a class="existingWikiWord" href="/nlab/show/Neil+Ross">Neil Ross</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Selinger">Peter Selinger</a>, <a class="existingWikiWord" href="/nlab/show/Beno%C3%AEt+Valiron">Benoît Valiron</a>, p. 5 of: <p><em>An Introduction to Quantum Programming in Quipper</em>, Lecture Notes in Computer Science 7948:110-124, Springer, 2013 (<a href="https://arxiv.org/abs/1304.5485">arXiv:1304.5485</a>)</p> </li> </ul> <p>and with focus on the <a class="existingWikiWord" href="/nlab/show/experiment">experimental</a> realization:</p> <ul> <li id="BouwmeesterEkertZeilinger20"><a class="existingWikiWord" href="/nlab/show/Dirk+Bouwmeester">Dirk Bouwmeester</a>, <a class="existingWikiWord" href="/nlab/show/Artur+Ekert">Artur Ekert</a>, <a class="existingWikiWord" href="/nlab/show/Anton+Zeilinger">Anton Zeilinger</a> (eds.), §3.3 of: <em>The Physics of Quantum Information – Quantum Cryptography, Quantum Teleportation, Quantum Computation</em>, Springer (2020) [<a href="https://doi.org/10.1007/978-3-662-04209-0">doi:10.1007/978-3-662-04209-0</a>]</li> </ul> <p>See also:</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Quantum_teleportation">Quantum teleportation</a></em></li> </ul> <p>Discussion via <a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>-calculus (<a class="existingWikiWord" href="/nlab/show/finite+quantum+mechanics+in+terms+of+dagger-compact+categories">finite quantum mechanics in terms of dagger-compact categories</a>):</p> <ul> <li id="Coecke04"> <p><a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, §4 of: <em>The logic of entanglement</em>, in: <em>Horizons of the Mind. A Tribute to Prakash Panangaden</em>, Lecture Notes in Computer Science <strong>8464</strong> (2014) [<a href="https://arxiv.org/abs/quant-ph/0402014">arXiv:quant-ph/0402014</a>, <a href="https://doi.org/10.1007/978-3-319-06880-0_13">doi:10.1007/978-3-319-06880-0_13</a>]</p> </li> <li id="AbramskyCoecke04"> <p><a class="existingWikiWord" href="/nlab/show/Samson+Abramsky">Samson Abramsky</a>, <a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, §2 in: <em>A categorical semantics of quantum protocols</em>, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS’04). IEEE Computer Science Press (2004) [<a href="https://arxiv.org/abs/quant-ph/0402130">arXiv:quant-ph/0402130</a>, <a href="https://doi.org/10.1109/LICS.2004.1319636">doi:10.1109/LICS.2004.1319636</a>]</p> </li> <li id="Coecke05"> <p><a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, §3c in: <em>Kindergarten Quantum Mechanics</em>, in <em>Quantum Theory: Reconsideration of Foundations</em> (QTRF 3) Vaxjo, Sweden, June 6-11, 2005, AIP Conf. Proc. <strong>810</strong> (2006) [<a href="https://arxiv.org/abs/quant-ph/0510032">arXiv:quant-ph/0510032</a>, <a href="https://doi.org/10.1063/1.2158713">doi:10.1063/1.2158713</a>]</p> </li> <li id="Coecke10"> <p><a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <em>Quantum Picturalism</em>, Contemporary Physics <strong>51</strong> (2010) 59-83 [<a href="https://arxiv.org/abs/0908.1787">arXiv:0908.1787</a>, <a href="https://doi.org/10.1080/00107510903257624">doi:10.1080/00107510903257624</a>]</p> </li> </ul> <p>Realization in <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirk+Bouwmeester">Dirk Bouwmeester</a>, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, <a class="existingWikiWord" href="/nlab/show/Harald+Weinfurter">Harald Weinfurter</a>, <a class="existingWikiWord" href="/nlab/show/Anton+Zeilinger">Anton Zeilinger</a>, <em>Experimental quantum teleportation</em>, Nature <strong>390</strong> 575–579 (1997) [<a href="https://doi.org/10.1038/37539">doi:10.1038/37539</a>]</p> </li> <li> <p>Syed Tahir Amin, <a class="existingWikiWord" href="/nlab/show/Aeysha+Khalique">Aeysha Khalique</a>, <em>Practical Quantum Teleportation of an Unknown Quantum State</em>, Can. J. Phys. <strong>95</strong> 5 (2017) 498 [<a href="https://arxiv.org/abs/1508.01141">arXiv:1508.01141</a>, <a href="https://doi.org/10.1139/cjp-2016-0758">doi:10.1139/cjp-2016-0758</a>]</p> </li> <li> <p>Dario Lago-Rivera, Jelena V. Rakonjac, Samuele Grandi, Hugues de Riedmatten, <em>Long-distance multiplexed quantum teleportation from a telecom photon to a solid-state qubit</em> [<a href="https://arxiv.org/abs/2209.06249">arXiv:2209.06249</a>]</p> </li> </ul> <p>Implementation of the quantum teleportation protocol in <a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a> via <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> of <a class="existingWikiWord" href="/nlab/show/Ising+anyons">Ising anyons</a>:</p> <ul> <li id="XuZhou22">Cheng-Qian Xu, D. L. Zhou, <em>Quantum teleportation using Ising anyons</em>, Phys. Rev. A <strong>106</strong> 012413 (2022) [<a href="https://arxiv.org/abs/2201.11923">arXiv:2201.11923</a>, <a href="https://doi.org/10.1103/PhysRevA.106.012413">doi:10.1103/PhysRevA.106.012413</a>]</li> </ul> <p>and via fusion of anyons:</p> <ul> <li id="Valera23"><a class="existingWikiWord" href="/nlab/show/Sachin+J.+Valera">Sachin J. Valera</a>, <em>Topological Quantum Teleportation and Superdense Coding Without Braiding</em> [<a href="https://arxiv.org/abs/2303.17700">arXiv:2303.17700</a>]</li> </ul> <p>Implementation in <a class="existingWikiWord" href="/nlab/show/quantum+programming+languages">quantum programming languages</a>:</p> <ul> <li><em><a href="https://qiskit.org/textbook/ch-algorithms/teleportation.html">Qiskit: Quantum Teleportation</a></em></li> </ul> <p>Argument that the image of quantum teleportation under <a class="existingWikiWord" href="/nlab/show/AdS-CFT+duality">AdS-CFT duality</a> is <em><a class="existingWikiWord" href="/nlab/show/wormhole">wormhole</a> traversal</em> (see also at <em><a class="existingWikiWord" href="/nlab/show/ER+%3D+EPR">ER = EPR</a></em>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Erik+Verlinde">Erik Verlinde</a>, <a class="existingWikiWord" href="/nlab/show/Herman+Verlinde">Herman Verlinde</a>, <em>A Conversation on ER = EPR</em> [<a href="https://arxiv.org/abs/2212.09389">arXiv:2212.09389</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ping+Gao">Ping Gao</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Louis+Jafferis">Daniel Louis Jafferis</a>, <a class="existingWikiWord" href="/nlab/show/Aron+C.+Wall">Aron C. Wall</a>, <em>Traversable Wormholes via a Double Trace Deformation</em>, Journal of High Energy Physics <strong>2017</strong> 151 (2017) [<a href="https://arxiv.org/abs/1608.05687">arXiv:1608.05687</a>, <a href="https://doi.org/10.1007/JHEP12(2017)151">doi:10.1007/JHEP12(2017)151</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Juan+Maldacena">Juan Maldacena</a>, <a class="existingWikiWord" href="/nlab/show/Douglas+Stanford">Douglas Stanford</a>, <a class="existingWikiWord" href="/nlab/show/Zhenbin+Yang">Zhenbin Yang</a>, <em>Diving into traversable wormholes</em>,Fortsch. Phys. <strong>65</strong> 5 (2017) 1700034 [<a href="https://arxiv.org/abs/1704.05333">arXiv:1704.05333</a>, <a href="https://doi.org/10.1002/prop.201700034">doi:10.1002/prop.201700034</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 9, 2023 at 07:36:43. 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