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Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 14 (from laksa113)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa014.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://ncatlab.org/nlab/show/quantum+superposition\')), getAhrefsUnparsedNoserviceFromURL(\'https://ncatlab.org/nlab/show/quantum+superposition\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/ncatlab.org\/nlab\/show\/quantum+superposition","crawl_time":1773553922,"first_indexed_time":1532793457,"http_code":200,"src_unparsed":"org,ncatlab!\/nlab\/show\/quantum+superposition s443","src_root_hash":"4727794731343683214","history_drop_reason":null,"meta_title":"quantum superposition in nLab","meta_descriptions":[],"attrs_boilerpipe_text":"Contents\nContext\nPhysics\nphysics\n,\nmathematical physics\n,\nphilosophy of physics\nSurveys, textbooks and lecture notes\n(higher) category theory and physics\ngeometry of physics\nbooks and reviews\n,\nphysics resources\ntheory (physics)\n,\nmodel (physics)\nexperiment\n,\nmeasurement\n,\ncomputable physics\nmechanics\nmass\n,\ncharge\n,\nmomentum\n,\nangular momentum\n,\nmoment of inertia\ndynamics on Lie groups\nrigid body dynamics\nfield (physics)\nLagrangian mechanics\nconfiguration space\n,\nstate\naction functional\n,\nLagrangian\ncovariant phase space\n,\nEuler-Lagrange equations\nHamiltonian mechanics\nphase space\nsymplectic geometry\nPoisson manifold\nsymplectic manifold\nsymplectic groupoid\nmultisymplectic geometry\nn-symplectic manifold\nspacetime\nsmooth Lorentzian manifold\nspecial relativity\ngeneral relativity\ngravity\nsupergravity\n,\ndilaton gravity\nblack hole\nClassical field theory\nclassical physics\nclassical mechanics\nwaves\nand\noptics\nthermodynamics\nQuantum Mechanics\nin terms of ∞-compact categories\nquantum information\nHamiltonian operator\ndensity matrix\nKochen-Specker theorem\nBell's theorem\nGleason's theorem\nQuantization\ngeometric quantization\ndeformation quantization\npath integral quantization\nsemiclassical approximation\nQuantum Field Theory\nAxiomatizations\nalgebraic QFT\nWightman axioms\nHaag-Kastler axioms\noperator algebra\nlocal net\nconformal net\nReeh-Schlieder theorem\nOsterwalder-Schrader theorem\nPCT theorem\nBisognano-Wichmann theorem\nmodular theory\nspin-statistics theorem\nboson\n,\nfermion\nfunctorial QFT\ncobordism\n(∞,n)-category of cobordisms\ncobordism hypothesis\n-theorem\nextended topological quantum field theory\nTools\nperturbative quantum field theory\n,\nvacuum\neffective quantum field theory\nrenormalization\nBV-BRST formalism\ngeometric ∞-function theory\nparticle physics\nphenomenology\nmodels\nstandard model of particle physics\nfields and quanta\nGrand Unified Theories\n,\nMSSM\nscattering amplitude\non-shell recursion\n,\nKLT relations\nStructural phenomena\nuniversality class\nquantum anomaly\nGreen-Schwarz mechanism\ninstanton\nspontaneously broken symmetry\nKaluza-Klein mechanism\nintegrable systems\nholonomic quantum fields\nTypes of quantum field thories\nTQFT\n2d TQFT\nDijkgraaf-Witten theory\nChern-Simons theory\nTCFT\nA-model\n,\nB-model\nhomological mirror symmetry\nQFT with defects\nconformal field theory\n(1,1)-dimensional Euclidean field theories and K-theory\n(2,1)-dimensional Euclidean field theory and elliptic cohomology\nCFT\nWZW model\n6d (2,0)-supersymmetric QFT\ngauge theory\nfield strength\ngauge group\n,\ngauge transformation\n,\ngauge fixing\nexamples\nelectromagnetic field\n,\nQED\nelectric charge\nmagnetic charge\nYang-Mills field\n,\nQCD\nYang-Mills theory\nspinors in Yang-Mills theory\ntopological Yang-Mills theory\nKalb-Ramond field\nsupergravity C-field\nRR field\nfirst-order formulation of gravity\ngeneral covariance\nsupergravity\nD'Auria-Fre formulation of supergravity\ngravity as a BF-theory\nsigma-model\nparticle\n,\nrelativistic particle\n,\nfundamental particle\n,\nspinning particle\n,\nsuperparticle\nstring\n,\nspinning string\n,\nsuperstring\nmembrane\nAKSZ theory\nString Theory\nstring theory results applied elsewhere\nnumber theory and physics\nRiemann hypothesis and physics\nContents\nIdea\nRelated concepts\nReferences\nIdea\nIn\nquantum theory\nthe\nstate\nof a\nphysical system\nis given by a\nvector\nin a\ncomplex vector space\n(usually a\nHilbert space\n). This means that given any two states\nψ\n1\n\\psi_1\nand\nψ\n2\n\\psi_2\n, their sum, the vector\nψ\n1\n+\nψ\n2\n\\psi_1 + \\psi_2\n, also represents a state. This is called the\nquantum superposition\nof the two states.