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theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a> in <a class="existingWikiWord" href="/nlab/show/quantum+information+theory+via+dagger-compact+categories">†-compact categories</a></p> <p><a class="existingWikiWord" href="/nlab/show/tensor+networks">tensor networks</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a></p> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a></p> </li> </ul> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+systems">quantum systems</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/parameterized+quantum+systems">parameterized</a>, <a class="existingWikiWord" href="/nlab/show/open+quantum+system">open</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+state+collapse">quantum state collapse</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+decoherence">quantum decoherence</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">quantum adiabatic theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/Berry+phases">Berry phases</a></p> <p><a class="existingWikiWord" href="/nlab/show/Dyson+formula">Dyson formula</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+many-body+physics">quantum many-body physics</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/functorial+quantum+field+theory">functorial quantum field theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-</a>)<a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+material">quantum material</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/topological+phases+of+matter">topological</a>) <a class="existingWikiWord" href="/nlab/show/phases+of+matter">phases of matter</a></p> </li> </ul> <p><br /></p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a> – <a class="existingWikiWord" href="/nlab/show/observables">observables</a> and <a class="existingWikiWord" href="/nlab/show/states">states</a></strong></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/states">states</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/quasi-state">quasi-state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a></p> <p><a class="existingWikiWord" href="/nlab/show/dimer">dimer</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">quantum state preparation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a>, <a class="existingWikiWord" href="/nlab/show/quantum+fluctuation">quantum fluctuation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+superposition">quantum superposition</a>, <a class="existingWikiWord" href="/nlab/show/quantum+interference">quantum interference</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a></p> <p><a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a></p> <p><a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superselection+sector">superselection sector</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> <p><a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coherent+quantum+state">coherent quantum state</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>, <a class="existingWikiWord" href="/nlab/show/excited+state">excited state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>, <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+amplitude">vacuum amplitude</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+fluctuation">vacuum fluctuation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+polarization">vacuum polarization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/thermal+vacuum">thermal vacuum</a>, <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/false+vacuum">false vacuum</a>, <a class="existingWikiWord" href="/nlab/show/tachyon">tachyon</a>, <a class="existingWikiWord" href="/nlab/show/Coleman-De+Luccia+instanton">Coleman-De Luccia instanton</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theta+vacuum">theta vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/non-geometric+string+theory+vacuum">non-geometric string theory vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entangled+state">entangled state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix+product+state">matrix product state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tree+tensor+network+state">tree tensor network state</a></p> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/observables">observables</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>, <a class="existingWikiWord" href="/nlab/show/beable">beable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator (in geometric quantization)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+effect">quantum effect</a>, <a class="existingWikiWord" href="/nlab/show/effect+algebra">effect algebra</a></p> </li> <li> <p>in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+observable">linear observable</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+observable">regular observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>, <a class="existingWikiWord" href="/nlab/show/retarded+product">retarded product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra+of+observables">interacting field algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorems">theorems</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nuiten%27s+lemma">Nuiten's lemma</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner%27s+theorem">Wigner's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> </ul> </li> </ul> </div> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information+via+dagger-compact+categories">quantum information via dagger-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+channel">quantum