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xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" 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> <a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a> <ul> <li> <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> <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> <a class="existingWikiWord" href="/nlab/show/tensor+networks">tensor networks</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+systems">quantum systems</a> (<a class="existingWikiWord" href="/nlab/show/parameterized+quantum+systems">parameterized</a>, <a class="existingWikiWord" href="/nlab/show/open+quantum+system">open</a>) <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> <a class="existingWikiWord" href="/nlab/show/quantum+state+collapse">quantum state collapse</a> <a class="existingWikiWord" href="/nlab/show/quantum+decoherence">quantum decoherence</a> <a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">quantum adiabatic theorem</a> <a class="existingWikiWord" href="/nlab/show/Berry+phases">Berry phases</a> <a class="existingWikiWord" href="/nlab/show/Dyson+formula">Dyson formula</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+many-body+physics">quantum many-body physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> <a class="existingWikiWord" href="/nlab/show/functorial+quantum+field+theory">functorial quantum field theory</a> <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> (<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> </li> <li> <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a> <a class="existingWikiWord" href="/nlab/show/quantum+material">quantum material</a> (<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> </li> </ul> <div> <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> <ul> <li> <a class="existingWikiWord" href="/nlab/show/states">states</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> <a class="existingWikiWord" href="/nlab/show/dimer">dimer</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">quantum state preparation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a>, <a class="existingWikiWord" href="/nlab/show/quantum+fluctuation">quantum fluctuation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+superposition">quantum superposition</a>, <a class="existingWikiWord" href="/nlab/show/quantum+interference">quantum interference</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> <a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a> <a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a> <a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a> <a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a> <a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/superselection+sector">superselection sector</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/coherent+quantum+state">coherent quantum state</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>, <a class="existingWikiWord" href="/nlab/show/excited+state">excited state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>, <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+polarization">vacuum polarization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/thermal+vacuum">thermal vacuum</a>, <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a> <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> <a class="existingWikiWord" href="/nlab/show/theta+vacuum">theta vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/non-geometric+string+theory+vacuum">non-geometric string theory vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/entangled+state">entangled state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/matrix+product+state">matrix product state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tree+tensor+network+state">tree tensor network state</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/observables">observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>, <a class="existingWikiWord" href="/nlab/show/beable">beable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator (in geometric quantization)</a> </li> <li> <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> </li> <li> in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/linear+observable">linear observable</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/regular+observable">regular observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> </li> <li> <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> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/theorems">theorems</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Nuiten%27s+lemma">Nuiten's lemma</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wigner%27s+theorem">Wigner's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> </ul> </li> </ul> </div> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information+via+dagger-compact+categories">quantum information via dagger-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+channel">quantum channel</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+teleportation">quantum teleportation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/topological+entanglement+entropy">topological entanglement entropy</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+technology">quantum technology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+sensing">quantum sensing</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+communication">quantum communication</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+cryptography">quantum cryptography</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+machine+learning">quantum machine learning</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+computing">quantum