\nIf these quantum states are thought of as\nwavefunctions\nsolving a\nlinear differential equation\nsuch as the\nSchroedinger equation\n,\nDirac equation\n,\nKlein-Gordon equation\nor\nTomonaga-Schwinger equation\n, their quantum superposition is an example of the general principle of\nsuperposition\nof solutions to\nlinear differential equations\n.\nThe concept of quantum superposition is at the heart of what makes\nquantum physics\npeculiar, notably via the phenomenon of\nentanglement\n.\nThe various fundamental quantum physics phenomena at play here may be identified with four\nmodal\n-\nunits\nof\nquantum modal logic\n(as discussed at\nquantum circuits via dependent linear types\n):\nquantum measurement\nquantum state preparation\nquantum superposition\nquantum parallelism\nquantum parallelism\nsuperposition\nquantum probability theory\n–\nobservables\nand\nstates\nstates\nclassical state\nquantum state\nspace of states (in geometric quantization)\nstate on a star-algebra\n,\nquasi-state\nqbit\n,\nBell state\ndimer\n,\ntensor network state\nquantum state preparation\nprobability amplitude\n,\nquantum fluctuation\npure state\nwave function\nbra-ket\nBell state\nquantum superposition\n,\nquantum interference\nquantum entanglement\nquantum measurement\nwave function collapse\nBorn rule\ndeferred measurement principle\nquantum reader monad\nmeasurement problem\nsuperselection sector\nmixed state\n,\ndensity matrix\nentanglement entropy\nholographic entanglement entropy\ncoherent quantum state\nground state\n,\nexcited state\nquasi-free state\nFock space\n,\nsecond quantization\nvacuum\n,\nvacuum state\nHadamard state\nvacuum diagram\nvacuum expectation value\n,\nvacuum amplitude\n,\nvacuum fluctuation\nvacuum energy\nvacuum polarization\ninteracting vacuum\nthermal vacuum\n,\nKMS state\nvacuum stability\nfalse vacuum\n,\ntachyon\n,\nColeman-De Luccia instanton\ntheta vacuum\nperturbative string theory vacuum\nnon-geometric string theory vacuum\nlandscape of string theory vacua\nentangled state\ntensor network state\nmatrix product state\ntree tensor network state\nobservables\nquantum observable\n,\nbeable\nalgebra of observables\n,\nstar-algebra\nBohr topos\nquantum operator (in geometric quantization)\nquantum operation\n,\nquantum effect\n,\neffect algebra\nin\nquantum field theory\nlocal observable\npolynomial observable\nlinear observable\nfield observable\nregular observable\nmicrocausal observable\nnormal-ordered product\n,\ntime-ordered products\n,\nretarded product\nWick algebra\nscattering amplitude\ninteracting field algebra of observables\n,\nBogoliubov's formula\nGNS construction\ntheorems\norder-theoretic structure in quantum mechanics\nGleason's theorem\nAlfsen-Shultz theorem\nHarding-Döring-Hamhalter theorem\nKochen-Specker theorem\nNuiten's lemma\nWigner's theorem\nno-cloning theorem\nBell's theorem\nReferences\nSee also\nWikipedia,\nQuantum superposition\nLast revised on January 13, 2023 at 22:36:23.\n    See the\nhistory\nof this page for a list of all contributions to it.","attrs_markdown":"# nLab quantum superposition\n[Skip the Navigation Links](https:\/\/ncatlab.org\/nlab\/show\/quantum+superposition#navEnd) \\|   [Home Page](https:\/\/ncatlab.org\/nlab\/show\/HomePage \"Home page\") \\| [All Pages](https:\/\/ncatlab.org\/nlab\/all_pages \"List of all pages\") \\| [Latest Revisions](https:\/\/ncatlab.org\/nlab\/latest_revisions \"Latest edits and page creations\") \\| [Discuss this page](https:\/\/nforum.ncatlab.org\/discussion\/14313\/#Item_5 \"Discuss this page in its dedicated thread on the nForum\") \\|\n\n### Context\n#### Physics\n**[physics](https:\/\/ncatlab.org\/nlab\/show\/physics)**, [mathematical physics](https:\/\/ncatlab.org\/nlab\/show\/mathematical+physics), [philosophy of physics](https:\/\/ncatlab.org\/nlab\/show\/philosophy+of+physics)\n\n## Surveys, textbooks and lecture notes\n- *[(higher) category theory and physics](https:\/\/ncatlab.org\/nlab\/show\/higher+category+theory+and+physics)*\n- *[geometry of physics](https:\/\/ncatlab.org\/nlab\/show\/geometry+of+physics)*\n- [books and reviews](https:\/\/ncatlab.org\/nlab\/show\/books+and+reviews+in+mathematical+physics), [physics resources](https:\/\/ncatlab.org\/nlab\/show\/physics+resources)\n***\n[theory (physics)](https:\/\/ncatlab.org\/nlab\/show\/theory+%28physics%29), [model (physics)](https:\/\/ncatlab.org\/nlab\/show\/model+%28physics%29)\n\n[experiment](https:\/\/ncatlab.org\/nlab\/show\/experiment), [measurement](https:\/\/ncatlab.org\/nlab\/show\/measurement), [computable physics](https:\/\/ncatlab.org\/nlab\/show\/computable+physics)\n\n- **[mechanics](https:\/\/ncatlab.org\/nlab\/show\/mechanics)**\n  - [mass](https:\/\/ncatlab.org\/nlab\/show\/mass), [charge](https:\/\/ncatlab.org\/nlab\/show\/charge), [momentum](https:\/\/ncatlab.org\/nlab\/show\/momentum), [angular