channel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+teleportation">quantum teleportation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+entanglement+entropy">topological entanglement entropy</a></p> </li> </ul> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adiabatic+quantum+computation">adiabatic quantum computation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measurement-based+quantum+computation">measurement-based quantum computation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+gate">quantum gate</a>, <a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+programming+language">quantum programming language</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+error+correction">quantum error correction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/HaPPY+code">HaPPY code</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Majorana+dimer+code">Majorana dimer code</a></p> </li> </ul> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/qbit">qbit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/spin+resonance+qbit">spin resonance qbit</a></li> </ul> <p>quantum algorithms:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grover%27s+algorithm">Grover's algorithm</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Shor%27s+algorithm">Shor's algorithm</a></p> </li> </ul> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+sensing">quantum sensing</a></strong></p> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+communication">quantum communication</a></strong></p> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#QuantumChannelsAndDecoherence'>Environmental representation of quantum channels</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> <p>In <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> <em>decoherence</em> refers to the disappearance of quantum <a class="existingWikiWord" href="/nlab/show/entanglement">entanglement</a> and <a class="existingWikiWord" href="/nlab/show/superposition">superposition</a> in the limit where small <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+systems">quantum mechanical systems</a> are coupled to large <span class="newWikiWord">thermal baths<a href="/nlab/new/thermal+baths">?</a></span>.</p> <p>This has been argued to resolve (and has been argued not to resolve) the problem with the <a class="existingWikiWord" href="/nlab/show/interpretation+of+quantum+mechanics">interpretation</a> of <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a>.</p> <h2 id="properties">Properties</h2> <div> <h3 id="QuantumChannelsAndDecoherence">Environmental representation of quantum channels</h3> <p>The crux of dynamical <a class="existingWikiWord" href="/nlab/show/quantum+decoherence">quantum decoherence</a> is that <em>fundamentally</em> the (time-)evolution of any <a class="existingWikiWord" href="/nlab/show/quantum+system">quantum system</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℋ</mi></mrow><annotation encoding="application/x-tex">\mathscr{H}</annotation></semantics></math> may be assumed <a class="existingWikiWord" href="/nlab/show/unitary+quantum+channel">unitary</a> (say via a <a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+equation">Schrödinger equation</a>) <em>when taking the whole evolution of its environment</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℬ</mi></mrow><annotation encoding="application/x-tex">\mathscr{B}</annotation></semantics></math> (the “bath”, ultimately the whole <a class="existingWikiWord" href="/nlab/show/observable+universe">observable universe</a>) into account, too, in that the evolution of the total system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℋ</mi><mo>⊗</mo><mi class="mathscript">ℬ</mi></mrow><annotation encoding="application/x-tex">\mathscr{H} \otimes \mathscr{B}</annotation></semantics></math> is given by a <a class="existingWikiWord" href="/nlab/show/unitary+operator">unitary operator</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><mi>evolve</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace></mrow></mpadded><mi class="mathscript">ℋ</mi><mo>⊗</mo><mi class="mathscript">ℬ</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi class="mathscript">ℋ</mi><mo>⊗</mo><mi class="mathscript">ℬ</mi></mtd></mtr> <mtr><mtd><mrow><mo>|</mo><mi>ψ</mi><mo>,</mo><mi>β</mi><mo>⟩</mo></mrow></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>U</mi> <mi>tot</mi></msub><mrow><mo>|</mo><mi>ψ</mi><mo>,</mo><mi>β</mi><mo>⟩</mo></mrow><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow></mpadded></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathllap{ evolve \;\colon\; } \mathscr{H} \otimes \mathscr{B} &\longrightarrow& \mathscr{H} \otimes \mathscr{B} \\ \left\vert \psi, \beta \right\rangle &\mapsto& U_{tot} \left\vert \psi, \beta \right\rangle \mathrlap{\,,} } </annotation></semantics></math></div> <p>after understanding the <a class="existingWikiWord" href="/nlab/show/mixed+states">mixed states</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi class="mathscript">ℋ</mi><mo>⊗</mo><msup><mi class="mathscript">ℋ</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\rho \,\colon\, \mathscr{H} \otimes \mathscr{H}^\ast</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/density+matrices">density matrices</a>) of the given <a class="existingWikiWord" href="/nlab/show/quantum+system">quantum system</a> as coupled to any given mixed state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>env</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi class="mathscript">ℬ</mi><mo>⊗</mo><msup><mi class="mathscript">ℬ</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">env \,\colon\, \mathscr{B} \otimes \mathscr{B}^\ast</annotation></semantics></math> of the bath (via <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><mi>couple</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace></mrow></mpadded><mi class="mathscript">ℋ</mi><mo>⊗</mo><msup><mi class="mathscript">ℋ</mi> <mo>*</mo></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo stretchy="false">(</mo><mi class="mathscript">ℋ</mi><mo>⊗</mo><mi class="mathscript">ℬ</mi><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mi class="mathscript">ℋ</mi><mo>⊗</mo><mi class="mathscript">ℬ</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mtd></mtr> <mtr><mtd><mi>ρ</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>ρ</mi><mo>⊗</mo><mi>env</mi><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>;</mo></mrow></mpadded></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathllap{ couple \;\colon\; } \mathscr{H} \otimes \mathscr{H}^\ast & \longrightarrow & (\mathscr{H} \otimes \mathscr{B}) \otimes (\mathscr{H} \otimes \mathscr{B})^\ast \\ \rho &\mapsto& \rho \otimes env \mathrlap{\,;} } </annotation></semantics></math></div> <p>…the only catch being that one cannot — and in any case does not (want or need to) — keep track of the precise <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> of the environment/bath, instead only of its <em>average</em> effect on the given <a class="existingWikiWord" href="/nlab/show/quantum+system">quantum system</a>, which by the rule of <a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a> is the <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a> that remains after the <a class="existingWikiWord" href="/nlab/show/partial+trace+quantum+channel">partial trace</a> over the environment:</p> <div class="maruku-equation" id="eq:PartialTraceOverBathTowardsQuantumChannels"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><mi>average</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace></mrow></mpadded><mo stretchy="false">(</mo><mi class="mathscript">ℋ</mi><mo>⊗</mo><mi class="mathscript">ℬ</mi><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mi class="mathscript">ℋ</mi><mo>⊗</mo><mi class="mathscript">ℬ</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi class="mathscript">ℋ</mi><mo>⊗</mo><msup><mi class="mathscript">ℋ</mi> <mo>*</mo></msup></mtd></mtr> <mtr><mtd><mover><mi>ρ</mi><mo>^</mo></mover></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>Tr</mi> <mi class="mathscript">ℬ</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mover><mi>ρ</mi><mo>^</mo></mover><mo maxsize="1.2em" minsize="1.2em">)</mo><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow></mpadded></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathllap{ average \;\colon\; } (\mathscr{H} \otimes \mathscr{B}) \otimes (\mathscr{H} \otimes \mathscr{B})^\ast &\longrightarrow& \mathscr{H} \otimes \mathscr{H}^\ast \\ \widehat{\rho} &\mapsto& Tr_{\mathscr{B}}\big(\widehat{\rho}\big) \mathrlap{\,.} } </annotation></semantics></math></div> <p>In summary this means <em>for practical purposes</em> that the probabilistic evolution of quantum systems <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℋ</mi></mrow><annotation encoding="application/x-tex">\mathscr{H}</annotation></semantics></math> is always of the composite form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi class="mathscript">ℋ</mi><mo>⊗</mo><msup><mi class="mathscript">ℋ</mi> <mo>*</mo></msup></mtd> <mtd><mover><mo>→</mo><mrow><mtable><mtr><mtd><mtext>couple to</mtext></mtd></mtr> <mtr><mtd><mtext>environment</mtext></mtd></mtr></mtable></mrow></mover></mtd> <mtd><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi class="mathscript">ℋ</mi></mtd></mtr> <mtr><mtd><mo>⊗</mo></mtd></mtr> <mtr><mtd><mi class="mathscript">ℬ</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>⊗</mo><msup><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi class="mathscript">ℋ</mi></mtd></mtr> <mtr><mtd><mo>⊗</mo></mtd></mtr> <mtr><mtd><mi class="mathscript">ℬ</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow> <mo>*</mo></msup></mtd> <mtd><mover><mo>→</mo><mrow><mtable><mtr><mtd><mtext>total unitary</mtext></mtd></mtr> <mtr><mtd><mtext>evolution</mtext></mtd></mtr></mtable></mrow></mover></mtd> <mtd><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi class="mathscript">ℋ</mi></mtd></mtr> <mtr><mtd><mo>⊗</mo></mtd></mtr> <mtr><mtd><mi class="mathscript">ℬ</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>⊗</mo><msup><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi class="mathscript">ℋ</mi></mtd></mtr> <mtr><mtd><mo>⊗</mo></mtd></mtr> <mtr><mtd><mi class="mathscript">ℬ</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow> <mo>*</mo></msup></mtd> <mtd><mover><mo>→</mo><mrow><mtable><mtr><mtd><mtext>average over</mtext></mtd></mtr> <mtr><mtd><mtext>environment</mtext></mtd></mtr></mtable></mrow></mover></mtd> <mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi class="mathscript">ℋ</mi><mo>⊗</mo><msup><mi class="mathscript">ℋ</mi> <mo>*</mo></msup></mtd></mtr> <mtr><mtd><mi>ρ</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>ρ</mi><mo>⊗</mo><mi>env</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mpadded width="0" lspace="-50%width"><mrow><msub><mi>U</mi> <mi>tot</mi></msub><mo>⋅</mo><mo stretchy="false">(</mo><mi>ρ</mi><mo>⊗</mo><mi>env</mi><mo stretchy="false">)</mo><mo>⋅</mo><msubsup><mi>U</mi> <mi>tot</mi> <mo>†</mo></msubsup></mrow></mpadded></mtd> <mtd><mo>↦</mo></mtd> <mtd><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mpadded width="0" lspace="-50%width"><mrow><msub><mi>Tr</mi> <mi