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xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="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> <h4 id="physics">Physics</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a> </li> </ul> <hr /> <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a> <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a> <ul> <li> Axiomatizations <ul> <li> <a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a> </li> </ul> </li> </ul> </li> <li> Tools <ul> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> Structural phenomena <ul> <li> <a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic 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</ul> <div> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#notation'>Notation</a></li> <li><a href='#the_density_operator_of_an_ensemble'>The density operator of an ensemble</a></li> <li><a href='#characterisation'>Characterisation</a></li> <li><a href='#coherence'>Coherence</a></li> <li><a href='#quantum_doubles'>Quantum doubles</a></li> <li><a href='#limitations'>Limitations</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> In <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> a density matrix is a linear endomorphism of a <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> of pure <a class="existingWikiWord" href="/nlab/show/quantum+states">quantum states</a> that represents a <a class="existingWikiWord" href="/nlab/show/statistical+ensemble">statistical ensemble</a> of <a class="existingWikiWord" href="/nlab/show/quantum+states">quantum states</a>, hence <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">states</a> which are not necessarily <a class="existingWikiWord" href="/nlab/show/pure+states">pure states</a> but <a class="existingWikiWord" href="/nlab/show/mixed+states">mixed states</a>. The space of density matrices inside all suitable endomorphisms is called the <a class="existingWikiWord" href="/nlab/show/Bloch+region">Bloch region</a>. <h2 id="notation">Notation</h2> In the following definition, we use Dirac’s “bra-ket” notation for vectors where a state vector, describing a <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> of a quantum system, is represented by a “ket” which is a column vector, <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><msub><mi>x</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd></mtr> <mtr><mtd><msub><mi>x</mi> <mi>n</mi></msub></mtd></mtr> <mtr><mtd><msub><mi>x</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex"> {|\psi_{\alpha}\rangle} = \left( \begin{aligned} \vdots \\ x_{n-1} \\ x_{n} \\ x_{n+1} \\ \vdots \end{aligned} \right) . </annotation></semantics></math></div> The <a class="existingWikiWord" href="/nlab/show/Hermitian+adjoint">Hermitian adjoint</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">⟨</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">|</mo></mrow><mo>=</mo><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo></mrow><msup><mo stretchy="false">)</mo> <mo>†</mo></msup></mrow><annotation encoding="application/x-tex">{\langle\psi_{\alpha}|} = ({|\psi_{\alpha}\rangle})^{\dagger}</annotation></semantics></math>, is called a “bra” (hence “bra(c)ket”) and is a row vector. <h2 id="the_density_operator_of_an_ensemble">The density operator of an ensemble</h2> Suppose we have a <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> that arises from some random process such that the state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo></mrow></mrow><annotation encoding="application/x-tex">{|\psi_{\alpha}\rangle}</annotation></semantics></math> has a probability <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>α</mi></msub><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">p_{\alpha} \in [0,1]</annotation></semantics></math> (we often speak of having ‘prepared’ the state with the associated probability). The possible states <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo></mrow></mrow><annotation encoding="application/x-tex">{|\psi_{\alpha}\rangle}</annotation></semantics></math> need not be orthogonal and we thus call such a collection of states a <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>. More specifically, a mixed state is often described as an <a class="existingWikiWord" href="/nlab/show/ensemble">ensemble</a> of quantum systems. Suppose we now measure some observable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>A</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{A}</annotation></semantics></math> on the system as a whole, i.e. on the ensemble. The expectation value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>A</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{A}</annotation></semantics></math> over the state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo></mrow></mrow><annotation encoding="application/x-tex">{|\psi_{\alpha}\rangle}</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>A</mi><msub><mo