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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="monoidal_categories">Monoidal categories</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/enriched+monoidal+category">enriched monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network">tensor network</a> </li> </ul> With braiding <ul> <li> <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/balanced+monoidal+category">balanced monoidal category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/twist">twist</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> </li> </ul> With duals for objects <ul> <li> <a class="existingWikiWord" href="/nlab/show/category+with+duals">category with duals</a> (list of them) </li> <li> <a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a> (what they have) </li> <li> <a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/autonomous+category">autonomous category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pivotal+category">pivotal category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spherical+category">spherical category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/ribbon+category">ribbon category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/tortile+category">tortile category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/compact+closed+category">compact closed category</a> </li> </ul> With duals for morphisms <ul> <li> monoidal dagger-category<a href="/nlab/new/monoidal+dagger-category">?</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+dagger-category">symmetric monoidal dagger-category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dagger+compact+category">dagger compact category</a> </li> </ul> With traces <ul> <li> <a class="existingWikiWord" href="/nlab/show/trace">trace</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/traced+monoidal+category">traced monoidal category</a> </li> </ul> Closed structure <ul> <li> <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/star-autonomous+category">star-autonomous category</a> </li> </ul> Special sorts of products <ul> <li> <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semicartesian+monoidal+category">semicartesian monoidal category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal category with diagonals</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a> </li> </ul> Semisimplicity <ul> <li> <a class="existingWikiWord" href="/nlab/show/semisimple+category">semisimple category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a> </li> </ul> Morphisms <ul> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a> (<a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax</a>, <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax</a>, <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong</a> <a class="existingWikiWord" href="/nlab/show/bilax+monoidal+functor">bilax</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+monoidal+functor">Frobenius</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a> </li> </ul> Internal monoids <ul> <li> <a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in a monoidal category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in a symmetric monoidal category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">module over a monoid</a> </li> </ul> Examples <ul> <li> <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a> </li> </ul> Theorems <ul> <li> <a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+monoidal+categories">coherence theorem for monoidal categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a> </li> </ul> In higher category theory <ul> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/braided+monoidal+2-category">braided monoidal 2-category</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/little+cubes+operad">little cubes operad</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/compact+double+category">compact double category</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='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#locc_and_slocc'>LOCC and SLOCC</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> While in <a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a> a (<a class="existingWikiWord" href="/nlab/show/pure+state">pure</a>) <a class="existingWikiWord" href="/nlab/show/state">state</a> is an <a class="existingWikiWord" href="/nlab/show/generalized+element">element</a> of an <a class="existingWikiWord" href="/nlab/show/object">object</a> in a <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a>, in contrast in <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> a pure state is an element of an object in a non-cartesian <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> (say of <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a>). As a result, in quantum mechanics a state of a compound <a class="existingWikiWord" href="/nlab/show/physical+system">physical system</a> may not come from a pair of states of the two <a class="existingWikiWord" href="/nlab/show/subsystems">subsystems</a>, but instead be a nontrivial <a class="existingWikiWord" href="/nlab/show/sum">sum</a> – a <a class="existingWikiWord" href="/nlab/show/superposition">superposition</a> – of such. These non-classical combinations of states of subsystems are called entangled states. <h2 id="definition">Definition</h2> In <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> a <a class="existingWikiWord" href="/nlab/show/state">state</a> of a <a class="existingWikiWord" href="/nlab/show/physical+system">physical system</a> is represented by a <a class="existingWikiWord" href="/nlab/show/vector">vector</a> in some (<a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert</a>-)<a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <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>. If the system is the composite of two <a class="existingWikiWord" href="/nlab/show/subsystems">subsystems</a> with state spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">H_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">H_2</annotation></semantics></math>, respectively, then the state space of the total system is the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>=</mo><msub><mi>H</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>H</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">H = H_1 \otimes H_2</annotation></semantics></math>. The <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> gives a <a class="existingWikiWord" href="/nlab/show/bilinear+map">bilinear map</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><msub><mi>H</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>H</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>H</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>H</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> p : H_1 \times H_2 \to H_1 \otimes H_2 </annotation></semantics></math></div> which sends a pair of states <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ψ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ψ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\psi_1, \psi_2)</annotation></semantics></math> to their tensor product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ψ</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>ψ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\psi_1 \otimes \psi_2</annotation></semantics></math>. States in the <a class="existingWikiWord" href="/nlab/show/image">image</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> are called product states or separable states. An entangled state is a state which is not a product state. Such a state corresponds to non-simple tensors. <h2 id="examples">Examples</h2> Consider two <a class="existingWikiWord" href="/nlab/show/quantum+systems">quantum systems</a>, <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, with state vectors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><msup><mi>Ψ</mi> <mrow><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">|\Psi^{(A)}\rangle</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><msup><mi>Ψ</mi> <mrow><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">|\Psi^{(B)}\rangle</annotation></semantics></math> respectively. The combined state of the system may be described by a single state vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><msup><mi>Ψ</mi> <mrow><mo stretchy="false">(</mo><mi>AB</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">|</mo><msup><mi>Ψ</mi> <mrow><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">⟩</mo><mo>⊗</mo><mo stretchy="false">|</mo><msup><mi>Ψ</mi> <mrow><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">|\Psi^{(AB)}\rangle=|\Psi^{(A)}\rangle \otimes |\Psi^{(B)}\rangle</annotation></semantics></math>. As an example, suppose that in the basis <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">|</mo><mn>0</mn><mo stretchy="false">⟩</mo><mo>,</mo><mo stretchy="false">|</mo><mn>1</mn><mo stretchy="false">⟩</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{|0\rangle ,|1\rangle\}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><msup><mi>Ψ</mi> <mrow><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">⟩</mo><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mrow><mo>(</mo><mo stretchy="false">|</mo><mn>0</mn><mo stretchy="false">⟩</mo><mo>+</mo><mo stretchy="false">|</mo><mn>1</mn><mo stretchy="false">⟩</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">|\Psi^{(A)}\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle +|1\rangle\right)</annotation></semantics></math>. This can be interpreted as system <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> being in state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mn>0</mn><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">|0\rangle</annotation></semantics></math> with probability 1/2 and state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mn>1</mn><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">|1\rangle</annotation></semantics></math> with probability 1/2. Suppose further that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><msup><mi>Ψ</mi> <mrow><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">|</mo><mn>0</mn><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">|\Psi^{(B)}\rangle = |0\rangle</annotation></semantics></math>. Then we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><msup><mi>Ψ</mi> <mrow><mo stretchy="false">(</mo><mi>AB</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">|</mo><msup><mi>Ψ</mi> <mrow><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">⟩</mo><mo>⊗</mo><mo stretchy="false">|</mo><msup><mi>Ψ</mi> <mrow><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">⟩</mo><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mrow><mo>(</mo><mo stretchy="false">|</mo><mn>0</mn><mo stretchy="false">⟩</mo><mo>+</mo><mo stretchy="false">|</mo><mn>1</mn><mo stretchy="false">⟩</mo><mo>)</mo></mrow><mo>⊗</mo><mo stretchy="false">|</mo><mn>0</mn><mo stretchy="false">⟩</mo><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mrow><mo>(</mo><mo stretchy="false">|</mo><mn>00</mn><mo stretchy="false">⟩</mo><mo>+</mo><mo stretchy="false">|</mo><mn>10</mn><mo stretchy="false">⟩</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">|\Psi^{(AB)}\rangle=|\Psi^{(A)}\rangle \otimes |\Psi^{(B)}\rangle=\frac{1}{\sqrt{2}}\left(|0\rangle +|1\rangle\right)\otimes|0\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle +|10\rangle\right)</annotation></semantics></math>. Such a state is said to be a product state because it is “factorable” or equivalently separable, i.e. it can be formed from some combination of individual states in the basis. Compare the above example to the state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><msup><mi>Ψ</mi> <mrow><mo stretchy="false">(</mo><mi>AB</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">⟩</mo><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mrow><mo>(</mo><mo stretchy="false">|</mo><mn>00</mn><mo stretchy="false">⟩</mo><mo>+</mo><mo stretchy="false">|</mo><mn>11</mn><mo stretchy="false">⟩</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">|\Psi^{(AB)}\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle +|11\rangle\right)</annotation></semantics></math>. This state is not a product state since it cannot be formed from any combination of individual states in the given basis. Such a state is known as an entangled state because it is said to be non-factorable or non-separable. The entangled states discussed above are, in fact, <a class="existingWikiWord" href="/nlab/show/pure+states">pure states</a> rather than <a class="existingWikiWord" href="/nlab/show/mixed+states">mixed states</a> because they cannot be broken down further. However, there is also a notion of entanglement for mixed states. <h2 id="properties">Properties</h2> <h3 id="locc_and_slocc">LOCC and SLOCC</h3> The following refers to (<a href="#CoeckeKissinger">Coecke-Kissinger</a>). Often if multi-party state<a href="/nlab/new/multi-party+state">?</a>s can be inter-converted via local operations, they are considered to be the same. This can be made formal by the following definition. <div class="num_defn"> <h6 id="definition_2">Definition</h6> Two states <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>Ψ</mi><mo stretchy="false">⟩</mo><mo>,</mo><mo stretchy="false">|</mo><mi>Φ</mi><mo stretchy="false">⟩</mo><mo>∈</mo><mo>⨂</mo><msub><mi>H</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">|\Psi\rangle,|\Phi\rangle \in \bigotimes H_i</annotation></semantics></math> are said to be equivalent up to local operations with classical communication (LOCC) if they can be inter-converted by a protocol involving any number of steps where (i) one party applies a local unitary operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>:</mo><msub><mi>H</mi> <mi>i</mi></msub><mo>→</mo><msub><mi>H</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U : H_i \rightarrow H_i</annotation></semantics></math> or (ii) one party sends some classical information to another. </div> Such a protocol is reversible, so since protocols compose, this generates an equivalence relation. While this removes a good deal of redundancy from the study of entanglement, it is often useful to use an even more course-grained relation. <div class="num_defn"> <h6 id="definition_3">Definition</h6> Two states <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>Ψ</mi><mo stretchy="false">⟩</mo><mo>,</mo><mo stretchy="false">|</mo><mi>Φ</mi><mo stretchy="false">⟩</mo><mo>∈</mo><mo>⨂</mo><msub><mi>H</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">|\Psi\rangle,|\Phi\rangle \in \bigotimes H_i</annotation></semantics></math> are said to be equivalent up to stochastic LOCC (SLOCC) if they can be inter-converted with some non-zero probability a protocol involving any number of steps where (i) one party applies a an arbitrary local operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>:</mo><msub><mi>H</mi> <mi>i</mi></msub><mo>→</mo><msub><mi>H</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">L : H_i \rightarrow H_i</annotation></semantics></math> or (ii) one party sends some classical information to another. </div> An example of a local stochastic operation is as follows. Suppose Alice and Bob share a state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>Ψ</mi><mo stretchy="false">⟩</mo><mo>∈</mo><msub><mi>H</mi> <mi>A</mi></msub><mo>⊗</mo><msub><mi>H</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">|\Psi\rangle \in H_A \otimes H_B</annotation></semantics></math> and Alice wishes to perform some operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>. Alice prepares an ancilla qubit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mn>0</mn><mo stretchy="false">⟩</mo><mo>∈</mo><msup><mi>ℂ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">|0\rangle \in \mathbb{C}^2</annotation></semantics></math> and performs a unitary operation <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>:</mo><msup><mi>ℂ</mi> <mn>2</mn></msup><mo>⊗</mo><msub><mi>H</mi> <mn>1</mn></msub><mo>→</mo><msup><mi>ℂ</mi> <mn>2</mn></msup><mo>⊗</mo><msub><mi>H</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> U : \mathbb{C}^2 \otimes H_1 \rightarrow \mathbb{C}^2 \otimes H_1 </annotation></semantics></math></div> on her qubit as well as her part of the state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>Ψ</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">|\Psi\rangle</annotation></semantics></math>. She then measures the ancilla qubit. If she gets an outcome of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mn>0</mn><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">|0\rangle</annotation></semantics></math>, she has performed some operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>:</mo><msub><mi>H</mi> <mi>A</mi></msub><mo>→</mo><msub><mi>H</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">L : H_A \rightarrow H_A</annotation></semantics></math> and if she gets