class="mathscript">ℬ</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>U</mi> <mi>tot</mi></msub><mo>⋅</mo><mo stretchy="false">(</mo><mi>ρ</mi><mo>⊗</mo><mi>env</mi><mo stretchy="false">)</mo><mo>⋅</mo><msubsup><mi>U</mi> <mi>tot</mi> <mo>†</mo></msubsup><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mpadded></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathscr{H} \otimes \mathscr{H}^\ast & \xrightarrow{ \array{ \text{couple to} \\ \text{environment} } } & \left( \array{ \mathscr{H} \\ \otimes \\ \mathscr{B} } \right) \otimes \left( \array{ \mathscr{H} \\ \otimes \\ \mathscr{B} } \right)^\ast & \xrightarrow{ \array{ \text{total unitary} \\ \text{evolution} } } & \left( \array{ \mathscr{H} \\ \otimes \\ \mathscr{B} } \right) \otimes \left( \array{ \mathscr{H} \\ \otimes \\ \mathscr{B} } \right)^\ast & \xrightarrow{ \array{ \text{average over} \\ \text{environment} } } & \;\;\;\; \mathscr{H} \otimes \mathscr{H}^\ast \\ \rho &\mapsto& \rho \otimes env &\mapsto& \mathclap{ U_{tot} \cdot (\rho \otimes env) \cdot U_{tot}^\dagger } &\mapsto& \;\;\;\;\;\;\;\;\;\;\;\; \mathclap{ Tr_{\mathscr{B}} \big( U_{tot} \cdot (\rho \otimes env) \cdot U_{tot}^\dagger \big) } } </annotation></semantics></math></div> <p>This composite turns out to be a “<em><a class="existingWikiWord" href="/nlab/show/quantum+channel">quantum channel</a></em>”</p> <p>The realization of a quantum channel in the form <a class="maruku-eqref" href="#eq:QuantumChannelVieDecoherence">(2)</a> is also called an <em>environmental representation</em> (eg. <a href="quantum+channel#ŻyczkowskiBengtsson04">Życzkowski & Bengtsson 2004 (3.5)</a>).</p> <p>In fact all quantum channels on a fixed Hilbert space have such an evironmental representation:</p> <p> <div class="num_prop" id="QuantumChannelsAsPartialTracesOfUnitariesOnTensors"> <h6>Proposition</h6> <p><strong>(environmental representation of quantum channels)</strong></p> <p>Every <a class="existingWikiWord" href="/nlab/show/quantum+channel">quantum channel</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>chan</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi class="mathscript">ℋ</mi><mo>⊗</mo><msup><mi class="mathscript">ℋ</mi> <mo>*</mo></msup><mo>⟶</mo><mi class="mathscript">ℋ</mi><mo>⊗</mo><msup><mi class="mathscript">ℋ</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> chan \;\;\colon\;\; \mathscr{H} \otimes \mathscr{H}^\ast \longrightarrow \mathscr{H} \otimes \mathscr{H}^\ast </annotation></semantics></math></div> <p>may be written as</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/unitary+quantum+channel">unitary quantum channel</a>, induced by a <a class="existingWikiWord" href="/nlab/show/unitary+operator">unitary operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>tot</mi></msub><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi class="mathscript">ℋ</mi><mo>⊗</mo><mi class="mathscript">ℬ</mi><mo>→</mo><mi class="mathscript">ℋ</mi><mo>⊗</mo><mi class="mathscript">ℬ</mi></mrow><annotation encoding="application/x-tex">U_{tot} \,\colon\, \mathscr{H} \otimes \mathscr{B} \to \mathscr{H} \otimes \mathscr{B} </annotation></semantics></math></p> </li> <li> <p>on a <a class="existingWikiWord" href="/nlab/show/compound+system">compound system</a> with some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℬ</mi></mrow><annotation encoding="application/x-tex">\mathscr{B}</annotation></semantics></math> (the “bath”), yielding a <em>total system</em> <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℋ</mi><mo>⊗</mo><mi class="mathscript">ℬ</mi></mrow><annotation encoding="application/x-tex">\mathscr{H} \otimes \mathscr{B}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/tensor+product+of+Hilbert+spaces">tensor product</a>),</p> </li> <li> <p>and acting on the given <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a> <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> coupled (tensored) with any <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> of the bath system,</p> </li> <li> <p>followed by <a class="existingWikiWord" href="/nlab/show/partial+trace+quantum+channel">partial trace</a> (averaging) over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℬ</mi></mrow><annotation encoding="application/x-tex">\mathscr{B}</annotation></semantics></math> (leading to <a class="existingWikiWord" href="/nlab/show/decoherence">decoherence</a> in the remaining state)</p> </li> </ol> <p>in that</p> <div class="maruku-equation" id="eq:QuantumChannelVieDecoherence"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>chan</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>Tr</mi> <mi class="mathscript">ℬ</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>U</mi> <mi>tot</mi></msub><mo>⋅</mo><mo stretchy="false">(</mo><mi>ρ</mi><mo>⊗</mo><mi>env</mi><mo stretchy="false">)</mo><mo>⋅</mo><msubsup><mi>U</mi> <mi>tot</mi> <mo>†</mo></msubsup><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> chan(\rho) \;\;=\;\; Tr_{\mathscr{B}} \big( U_{tot} \cdot (\rho \otimes env) \cdot U_{tot}^\dagger \big) \,. </annotation></semantics></math></div> <p>Conversely, every operation of the form <a class="maruku-eqref" href="#eq:QuantumChannelVieDecoherence">(2)</a> is a quantum channel.</p> </div> </p> <p>This is originally due to <a href="quantum+channel#Lindblad75">Lindblad 1975</a> (see top of p. 149 and inside the proof of Lem. 5). For exposition and review see: <a href="quantum+channel#NielsenChuang00">Nielsen & Chuang 2000 §8.2.2-8.2.3</a>. An account of the infinite-dimensional case is in <a href="quantum+channel#AttalLectureNotes">Attal, Thm. 6.5 & 6.7</a>. These authors focus on the case that the environment is in a <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a>, the (parital) generalization to <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed</a> environment states is discussed in <a href="quantum+channel#BengtssonŻyczkowski06">Bengtsson & Życzkowski 2006 pp. 258</a>.