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wave function collapse in nLab
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content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="quantum_systems">Quantum systems</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>, <a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">dependent</a> <a class="existingWikiWord" href="/nlab/show/linear+type+theory">linear type theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a> in <a class="existingWikiWord" href="/nlab/show/quantum+information+theory+via+dagger-compact+categories">†-compact categories</a></p> <p><a class="existingWikiWord" href="/nlab/show/tensor+networks">tensor networks</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a></p> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a></p> </li> </ul> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+systems">quantum systems</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/parameterized+quantum+systems">parameterized</a>, <a class="existingWikiWord" href="/nlab/show/open+quantum+system">open</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+state+collapse">quantum state collapse</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+decoherence">quantum decoherence</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">quantum adiabatic theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/Berry+phases">Berry phases</a></p> <p><a class="existingWikiWord" href="/nlab/show/Dyson+formula">Dyson formula</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+many-body+physics">quantum many-body physics</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/functorial+quantum+field+theory">functorial quantum field theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-</a>)<a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+material">quantum material</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/topological+phases+of+matter">topological</a>) <a class="existingWikiWord" href="/nlab/show/phases+of+matter">phases of matter</a></p> </li> </ul> <p><br /></p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a> – <a class="existingWikiWord" href="/nlab/show/observables">observables</a> and <a class="existingWikiWord" href="/nlab/show/states">states</a></strong></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/states">states</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/quasi-state">quasi-state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a></p> <p><a class="existingWikiWord" href="/nlab/show/dimer">dimer</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">quantum state preparation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a>, <a class="existingWikiWord" href="/nlab/show/quantum+fluctuation">quantum fluctuation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+superposition">quantum superposition</a>, <a class="existingWikiWord" href="/nlab/show/quantum+interference">quantum interference</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a></p> <p><a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a></p> <p><a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superselection+sector">superselection sector</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> <p><a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coherent+quantum+state">coherent quantum state</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>, <a class="existingWikiWord" href="/nlab/show/excited+state">excited state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>, <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+amplitude">vacuum amplitude</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+fluctuation">vacuum fluctuation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+polarization">vacuum polarization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/thermal+vacuum">thermal vacuum</a>, <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/false+vacuum">false vacuum</a>, <a class="existingWikiWord" href="/nlab/show/tachyon">tachyon</a>, <a class="existingWikiWord" href="/nlab/show/Coleman-De+Luccia+instanton">Coleman-De Luccia instanton</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theta+vacuum">theta vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/non-geometric+string+theory+vacuum">non-geometric string theory vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entangled+state">entangled state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix+product+state">matrix product state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tree+tensor+network+state">tree tensor network state</a></p> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/observables">observables</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>, <a class="existingWikiWord" href="/nlab/show/beable">beable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator (in geometric quantization)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+effect">quantum effect</a>, <a class="existingWikiWord" href="/nlab/show/effect+algebra">effect algebra</a></p> </li> <li> <p>in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+observable">linear observable</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+observable">regular observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>, <a class="existingWikiWord" href="/nlab/show/retarded+product">retarded product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra+of+observables">interacting field algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorems">theorems</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nuiten%27s+lemma">Nuiten's lemma</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner%27s+theorem">Wigner's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> </ul> </li> </ul> </div> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information+via+dagger-compact+categories">quantum information via dagger-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+channel">quantum channel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+teleportation">quantum