stretchy="false">⟩</mo> <mi>α</mi></msub><mo>=</mo><mo stretchy="false">⟨</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>A</mi></mstyle><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle{A}\rangle_{\alpha} = \langle\psi_{\alpha} | \mathbf{A} |\psi_{\alpha}\rangle</annotation></semantics></math>. Over the entire ensemble, this becomes <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>A</mi><mo stretchy="false">⟩</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>α</mi></munder><msub><mi>p</mi> <mi>α</mi></msub><mo stretchy="false">⟨</mo><mi>A</mi><msub><mo stretchy="false">⟩</mo> <mi>α</mi></msub><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>α</mi></munder><msub><mi>p</mi> <mi>α</mi></msub><mi>tr</mi><mrow><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo></mrow><mrow><mo stretchy="false">⟨</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">|</mo></mrow><mstyle mathvariant="bold"><mi>A</mi></mstyle><mo>.</mo></mrow><annotation encoding="application/x-tex"> \langle{A}\rangle = \sum_{\alpha} p_{\alpha} \langle{A}\rangle_{\alpha} = \sum_{\alpha} p_{\alpha} tr {|\psi_{\alpha}\rangle} {\langle\psi_{\alpha}|} \mathbf{A} .</annotation></semantics></math></div> Given the above, we define the density operator to be <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>ρ</mi></mstyle><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>α</mi></munder><msub><mi>ρ</mi> <mi>α</mi></msub><mrow><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo></mrow><mrow><mo stretchy="false">⟨</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">|</mo></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\rho} = \sum_{\alpha} \rho_\alpha {|\psi_{\alpha}\rangle}{\langle\psi_{\alpha}|} .</annotation></semantics></math></div> We call the <a class="existingWikiWord" href="/nlab/show/matrix">matrix</a> representation of the density operator, relative to a given <a class="existingWikiWord" href="/nlab/show/basis">basis</a>, the density matrix. <h2 id="characterisation">Characterisation</h2> An operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> is the density operator associated to some ensemble if and only if it is a positive operator with trace 1. (Nielsen and Chuang Theorem 2.5, p. 101) <h2 id="coherence">Coherence</h2> Diagonal density matrices with at least two non-zero terms on the diagonal represent <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>s. Density matrices that possess non-zero off-diagonal terms represent <a class="existingWikiWord" href="/nlab/show/superposition+state">superposition state</a>s. Such states are referred to as coherent and the off-diagonal entries are called the coherences. Any physical process that has the effect of suppressing the coherences is known as <a class="existingWikiWord" href="/nlab/show/decoherence">decoherence</a>. <h2 id="quantum_doubles">Quantum doubles</h2> Given a pure state on the tensor product of Hilbert spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>⊗</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">V\otimes W</annotation></semantics></math>, the partial trace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">tr</mo> <mi>W</mi></msub></mrow><annotation encoding="application/x-tex">\tr_W</annotation></semantics></math> specifies a mixed state on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>. Moreover, every mixed state comes from the partial trace of a pure state. Hence, this gives an alternate characterization of what a density matrix is: The image of the “pure state” outer products <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>ψ</mi><mo stretchy="false">⟩</mo><mo stretchy="false">⟨</mo><mi>ψ</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|\psi\rangle\langle\psi|</annotation></semantics></math> under partial trace. Even better, given a density matrix <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">\rho\in V</annotation></semantics></math> one can explicitly construct a pure state which traces to it. Expand <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> as <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>α</mi></munder><msub><mi>ρ</mi> <mi>α</mi></msub><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo><mo stretchy="false">⟨</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">|</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">\rho=\sum_{\alpha}\rho_\alpha |\psi_\alpha \rangle \langle \psi_\alpha |.</annotation></semantics></math></div> The states <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">| \psi_{\alpha}\rangle</annotation></semantics></math> serve as a basis for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, and the states <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">\langle\psi_\alpha |</annotation></semantics></math> serve as a basis for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">V^*</annotation></semantics></math>. Hence, we can consider the state <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>ρ</mi><mo stretchy="false">˜</mo></mover><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>α</mi></munder><msqrt><mrow><msub><mi>ρ</mi> <mi>α</mi></msub></mrow></msqrt><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo><mo>⊗</mo><mo stretchy="false">⟨</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">|</mo><mo>∈</mo><mi>V</mi><mo>⊗</mo><msup><mi>V</mi> <mo>*</mo></msup><mo>.