</p> <p> <div class="proof" id="ProofOfEnvironmentalRepresentationOfQuantumChannels"> <h6>Proof</h6> <p>We spell out the proof assuming <a class="existingWikiWord" href="/nlab/show/finite-dimensional+Hilbert+spaces">finite-dimensional Hilbert spaces</a>. (The general case follows the same idea, supplemented by arguments that the following <a class="existingWikiWord" href="/nlab/show/sums">sums</a> <a class="existingWikiWord" href="/nlab/show/convergence">converge</a>.)</p> <p>Now given a <a class="existingWikiWord" href="/nlab/show/completely+positive+map">completely positive map</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>chan</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi class="mathscript">ℋ</mi><mo>⊗</mo><msup><mi class="mathscript">ℋ</mi> <mo>*</mo></msup><mo>⟶</mo><mi class="mathscript">ℋ</mi><mo>⊗</mo><msup><mi class="mathscript">ℋ</mi> <mo>*</mo></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> chan \,\colon\, \mathscr{H} \otimes \mathscr{H}^\ast \longrightarrow \mathscr{H} \otimes \mathscr{H}^\ast \,, </annotation></semantics></math></div> <p>then by <a href="quantum+channel#OperatorSumDecompositionOfQuantumChannels">operator-sum decomposition</a> there exists a <a class="existingWikiWord" href="/nlab/show/set">set</a> (<a class="existingWikiWord" href="/nlab/show/finite+set">finite</a>, under our assumptions) <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited</a> by at least one element</p> <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>ini</mi></msub><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>S</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> s_{ini} \,\colon\, S \,, </annotation></semantics></math></div> <p>and an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/indexed+set">indexed set</a> of <a class="existingWikiWord" href="/nlab/show/linear+operators">linear operators</a></p> <div class="maruku-equation" id="eq:KrausDecompositionInConstructionOfEnvRep"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>s</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>S</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>E</mi> <mi>s</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi class="mathscript">ℋ</mi><mo>⟶</mo><mi class="mathscript">ℋ</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mtext>with</mtext><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>s</mi></munder><msubsup><mi>E</mi> <mi>s</mi> <mo>†</mo></msubsup><mo>⋅</mo><msub><mi>E</mi> <mi>s</mi></msub><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mi>Id</mi><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow></mpadded></mrow><annotation encoding="application/x-tex"> s \,\colon\, S \;\;\; \vdash \;\;\; E_s \;\colon\; \mathscr{H} \longrightarrow \mathscr{H} \,,\;\;\;\; \text{with} \;\;\;\; \underset{s}{\sum} E_s^\dagger \cdot E_s \,=\, Id \mathrlap{\,,} </annotation></semantics></math></div> <p>such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>chan</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>s</mi></munder><mspace width="thinmathspace"></mspace><msub><mi>E</mi> <mi>s</mi></msub><mo>⋅</mo><mi>ρ</mi><mo>⋅</mo><msubsup><mi>E</mi> <mi>s</mi> <mo>†</mo></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> chan(\rho) \;=\; \underset{s}{\sum} \, E_s \cdot \rho \cdot E_s^\dagger \,. </annotation></semantics></math></div> <p>Now take</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℬ</mi><mspace width="thinmathspace"></mspace><mo>≡</mo><mspace width="thinmathspace"></mspace><munder><mo>⊕</mo><mi>S</mi></munder><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> \mathscr{B} \,\equiv\, \underset{S}{\oplus} \mathbb{C} </annotation></semantics></math></div> <p>with its canonical <a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian</a> <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a>-structure with <a class="existingWikiWord" href="/nlab/show/orthonormal+linear+basis">orthonormal linear basis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mrow><mo>|</mo><mi>s</mi><mo>⟩</mo></mrow><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mrow><mi>s</mi><mo lspace="verythinmathspace">:</mo><mi>S</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\big(\left\vert s \right\rangle\big)_{s \colon S}</annotation></semantics></math> and consider the linear map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><mi>V</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mpadded><mi class="mathscript">ℋ</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi class="mathscript">ℋ</mi><mo>⊗</mo><mi class="mathscript">ℬ</mi></mtd></mtr> <mtr><mtd><mrow><mo>|</mo><mi>ψ</mi><mo>⟩</mo></mrow></mtd> <mtd><mo>↦</mo></mtd> <mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>s</mi></munder><mspace width="thinmathspace"></mspace><msub><mi>E</mi> <mi>s</mi></msub><mrow><mo>|</mo><mi>ψ</mi><mo>⟩</mo></mrow><mo>⊗</mo><mrow><mo>|</mo><mi>s</mi><mo>⟩</mo></mrow><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow></mpadded></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathllap{ V \;\colon\;\; } \mathscr{H} &\longrightarrow& \mathscr{H} \otimes \mathscr{B} \\ \left\vert \psi \right\rangle &\mapsto& \underset{s}{\sum} \, E_s \left\vert \psi \right\rangle \otimes \left\vert s \right\rangle \mathrlap{\,.