computing</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/qdit">qdit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+gate">quantum gate</a>, <a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adiabatic+quantum+computation">adiabatic quantum computation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/measurement-based+quantum+computation">measurement-based quantum computation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+programming+language">quantum programming language</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+error+correction">quantum error correction</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/HaPPY+code">HaPPY code</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Majorana+dimer+code">Majorana dimer code</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin+resonance+qbit">spin resonance qbit</a> </li> <li> quantum algorithms: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Grover%27s+algorithm">Grover's algorithm</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Shor%27s+algorithm">Shor's algorithm</a> </li> </ul> </li> </ul> </div></div> <h4 id="measure_and_probability_theory">Measure and probability theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a> <a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a> (<a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a>) <h2 id="measure_theory">Measure theory</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/measurable+space">measurable space</a>, <a class="existingWikiWord" href="/nlab/show/measurable+locale">measurable locale</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/measure">measure</a>, <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+measure+theory">geometric measure theory</a> </li> </ul> <h2 id="probability_theory">Probability theory</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/probability+distribution">probability distribution</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/state">state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/states+in+AQFT+and+operator+algebra">in AQFT and operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/entropy">entropy</a>, <a class="existingWikiWord" href="/nlab/show/relative+entropy">relative entropy</a> </li> </ul> <h2 id="information_geometry">Information geometry</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/information+geometry">information geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/information+metric">information metric</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wasserstein+metric">Wasserstein metric</a> </li> </ul> <h2 id="thermodynamics">Thermodynamics</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/second+law+of+thermodynamics">second law of thermodynamics</a>, <a class="existingWikiWord" href="/nlab/show/generalized+second+law+of+theormodynamics">generalized second law</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/ergodic+theory">ergodic theory</a> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/de+Finetti%27s+theorem">de Finetti's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/law+of+large+numbers">law of large numbers</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kolmogorov+extension+theorem">Kolmogorov extension theorem</a> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/machine+learning">machine learning</a>, <a class="existingWikiWord" href="/nlab/show/neural+networks">neural networks</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesTopologicalEntanglementEntropy'>Topological entanglement entropy</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesLongRangeEntanglementRelationToStrongInteraction'>Relation to strong interaction</a></li> <li><a href='#CharacterizingTopologicalOrder'>Characterizing topological order</a></li> <li><a href='#simulation_and_experiment'>Simulation and experiment</a></li> </ul> </ul> </ul> </div> <h2 id="idea">Idea</h2> <a class="existingWikiWord" href="/nlab/show/entropy">entropy</a> induced by <a class="existingWikiWord" href="/nlab/show/entanglement">entanglement</a> in <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a>, essentially a synonym for <a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a> entropy. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> (<a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a>) of some <a class="existingWikiWord" href="/nlab/show/quantum+systems">quantum systems</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a> with complementary subsystem <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{A}</annotation></semantics></math>, then its entanglement entropy is <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>A</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo>−</mo><msub><mi>Tr</mi> <mi>A</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>ρ</mi> <mi>A</mi></msub><mi>ln</mi><msub><mi>ρ</mi> <mi>A</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mtext>where</mtext><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>ρ</mi> <mi>A</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>Tr</mi> <mover><mi>A</mi><mo stretchy="false">¯</mo></mover></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S_A \;\coloneqq\; - Tr_{A}\big( \rho_A \ln \rho_A \big) \,, \;\;\; \text{where} \;\;\; \rho_A \;\coloneqq\; Tr_{\bar A}(\rho) \,. </annotation></semantics></math></div> Special aspects: A constant contribution to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">S_A</annotation></semantics></math> (i.e. independent of the choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, in a suitable sense) is called a <a class="existingWikiWord" href="/nlab/show/topological+entanglement+entropy">topological entanglement entropy</a> indicating <a class="existingWikiWord" href="/nlab/show/long-range+entanglement">long-range