</mo></mrow><annotation encoding="application/x-tex">\tilde{\rho}=\sum_{\alpha}\sqrt{\rho_\alpha}|\psi_\alpha \rangle\otimes \langle \psi_\alpha |\in V\otimes V^*.</annotation></semantics></math></div> Tracing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">V^*</annotation></semantics></math> out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ρ</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde{\rho}</annotation></semantics></math> allows us to recover <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>. This can be seen in the following computation: <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><msub><mo lspace="0em" rspace="thinmathspace">tr</mo> <mrow><msup><mi>V</mi> <mo>*</mo></msup></mrow></msub><mo stretchy="false">(</mo><mover><mi>ρ</mi><mo stretchy="false">˜</mo></mover><msup><mover><mi>ρ</mi><mo stretchy="false">˜</mo></mover> <mo>†</mo></msup><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>α</mi></munder><msqrt><mrow><msub><mi>ρ</mi> <mi>α</mi></msub></mrow></msqrt><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo><mo>⊗</mo><mo stretchy="false">⟨</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">|</mo><mo>)</mo></mrow><mo>⊗</mo><msup><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>α</mi></munder><msqrt><mrow><msub><mi>ρ</mi> <mi>α</mi></msub></mrow></msqrt><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo><mo>⊗</mo><mo stretchy="false">⟨</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">|</mo><mo>)</mo></mrow> <mo>†</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>α</mi></munder><msqrt><mrow><msub><mi>ρ</mi> <mi>α</mi></msub></mrow></msqrt><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo><mo>⊗</mo><mo stretchy="false">⟨</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">|</mo><mo>)</mo></mrow><mo>⊗</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>α</mi></munder><msqrt><mrow><msub><mi>ρ</mi> <mi>α</mi></msub></mrow></msqrt><mo stretchy="false">⟨</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">|</mo><mo>⊗</mo><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow></munder><msqrt><mrow><msub><mi>ρ</mi> <mi>α</mi></msub></mrow></msqrt><msqrt><mrow><msub><mi>ρ</mi> <mi>β</mi></msub></mrow></msqrt><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo><mo stretchy="false">⟨</mo><msub><mi>ψ</mi> <mi>β</mi></msub><mo stretchy="false">|</mo><mo>⊗</mo><mo stretchy="false">⟨</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>β</mi></msub><mo stretchy="false">⟩</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>α</mi></munder><msub><mi>ρ</mi> <mi>α</mi></msub><mo stretchy="false">|</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">⟩</mo><mo stretchy="false">⟨</mo><msub><mi>ψ</mi> <mi>α</mi></msub><mo stretchy="false">|</mo><mo>=</mo><mi>ρ</mi><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \tr_{V^*}(\tilde{\rho} \tilde{\rho}^{\dagger})&=\left(\sum_{\alpha}\sqrt{\rho_\alpha}|\psi_\alpha \rangle\otimes \langle \psi_\alpha |\right) \otimes \left(\sum_{\alpha}\sqrt{\rho_\alpha}|\psi_\alpha \rangle\otimes \langle \psi_\alpha |\right)^{\dagger}\\ &=\left(\sum_{\alpha}\sqrt{\rho_\alpha}|\psi_\alpha \rangle\otimes \langle \psi_\alpha |\right) \otimes \left(\sum_{\alpha}\sqrt{\rho_\alpha}\langle\psi_\alpha |\otimes |\psi_\alpha \rangle\right)\\ &=\sum_{\alpha,\beta}\sqrt{\rho_{\alpha}}\sqrt{\rho_{\beta}}|\psi_\alpha \rangle \langle \psi_\beta | \otimes \langle \psi_\alpha | \psi_\beta \rangle\\ &=\sum_{\alpha}\rho_\alpha |\psi_\alpha \rangle \langle \psi_\alpha |=\rho. \end{aligned} </annotation></semantics></math></div> <h2 id="limitations">Limitations</h2> Note that a density operator, as the representation of the state of a quantum system, is less restrictive than a state vector which specifies the wavefunction. On the other hand, two different state vectors can give rise to the same density operator. However, in that case, the two vectors are the same up to a phase, so arguably the density operator still describes the <a class="existingWikiWord" href="/nlab/show/physical+state">physical state</a> unambiguously. More controversially, two entirely different probabilisitic combinations of state vectors can give rise to the same density operator. <a class="existingWikiWord" href="/nlab/show/Roger+Penrose">Roger Penrose</a>, for one, has argued that this means that that the density operator does not describe mixed states unambiguously. But one can also argue the reverse: that mixed states with the same operator really are the same physical state, since they are observationally