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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> nLab quantum probability theory </h1> <div class="navigation"> <a href='#navEnd'>Skip the Navigation Links</a> | <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/8192/#Item_33" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="quantum_systems">Quantum systems</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>, <a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">dependent</a> <a class="existingWikiWord" href="/nlab/show/linear+type+theory">linear type theory</a> <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a> in <a class="existingWikiWord" href="/nlab/show/quantum+information+theory+via+dagger-compact+categories">†-compact categories</a> <a class="existingWikiWord" href="/nlab/show/tensor+networks">tensor networks</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+systems">quantum systems</a> (<a class="existingWikiWord" href="/nlab/show/parameterized+quantum+systems">parameterized</a>, <a class="existingWikiWord" href="/nlab/show/open+quantum+system">open</a>) <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> <a class="existingWikiWord" href="/nlab/show/quantum+state+collapse">quantum state collapse</a> <a class="existingWikiWord" href="/nlab/show/quantum+decoherence">quantum decoherence</a> <a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">quantum adiabatic theorem</a> <a class="existingWikiWord" href="/nlab/show/Berry+phases">Berry phases</a> <a class="existingWikiWord" href="/nlab/show/Dyson+formula">Dyson formula</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+many-body+physics">quantum many-body physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> <a class="existingWikiWord" href="/nlab/show/functorial+quantum+field+theory">functorial quantum field theory</a> <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> (<a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-</a>)<a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a> <a class="existingWikiWord" href="/nlab/show/quantum+material">quantum material</a> (<a class="existingWikiWord" href="/nlab/show/topological+phases+of+matter">topological</a>) <a class="existingWikiWord" href="/nlab/show/phases+of+matter">phases of matter</a> </li> </ul> <div> <a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a> – <a class="existingWikiWord" href="/nlab/show/observables">observables</a> and <a class="existingWikiWord" href="/nlab/show/states">states</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/states">states</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/quasi-state">quasi-state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> <a class="existingWikiWord" href="/nlab/show/dimer">dimer</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">quantum state preparation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a>, <a class="existingWikiWord" href="/nlab/show/quantum+fluctuation">quantum fluctuation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+superposition">quantum superposition</a>, <a class="existingWikiWord" href="/nlab/show/quantum+interference">quantum interference</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> <a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a> <a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a> <a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a> <a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a> <a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/superselection+sector">superselection sector</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/coherent+quantum+state">coherent quantum state</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>, <a class="existingWikiWord" href="/nlab/show/excited+state">excited state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>, <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+amplitude">vacuum amplitude</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+fluctuation">vacuum fluctuation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+polarization">vacuum polarization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/thermal+vacuum">thermal vacuum</a>, <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/false+vacuum">false vacuum</a>, <a class="existingWikiWord" href="/nlab/show/tachyon">tachyon</a>, <a class="existingWikiWord" href="/nlab/show/Coleman-De+Luccia+instanton">Coleman-De Luccia instanton</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/theta+vacuum">theta vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/non-geometric+string+theory+vacuum">non-geometric string theory vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/entangled+state">entangled state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/matrix+product+state">matrix product state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tree+tensor+network+state">tree tensor network state</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/observables">observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>, <a class="existingWikiWord" href="/nlab/show/beable">beable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator (in geometric quantization)</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+effect">quantum effect</a>, <a class="existingWikiWord" href="/nlab/show/effect+algebra">effect algebra</a> </li> <li> in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/linear+observable">linear observable</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/regular+observable">regular observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>, <a class="existingWikiWord" href="/nlab/show/retarded+product">retarded product</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+field+algebra+of+observables">interacting field algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/theorems">theorems</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Nuiten%27s+lemma">Nuiten's lemma</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wigner%27s+theorem">Wigner's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> </ul> </li> </ul> </div> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information+via+dagger-compact+categories">quantum information via dagger-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+channel">quantum channel</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+teleportation">quantum teleportation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/topological+entanglement+entropy">topological entanglement entropy</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+technology">quantum technology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+sensing">quantum sensing</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+communication">quantum communication</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+cryptography">quantum cryptography</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+machine+learning">quantum machine learning</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+computing">quantum computing</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/qdit">qdit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+gate">quantum gate</a>, <a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adiabatic+quantum+computation">adiabatic quantum computation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/measurement-based+quantum+computation">measurement-based quantum computation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+programming+language">quantum programming language</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+error+correction">quantum error correction</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/HaPPY+code">HaPPY code</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Majorana+dimer+code">Majorana dimer code</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin+resonance+qbit">spin resonance qbit</a> </li> <li> quantum algorithms: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Grover%27s+algorithm">Grover's algorithm</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Shor%27s+algorithm">Shor's algorithm</a> </li> </ul> </li> </ul> </div></div> <h4 id="measure_and_probability_theory">Measure and probability theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/measure+theory">measure theory</a> <a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a> (<a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a>) <h2 id="measure_theory">Measure theory</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/measurable+space">measurable space</a>, <a class="existingWikiWord" href="/nlab/show/measurable+locale">measurable locale</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/measure">measure</a>, <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+measure+theory">geometric measure theory</a> </li> </ul> <h2 id="probability_theory">Probability theory</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/probability+distribution">probability distribution</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/state">state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/states+in+AQFT+and+operator+algebra">in AQFT and operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/entropy">entropy</a>, <a class="existingWikiWord" href="/nlab/show/relative+entropy">relative entropy</a> </li> </ul> <h2 id="information_geometry">Information geometry</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/information+geometry">information geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/information+metric">information metric</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wasserstein+metric">Wasserstein metric</a> </li> </ul> <h2 id="thermodynamics">Thermodynamics</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/second+law+of+thermodynamics">second law of thermodynamics</a>, <a class="existingWikiWord" href="/nlab/show/generalized+second+law+of+theormodynamics">generalized second law</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/ergodic+theory">ergodic theory</a> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/de+Finetti%27s+theorem">de Finetti's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/law+of+large+numbers">law of large numbers</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kolmogorov+extension+theorem">Kolmogorov extension theorem</a> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/machine+learning">machine learning</a>, <a class="existingWikiWord" href="/nlab/show/neural+networks">neural networks</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#AsClassicalProbabilityInBohrTopos'>As classical probability theory internal to a Bohr topos</a></li> <li><a href='#ConditionalExpectationAndWaveFunctionCollapse'>Conditional expectation and Wave function collapse</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> In <a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a>, the concept of noncommutative probability space or quantum probability space is the generalization of that of <a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a> as