teleportation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+entanglement+entropy">topological entanglement entropy</a></p> </li> </ul> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+technology">quantum technology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+sensing">quantum sensing</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+communication">quantum communication</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+cryptography">quantum cryptography</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+machine+learning">quantum machine learning</a></p> </li> </ul> <p><br /></p> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+computing">quantum computing</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/qdit">qdit</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/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+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> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+resonance+qbit">spin resonance qbit</a></p> </li> <li> <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> </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='#RelationToConditionalExpectationValues'>Relation to conditional expectation values</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In the context of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>, the <strong>collapse of the wave function</strong>, also known as the <strong>reduction of the wave packet</strong>, is said to occur after <a class="existingWikiWord" href="/nlab/show/observation">observation</a> or <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, when a wave function expressed as the sum of <a class="existingWikiWord" href="/nlab/show/eigenfunctions">eigenfunctions</a> of the observable is projected randomly onto one of them. Different <a class="existingWikiWord" href="/nlab/show/interpretations+of+quantum+mechanics">interpretations of quantum mechanics</a> understand this process differently.</p> <p>The perspective associated with the <em><a class="existingWikiWord" href="/nlab/show/Bayesian+interpretation+of+quantum+mechanics">Bayesian interpretation of quantum mechanics</a></em> observes (see <a href="#RelationToConditionalExpectationValues">below</a>) that the apparent collapse is just the mathematical reflection of the formula for <a class="existingWikiWord" href="/nlab/show/conditional+expectation+values">conditional expectation values</a> in <a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a>.</p> <h2 id="RelationToConditionalExpectationValues">Relation to conditional expectation values</h2> <p>There is a close relation between wave function collapse and <a class="existingWikiWord" href="/nlab/show/conditional+expectation+values">conditional expectation values</a> in <a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a> (e.g. <a href="#Kuperberg05">Kuperberg 05, section 1.2</a>, <a href="#Yuan12">Yuan 12</a>):</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒜</mi><mo>,</mo><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{A},\langle -\rangle)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/quantum+probability+space">quantum probability space</a>, hence a <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex</a> <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</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">\mathcal{A}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/quantum+observables">quantum observables</a>, and a <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒜</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\langle -\rangle \;\colon\; \mathcal{A} \to \mathbb{C}</annotation></semantics></math>.</p> <p>This means that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{A}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/observable">observable</a>, its <em><a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a></em> in the given <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔼</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">⟨</mo><mi>A</mi><mo stretchy="false">⟩</mo><mo>∈</mo><mi>ℂ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{E}(A) \;\coloneqq\; \langle A \rangle \in \mathbb{C} \,. </annotation></semantics></math></div> <p>More generally, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">P \in \mathcal{A}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/real+part">real</a> <a class="existingWikiWord" href="/nlab/show/idempotent">idempotent</a>/<a class="existingWikiWord" href="/nlab/show/projector">projector</a></p> <div class="maruku-equation" id="eq:RealIdempotent"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>P</mi> <mo>*</mo></msup><mo>=</mo><mi>P</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mphantom><mi>AAA</mi></mphantom><mi>P</mi><mi>P</mi><mo>=</mo><mi>P</mi></mrow><annotation encoding="application/x-tex"> P^\ast = P \,, \phantom{AAA} P P = P </annotation></semantics></math></div> <p>thought of as an event, then for any observable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{A}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/conditional+expectation+value">conditional expectation value</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, conditioned on the observation 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>, is (e.g. <a href="#RedeiSummers06">Redei-Summers 06, section 7.3</a>, see also <a href="#FroehlichSchubnel15">Fröhlich-Schubnel 15, (5.49)</a>, <a href="#Froehlich16">Fröhlich 19 (45)</a>)</p> <div class="maruku-equation" id="eq:ConditionalExpectation"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔼</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">|</mo><mi>P</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mfrac><mrow><mo>⟨</mo><mi>P</mi><mi>A</mi><mi>P</mi><mo>⟩</mo></mrow><mrow><mo>⟨</mo><mi>P</mi><mo>⟩</mo></mrow></mfrac><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{E}(A \vert P) \;\coloneqq\; \frac{ \left \langle P A P \right\rangle }{ \left\langle P \right\rangle } \,. </annotation></semantics></math></div> <p>Now assume a <a