} } </annotation></semantics></math></div> <p>Observe that this is a <a class="existingWikiWord" href="/nlab/show/linear+isometry">linear isometry</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left left"><mtr><mtd><mrow><mo>⟨</mo><mi>ψ</mi><mo>|</mo></mrow><msup><mi>V</mi> <mo>†</mo></msup><mi>V</mi><mrow><mo>|</mo><mi>ψ</mi><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>s</mi><mo>,</mo><mi>s</mi><mo>′</mo></mrow></munder><mrow><mo>⟨</mo><mi>ψ</mi><mo>|</mo></mrow><msubsup><mi>E</mi> <mrow><mi>s</mi><mo>′</mo></mrow> <mo>†</mo></msubsup><msub><mi>E</mi> <mi>s</mi></msub><mrow><mo>|</mo><mi>ψ</mi><mo>⟩</mo></mrow><munder><munder><mrow><mo>⟨</mo><mi>s</mi><mo>′</mo><mo stretchy="false">|</mo><mi>s</mi><mo>⟩</mo></mrow><mo>⏟</mo></munder><mrow><msubsup><mi>δ</mi> <mi>s</mi> <mrow><mi>s</mi><mo>′</mo></mrow></msubsup></mrow></munder></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>⟨</mo><mi>ψ</mi><mo>|</mo></mrow><munder><munder><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>s</mi></munder><msubsup><mi>E</mi> <mi>s</mi> <mo>†</mo></msubsup><msub><mi>E</mi> <mi>s</mi></msub></mrow><mo>⏟</mo></munder><mi>Id</mi></munder><mrow><mo>|</mo><mi>ψ</mi><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>⟨</mo><mi>ψ</mi><mo stretchy="false">|</mo><mi>ψ</mi><mo>⟩</mo></mrow><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow></mpadded></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{ll} \left\langle \psi \right\vert V^\dagger V \left\vert \psi \right\rangle \\ \;=\; \underset{s,s'}{\sum} \left\langle \psi \right\vert E_{s'}^\dagger E_s \left\vert \psi \right\rangle \underset{ \delta_s^{s'} }{ \underbrace{ \left\langle s' \vert s \right\rangle } } \\ \;=\; \left\langle \psi \right\vert \underset{ Id }{ \underbrace{ \underset{s}{\sum} E_{s}^\dagger E_s } } \left\vert \psi \right\rangle \\ \;=\; \left\langle \psi \vert \psi \right\rangle \mathrlap{\,.} \end{array} </annotation></semantics></math></div> <p>This <a href="linear+isometryLinearIsometriesAreInjective">implies</a> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/injective+map">injective</a> so that we have a <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a>-decomposition of its <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a> into its <a class="existingWikiWord" href="/nlab/show/image">image</a> and its <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a> <a class="existingWikiWord" href="/nlab/show/orthogonal+complement">orthogonal complement</a>, which is unitarily isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo stretchy="false">(</mo><mi class="mathscript">ℬ</mi><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">dim(\mathscr{B})-1</annotation></semantics></math> summands of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℋ</mi></mrow><annotation encoding="application/x-tex">\mathscr{H}</annotation></semantics></math> that we may identify as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℋ</mi><mo>⊗</mo><mi class="mathscript">ℬ</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi class="mathscript">ℋ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>⊕</mo><mo maxsize="1.8em" minsize="1.8em">(</mo><mi class="mathscript">ℋ</mi><mo>⊗</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi class="mathscript">ℬ</mi><mo>⊖</mo><mi>ℂ</mi><mrow><mo>|</mo><msub><mi>s</mi> <mn>0</mn></msub><mo>⟩</mo></mrow><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathscr{H} \otimes \mathscr{B} \;\simeq\; V\big( \mathscr{H} \big) \oplus \Big( \mathscr{H} \otimes \big( \mathscr{B} \ominus \mathbb{C}\left\vert s_0 \right\rangle \big) \Big) \,. </annotation></semantics></math></div> <p>In total this yields a unitary operator</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi class="mathscript">ℋ</mi><mo>⊗</mo><mi class="mathscript">ℬ</mi><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><mi class="mathscript">ℋ</mi><mo>⊕</mo><mo maxsize="1.8em" minsize="1.8em">(</mo><mi class="mathscript">ℋ</mi><mo>⊗</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi class="mathscript">ℬ</mi><mo>⊖</mo><mi>ℂ</mi><mrow><mo>|</mo><msub><mi>s</mi> <mi>ini</mi></msub><mo>⟩</mo></mrow><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><munderover><mo>⟶</mo><mrow></mrow><mrow></mrow></munderover><mi>V</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi class="mathscript">ℋ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>⊕</mo><mo maxsize="1.8em" minsize="1.8em">(</mo><mi class="mathscript">ℋ</mi><mo>⊗</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi class="mathscript">ℬ</mi><mo>⊖</mo><mi>ℂ</mi><mrow><mo>|</mo><msub><mi>s</mi> <mi>ini</mi></msub><mo>⟩</mo></mrow><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi class="mathscript">ℋ</mi><mo>⊗</mo><mi class="mathscript">ℬ</mi></mrow><annotation encoding="application/x-tex"> U \;\colon\; \mathscr{H} \otimes \mathscr{B} \,\simeq\, \mathscr{H} \oplus \Big( \mathscr{H} \otimes \big( \mathscr{B} \ominus \mathbb{C}\left\vert s_{ini} \right\rangle \big) \Big) \underoverset{}{}{\longrightarrow} V\big( \mathscr{H} \big) \oplus \Big( \mathscr{H} \otimes \big( \mathscr{B} \ominus \mathbb{C}\left\vert s_{ini} \right\rangle \big) \Big) \;\simeq\; \mathscr{H} \otimes \mathscr{B} </annotation></semantics></math></div> <p>and we claim that this has the desired action on couplings of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">ℋ</mi></mrow><annotation encoding="application/x-tex">\mathscr{H}</annotation></semantics></math>-system to the pure bath state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>|</mo><msub><mi>s</mi> <mi>ini</mi></msub><mo>⟩</mo></mrow></mrow><annotation