the concept of “space” is generalized to <a class="existingWikiWord" href="/nlab/show/non-commutative+geometry">non-commutative geometry</a>. The basic idea is to encode a would-be <a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a> <a class="existingWikiWord" href="/nlab/show/Isbell+duality">dually</a> in its <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of functions</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>, typically regarded as a <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>, and encode the <a class="existingWikiWord" href="/nlab/show/probability+measure">probability measure</a> as a <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on this star algebra</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" 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><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \langle - \rangle \;\colon\; \mathcal{A} \longrightarrow \mathbb{C} \,. </annotation></semantics></math></div> Hence this primarily <a class="existingWikiWord" href="/nlab/show/axiom">axiomatizes</a> the concept of <a class="existingWikiWord" href="/nlab/show/expectation+values">expectation values</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>A</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle A\rangle</annotation></semantics></math> (<a href="#Segal65">Segal 65</a>, <a href="#Whittle92">Whittle 92</a>) while leaving the nature of the underlying <a class="existingWikiWord" href="/nlab/show/probability+measure">probability measure</a> implicit (in contrast to the classical formalization of <a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a> by <a class="existingWikiWord" href="/nlab/show/Andrey+Kolmogorov">Andrey Kolmogorov</a>). Often <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> is assumed/required to be a <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a> (e.g. <a href="#Kuperberg05">Kuperberg 05, section 1.8</a>). Often <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> is taken to be the full algebra of <a class="existingWikiWord" href="/nlab/show/bounded+operators">bounded operators</a> on some <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> (e.g. <a href="#Attal">Attal, def. 7.1</a>). In <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</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> is an <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a> (or a <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net</a> thereof) and <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 stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle (-)\rangle</annotation></semantics></math> is a particular <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a>, for instance a <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a>. The formulation of <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative quantum field theory</a> from the algebraic perspective of quantum probability is known as <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> (<a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a>). The formulation of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> from the algebraic perspective of quantum probability is known as <a class="existingWikiWord" href="/nlab/show/perturbative+algebraic+quantum+field+theory">perturbative algebraic quantum field theory</a> (<a class="existingWikiWord" href="/nlab/show/pAQFT">pAQFT</a>). The sentiment that <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> is quantum probability theory is also referred to as the <a class="existingWikiWord" href="/nlab/show/Bayesian+interpretation+of+quantum+mechanics">Bayesian interpretation of quantum mechanics</a> (“QBism”). <h2 id="properties">Properties</h2> <h3 id="AsClassicalProbabilityInBohrTopos">As classical probability theory internal to a Bohr topos</h3> The idea that <blockquote> <a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a> is “just as” classical <a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a> but generalized to <a class="existingWikiWord" href="/nlab/show/non-commutative+geometry">non-commutative</a> <a class="existingWikiWord" href="/nlab/show/probability+spaces">probability spaces</a>, hence, for <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a>, to quantized <a class="existingWikiWord" href="/nlab/show/phase+spaces">phase spaces</a> </blockquote> may be made precise and fully manifest by understanding quantum probability theory as being classical <a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to the <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> of the given <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a>. For details see at <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> the section <a href="Bohr+topos#KinematicsOnBohrTopos">Kinematics in a Bohr topos</a>. For going deeper, see at <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a>. <h3 id="ConditionalExpectationAndWaveFunctionCollapse">Conditional expectation and Wave function collapse</h3> Quantum probability theory shows that “<a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a>” is just part 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> (e.g. <a href="#Kuperberg05">Kuperberg 05, section 1.2</a>, <a href="#Yuan12">Yuan 12</a>): 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>. 