momentum](https:\/\/ncatlab.org\/nlab\/show\/angular+momentum), [moment of inertia](https:\/\/ncatlab.org\/nlab\/show\/moment+of+inertia)\n  - [dynamics on Lie groups](https:\/\/ncatlab.org\/nlab\/show\/dynamics+on+Lie+groups)\n    - [rigid body dynamics](https:\/\/ncatlab.org\/nlab\/show\/rigid+body+dynamics)\n- [field (physics)](https:\/\/ncatlab.org\/nlab\/show\/field+%28physics%29)\n  - [Lagrangian mechanics](https:\/\/ncatlab.org\/nlab\/show\/Lagrangian+mechanics)\n    - [configuration space](https:\/\/ncatlab.org\/nlab\/show\/configuration+space), [state](https:\/\/ncatlab.org\/nlab\/show\/state)\n    - [action functional](https:\/\/ncatlab.org\/nlab\/show\/action+functional), [Lagrangian](https:\/\/ncatlab.org\/nlab\/show\/Lagrangian)\n    - [covariant phase space](https:\/\/ncatlab.org\/nlab\/show\/covariant+phase+space), [Euler-Lagrange equations](https:\/\/ncatlab.org\/nlab\/show\/Euler-Lagrange+equations)\n  - [Hamiltonian mechanics](https:\/\/ncatlab.org\/nlab\/show\/Hamiltonian+mechanics)\n    - [phase space](https:\/\/ncatlab.org\/nlab\/show\/phase+space)\n    - [symplectic geometry](https:\/\/ncatlab.org\/nlab\/show\/symplectic+geometry)\n      - [Poisson manifold](https:\/\/ncatlab.org\/nlab\/show\/Poisson+manifold)\n      - [symplectic manifold](https:\/\/ncatlab.org\/nlab\/show\/symplectic+manifold)\n      - [symplectic groupoid](https:\/\/ncatlab.org\/nlab\/show\/symplectic+groupoid)\n    - [multisymplectic geometry](https:\/\/ncatlab.org\/nlab\/show\/multisymplectic+geometry)\n      - [n-symplectic manifold](https:\/\/ncatlab.org\/nlab\/show\/n-symplectic+manifold)\n  - [spacetime](https:\/\/ncatlab.org\/nlab\/show\/spacetime)\n    - [smooth Lorentzian manifold](https:\/\/ncatlab.org\/nlab\/show\/smooth+Lorentzian+manifold)\n    - [special relativity](https:\/\/ncatlab.org\/nlab\/show\/special+relativity)\n    - [general relativity](https:\/\/ncatlab.org\/nlab\/show\/general+relativity)\n    - [gravity](https:\/\/ncatlab.org\/nlab\/show\/gravity)\n      - [supergravity](https:\/\/ncatlab.org\/nlab\/show\/supergravity), [dilaton gravity](https:\/\/ncatlab.org\/nlab\/show\/dilaton+gravity)\n      - [black hole](https:\/\/ncatlab.org\/nlab\/show\/black+hole)\n- **[Classical field theory](https:\/\/ncatlab.org\/nlab\/show\/classical+field+theory)**\n  - [classical physics](https:\/\/ncatlab.org\/nlab\/show\/classical+physics)\n    - [classical mechanics](https:\/\/ncatlab.org\/nlab\/show\/classical+mechanics)\n    - [waves](https:\/\/ncatlab.org\/nlab\/show\/waves) and [optics](https:\/\/ncatlab.org\/nlab\/show\/optics)\n    - [thermodynamics](https:\/\/ncatlab.org\/nlab\/show\/thermodynamics)\n- **[Quantum Mechanics](https:\/\/ncatlab.org\/nlab\/show\/quantum+mechanics)**\n  - [in terms of ∞-compact categories](https:\/\/ncatlab.org\/nlab\/show\/quantum+mechanics+in+terms+of+dagger-compact+categories)\n  - [quantum information](https:\/\/ncatlab.org\/nlab\/show\/quantum+information)\n  - [Hamiltonian operator](https:\/\/ncatlab.org\/nlab\/show\/Hamiltonian+operator)\n  - [density matrix](https:\/\/ncatlab.org\/nlab\/show\/density+matrix)\n  - [Kochen-Specker theorem](https:\/\/ncatlab.org\/nlab\/show\/Kochen-Specker+theorem)\n  - [Bell's theorem](https:\/\/ncatlab.org\/nlab\/show\/Bell%27s+theorem)\n  - [Gleason's theorem](https:\/\/ncatlab.org\/nlab\/show\/Gleason%27s+theorem)\n- **[Quantization](https:\/\/ncatlab.org\/nlab\/show\/quantization)**\n  - [geometric quantization](https:\/\/ncatlab.org\/nlab\/show\/geometric+quantization)\n  - [deformation quantization](https:\/\/ncatlab.org\/nlab\/show\/deformation+quantization)\n  - [path integral quantization](https:\/\/ncatlab.org\/nlab\/show\/path+integral)\n  - [semiclassical approximation](https:\/\/ncatlab.org\/nlab\/show\/semiclassical+approximation)\n- **[Quantum Field Theory](https:\/\/ncatlab.org\/nlab\/show\/quantum+field+theory)**\n  - Axiomatizations\n    - [algebraic QFT](https:\/\/ncatlab.org\/nlab\/show\/AQFT)\n      - [Wightman axioms](https:\/\/ncatlab.org\/nlab\/show\/Wightman+axioms)\n      - [Haag-Kastler axioms](https:\/\/ncatlab.org\/nlab\/show\/Haag-Kastler+axioms)\n        - [operator algebra](https:\/\/ncatlab.org\/nlab\/show\/operator+algebra)\n        - [local net](https:\/\/ncatlab.org\/nlab\/show\/local+net)\n      - [conformal net](https:\/\/ncatlab.org\/nlab\/show\/conformal+net)\n      - [Reeh-Schlieder theorem](https:\/\/ncatlab.org\/nlab\/show\/Reeh-Schlieder+theorem)\n      - [Osterwalder-Schrader theorem](https:\/\/ncatlab.org\/nlab\/show\/Osterwalder-Schrader+theorem)\n      - [PCT theorem](https:\/\/ncatlab.org\/nlab\/show\/PCT+theorem)\n      - [Bisognano-Wichmann theorem](https:\/\/ncatlab.org\/nlab\/show\/Bisognano-Wichmann+theorem)\n        - [modular theory](https:\/\/ncatlab.org\/nlab\/show\/modular+theory)\n      - [spin-statistics theorem](https:\/\/ncatlab.org\/nlab\/show\/spin-statistics+theorem)\n        - [boson](https:\/\/ncatlab.org\/nlab\/show\/boson), [fermion](https:\/\/ncatlab.org\/nlab\/show\/fermion)\n    - [functorial QFT](https:\/\/ncatlab.org\/nlab\/show\/FQFT)\n      - [cobordism](https:\/\/ncatlab.org\/nlab\/show\/cobordism)\n      - [(∞,n)-category of cobordisms](https:\/\/ncatlab.org\/nlab\/show\/%28%E2%88%9E%2Cn%29-category+of+cobordisms)\n      - [cobordism hypothesis](https:\/\/ncatlab.org\/nlab\/show\/cobordism+hypothesis)\\-theorem\n      - [extended topological quantum field theory](https:\/\/ncatlab.org\/nlab\/show\/extended+topological+quantum+field+theory)\n  - Tools\n    - [perturbative quantum field theory](https:\/\/ncatlab.org\/nlab\/show\/perturbative+quantum+field+theory), [vacuum](https:\/\/ncatlab.org\/nlab\/show\/vacuum)\n    - [effective quantum field theory](https:\/\/ncatlab.org\/nlab\/show\/effective+quantum+field+theory)\n    - [renormalization](https:\/\/ncatlab.org\/nlab\/show\/renormalization)\n    - [BV-BRST formalism](https:\/\/ncatlab.org\/nlab\/show\/BV-BRST+formalism)\n    - [geometric ∞-function theory](https:\/\/ncatlab.org\/nlab\/show\/geometric+%E2%88%9E-function+theory)\n  - [particle physics](https:\/\/ncatlab.org\/nlab\/show\/particle+physics)\n    - [phenomenology](https:\/\/ncatlab.org\/nlab\/show\/phenomenology)\n    - [models](https:\/\/ncatlab.org\/nlab\/show\/model+%28in+particle+phyiscs%29)\n      - [standard model of particle physics](https:\/\/ncatlab.org\/nlab\/show\/standard+model+of+particle+physics)\n      - [fields and quanta](https:\/\/ncatlab.org\/nlab\/show\/fields+and+quanta+-+table)\n      - [Grand Unified Theories](https:\/\/ncatlab.org\/nlab\/show\/GUT), [MSSM](https:\/\/ncatlab.org\/nlab\/show\/MSSM)\n    - [scattering amplitude](https:\/\/ncatlab.org\/nlab\/show\/scattering+amplitude)\n      - [on-shell recursion](https:\/\/ncatlab.org\/nlab\/show\/on-shell+recursion), [KLT relations](https:\/\/ncatlab.org\/nlab\/show\/KLT+relations)\n  - Structural phenomena\n    - [universality class](https:\/\/ncatlab.org\/nlab\/show\/universality+class)\n    - [quantum anomaly](https:\/\/ncatlab.org\/nlab\/show\/quantum+anomaly)\n      - [Green-Schwarz mechanism](https:\/\/ncatlab.org\/nlab\/show\/Green-Schwarz+mechanism)\n    - [instanton](https:\/\/ncatlab.org\/nlab\/show\/instanton)\n    - [spontaneously broken symmetry](https:\/\/ncatlab.org\/nlab\/show\/spontaneously+broken+symmetry)\n    - [Kaluza-Klein mechanism](https:\/\/ncatlab.org\/nlab\/show\/Kaluza-Klein+mechanism)\n    - [integrable systems](https:\/\/ncatlab.org\/nlab\/show\/integrable+systems)\n    - [holonomic quantum fields](https:\/\/ncatlab.org\/nlab\/show\/holonomic+quantum+fields)\n  - Types of quantum field thories\n    - [TQFT](https:\/\/ncatlab.org\/nlab\/show\/TQFT)\n      - [2d TQFT](https:\/\/ncatlab.org\/nlab\/show\/2d+TQFT)\n      - [Dijkgraaf-Witten theory](https:\/\/ncatlab.org\/nlab\/show\/Dijkgraaf-Witten+theory)\n      - [Chern-Simons theory](https:\/\/ncatlab.org\/nlab\/show\/Chern-Simons+theory)\n    - [TCFT](https:\/\/ncatlab.org\/nlab\/show\/TCFT)\n      - [A-model](https:\/\/ncatlab.org\/nlab\/show\/A-model), [B-model](https:\/\/ncatlab.org\/nlab\/show\/B-model)\n      - [homological mirror symmetry](https:\/\/ncatlab.org\/nlab\/show\/homological+mirror+symmetry)\n    - [QFT with defects](https:\/\/ncatlab.org\/nlab\/show\/QFT+with+defects)\n    - [conformal field theory](https:\/\/ncatlab.org\/nlab\/show\/conformal+field+theory)\n    - [(1,1)-dimensional Euclidean field theories and K-theory](https:\/\/ncatlab.org\/nlab\/show\/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory)\n    - [(2,1)-dimensional Euclidean field theory and elliptic cohomology](https:\/\/ncatlab.org\/nlab\/show\/%282%2C1%29-dimensional+Euclidean+field+theory)\n    - [CFT](https:\/\/ncatlab.org\/nlab\/show\/CFT)\n      - [WZW model](https:\/\/ncatlab.org\/nlab\/show\/WZW+model)\n      - [6d (2,0)-supersymmetric QFT](https:\/\/ncatlab.org\/nlab\/show\/6d+%282%2C0%29-supersymmetric+QFT)\n    - [gauge theory](https:\/\/ncatlab.org\/nlab\/show\/gauge+theory)\n      - [field strength](https:\/\/ncatlab.org\/nlab\/show\/field+strength)\n      - [gauge group](https:\/\/ncatlab.org\/nlab\/show\/gauge+group), [gauge transformation](https:\/\/ncatlab.org\/nlab\/show\/gauge+transformation), [gauge fixing](https:\/\/ncatlab.org\/nlab\/show\/gauge+fixing)\n      - examples\n        - [electromagnetic