encoding="application/x-tex">\left\vert s_{ini} \right\rangle</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left"><mtr><mtd><msup><mi>trace</mi> <mi class="mathscript">ℬ</mi></msup><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>U</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mrow><mo>|</mo><msub><mi>s</mi> <mi>ini</mi></msub><mo>⟩</mo></mrow><mi>ρ</mi><mrow><mo>⟨</mo><msub><mi>s</mi> <mi>ini</mi></msub><mo>|</mo></mrow><mo maxsize="1.2em" minsize="1.2em">)</mo><msup><mi>U</mi> <mo>†</mo></msup><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>s</mi><mo>,</mo><mi>s</mi><mo>′</mo></mrow></munder><msup><mi>trace</mi> <mi class="mathscript">ℬ</mi></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mrow><mo>|</mo><mi>s</mi><mo>⟩</mo></mrow><msub><mi>E</mi> <mi>s</mi></msub><mo>⋅</mo><mi>ρ</mi><mo>⋅</mo><msubsup><mi>E</mi> <mrow><mi>s</mi><mo>′</mo></mrow> <mo>†</mo></msubsup><mrow><mo>⟨</mo><mi>s</mi><mo>′</mo><mo>|</mo></mrow><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>s</mi><mo>,</mo><mi>s</mi><mo>′</mo></mrow></munder><munder><munder><mrow><mo>⟨</mo><mi>s</mi><mo>′</mo><mo stretchy="false">|</mo><mi>s</mi><mo>⟩</mo></mrow><mo>⏟</mo></munder><mrow><msubsup><mi>δ</mi> <mi>s</mi> <mrow><mi>s</mi><mo>′</mo></mrow></msubsup></mrow></munder><msub><mi>E</mi> <mi>s</mi></msub><mo>⋅</mo><mi>ρ</mi><mo>⋅</mo><msubsup><mi>E</mi> <mrow><mi>s</mi><mo>′</mo></mrow> <mo>†</mo></msubsup></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>s</mi></munder><msub><mi>E</mi> <mi>s</mi></msub><mo>⋅</mo><mi>ρ</mi><mo>⋅</mo><msubsup><mi>E</mi> <mi>s</mi> <mo>†</mo></msubsup></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>chan</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{l} trace^{\mathscr{B}} \Big( U \big( \left\vert s_{ini} \right\rangle \rho \left\langle s_{ini} \right\vert \big) U^\dagger \Big) \\ \;=\; \underset{s,s'}{\sum} trace^{\mathscr{B}} \big( \left\vert s \right\rangle E_s \cdot \rho \cdot E_{s'}^\dagger \left\langle s' \right\vert \big) \\ \;=\; \underset{s,s'}{\sum} \underset{ \delta_{s}^{s'} }{ \underbrace{ \left\langle s' \vert s \right\rangle } } E_s \cdot \rho \cdot E_{s'}^\dagger \\ \;=\; \underset{s}{\sum} E_s \cdot \rho \cdot E_{s}^\dagger \\ \;=\; chan(\rho) \,. \end{array} </annotation></semantics></math></div> <p>This concludes the construction of an environmental representation where the environment is in a pure state.</p> </div> </p> <p> <div class="num_remark"> <h6>Remark</h6> <p>The above theorem is often phrased as “… and the environment can be assumed to be in a pure state”. But in fact the proof crucially uses the assumption that the environment is in a pure state. It is not clear that there is a proof that works more generally.</p> <p>In fact, if the environment is taken to be in the maximally mixed state, then the resulting quantum channels are called <em>noisy operations</em> or <em><a class="existingWikiWord" href="/nlab/show/unistochastic+quantum+channel">unistochastic quantum channels</a></em> and are not expected to exhaust all quantum channels.</p> </div> </p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+limit">classical limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a>, <a class="existingWikiWord" href="/nlab/show/interpretation+of+quantum+mechanics">interpretation of quantum mechanics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/observable+universe">observable universe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/consistent+histories">consistent histories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/einselection">einselection</a></p> </li> </ul> <h2 id="references">References</h2> <p>Precursor discussion:</p> <ul> <li id="vonNeumann32"> <p><a class="existingWikiWord" href="/nlab/show/John+von+Neumann">John von Neumann</a>, §VI.3 of:</p> <p><em>Mathematische Grundlagen der Quantenmechanik</em> (German) (1932, 1971) [<a href="https://link.springer.com/book/10.1007/978-3-642-96048-2">doi:10.1007/978-3-642-96048-2</a>]</p> <p><em>Mathematical Foundations of Quantum Mechanics</em> Princeton University Press (1955) [<a href="https://doi.org/10.1515/9781400889921">doi:10.1515/9781400889921</a>, <a href="https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics">Wikipedia entry</a>]</p> </li> </ul> <p>Original discusssion identifying quantum decoherence as interaction with an averaged environment (“bath”):</p> <ul> <li id="Zeh70"> <p><a class="existingWikiWord" href="/nlab/show/H.+Dieter+Zeh">H. Dieter Zeh</a>, <em>On the interpretation of measurement in quantum theory</em>, Found Phys <strong>1</strong> (1970) 69–76 [<a href="https://doi.org/10.1007/BF00708656">doi:10.1007/BF00708656</a>]</p> </li> <li id="Zurek81"> <p><a class="existingWikiWord" href="/nlab/show/Wojciech+Zurek">Wojciech Zurek</a>, <em>Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?</em>, Phys. Rev. D <strong>24</strong> (1981) 1516-1525 [<a href="https://doi.org/10.1103/PhysRevD.24.1516">doi:10.1103/PhysRevD.24.1516</a>]</p> </li> <li id="Zurek82"> <p><a class="existingWikiWord" href="/nlab/show/Wojciech+Zurek">Wojciech Zurek</a>, <em>Environment-induced superselection rules</em>, Phys. Rev. D <strong>26</strong> (1982) 1862-1880 [<a href="https://doi.org/10.1103/PhysRevD.26.1862">doi:10.1103/PhysRevD.26.1862</a>]</p> </li> <li id="JoosZeh85"> <p><a class="existingWikiWord" href="/nlab/show/Erich+Joos">Erich Joos</a>, <a class="existingWikiWord" href="/nlab/show/H.