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 <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a> in the given <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state</a> is <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> 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> <div class="maruku-equation" id="eq:RealIdempotent">(1)<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> 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>) <div class="maruku-equation" id="eq:ConditionalExpectation">(2)<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> 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 an <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 <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> 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>) <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> where in the last step we used <a class="maruku-eqref" href="#eq:RealIdempotent">(1)</a>. This says that assuming 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>. This is the statement of “<a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a>”: <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> 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). <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/noncommutative+measure+theory">noncommutative measure theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+statistical+mechanics">quantum statistical mechanics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a>, <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+inequalities">Bell's inequalities</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bayesian+interpretation+of+quantum+mechanics">Bayesian interpretation of quantum mechanics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> </li> </ul> <div> <a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a> – <a class="existingWikiWord" href="/nlab/show/observables">observables</a> and <a class="existingWikiWord" href="/nlab/show/states">states</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/states">states</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/quasi-state">quasi-state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> <a class="existingWikiWord" href="/nlab/show/dimer">dimer</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">quantum state preparation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a>, <a class="existingWikiWord" href="/nlab/show/quantum+fluctuation">quantum fluctuation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+superposition">quantum superposition</a>, <a class="existingWikiWord" href="/nlab/show/quantum+interference">quantum interference</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> <a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a> <a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a> <a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a> <a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a> <a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/superselection+sector">superselection sector</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/coherent+quantum+state">coherent quantum state</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>, <a class="existingWikiWord" href="/nlab/show/excited+state">excited state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>, <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+amplitude">vacuum amplitude</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+fluctuation">vacuum fluctuation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+polarization">vacuum polarization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/thermal+vacuum">thermal vacuum</a>, <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/false+vacuum">false vacuum</a>, <a class="existingWikiWord" href="/nlab/show/tachyon">tachyon</a>, <a class="existingWikiWord" href="/nlab/show/Coleman-De+Luccia+instanton">Coleman-De Luccia instanton</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/theta+vacuum">theta vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/non-geometric+string+theory+vacuum">non-geometric string theory vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/entangled+state">entangled state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/matrix+product+state">matrix product state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tree+tensor+network+state">tree tensor network state</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/observables">observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>, <a class="existingWikiWord" href="/nlab/show/beable">beable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator (in geometric quantization)</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+effect">quantum effect</a>, <a class="existingWikiWord" href="/nlab/show/effect+algebra">effect algebra</a> </li> <li> in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/linear+observable">linear observable</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/regular+observable">regular observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>, <a class="existingWikiWord" href="/nlab/show/retarded+product">retarded product</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+field+algebra+of+observables">interacting field algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/theorems">theorems</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Nuiten%27s+lemma">Nuiten's lemma</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wigner%27s+theorem">Wigner's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> </ul> </li> </ul> </div> <h2 id="references">References</h2> The understanding of <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> as