field](https:\/\/ncatlab.org\/nlab\/show\/electromagnetic+field), [QED](https:\/\/ncatlab.org\/nlab\/show\/QED)\n      - [electric charge](https:\/\/ncatlab.org\/nlab\/show\/electric+charge)\n      - [magnetic charge](https:\/\/ncatlab.org\/nlab\/show\/magnetic+charge)\n        - [Yang-Mills field](https:\/\/ncatlab.org\/nlab\/show\/Yang-Mills+field), [QCD](https:\/\/ncatlab.org\/nlab\/show\/QCD)\n      - [Yang-Mills theory](https:\/\/ncatlab.org\/nlab\/show\/Yang-Mills+theory)\n      - [spinors in Yang-Mills theory](https:\/\/ncatlab.org\/nlab\/show\/spinors+in+Yang-Mills+theory)\n      - [topological Yang-Mills theory](https:\/\/ncatlab.org\/nlab\/show\/topological+Yang-Mills+theory)\n        - [Kalb-Ramond field](https:\/\/ncatlab.org\/nlab\/show\/Kalb-Ramond+field)\n        - [supergravity C-field](https:\/\/ncatlab.org\/nlab\/show\/supergravity+C-field)\n        - [RR field](https:\/\/ncatlab.org\/nlab\/show\/RR+field)\n        - [first-order formulation of gravity](https:\/\/ncatlab.org\/nlab\/show\/first-order+formulation+of+gravity)\n      - [general covariance](https:\/\/ncatlab.org\/nlab\/show\/general+covariance)\n      - [supergravity](https:\/\/ncatlab.org\/nlab\/show\/supergravity)\n      - [D'Auria-Fre formulation of supergravity](https:\/\/ncatlab.org\/nlab\/show\/D%27Auria-Fre+formulation+of+supergravity)\n      - [gravity as a BF-theory](https:\/\/ncatlab.org\/nlab\/show\/gravity+as+a+BF-theory)\n    - [sigma-model](https:\/\/ncatlab.org\/nlab\/show\/sigma-model)\n      - [particle](https:\/\/ncatlab.org\/nlab\/show\/particle), [relativistic particle](https:\/\/ncatlab.org\/nlab\/show\/relativistic+particle), [fundamental particle](https:\/\/ncatlab.org\/nlab\/show\/fundamental+particle), [spinning particle](https:\/\/ncatlab.org\/nlab\/show\/spinning+particle), [superparticle](https:\/\/ncatlab.org\/nlab\/show\/superparticle)\n      - [string](https:\/\/ncatlab.org\/nlab\/show\/string), [spinning string](https:\/\/ncatlab.org\/nlab\/show\/spinning+string), [superstring](https:\/\/ncatlab.org\/nlab\/show\/superstring)\n      - [membrane](https:\/\/ncatlab.org\/nlab\/show\/membrane)\n      - [AKSZ theory](https:\/\/ncatlab.org\/nlab\/show\/AKSZ+theory)\n- [String Theory](https:\/\/ncatlab.org\/nlab\/show\/string+theory)\n  - [string theory results applied elsewhere](https:\/\/ncatlab.org\/nlab\/show\/string+theory+results+applied+elsewhere)\n- [number theory and physics](https:\/\/ncatlab.org\/nlab\/show\/number+theory+and+physics)\n  - [Riemann hypothesis and physics](https:\/\/ncatlab.org\/nlab\/show\/Riemann+hypothesis+and+physics)\n\n[Edit this sidebar](https:\/\/ncatlab.org\/nlab\/edit\/physicscontents)\n# Contents\n- [Idea](https:\/\/ncatlab.org\/nlab\/show\/quantum+superposition#idea)\n- [Related concepts](https:\/\/ncatlab.org\/nlab\/show\/quantum+superposition#related_concepts)\n- [References](https:\/\/ncatlab.org\/nlab\/show\/quantum+superposition#references)\n## Idea\nIn [quantum theory](https:\/\/ncatlab.org\/nlab\/show\/quantum+theory) the [state](https:\/\/ncatlab.org\/nlab\/show\/state) of a [physical system](https:\/\/ncatlab.org\/nlab\/show\/physical+system) is given by a [vector](https:\/\/ncatlab.org\/nlab\/show\/vector) in a [complex vector space](https:\/\/ncatlab.org\/nlab\/show\/complex+vector+space) (usually a [Hilbert space](https:\/\/ncatlab.org\/nlab\/show\/Hilbert+space)). This means that given any two states ψ 1 \\\\psi\\_1 and ψ 2 \\\\psi\\_2, their sum, the vector ψ 1 \\+ ψ 2 \\\\psi\\_1 + \\\\psi\\_2, also represents a state. This is called the *quantum superposition* of the two states.\n\nIf these quantum states are thought of as [wavefunctions](https:\/\/ncatlab.org\/nlab\/show\/wavefunctions) solving a [linear differential equation](https:\/\/ncatlab.org\/nlab\/show\/linear+differential+equation) such as the [Schroedinger equation](https:\/\/ncatlab.org\/nlab\/show\/Schroedinger+equation), [Dirac equation](https:\/\/ncatlab.org\/nlab\/show\/Dirac+equation), [Klein-Gordon equation](https:\/\/ncatlab.org\/nlab\/show\/Klein-Gordon+equation) or [Tomonaga-Schwinger equation](https:\/\/ncatlab.org\/nlab\/show\/Tomonaga-Schwinger+equation), their quantum superposition is an example of the general principle of *[superposition](https:\/\/ncatlab.org\/nlab\/show\/superposition)* of solutions to [linear differential equations](https:\/\/ncatlab.org\/nlab\/show\/linear+differential+equations).\n\nThe concept of quantum superposition is at the heart of what makes [quantum physics](https:\/\/ncatlab.org\/nlab\/show\/quantum+physics) peculiar, notably via the phenomenon of [entanglement](https:\/\/ncatlab.org\/nlab\/show\/entanglement).