indistinguishable. <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/maximally+mixed+quantum+state">maximally mixed quantum state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+channel">quantum channel</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</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 notion of density matrices originates with <ul> <li id="vonNeumann32"> <a class="existingWikiWord" href="/nlab/show/John+von+Neumann">John von Neumann</a>, §IV, §V 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> (in precursor discussion of what later came to be called <a class="existingWikiWord" href="/nlab/show/quantum+measurement+channels">quantum measurement channels</a>). Most accounts of <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> discuss density matrices in some form. See for instance: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Jun+John+Sakurai">Jun John Sakurai</a>, Jim Napolitano, §3.4 in: Modern Quantum Mechanics, Cambridge University Press (1985, 1994, 2020) [<a href="https://doi.org/10.1017/9781108587280">doi:10.1017/9781108587280</a>, <a href="https://en.wikipedia.org/wiki/Modern_Quantum_Mechanics">Wikipedia</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Chris+Isham">Chris Isham</a>, §6.1 in: Lectures on Quantum Theory – Mathematical and Structural Foundations, World Scientific (1995) [<a href="https://doi.org/10.1142/p001">doi:10.1142/p001</a>, <a href="https://archive.org/details/lecturesonquantu0000isha">ark:/13960/t4xh7cs99</a>] </li> </ul> In the context of <a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a>: <ul> <li> <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>, §2.4 in: Quantum computation and quantum information, 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>] </li> <li> <a class="existingWikiWord" href="/nlab/show/John+Preskill">John Preskill</a>, §2.3 in: States and Ensembles, chapter 2 of: Quantum Computation, lecture notes, since 2004 [<a href="http://www.theory.caltech.edu/~preskill/ph219/chap2_15.pdf">pdf</a>, <a href="http://theory.caltech.edu/~preskill/ph229/">web</a>] </li> <li id="Kuperberg05"> <a class="existingWikiWord" href="/nlab/show/Greg+Kuperberg">Greg Kuperberg</a>, §1.4 of: 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>] </li> </ul> In the context of <a class="existingWikiWord" href="/nlab/show/entanglement">entanglement</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Ingemar+Bengtsson">Ingemar Bengtsson</a>, <a class="existingWikiWord" href="/nlab/show/Karol+%C5%BByczkowski">Karol Życzkowski</a>, Chapters 8 and 12 of: Geometry of Quantum States — An Introduction to Quantum Entanglement, Cambridge University Press (2006) [<a href="https://doi.org/10.1017/CBO9780511535048">doi:10.1017/CBO9780511535048</a>]</li> </ul> In the context of <a class="existingWikiWord" href="/nlab/show/decoherence">decoherence</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Maximilian+Schlosshauer">Maximilian Schlosshauer</a>, §2.4 in: Decoherence and the Quantum-To-Classical Transition, The Frontiers Collection, Springer (2007) [<a href="https://doi.org/10.1007/978-3-540-35775-9">doi:10.1007/978-3-540-35775-9</a>]</li> </ul> In the context of <a class="existingWikiWord" href="/nlab/show/open+quantum+systems">open quantum systems</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Heinz-Peter+Breuer">Heinz-Peter Breuer</a>, <a class="existingWikiWord" href="/nlab/show/Francesco+Petruccione">Francesco Petruccione</a>, §2.1.3 and Part II of: The Theory of Open Quantum Systems, Oxford University Press (2007) [<a href="https://doi.org/10.1093/acprof:oso/9780199213900.001.0001">doi:10.1093/acprof:oso/9780199213900.001.0001</a>]</li> </ul> Further: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Franco+Strocchi">Franco Strocchi</a>, pp. 46 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> <a class="existingWikiWord" href="/nlab/show/Benjamin+Schumacher">Benjamin Schumacher</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Westmoreland">Michael Westmoreland</a>, §8 of: Quantum Processes, Systems, and Information, Cambridge University Press (2010) [<a href="https://doi.org/10.1017/CBO9780511814006">doi:10.1017/CBO9780511814006</a>] </li> </ul> On <a class="existingWikiWord" href="/nlab/show/quantum+circuits">quantum circuits</a> with density matrices: <ul> <li><a class="existingWikiWord" href="/nlab/show/Dorit+Aharonov">Dorit Aharonov</a>, <a class="existingWikiWord" href="/nlab/show/Alexei+Kitaev">Alexei Kitaev</a>, <a class="existingWikiWord" href="/nlab/show/Noam+Nisan">Noam Nisan</a>, Quantum Circuits with Mixed States, Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computation (STOC) (1998) 20-30 [<a href="https://arxiv.org/abs/quant-ph/9806029">arXiv:quant-ph/9806029</a>, <a href="https://doi.org/10.1145/276698.276708">doi:10.1145/276698.276708</a>]</li> </ul> </body></html> </div> <div class="revisedby"> Last revised on October 8, 2024 at 12:35:36. 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