a <a class="existingWikiWord" href="/nlab/show/probability+theory">probabilistic</a> <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a> originates with the formulation of the <a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a> and was made fully explicit in: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Pascual+Jordan">Pascual Jordan</a>, Über eine neue Begründung der Quantenmechanik, Zeitschrift für Physik 40 (1927) 809–838 [<a href="https://doi.org/10.1007/BF01390903">doi:10.1007/BF01390903</a>] </li> <li id="vonNeumann32"> <a class="existingWikiWord" href="/nlab/show/John+von+Neumann">John von Neumann</a>, Die quantenmechanische Statistik, Part III in: Mathematische Grundlagen der Quantenmechanik, 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>] Mathematical Foundations of Quantum Mechanics 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>] </li> </ul> see also: <ul> <li><a class="existingWikiWord" href="/nlab/show/Jagdish+Mehra">Jagdish Mehra</a>, Helmut Rechenberg, The Probability Interpretation and the Statistical Transformation Theory, the Physical Interpretation, and the Empirical and Mathematical Foundations of Quantum Mechanics 1926-1932, Part 1 in: The Historical Development of Quantum Theory. Volume 6: The Completion of Quantum Mechanics, 1926-1941, Springer (2001) [<a href="https://link.springer.com/book/9780387989716">ISBN:978-0-387-98971-6</a>]</li> </ul> with emphasis on <a class="existingWikiWord" href="/nlab/show/POVMs">POVMs</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Paul+Busch">Paul Busch</a>, Marian Grabowski, <a class="existingWikiWord" href="/nlab/show/Pekka+J.+Lahti">Pekka J. Lahti</a>, Operational Quantum Physics, Lecture Notes in Physics Monographs 31, Springer (1995) [<a href="https://doi.org/10.1007/978-3-540-49239-9">doi:10.1007/978-3-540-49239-9</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Paul+Busch">Paul Busch</a>, <a class="existingWikiWord" href="/nlab/show/Pekka+J.+Lahti">Pekka J. Lahti</a>, Juha-Pekka Pellonpää, Kari Ylinen, Quantum Measurement, Springer (2016) [<a href="https://doi.org/10.1007/978-3-319-43389-9">doi:10.1007/978-3-319-43389-9</a>] </li> </ul> The axiomatization of <a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a> in terms of the concept of <a class="existingWikiWord" href="/nlab/show/expectation+values">expectation values</a> (instead of <a class="existingWikiWord" href="/nlab/show/probability+measures">probability measures</a>) is amplified in: <ul> <li> <a href="#vonNeumann32">vonNeumann 1932, §IV.1</a> </li> <li id="Segal65"> <a class="existingWikiWord" href="/nlab/show/Irving+Segal">Irving Segal</a>, Algebraic integration theory, Bull. Amer. Math. Soc. Volume 71, Number 3, Part 1 (1965), 419-489 (<a href="https://projecteuclid.org/euclid.bams/1183526903">eculid:1183526903</a>) </li> <li id="Whittle92"> <a class="existingWikiWord" href="/nlab/show/Peter+Whittle">Peter Whittle</a>, Probability via expectation, Springer 1992 (<a href="https://link.springer.com/book/10.1007/978-1-4612-0509-8">doi:10.1007/978-1-4612-0509-8</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Rolando+Rebolledo">Rolando Rebolledo</a>, An algebraic view on Probability, Section 1.2 in: Complete Positivity and the Markov structure of Open Quantum Systems, chapter in: <a class="existingWikiWord" href="/nlab/show/St%C3%A9phane+Attal">Stéphane Attal</a>, <a class="existingWikiWord" href="/nlab/show/Alain+Joye">Alain Joye</a>, <a class="existingWikiWord" href="/nlab/show/Claude-Alain+Pillet">Claude-Alain Pillet</a> (eds.), Open Quantum Systems II – The Markovian approach, Lecture Notes in Mathematics 1881, Springer (2006) 149-182 [<a href="https://doi.org/10.1007/b128451">doi:10.1007/b128451</a>] </li> </ul> Dedicated discussion of quantum probability theory: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Itamar+Pitowsky">Itamar Pitowsky</a>, Quantum Probability – Quantum Logic, Lecture Notes in Physics 321, Springer (1989) [<a href="https://doi.org/10.1007/BFb0021186">doi:10.1007/BFb0021186</a>] <blockquote> (emphasis on the <a class="existingWikiWord" href="/nlab/show/Bell%27s+inequalities">Bell's inequalities</a> and <a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a>) </blockquote> </li> <li id="Meyer95"> <a class="existingWikiWord" href="/nlab/show/Paul-Andr%C3%A9+Meyer">Paul-André Meyer</a>, Quantum Probability for Probabilists, Lecture Notes in Mathematics 1538, Springer 1995 (<a href="https://link.springer.com/book/10.1007/BFb0084701">doi:10.1007/BFb0084701</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Hans+Maassen">Hans Maassen</a>, Quantum Probability Theory, Lecture notes 1998 (<a href="https://www.math.ru.nl/~maassen/lectures/qp.