\n\nThe various fundamental quantum physics phenomena at play here may be identified with four [modal](https:\/\/ncatlab.org\/nlab\/show\/modality)\\-[units](https:\/\/ncatlab.org\/nlab\/show\/unit+of+a+monad) of [quantum modal logic](https:\/\/ncatlab.org\/nlab\/show\/necessity+and+possibility#ModalQuantumLogic) (as discussed at *[quantum circuits via dependent linear types](https:\/\/ncatlab.org\/nlab\/show\/quantum+circuits+via+dependent+linear+types)*):\n\n![](https:\/\/ncatlab.org\/nlab\/files\/QCwthLHT-QuantumModalUnits-221110.jpg)\n\n|   |   |\n|---|---|\n| [quantum measurement](https:\/\/ncatlab.org\/nlab\/show\/quantum+measurement) | [quantum state preparation](https:\/\/ncatlab.org\/nlab\/show\/quantum+state+preparation) |\n| quantum superposition | [quantum parallelism](https:\/\/ncatlab.org\/nlab\/show\/quantum+parallelism) |\n## Related concepts\n- [quantum parallelism](https:\/\/ncatlab.org\/nlab\/show\/quantum+parallelism)\n- [superposition](https:\/\/ncatlab.org\/nlab\/show\/superposition)\n\n**[quantum probability theory](https:\/\/ncatlab.org\/nlab\/show\/quantum+probability+theory) – [observables](https:\/\/ncatlab.org\/nlab\/show\/observables) and [states](https:\/\/ncatlab.org\/nlab\/show\/states)**\n\n- **[states](https:\/\/ncatlab.org\/nlab\/show\/states)**\n  - [classical state](https:\/\/ncatlab.org\/nlab\/show\/classical+state)\n  - [quantum state](https:\/\/ncatlab.org\/nlab\/show\/quantum+state)\n    - [space of states (in geometric quantization)](https:\/\/ncatlab.org\/nlab\/show\/space+of+states+%28in+geometric+quantization%29)\n    - [state on a star-algebra](https:\/\/ncatlab.org\/nlab\/show\/state+on+a+star-algebra), [quasi-state](https:\/\/ncatlab.org\/nlab\/show\/quasi-state)\n    - [qbit](https:\/\/ncatlab.org\/nlab\/show\/qbit), [Bell state](https:\/\/ncatlab.org\/nlab\/show\/Bell+state)\n      [dimer](https:\/\/ncatlab.org\/nlab\/show\/dimer), [tensor network state](https:\/\/ncatlab.org\/nlab\/show\/tensor+network+state)\n    - [quantum state preparation](https:\/\/ncatlab.org\/nlab\/show\/quantum+state+preparation)\n    - [probability amplitude](https:\/\/ncatlab.org\/nlab\/show\/probability+amplitude), [quantum fluctuation](https:\/\/ncatlab.org\/nlab\/show\/quantum+fluctuation)\n    - [pure state](https:\/\/ncatlab.org\/nlab\/show\/pure+state)\n      - [wave function](https:\/\/ncatlab.org\/nlab\/show\/wave+function)\n      - [bra-ket](https:\/\/ncatlab.org\/nlab\/show\/bra-ket)\n      - [Bell state](https:\/\/ncatlab.org\/nlab\/show\/Bell+state)\n    - [quantum superposition](https:\/\/ncatlab.org\/nlab\/show\/quantum+superposition), [quantum interference](https:\/\/ncatlab.org\/nlab\/show\/quantum+interference)\n    - [quantum entanglement](https:\/\/ncatlab.org\/nlab\/show\/quantum+entanglement)\n    - [quantum measurement](https:\/\/ncatlab.org\/nlab\/show\/quantum+measurement)\n      [wave function collapse](https:\/\/ncatlab.org\/nlab\/show\/wave+function+collapse)\n      [Born rule](https:\/\/ncatlab.org\/nlab\/show\/Born+rule)\n      [deferred measurement principle](https:\/\/ncatlab.org\/nlab\/show\/deferred+measurement+principle)\n      [quantum reader monad](https:\/\/ncatlab.org\/nlab\/show\/quantum+reader+monad)\n      [measurement problem](https:\/\/ncatlab.org\/nlab\/show\/measurement+problem)\n    - [superselection sector](https:\/\/ncatlab.org\/nlab\/show\/superselection+sector)\n    - [mixed state](https:\/\/ncatlab.org\/nlab\/show\/mixed+state), [density matrix](https:\/\/ncatlab.org\/nlab\/show\/density+matrix)\n      [entanglement entropy](https:\/\/ncatlab.org\/nlab\/show\/entanglement+entropy)\n      [holographic entanglement entropy](https:\/\/ncatlab.org\/nlab\/show\/holographic+entanglement+entropy)\n    - [coherent quantum state](https:\/\/ncatlab.org\/nlab\/show\/coherent+quantum+state)\n  - [ground state](https:\/\/ncatlab.org\/nlab\/show\/ground+state), [excited state](https:\/\/ncatlab.org\/nlab\/show\/excited+state)\n  - [quasi-free state](https:\/\/ncatlab.org\/nlab\/show\/quasi-free+state)\n  - [Fock space](https:\/\/ncatlab.org\/nlab\/show\/Fock+space), [second quantization](https:\/\/ncatlab.org\/nlab\/show\/second+quantization)\n  - [vacuum](https:\/\/ncatlab.org\/nlab\/show\/vacuum), [vacuum state](https:\/\/ncatlab.org\/nlab\/show\/vacuum+state)\n    - [Hadamard state](https:\/\/ncatlab.org\/nlab\/show\/Hadamard+state)\n    - [vacuum diagram](https:\/\/ncatlab.org\/nlab\/show\/vacuum+diagram)\n    - [vacuum expectation value](https:\/\/ncatlab.org\/nlab\/show\/vacuum+expectation+value), [vacuum amplitude](https:\/\/ncatlab.org\/nlab\/show\/vacuum+amplitude), [vacuum fluctuation](https:\/\/ncatlab.org\/nlab\/show\/vacuum+fluctuation)\n    - [vacuum energy](https:\/\/ncatlab.org\/nlab\/show\/vacuum+energy)\n    - [vacuum polarization](https:\/\/ncatlab.org\/nlab\/show\/vacuum+polarization)\n    - [interacting vacuum](https:\/\/ncatlab.org\/nlab\/show\/interacting+vacuum)\n    - [thermal vacuum](https:\/\/ncatlab.org\/nlab\/show\/thermal+vacuum), [KMS state](https:\/\/ncatlab.org\/nlab\/show\/KMS+state)\n    - [vacuum stability](https:\/\/ncatlab.org\/nlab\/show\/vacuum+stability)\n      - [false vacuum](https:\/\/ncatlab.org\/nlab\/show\/false+vacuum), [tachyon](https:\/\/ncatlab.org\/nlab\/show\/tachyon), [Coleman-De Luccia instanton](https:\/\/ncatlab.org\/nlab\/show\/Coleman-De+Luccia+instanton)\n    - [theta vacuum](https:\/\/ncatlab.org\/nlab\/show\/theta+vacuum)\n    - [perturbative string theory vacuum](https:\/\/ncatlab.org\/nlab\/show\/perturbative+string+theory+vacuum)\n      - [non-geometric string theory vacuum](https:\/\/ncatlab.org\/nlab\/show\/non-geometric+string+theory+vacuum)\n      - [landscape of string theory vacua](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua)\n  - [entangled state](https:\/\/ncatlab.org\/nlab\/show\/entangled+state)\n  - [tensor network state](https:\/\/ncatlab.org\/nlab\/show\/tensor+network+state)\n    - [matrix product state](https:\/\/ncatlab.org\/nlab\/show\/matrix+product+state)\n    - [tree tensor network state](https:\/\/ncatlab.org\/nlab\/show\/tree+tensor+network+state)\n- **[observables](https:\/\/ncatlab.org\/nlab\/show\/observables)**\n  - [quantum observable](https:\/\/ncatlab.org\/nlab\/show\/quantum+observable), [beable](https:\/\/ncatlab.org\/nlab\/show\/beable)\n  - [algebra of observables](https:\/\/ncatlab.org\/nlab\/show\/algebra+of+observables), [star-algebra](https:\/\/ncatlab.org\/nlab\/show\/star-algebra)\n  - [Bohr topos](https:\/\/ncatlab.org\/nlab\/show\/Bohr+topos)\n  - [quantum operator (in geometric quantization)](https:\/\/ncatlab.org\/nlab\/show\/quantum+operator+%28in+geometric+quantization%29)\n  - [quantum operation](https:\/\/ncatlab.org\/nlab\/show\/quantum+operation), [quantum effect](https:\/\/ncatlab.org\/nlab\/show\/quantum+effect), [effect algebra](https:\/\/ncatlab.org\/nlab\/show\/effect+algebra)\n  - in [quantum field theory](https:\/\/ncatlab.org\/nlab\/show\/quantum+field+theory)\n    - [local observable](https:\/\/ncatlab.org\/nlab\/show\/local+observable)\n    - [polynomial observable](https:\/\/ncatlab.org\/nlab\/show\/polynomial+observable)\n      - [linear observable](https:\/\/ncatlab.org\/nlab\/show\/linear+observable)\n        - [field observable](https:\/\/ncatlab.org\/nlab\/show\/field+observable)\n    - [regular observable](https:\/\/ncatlab.org\/nlab\/show\/regular+observable)\n    - [microcausal observable](https:\/\/ncatlab.org\/nlab\/show\/microcausal+observable)\n    - [normal-ordered product](https:\/\/ncatlab.org\/nlab\/show\/normal-ordered+product), [time-ordered products](https:\/\/ncatlab.org\/nlab\/show\/time-ordered+products), [retarded product](https:\/\/ncatlab.org\/nlab\/show\/retarded+product)\n    - [Wick algebra](https:\/\/ncatlab.org\/nlab\/show\/Wick+algebra)\n    - [scattering amplitude](https:\/\/ncatlab.org\/nlab\/show\/scattering+amplitude)\n    - [interacting field algebra of observables](https:\/\/ncatlab.org\/nlab\/show\/interacting+field+algebra+of+observables), [Bogoliubov's formula](https:\/\/ncatlab.org\/nlab\/show\/Bogoliubov%27s+formula)\n- [GNS construction](https:\/\/ncatlab.org\/nlab\/show\/GNS+construction)\n- [theorems](https:\/\/ncatlab.org\/nlab\/show\/theorems)\n  - [order-theoretic structure in quantum mechanics](https:\/\/ncatlab.org\/nlab\/show\/order-theoretic+structure+in+quantum+mechanics)\n    - [Gleason's theorem](https:\/\/ncatlab.org\/nlab\/show\/Gleason%27s+theorem)\n    - [Alfsen-Shultz theorem](https:\/\/ncatlab.org\/nlab\/show\/Alfsen-Shultz+theorem)\n    - [Harding-Döring-Hamhalter theorem](https:\/\/ncatlab.org\/nlab\/show\/Harding-D%C3%B6ring-Hamhalter+theorem)\n    - [Kochen-Specker theorem](https:\/\/ncatlab.org\/nlab\/show\/Kochen-Specker+theorem)\n    - [Nuiten's lemma](https:\/\/ncatlab.org\/nlab\/show\/Nuiten%27s+lemma)\n  - [Wigner's theorem](https:\/\/ncatlab.org\/nlab\/show\/Wigner%27s+theorem)\n  - [no-cloning theorem](https:\/\/ncatlab.org\/nlab\/show\/no-cloning+theorem)\n  - [Bell's theorem](https:\/\/ncatlab.org\/nlab\/show\/Bell%27s+theorem)\n## References\nSee also\n\n- Wikipedia, *[Quantum superposition](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition)*\n\nLast revised on January 13, 2023 at 22:36:23. See the [history](https:\/\/ncatlab.org\/nlab\/history\/quantum+superposition) of this page for a list of all contributions to it.