+Dieter+Zeh">H. Dieter Zeh</a>, <em>The emergence of classical properties through interaction with the environment</em>, Z. Physik B – Condensed Matter <strong>59</strong> (1985) 223–243 [<a href="https://doi.org/10.1007/BF01725541">doi:10.1007/BF01725541</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Erich+Joos">Erich Joos</a>, <em><a href="http://www.decoherence.de/">www.decoherence.de</a></em></p> </li> <li id="Zurek03"> <p><a class="existingWikiWord" href="/nlab/show/Wojciech+Zurek">Wojciech Zurek</a>, <em>Decoherence, einselection, and the quantum origins of the classical</em>, Rev. Mod. Phys. <strong>75</strong> (2003) 715-775 [<a href="http://arxiv.org/abs/quant-ph/0105127">quant-ph/0105127</a>, <a href="http://dx.doi.org/10.1103/RevModPhys.75.715">doi:10.1103/RevModPhys.75.715</a>]</p> </li> </ul> <p>With regards to the <a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a> and <a class="existingWikiWord" href="/nlab/show/interpretations+of+quantum+mechanics">interpretations of quantum mechanics</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Roland+Omn%C3%A8s">Roland Omnès</a>, §7 of: <em><a class="existingWikiWord" href="/nlab/show/The+Interpretation+of+Quantum+Mechanics">The Interpretation of Quantum Mechanics</a></em>, Princeton University Press (1994) [<a href="http://press.princeton.edu/titles/5525.html">ISBN:9780691036694</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Maximilian+Schlosshauer">Maximilian Schlosshauer</a>, <em>Decoherence, the measurement problem, and interpretations of quantum mechanics</em>, Rev. Mod. Phys. <strong>76</strong> (2004) 1267-1305 [<a href="https://arxiv.org/abs/quant-ph/0312059">arXiv;quant-ph/0312059</a>, <a href="https://doi.org/10.1103/RevModPhys.76.1267">doi:10.1103/RevModPhys.76.1267</a>]</p> </li> <li> <p>Guido Bacciagaluppi, <em>The Role of Decoherence in Quantum Mechanics</em>, Stanf. Enc. of Phil. (2020) [<a href="https://plato.stanford.edu/entries/qm-decoherence/">web</a>]</p> </li> </ul> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heinz-Peter+Breuer">Heinz-Peter Breuer</a>, <a class="existingWikiWord" href="/nlab/show/Francesco+Petruccione">Francesco Petruccione</a>, <em>Decoherence</em>, Chapter 4 in: <em>The Theory of Open Quantum Systems</em>, Oxford University Press (2007) [book:<a href="https://doi.org/10.1093/acprof:oso/9780199213900.001.0001">doi:10.1093/acprof:oso/9780199213900.001.0001</a>, chapter:<a href="https://doi.org/10.1093/acprof:oso/9780199213900.003.04">.003.04</a>]</p> </li> <li id="Schlosshauer07"> <p><a class="existingWikiWord" href="/nlab/show/Maximilian+Schlosshauer">Maximilian Schlosshauer</a>, <em>Decoherence and the Quantum-To-Classical Transition</em>, The Frontiers Collection, Springer (2007) [<a href="https://doi.org/10.1007/978-3-540-35775-9">doi:10.1007/978-3-540-35775-9</a>]</p> </li> </ul> <p>Further review:</p> <ul> <li> <p>Claus Kiefer, <a class="existingWikiWord" href="/nlab/show/Erich+Joos">Erich Joos</a>, <em>Decoherence: Concepts and Examples</em>, in: <em>Quantum Future From Volta and Como to the Present and Beyond</em>, Lecture Notes in Physics <em>517*</em> (1999) [<a href="https://arxiv.org/abs/quant-ph/9803052">arXiv:quant-ph/9803052</a>, <a href="https://doi.org/10.1007/BFb0105342">doi:10.1007/BFb0105342</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Maximilian+Schlosshauer">Maximilian Schlosshauer</a>, <em>Quantum Decoherence</em>, Phys. Rep. <strong>831</strong> (2019) 1-57 [<a href="https://arxiv.org/abs/1911.06282">arXiv:1911.06282</a>, <a href="https://doi.org/10.1016/j.physrep.2019.10.001">doi:10.1016/j.physrep.2019.10.001</a>]</p> </li> </ul> <p>See also:</p> <ul> <li> <p>Chris Nagele, Oliver Janssen, Matthew Kleban, <em>Decoherence: A Numerical Study</em> [<a href="https://arxiv.org/abs/2010.04803">arXiv:2010.04803</a>]</p> </li> <li> <p>Wikipedia, <em><a href="http://en.wikipedia.org/wiki/Quantum_decoherence">Quantum decoherence</a></em></p> </li> </ul> <p>A proposal for mathematical quantification of coherence:</p> <ul> <li>Fedor Herbut, <em>A quantum measure of coherence and incompatibility</em>, Journal of Physics A: Mathematical and General 38, 2959 (2005) (<a href="https://arxiv.org/abs/quant-ph/0503077">arXiv:quant-ph/0503077</a>, <a href="https://iopscience.iop.org/article/10.1088/0305-4470/38/13/010">arXiv:10.1088/0305-4470/38/13/010</a>)</li> </ul> <p>which was rediscovered and then became famous with:</p> <ul> <li id="BaumgratzCramerPlenio14">T. Baumgratz, M. Cramer, M. B. Plenio, <em>Quantifying Coherence</em>, Phys. Rev. Lett. <strong>113</strong> 140401 (2014) [<a href="https://arxiv.org/abs/1311.0275">arXiv:1311.0275</a>, <a href="https://doi.org/10.1103/PhysRevLett.113.140401">doi:10.1103/PhysRevLett.113.140401</a>]</li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Radhakrishnan+Chandrashekar">R. Chandrashekar</a>, P. Manikandan, J. Segar, <a class="existingWikiWord" href="/nlab/show/Tim+Byrnes">Tim Byrnes</a>, <em>Distribution of quantum coherence in multipartite systems</em>, Phys. Rev. Lett. <strong>116</strong> 150504 (2016) [<a href="https://arxiv.org/abs/1602.00286">arXiv:1602.00286</a>, <a href="https://doi.org/10.1103/PhysRevLett.116.150504">doi:10.1103/PhysRevLett.116.150504</a>]</p> </li> <li> <p>Md. Manirul Ali, Po-Wen Chen, <a class="existingWikiWord" href="/nlab/show/Radhakrishnan+Chandrashekar">R. Chandrashekar</a>, <em>Detecting quantum phase localization using Arnold tongue</em>, Physica A: Statistical Mechanics and its Applications <strong>633</strong> (2024) 129436 [<a href="https://doi.org/10.1016/j.physa.2023.129436">doi:10.1016/j.physa.2023.129436</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 12, 2024 at 11:23:39. 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