outcome <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mn>1</mn><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">|1\rangle</annotation></semantics></math> she has performed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>′</mo><mo>:</mo><msub><mi>H</mi> <mi>A</mi></msub><mo>→</mo><msub><mi>H</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">L' : H_A \rightarrow H_A</annotation></semantics></math>. The probability of Alice successfully performing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is then the probability of getting the outcome of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mn>0</mn><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">|0\rangle</annotation></semantics></math> when she performed her measurement. <div class="num_theorem"> <h6 id="theorem">Theorem</h6> Two states are SLOCC-equivalent iff they can be inter-converted by applying arbitrary invertible local operations (ILOs). </div> Its easy to show using the Schur decomposition that there are only two SLOCC-equivalence classes in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mn>2</mn></msup><mo>⊗</mo><msup><mi>ℂ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}^2 \otimes \mathbb{C}^2</annotation></semantics></math>, namely the product state class and the <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> class. Perhaps more surprising is the following result to to Dur, Vidal, and Cirac. [2] <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> Any genuine tripartite 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>> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∈</mo><msup><mi>ℂ</mi> <mn>2</mn></msup><mo>⊗</mo><msup><mi>ℂ</mi> <mn>2</mn></msup><mo>⊗</mo><msup><mi>ℂ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\in \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2</annotation></semantics></math> is SLOCC-equivalent to either |W> or |GHZ>;. </div> By genuine, they mean a state that is not a product of smaller states. The two states are defined as: <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>W</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">|</mo><mn>100</mn><mo stretchy="false">⟩</mo><mo>+</mo><mo stretchy="false">|</mo><mn>010</mn><mo stretchy="false">⟩</mo><mo>+</mo><mo stretchy="false">|</mo><mn>001</mn><mo stretchy="false">⟩</mo><mspace width="2em"></mspace><mspace width="2em"></mspace><mo stretchy="false">|</mo><mi>GHZ</mi><mo stretchy="false">⟩</mo><mo>=</mo><mo stretchy="false">|</mo><mn>000</mn><mo stretchy="false">⟩</mo><mo>+</mo><mo stretchy="false">|</mo><mn>111</mn><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> |W\rangle = |100\rangle + |010\rangle + |001\rangle \qquad\qquad |GHZ\rangle = |000\rangle + |111\rangle </annotation></semantics></math></div> Each of these states yields the structure of a commutative <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>GHZ</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">|GHZ\rangle</annotation></semantics></math> yields a special CFA and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>W</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">|W\rangle</annotation></semantics></math> yields an “anti-special” CFA. This structure serves to uniquely identity these states (up to SLOCC) in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}^2</annotation></semantics></math>. [1] <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/short-range+entanglement">short-range entanglement</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+embezzlement">quantum embezzlement</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tensor+network">tensor network</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+teleportation">quantum teleportation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+error+correction">quantum error correction</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> 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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> Early discussion of <a class="existingWikiWord" href="/nlab/show/composite+quantum+systems">composite quantum systems</a> and their <a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Erwin+Schr%C3%B6dinger">Erwin Schrödinger</a>, Discussion of Probability Relations between Separated Systems, Mathematical Proceedings of the Cambridge Philosophical Society, 31 4 (1935) 555-563 [<a href="https://doi.org/10.1017/S0305004100013554">doi:10.1017/S0305004100013554</a>]</li> </ul> Introduction and review: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Teiko+Heinosaari">Teiko Heinosaari</a>, <a class="existingWikiWord" href="/nlab/show/M%C3%A1rio+Ziman">Mário Ziman</a>, Section 6 of: The Mathematical Language of Quantum Theory – From Uncertainty to Entanglement, Cambridge University Press (2011) [<a href="https://doi.org/10.1017/CBO9781139031103">doi:10.1017/CBO9781139031103</a>] </li> <li> Dagmar Bruss, Characterizing entanglement, J. Math. Phys. 43, 4237 (2002) <a href="http://arxiv.org/abs/quant-ph/0110078">arXiv:quant-ph/0110078</a> <a href="http://dx.doi.org/10.1063/1.1494474">doi</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Ingemar+Bengtsson">Ingemar Bengtsson</a>, <a class="existingWikiWord" href="/nlab/show/Karol+%C5%BByczkowski">Karol Życzkowski</a>, Chapter 15 of: Geometry of Quantum States — An Introduction to Quantum Entanglement, Cambridge University Press (2006) [<a href="https://doi.org/10.1017/CBO9780511535048">doi:10.1017/CBO9780511535048</a>] </li> </ul> Extensive survey of the field: <ul> <li>Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki: Quantum entanglement, Rev. Mod. Phys. 81 (2009) 865 [<a