entanglement</a> and “<a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a>”, being proportional to the total <a class="existingWikiWord" href="/nlab/show/quantum+dimension">quantum dimension</a> of <a class="existingWikiWord" href="/nlab/show/anyon">anyon</a>-excitations. (See the references <a href="#ReferencesTopologicalEntanglementEntropy">below</a>.) On the other extreme, for <a class="existingWikiWord" href="/nlab/show/short+range+entanglement">short range entanglement</a> the entanglement entropy is thought to scale with the surface <a class="existingWikiWord" href="/nlab/show/area">area</a> of the subsystem <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> (to the extent that this makes sense, say in say in <a class="existingWikiWord" href="/nlab/show/lattice+models">lattice models</a>), a behaviour reminiscent of <a class="existingWikiWord" href="/nlab/show/Bekenstein-Hawking+entropy">Bekenstein-Hawking entropy</a> of <a class="existingWikiWord" href="/nlab/show/black+holes">black holes</a>. For more on this see at <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a>. <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/tensor+network">tensor network</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+hole+entropy">black hole entropy</a> via <a class="existingWikiWord" href="/nlab/show/AdS-CFT+duality">AdS-CFT duality</a> </li> </ul> <div> <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> <ul> <li> <a class="existingWikiWord" href="/nlab/show/states">states</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> <a class="existingWikiWord" href="/nlab/show/dimer">dimer</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">quantum state preparation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a>, <a class="existingWikiWord" href="/nlab/show/quantum+fluctuation">quantum fluctuation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+superposition">quantum superposition</a>, <a class="existingWikiWord" href="/nlab/show/quantum+interference">quantum interference</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> <a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a> <a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a> <a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a> <a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a> <a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/superselection+sector">superselection sector</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/coherent+quantum+state">coherent quantum state</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>, <a class="existingWikiWord" href="/nlab/show/excited+state">excited state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>, <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+polarization">vacuum polarization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/thermal+vacuum">thermal vacuum</a>, <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a> <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> <a class="existingWikiWord" href="/nlab/show/theta+vacuum">theta vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/non-geometric+string+theory+vacuum">non-geometric string theory vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/entangled+state">entangled state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/matrix+product+state">matrix product state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tree+tensor+network+state">tree tensor network state</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/observables">observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>, <a class="existingWikiWord" href="/nlab/show/beable">beable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator (in geometric quantization)</a> </li> <li> <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> </li> <li> in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/linear+observable">linear observable</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/regular+observable">regular observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> </li> <li> <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> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/theorems">theorems</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Nuiten%27s+lemma">Nuiten's lemma</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wigner%27s+theorem">Wigner's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> </ul> </li> </ul> </div> <h2 id="references">References</h2> <h3 id="general">General</h3> <ul> <li>Matthew Headrick, Lectures on entanglement entropy in field theory and holography (<a href="https://arxiv.org/abs/1907.08126">arXiv:1907.08126</a>)</li> </ul> In <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Roberto+Longo">Roberto Longo</a>, Feng Xu, Von Neumann Entropy in QFT (<a href="https://arxiv.org/abs/1911.09390">arxiv:1911.09390</a>)</li> </ul> See also <ul> <li>Wikipedia, <a href="http://en.wikipedia.org/wiki/Entropy_of_entanglement">Entropy of entanglement</a></li> </ul> <div> <h3 id="ReferencesTopologicalEntanglementEntropy">Topological entanglement entropy</h3> On <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> in <a class="existingWikiWord" href="/nlab/show/arithmetic+Chern-Simons+theory">arithmetic Chern-Simons theory</a>: <ul> <li>Hee-Joong Chung, Dohyeong Kim, <a class="existingWikiWord" href="/nlab/show/Minhyong+Kim">Minhyong Kim</a>, Jeehoon Park, Hwajong Yoo, Entanglement entropies in the abelian arithmetic Chern-Simons theory &lbrack;<a href="https://arxiv.org/abs/2312.17138">arXiv:2312.17138</a>&rbrack;</li> </ul> <h4 id="general">General</h4> Identification of a contribution to <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> at <a class="existingWikiWord" href="/nlab/show/absolute+zero">absolute