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/MaassenQuantumProbability.pdf" title="pdf">pdf</a>) </li> <li id="Kuperberg05"> <a class="existingWikiWord" href="/nlab/show/Greg+Kuperberg">Greg Kuperberg</a>, A concise introduction to quantum probability, quantum mechanics, and quantum computation (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>] </li> <li id="RedeiSummers06"> <a class="existingWikiWord" href="/nlab/show/Miklos+Redei">Miklos Redei</a>, <a class="existingWikiWord" href="/nlab/show/Stephen+Summers">Stephen Summers</a>, Quantum Probability Theory, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics Volume 38, Issue 2, June 2007, Pages 390-417 (<a href="https://arxiv.org/abs/quant-ph/0601158">arXiv:quant-ph/0601158</a>, <a href="https://doi.org/10.1016/j.shpsb.2006.05.006">doi:10.1016/j.shpsb.2006.05.006</a>) </li> <li id="Attal"> <a class="existingWikiWord" href="/nlab/show/St%C3%A9phane+Attal">Stéphane Attal</a>, Quantum Probability, Lecture 7 in: Lectures on Quantum Noises [<a href="http://math.univ-lyon1.fr/~attal/Quantum_Probability.pdf">pdf</a>, <a href="http://math.univ-lyon1.fr/~attal/chapters.html">webpage</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Franco+Strocchi">Franco Strocchi</a>, Section 2.4 in: An introduction to the mathematical structure of quantum mechanics, Advanced Series in Mathematical Physics 28, World Scientific (2008) [<a href="https://doi.org/10.1142/7038">doi:10.1142/7038</a>] </li> <li id="Gleason09"> <a class="existingWikiWord" href="/nlab/show/Jonathan+Gleason">Jonathan Gleason</a>, The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebraic formalism of quantum mechanics (2009) [<a class="existingWikiWord" href="/nlab/files/Gleason09.pdf" title="pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/GleasonAlgebraic.pdf" title="pdf">pdf</a>] </li> <li id="Gleason11"> <a class="existingWikiWord" href="/nlab/show/Jonathan+Gleason">Jonathan Gleason</a>, From Classical to Quantum: The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">F^\ast</annotation></semantics></math>-Algebraic Approach, contribution to <a href="http://www.math.uchicago.edu/~may/VIGRE/VIGREREU2011.html">VIGRE REU 2011</a>, Chicago (2011) [<a href="https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Gleason.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/GleasonFAlgebraic.pdf" title="pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Qiaochu+Yuan">Qiaochu Yuan</a>, <a href="https://qchu.wordpress.com/2012/08/18/noncommutative-probability/">Noncommutative probability</a>, 2012 </li> <li id="Yuan12"> <a class="existingWikiWord" href="/nlab/show/Qiaochu+Yuan">Qiaochu Yuan</a>, <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>, 2012 </li> <li> A. Ibort, V.I. Manko, G. Marmo, A. Simoni, F. Ventriglia, A pedagogical presentation of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebraic approach to quantum tomography, Phys. Scr., 84 (2011) 065006 (<a href="https://arxiv.org/abs/1204.5231">arXiv:1204.5231</a>) </li> <li id="FroehlichSchubnel15"> <a class="existingWikiWord" href="/nlab/show/J%C3%BCrg+Fr%C3%B6hlich">Jürg Fröhlich</a>, B. Schubnel, Quantum Probability Theory and the Foundations of Quantum Mechanics. In: Blanchard P., Fröhlich J. (eds.) The Message of Quantum Science. 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>) </li> <li id="Froehlich16"> <a class="existingWikiWord" href="/nlab/show/J%C3%BCrg+Fr%C3%B6hlich">Jürg Fröhlich</a>, The structure of quantum theory, Chapter 6 in The quest for laws and structure, 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>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Jan+Swart">Jan Swart</a>, Introduction to Quantum Probability, 2017 (<a href="http://staff.utia.cas.cz/swart/dict.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/SwartQuantumProbability.pdf" title="pdf">pdf</a>) </li> <li id="Landsman17"> <a class="existingWikiWord" href="/nlab/show/Klaas+Landsman">Klaas Landsman</a>, Foundations of quantum theory – From classical concepts to Operator algebras, Springer Open 2017 (<a href="https://link.springer.com/book/10.1007/978-3-319-51777-3">doi:10.1007/978-3-319-51777-3</a>, <a href="https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf">pdf</a>) </li> </ul> Further discussion of quantum probability: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Dror+Bar-Natan">Dror Bar-Natan</a>, Two examples in noncommutative probability, Foundations of Physics, 19 (1989) 97–104 (<a href="https://doi.org/10.1007/BF00737769">doi:10.1007/BF00737769</a>) </li> <li> Nicolò Drago, <a class="existingWikiWord" href="/nlab/show/Valter+Moretti">Valter Moretti</a>, The notion of observable and the moment problem for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math>-algebras and their GNS representations (<a href="https://arxiv.org/abs/1903.07496">doi:1903.07496</a>, <a href="http://inspirehep.net/record/1725528">spire:1725528</a>) </li> </ul> Discussion of <a class="existingWikiWord" href="/nlab/show/density+matrices">density matrices</a>, <a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a> and <a class="existingWikiWord" href="/nlab/show/entropy">entropy</a> in quantum probability/<a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum theory</a>, via the <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/A.+P.+Balachandran">A. P. Balachandran</a>, T. R. Govindarajan, Amilcar R. de Queiroz, A. F. Reyes-Lega, Section II of: Algebraic approach to entanglement and entropy, Phys. Rev. A 88 (2013) 022301 [<a href="http://arxiv.org/abs/1301.1300">arXiv:1301.1300</a>, <a href="https://doi.org/10.1103/PhysRevA.88.022301">doi:10.1103/PhysRevA.88.022301</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/A.+P.+Balachandran">A. P. Balachandran</a>, A. R. de Queiroz, S. Vaidya, Entropy of Quantum States: Ambiguities, Eur. Phys. J. Plus 128 (2013) 112 [<a href="https://arxiv.org/abs/1212.1239">arXiv:1212.1239</a>, <a href="https://doi.org/10.1140/epjp/i2013-13112-3">doi:10.1140/epjp/i2013-13112-3</a>] </li> <li> Paolo Facchi, Giovanni Gramegna, Arturo Konderak, Entropy of quantum states (<a href="https://arxiv.org/abs/2104.12611">arXiv:2104.12611</a>) </li> </ul> </body></html> </div> <div class="revisedby"> Last revised on December 29, 2024 at 22:21:13. 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