class="existingWikiWord" href="/nlab/show/star-representation">star-representation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒜</mi><mo>→</mo><mi>End</mi><mo stretchy="false">(</mo><mi>ℋ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho \;\colon\; \mathcal{A} \to End(\mathcal{H})</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a> by <a class="existingWikiWord" href="/nlab/show/linear+operators">linear operators</a> on a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math> is given, and that the state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle -\rangle</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a>, hence given by a <a class="existingWikiWord" href="/nlab/show/vector">vector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>∈</mo><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\psi \in \mathcal{H}</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a>”) via the <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">⟩</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ℋ</mi><mo>⊗</mo><mi>ℋ</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\langle (-), (-)\rangle \;\colon\; \mathcal{H} \otimes \mathcal{H} \to \mathbb{C}</annotation></semantics></math> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">⟨</mo><mi>A</mi><mo stretchy="false">⟩</mo></mtd> <mtd><mo>≔</mo><mrow><mo>⟨</mo><mi>ψ</mi><mo stretchy="false">|</mo><mi>A</mi><mo stretchy="false">|</mo><mi>ψ</mi><mo>⟩</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≔</mo><mrow><mo>⟨</mo><mi>ψ</mi><mo>,</mo><mi>A</mi><mi>ψ</mi><mo>⟩</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \langle A \rangle & \coloneqq \left\langle\psi \vert A \vert \psi \right\rangle \\ & \coloneqq \left\langle\psi, A \psi \right\rangle \end{aligned} \,. </annotation></semantics></math></div> <p>In this case the expression for the <a class="existingWikiWord" href="/nlab/show/conditional+expectation+value">conditional expectation value</a> <a class="maruku-eqref" href="#eq:ConditionalExpectation">(2)</a> of an observable <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> conditioned on an idempotent observable <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> becomes (notationally suppressing the <a class="existingWikiWord" href="/nlab/show/representation">representation</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>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>𝔼</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">|</mo><mi>P</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mfrac><mrow><mo>⟨</mo><mi>ψ</mi><mo stretchy="false">|</mo><mi>P</mi><mi>A</mi><mi>P</mi><mo stretchy="false">|</mo><mi>ψ</mi><mo>⟩</mo></mrow><mrow><mo>⟨</mo><mi>ψ</mi><mo stretchy="false">|</mo><mi>P</mi><mo stretchy="false">|</mo><mi>ψ</mi><mo>⟩</mo></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mrow><mo>⟨</mo><mi>P</mi><mi>ψ</mi><mo stretchy="false">|</mo><mi>A</mi><mo stretchy="false">|</mo><mi>P</mi><mi>ψ</mi><mo>⟩</mo></mrow><mrow><mo>⟨</mo><mi>P</mi><mi>ψ</mi><mo stretchy="false">|</mo><mi>P</mi><mi>ψ</mi><mo>⟩</mo></mrow></mfrac><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbb{E}(A\vert P) & = \frac{ \left\langle \psi \vert P A P\vert \psi \right\rangle }{ \left\langle \psi \vert P \vert \psi \right\rangle } \\ & = \frac{ \left\langle P \psi \vert A \vert P \psi \right\rangle }{ \left\langle P \psi \vert P \psi \right\rangle } \,, \end{aligned} </annotation></semantics></math></div> <p>where in the last step we used <a class="maruku-eqref" href="#eq:ConditionalExpectation">(2)</a>.</p> <p>This says that <em>assuming</em> that <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> has been observed in the <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> <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">\vert \psi\rangle</annotation></semantics></math>, then the corresponding <a class="existingWikiWord" href="/nlab/show/conditional+expectation+values">conditional expectation values</a> are the same as actual <a class="existingWikiWord" href="/nlab/show/expectation+values">expectation values</a> but for the new pure state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>P</mi><mi>ψ</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\vert P \psi \rangle</annotation></semantics></math>.</p> <p>This is the statement of “wave function collapse”</p> <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>ψ</mi><mo stretchy="false">⟩</mo><mo>↦</mo><mi>P</mi><mo stretchy="false">|</mo><mi>ψ</mi><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \vert \psi \rangle \mapsto P \vert \psi \rangle \,. </annotation></semantics></math></div> <p>The original <a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>∈</mo><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\psi \in \mathcal{H}</annotation></semantics></math>, and after observing <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> it “collapses” to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>ψ</mi><mo>∈</mo><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">P \psi \in \mathcal{H}</annotation></semantics></math> (up to normalization).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><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/Bayesian+interpretation+of+quantum+mechanics">Bayesian interpretation of quantum mechanics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/propositions+as+projections">propositions as projections</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a></p> </li> </ul> <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> <h2 id="References">References</h2> <p>Early explicit discussion of wavefunction collapse in <a class="existingWikiWord" href="/nlab/show/quantum+measurements">quantum measurements</a>, including discussion relating to historical experiments:</p> <ul> <li id="vonNeumann32"> <p><a class="existingWikiWord" href="/nlab/show/John+von+Neumann">John von Neumann</a>, §III.3 and §VI of:</p> <p><em>Mathematische Grundlagen der Quantenmechanik</em>, Springer (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>But the formulation in <a href="#vonNeumann32">von Neumann 1932</a> postulated that a <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> would collapse to a <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a> when the <a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>‘s <a class="existingWikiWord" href="/nlab/show/eigenvalues">eigenvalues</a> are degenerate.