href="https://arxiv.org/abs/quant-ph/0702225">arXiv:quant-ph/0702225</a>, <a href="https://doi.org/10.1103/RevModPhys.81.865">doi:10.1103/RevModPhys.81.865</a>]</li> </ul> As a notion in <a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Watrous">John Watrous</a>, Chapter 6 of: The Theory of Quantum Information, Cambridge University Press (2018) [<a href="https://doi.org/10.1017/9781316848142">doi:10.1017/9781316848142</a>, <a href="https://cs.uwaterloo.ca/~watrous/TQI/">webpage</a>, <a href="https://cs.uwaterloo.ca/~watrous/TQI/TQI.pdf">pdf</a>]</li> </ul> In relation to <a class="existingWikiWord" href="/nlab/show/correlation">correlation</a>: <ul> <li id="LuoLuo03">Shun-long Luo, You-feng Luo, Correlation and Entanglement, Acta Mathematicae Applicatae Sinica 19 (2003) 581–598 <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/10.1007/s10255-003-0133-z">doi:10.1007/s10255-003-0133-z</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> and in relation to <a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a>/<a class="existingWikiWord" href="/nlab/show/quantum+supremacy">quantum supremacy</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Preskill">John Preskill</a>: Quantum computing and the entanglement frontier: pp. 63-80 in: The Theory of the Quantum World – Proceedings of the 25th Solvay Conference on Physics, World Scientific (2013) [<a href="https://arxiv.org/abs/1203.5813">arXiv:1203.5813</a>, <a href="https://doi.org/10.1142/8674">doi:10.1142/8674</a>, slides: <a href="https://simons.berkeley.edu/sites/default/files/docs/394/preskilljohn.pdf">pdf</a>]</li> </ul> and in the context of <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> of <a class="existingWikiWord" href="/nlab/show/topological+phases+of+matter">topological phases of matter</a>: <ul> <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. 1 of 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> Exposition of entanglement as a phenomenon of non-<a class="existingWikiWord" href="/nlab/show/Cartesian+monoidal+category">Cartesian</a> <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a>: <ul> <li id="Baez04"><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, Quantum Quandaries: a Category-Theoretic Perspective, in D. Rickles et al. (ed.) The structural foundations of quantum gravity, Clarendon Press (2006) 240-265 [<a href="https://arxiv.org/abs/quant-ph/0404040">arXiv:quant-ph/0404040</a>, <a href="https://global.oup.com/academic/product/the-structural-foundations-of-quantum-gravity-9780199269693">ISBN:9780199269693</a>]</li> </ul> A discussion in <a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">quantum mechanics in terms of dagger-compact categories</a> is in <ul> <li id="CoeckeKissinger"><a class="existingWikiWord" href="/nlab/show/Bob+Coecke">Bob Coecke</a>, <a class="existingWikiWord" href="/nlab/show/Aleks+Kissinger">Aleks Kissinger</a>, The compositional structure of multipartite quantum entanglement (<a href="http://arxiv.org/abs/1002.2540">arXiv:1002.2540</a>)</li> </ul> See also <ul> <li>W. Dür, G. Vidal, J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A. 62, 062314</li> </ul> Discussion in <a class="existingWikiWord" href="/nlab/show/quantum+optics">quantum optics</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Tim+Byrnes">Tim Byrnes</a>, <a class="existingWikiWord" href="/nlab/show/Ebubechukwu+Ilo-Okeke">Ebubechukwu Ilo-Okeke</a>, Section 10 of: Quantum Atom Optics: Theory and Applications to Quantum Technology, Cambridge University Press 2021 (<a href="https://arxiv.org/abs/2007.14601">arXiv:2007.14601</a>, <a href="https://www.cambridge.org/core/books/quantum-atom-optics/2D867888B5C666D3A936F1C942C99568">CUP</a>)</li> </ul> A connection to algebraic geometry is proposed in <ul> <li>Frédéric Holweck, Jean-Gabriel Luque, Jean-Yves Thibon, Geometric descriptions of entangled states by auxiliary varieties, J. Math. Phys. 53, 102203 (2012); <a href="http://dx.doi.org/10.1063/1.4753989">doi</a></li> </ul> The following work included the consideration of identical particles into the study of quantum entanglement. In this case, the usage of partial trace may not be suitable and instead subsystems are described in terms of subalgebras. The work is in operator algebraic framework, based on usage of <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> and related to the consideration of von Neumann entropy. <ul> <li><a class="existingWikiWord" href="/nlab/show/A.+P.+Balachandran">A. P. Balachandran</a>, <a class="existingWikiWord" href="/nlab/show/T.+R.+Govindarajan">T. R. Govindarajan</a>, Amilcar R. de Queiroz, A. F. Reyes-Lega, Entanglement and particle identity: a unifying approach, Phys. Rev. Lett. 110, 080503 (2013) <a href="http://arxiv.org/abs/1303.0688">arxiv/1303.0688</a>; Algebraic approach to entanglement and entropy, <a href="http://arxiv.org/abs/1301.1300">arxiv/1301.1300</a>; Entanglement, particle identity and the GNS construction: a unifying approach <a href="http://arxiv.org/pdf/1205.2882">arxiv/1205.2882</a> (earlier, longer version, overlapping with 1303.0688)</li> </ul> Use of <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> for quantifying entanglement: <ul> <li id="Mainiero2019"><a class="existingWikiWord" href="/nlab/show/Tom+Mainiero">Tom Mainiero</a>, Homological Tools for the Quantum Mechanic (2019) [<a href="https://arxiv.org/abs/1901.02011">arXiv:1901.02011</a>, <a href="http://10.10.11.6/handle/1/1339">10.10.11.6/handle/1/1339</a>]</li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/physics">physics</a></div></body></html> </div> <div class="revisedby"> Last revised on November 8, 2024 at 22:23:12. 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