zero</a> which is independent of the <a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a>‘s size (“<a class="existingWikiWord" href="/nlab/show/topological+entanglement+entropy">topological entanglement entropy</a>”, “<a class="existingWikiWord" href="/nlab/show/long-range+entanglement">long-range entanglement</a>”), reflecting <a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a> and proportional to the total <a class="existingWikiWord" href="/nlab/show/quantum+dimension">quantum dimension</a> of <a class="existingWikiWord" href="/nlab/show/anyon">anyon excitations</a>: <ul> <li id="KitaevPreskill06"> <a class="existingWikiWord" href="/nlab/show/Alexei+Kitaev">Alexei Kitaev</a>, <a class="existingWikiWord" href="/nlab/show/John+Preskill">John Preskill</a>, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 (<a href="https://arxiv.org/abs/hep-th/0510092">arXiv:hep-th/0510092</a>) </li> <li id="LevinWen06"> <a class="existingWikiWord" href="/nlab/show/Michael+Levin">Michael Levin</a>, <a class="existingWikiWord" href="/nlab/show/Xiao-Gang+Wen">Xiao-Gang Wen</a>, Detecting topological order in a ground state wave function, Phys. Rev. Lett., 96, 110405 (2006) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/cond-mat/0510613">arXiv:cond-mat/0510613</a>, <a href="https://doi.org/10.1103/PhysRevLett.96.110405">doi:10.1103/PhysRevLett.96.110405</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> <blockquote> (in view of <a class="existingWikiWord" href="/nlab/show/string-net+model">string-net models</a>) </blockquote> </li> </ul> Review: <ul> <li id="Furukawa08"> Shunsuke Furukawa, Entanglement Entropy in Conventional and Topological Orders, talk at <a href="http://www.issp.u-tokyo.ac.jp/public/tassp/">Topological Aspects of Solid State Physics</a> 2008 (<a href="http://www.issp.u-tokyo.ac.jp/public/tassp/pdf_presentation/SY_S_Furukawa.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/FurukawaEntanglementEntropy.pdf" title="pdf">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Tarun+Grover">Tarun Grover</a>, Entanglement entropy and strongly correlated topological matter, Modern Physics Letters A 28 05 (2013) 1330001 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1142/S0217732313300012">doi:10.1142/S0217732313300012</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bei+Zeng">Bei Zeng</a>, <a class="existingWikiWord" href="/nlab/show/Xie+Chen">Xie Chen</a>, <a class="existingWikiWord" href="/nlab/show/Duan-Lu+Zhou">Duan-Lu Zhou</a>, <a class="existingWikiWord" href="/nlab/show/Xiao-Gang+Wen">Xiao-Gang Wen</a>: Sec. 5 of: <a class="existingWikiWord" href="/nlab/show/Quantum+Information+Meets+Quantum+Matter">Quantum Information Meets Quantum Matter</a> – From Quantum Entanglement to Topological Phases of Many-Body Systems, Quantum Science and Technology (QST), Springer (2019) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1508.02595">arXiv:1508.02595</a>, <a href="https://doi.org/10.1007/978-1-4939-9084-9">doi:10.1007/978-1-4939-9084-9</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> In terms of <a class="existingWikiWord" href="/nlab/show/Renyi+entropy">Renyi entropy</a> (it’s independent of the Renyi entropy parameter): <ul> <li>Ulrich Schollwöck, (Almost) 25 Years of DMRG - What Is It About? (<a href="https://confs.physics.ox.ac.uk/pauli2016/files/slides/Schollwoeck.pdf">pdf</a>)</li> </ul> and in the example of <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>: <ul> <li>Aditya Dwivedi, Siddharth Dwivedi, Bhabani Prasad Mandal, Pichai Ramadevi, <a class="existingWikiWord" href="/nlab/show/Vivek+Kumar+Singh">Vivek Kumar Singh</a>, Topological entanglement and hyperbolic volume, J. High Energy Phys. 21021 172 (2021) 172 &lbrack;<a href="https://arxiv.org/abs/2106.03396">arXiv:2106.03396</a>, <a href="https://doi.org/10.1007/JHEP10(2021)172">doi:10.1007/JHEP10(2021)172</a>&rbrack;</li> </ul> Discussion in the <a class="existingWikiWord" href="/nlab/show/dimer+model">dimer model</a>: <ul> <li>Shunsuke Furukawa, Gregoire Misguich, Topological Entanglement Entropy in the Quantum Dimer Model on the Triangular Lattice, Phys. Rev. B 75, 214407 (2007) (<a href="https://arxiv.org/abs/cond-mat/0612227">arXiv:cond-mat/0612227</a>)</li> </ul> Discussion via <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a>: <ul> <li> Ari Pakman, <a class="existingWikiWord" href="/nlab/show/Andrei+Parnachev">Andrei Parnachev</a>, Topological Entanglement Entropy and Holography, JHEP 0807: 097 (2008) (<a href="https://arxiv.org/abs/0805.1891">arXiv:0805.1891</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Andrei+Parnachev">Andrei Parnachev</a>, Napat Poovuttikul, Topological Entanglement Entropy, Ground State Degeneracy and Holography, Journal of High Energy Physics volume 2015, Article number: 92 (2015) (<a href="https://arxiv.org/abs/1504.08244">arXiv:1504.08244</a>) </li> </ul> See also: <ul> <li>Tatsuma Nishioka, Tadashi Takayanagi, Yusuke Taki, Topological pseudo entropy (<a href="https://arxiv.org/abs/2107.01797">arXiv:2107.01797</a>)</li> </ul> <h4 id="ReferencesLongRangeEntanglementRelationToStrongInteraction">Relation to strong interaction</h4> Relation of <a class="existingWikiWord" href="/nlab/show/long-range+entanglement">long-range entanglement</a> to strong <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a>: <ul> <li id="ZaanenLiuSunSchalm15"> <a class="existingWikiWord" href="/nlab/show/Jan+Zaanen">Jan Zaanen</a>, Yan Liu, Ya-Wen Sun, <a class="existingWikiWord" href="/nlab/show/Koenraad+Schalm">Koenraad Schalm</a>, Holographic Duality in Condensed Matter Physics, Cambridge University Press 2015 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1017/CBO9781139942492">doi:10.1017/CBO9781139942492</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> <blockquote> In a way it appears obvious that the strongly interacting bosonic quantum critical state is subject to long-range entanglement. Nonetheless, the status of this claim is conjectural. It is at present impossible to arrive at more solid conclusions that are based on rigorous mathematical procedures. It does illustrate emphatically the central challenge in the pursuit of field-theoretical quantum information: there are as yet not general measures available to precisely enumerate the meaning of long-range entanglement in such seriously quantum field-theoretical systems. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math>p. 527<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </blockquote> </li> <li id="LuVijay22"> Tsung-Cheng Lu, Sagar Vijay, Characterizing Long-Range Entanglement in a Mixed State Through an Emergent Order on the Entangling Surface <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2201.07792">arXiv:2201.07792</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> <blockquote> strongly interacting quantum phases of matter at zero temperature can exhibit universal patterns of long-range entanglement </blockquote> </li> </ul> <h4 id="CharacterizingTopologicalOrder">Characterizing topological order</h4> On characterizing <a class="existingWikiWord" href="/nlab/show/anyon">anyon</a> <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> / <a class="existingWikiWord" href="/nlab/show/modular+transformations">modular transformations</a> on <a class="existingWikiWord" href="/nlab/show/topological+order">topologically ordered</a> <a class="existingWikiWord" href="/nlab/show/ground+states">ground states</a> by analysis of (<a class="existingWikiWord" href="/nlab/show/topological+entanglement+entropy">topological</a>) <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> of subregions: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Yi+Zhang">Yi Zhang</a>, <a class="existingWikiWord" href="/nlab/show/Tarun+Grover">Tarun Grover</a>, <a class="existingWikiWord" href="/nlab/show/Ari+M.+Turner">Ari M. Turner</a>, <a class="existingWikiWord" href="/nlab/show/Masaki+Oshikawa">Masaki Oshikawa</a>, <a class="existingWikiWord" href="/nlab/show/Ashvin+Vishwanath">Ashvin Vishwanath</a>, Quasiparticle statistics and braiding from ground-state entanglement, Phys. Rev. B 85 (2012) 235151 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1103/PhysRevB.85.235151">doi:10.1103/PhysRevB.85.235151</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Yi+Zhang">Yi Zhang</a>, <a class="existingWikiWord" href="/nlab/show/Tarun+Grover">Tarun Grover</a>, <a class="existingWikiWord" href="/nlab/show/Ashvin+Vishwanath">Ashvin Vishwanath</a>, General procedure for determining braiding and statistics of anyons using entanglement interferometry, Phys. Rev. B 91 (2015) 035127 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1412.0677">arXiv:1412.0677</a>, <a href="https://doi.org/10.1103/PhysRevB.91.035127">doi:10.1103/PhysRevB.91.035127</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Zhuan+Li">Zhuan Li</a>, <a class="existingWikiWord" href="/nlab/show/Roger+S.+K.+Mong">Roger S. K. Mong</a>, Detecting topological order from modular transformations of ground states on the torus, Phys. Rev. B 106 (2022) 235115 &lbrack;<a href="https://doi.org/10.1103/PhysRevB.106.235115">doi:10.1103/PhysRevB.106.235115</a>, <a href="https://arxiv.org/abs/2203.04329">arXiv:2203.04329</a>&rbrack; </li> </ul> Relation to irreducible correlation: <ul> <li>Kohtaro Kato, Fabian Furrer, Mio murao: Information-theoretical analysis of topological entanglement entropy and multipartite correlations, Phys. Rev. A 93 022317 (2016) [<a href="https://doi.org/10.1103/PhysRevA.93.022317">doi:10.1103/PhysRevA.93.022317</a>, <a href="https://arxiv.org/abs/1505.01917">arXiv:1505.01917</a>]</li> </ul> <h4 id="simulation_and_experiment">Simulation and experiment</h4> Experimental observation: <ul> <li> A. Hamma, W. Zhang, S. Haas, and D. A. Lidar, Entanglement, fidelity, and topological entropy in a quantum phase transition to topological order, Phys. Rev. B 77, 155111 (2008) (<a href="https://doi.org/10.1103/PhysRevB.77.155111">doi:10.1103/PhysRevB.77.155111</a>, <a href="https://arxiv.org/abs/0705.0026">arXiv:0705.0026</a>) </li> <li> Hong-Chen Jiang, <a class="existingWikiWord" href="/nlab/show/Zhenghan+Wang">Zhenghan Wang</a>, <a class="existingWikiWord" href="/nlab/show/Leon+Balents">Leon Balents</a>, Identifying Topological Order by Entanglement Entropy, Nature Physics 8 902-905 (2012) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1205.4289">arXiv:1205.4289</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> Detection of long-range entanglement entropy in quantum simulations on <a class="existingWikiWord" href="/nlab/show/quantum+computers">quantum computers</a>: <ul> <li> Realizing topologically ordered states on a quantum processor, Science 374 6572 (2021) 1237-1241 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1126/science.abi8378">doi:10.1126/science.abi8378</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> Probing topological spin liquids on a programmable quantum simulator, Science 374 6572 (2021) 1242-1247 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1126/science.abi8794">doi:10.1126/science.abi8794</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> exposition in: <ul> <li>Isabelle Dumé, <a href="https://physicsworld.com/a/long-range-quantum-entanglement-measured-at-last/">Long-range quantum entanglement measured at last</a>, PhysicsWorld (Jan 2022)</li> </ul> </div></body></html> </div> <div class="revisedby"> Last revised on May 26, 2022 at 10:12:35. 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