</p> <p>This problem was pointed out and the modern formulation of the collapse postulate was formulated by:</p> <ul> <li id="Lüders51"> <p><a class="existingWikiWord" href="/nlab/show/Gerhart+L%C3%BCders">Gerhart Lüders</a>:</p> <p><em>Über die Zustandsänderung durch den Meßprozeß</em>, Ann. Phys. <strong>8</strong> (1951) 322–328 [<a href="https://doi.org/10.1002/andp.19504430510">doi:10.1002/andp.19504430510</a>]</p> <p><em>Concerning the state-change due to the measurement process</em>, Ann. Phys. <strong>15</strong> 9 (2006) 663-670 [<a href="http://myweb.rz.uni-augsburg.de/~eckern/adp/history/historic-papers/2006_518_663-670.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Lueders-StateChange.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>Early review:</p> <ul> <li id="Scheibe73"><a class="existingWikiWord" href="/nlab/show/Erhard+Scheibe">Erhard Scheibe</a>, pp 137 in: <em><em>The logical analysis of quantum mechanics&, Pergamon Press Oxford (1973)</em></em></li> </ul> <p>Textbook accounts in <a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+A.+Nielsen">Michael A. Nielsen</a>, <a class="existingWikiWord" href="/nlab/show/Isaac+L.+Chuang">Isaac L. Chuang</a>, §2.2.3 of: <em>Quantum computation and quantum information</em>, Cambridge University Press (2000) [<a href="https://doi.org/10.1017/CBO9780511976667">doi:10.1017/CBO9780511976667</a>, <a href="http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/NielsenChuangQuantumComputation.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Joseph+M.+Renes">Joseph M. Renes</a>, eq. (3.2) in: <em>Quantum Information Theory</em> (2015) [<a href="https://edu.itp.phys.ethz.ch/hs15/QIT/renes_lecture_notes14.pdf">pdf</a>], (A.2) in: De Gruyter (2022) [<a href="https://doi.org/10.1515/9783110570250">doi:10.1515/9783110570250</a>]</p> </li> </ul> <p>Discussion from the point of view of <a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a>:</p> <ul> <li id="Kuperberg05"> <p><a class="existingWikiWord" href="/nlab/show/Greg+Kuperberg">Greg Kuperberg</a>, <em>A concise introduction to quantum probability, quantum mechanics, and quantum computation</em> (2005) [<a href="http://www.math.ucdavis.edu/~greg/intro-2005.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Kuperberg-ConciseQuantum.pdf" title="pdf">pdf</a>]</p> </li> <li id="RedeiSummers06"> <p><a class="existingWikiWord" href="/nlab/show/Miklos+Redei">Miklos Redei</a>, <a class="existingWikiWord" href="/nlab/show/Stephen+Summers">Stephen Summers</a>, <em>Quantum Probability Theory</em> (<a href="https://arxiv.org/abs/quant-ph/0601158">arXiv:quant-ph/0601158</a>)</p> </li> <li id="Yuan12"> <p><a class="existingWikiWord" href="/nlab/show/Qiaochu+Yuan">Qiaochu Yuan</a>, <em><a href="https://qchu.wordpress.com/2012/09/09/finite-noncommutative-probability-the-born-rule-and-wave-function-collapse/">Finite noncommutative probability, the Born rule, and wave function collapse</a></em>, 2012</p> </li> <li id="FroehlichSchubnel15"> <p><a class="existingWikiWord" href="/nlab/show/J%C3%BCrg+Fr%C3%B6hlich">Jürg Fröhlich</a>, B. Schubnel, <em>Quantum Probability Theory and the Foundations of Quantum Mechanics</em>. In: Blanchard P., Fröhlich J. (eds.) <em>The Message of Quantum Science</em>. Lecture Notes in Physics, vol 899. Springer 2015 (<a href="https://arxiv.org/abs/1310.1484">arXiv:1310.1484</a>, <a href="https://doi.org/10.1007/978-3-662-46422-9_7">doi:10.1007/978-3-662-46422-9_7</a>)</p> </li> <li id="Froehlich16"> <p><a class="existingWikiWord" href="/nlab/show/J%C3%BCrg+Fr%C3%B6hlich">Jürg Fröhlich</a>, <em>The structure of quantum theory</em>, Chapter 6 in <em>The quest for laws and structure</em>, EMS 2016 (<a href="https://www.researchgate.net/publication/308595814_The_Quest_for_Laws_and_Structure">doi</a>, <a href="https://www.ems-ph.org/books/show_abstract.php?proj_nr=207&vol=1&rank=8">doi:10.4171/164-1/8</a>)</p> </li> </ul> <p>See also</p> <ul> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Wave_function_collapse">Wave function collapse</a></em></p> </li> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Born_rule">Born rule</a></em></p> </li> </ul> <p>More speculative discussion of state collapse as caused by stochastic terms added to the <a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+equation">Schrödinger equation</a>:</p> <p>from the point of view of the <a class="existingWikiWord" href="/nlab/show/symplectic+formulation+of+quantum+mechanics">symplectic formulation of quantum mechanics</a>:</p> <ul> <li id="Hughston96"><a class="existingWikiWord" href="/nlab/show/Lane+P.+Hughston">Lane P. Hughston</a>, <em>Geometry of stochastic state vector reduction</em>, Proceedings of the Royal Society A <strong>452</strong> 1947 (1996) [<a href="https://doi.org/10.1098/rspa.1996.0048">doi:10.1098/rspa.1996.0048</a>, <a href="https://www.jstor.org/stable/52944">jstor:52944</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 3, 2023 at 09:35:51. See the <a href="/nlab/history/wave+function+collapse" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/wave+function+collapse" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/8275/#Item_6">Discuss</a><span class="backintime"><a href="/nlab/revision/wave+function+collapse/20" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/wave+function+collapse" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/wave+function+collapse" accesskey="S" class="navlink" id="history" rel="nofollow">History (20 revisions)</a> <a href="/nlab/show/wave+function+collapse/cite" style="color: black">Cite</a> <a href="/nlab/print/wave+function+collapse" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/wave+function+collapse" id="view_source" rel="nofollow">Source</a> </div> </div> <!-- Content --> </div> <!-- Container --> </body> </html>