\n\n[Edit](https:\/\/ncatlab.org\/nlab\/edit\/quantum+superposition)[Discuss](https:\/\/nforum.ncatlab.org\/discussion\/14313\/#Item_5)[Previous revision](https:\/\/ncatlab.org\/nlab\/revision\/quantum+superposition\/12)[Changes from previous revision](https:\/\/ncatlab.org\/nlab\/show\/diff\/quantum+superposition)[History (12 revisions)](https:\/\/ncatlab.org\/nlab\/history\/quantum+superposition) [Cite](https:\/\/ncatlab.org\/nlab\/show\/quantum+superposition\/cite) [Print](https:\/\/ncatlab.org\/nlab\/print\/quantum+superposition) [Source](https:\/\/ncatlab.org\/nlab\/source\/quantum+superposition)","attrs_readable_markdown":"### Context\n#### Physics\n**[physics](https:\/\/ncatlab.org\/nlab\/show\/physics)**, [mathematical physics](https:\/\/ncatlab.org\/nlab\/show\/mathematical+physics), [philosophy of physics](https:\/\/ncatlab.org\/nlab\/show\/philosophy+of+physics)\n\n## Surveys, textbooks and lecture notes\n- *[(higher) category theory and physics](https:\/\/ncatlab.org\/nlab\/show\/higher+category+theory+and+physics)*\n- *[geometry of physics](https:\/\/ncatlab.org\/nlab\/show\/geometry+of+physics)*\n- [books and reviews](https:\/\/ncatlab.org\/nlab\/show\/books+and+reviews+in+mathematical+physics), [physics resources](https:\/\/ncatlab.org\/nlab\/show\/physics+resources)\n***\n[theory (physics)](https:\/\/ncatlab.org\/nlab\/show\/theory+%28physics%29), [model (physics)](https:\/\/ncatlab.org\/nlab\/show\/model+%28physics%29)\n\n[experiment](https:\/\/ncatlab.org\/nlab\/show\/experiment), [measurement](https:\/\/ncatlab.org\/nlab\/show\/measurement), [computable physics](https:\/\/ncatlab.org\/nlab\/show\/computable+physics)\n\n- **[mechanics](https:\/\/ncatlab.org\/nlab\/show\/mechanics)**\n  - [mass](https:\/\/ncatlab.org\/nlab\/show\/mass), [charge](https:\/\/ncatlab.org\/nlab\/show\/charge), [momentum](https:\/\/ncatlab.org\/nlab\/show\/momentum), [angular momentum](https:\/\/ncatlab.org\/nlab\/show\/angular+momentum), [moment of inertia](https:\/\/ncatlab.org\/nlab\/show\/moment+of+inertia)\n  - [dynamics on Lie groups](https:\/\/ncatlab.org\/nlab\/show\/dynamics+on+Lie+groups)\n    - [rigid body dynamics](https:\/\/ncatlab.org\/nlab\/show\/rigid+body+dynamics)\n- [field (physics)](https:\/\/ncatlab.org\/nlab\/show\/field+%28physics%29)\n  - [Lagrangian mechanics](https:\/\/ncatlab.org\/nlab\/show\/Lagrangian+mechanics)\n    - [configuration space](https:\/\/ncatlab.org\/nlab\/show\/configuration+space), [state](https:\/\/ncatlab.org\/nlab\/show\/state)\n    - [action functional](https:\/\/ncatlab.org\/nlab\/show\/action+functional), [Lagrangian](https:\/\/ncatlab.org\/nlab\/show\/Lagrangian)\n    - [covariant phase space](https:\/\/ncatlab.org\/nlab\/show\/covariant+phase+space), [Euler-Lagrange equations](https:\/\/ncatlab.org\/nlab\/show\/Euler-Lagrange+equations)\n  - [Hamiltonian mechanics](https:\/\/ncatlab.org\/nlab\/show\/Hamiltonian+mechanics)\n    - [phase space](https:\/\/ncatlab.org\/nlab\/show\/phase+space)\n    - [symplectic geometry](https:\/\/ncatlab.org\/nlab\/show\/symplectic+geometry)\n      - [Poisson manifold](https:\/\/ncatlab.org\/nlab\/show\/Poisson+manifold)\n      - [symplectic manifold](https:\/\/ncatlab.org\/nlab\/show\/symplectic+manifold)\n      - [symplectic groupoid](https:\/\/ncatlab.org\/nlab\/show\/symplectic+groupoid)\n    - [multisymplectic geometry](https:\/\/ncatlab.org\/nlab\/show\/multisymplectic+geometry)\n      - [n-symplectic manifold](https:\/\/ncatlab.org\/nlab\/show\/n-symplectic+manifold)\n  - [spacetime](https:\/\/ncatlab.org\/nlab\/show\/spacetime)\n    - [smooth Lorentzian manifold](https:\/\/ncatlab.org\/nlab\/show\/smooth+Lorentzian+manifold)\n    - [special relativity](https:\/\/ncatlab.org\/nlab\/show\/special+relativity)\n    - [general relativity](https:\/\/ncatlab.org\/nlab\/show\/general+relativity)\n    - [gravity](https:\/\/ncatlab.org\/nlab\/show\/gravity)\n      - [supergravity](https:\/\/ncatlab.org\/nlab\/show\/supergravity), [dilaton gravity](https:\/\/ncatlab.org\/nlab\/show\/dilaton+gravity)\n      - [black hole](https:\/\/ncatlab.org\/nlab\/show\/black+hole)\n- **[Classical field theory](https:\/\/ncatlab.org\/nlab\/show\/classical+field+theory)**\n  - [classical physics](https:\/\/ncatlab.org\/nlab\/show\/classical+physics)\n    - [classical mechanics](https:\/\/ncatlab.org\/nlab\/show\/classical+mechanics)\n    - [waves](https:\/\/ncatlab.org\/nlab\/show\/waves) and [optics](https:\/\/ncatlab.org\/nlab\/show\/optics)\n    - [thermodynamics](https:\/\/ncatlab.org\/nlab\/show\/thermodynamics)\n- **[Quantum Mechanics](https:\/\/ncatlab.org\/nlab\/show\/quantum+mechanics)**\n  - [in terms of ∞-compact categories](https:\/\/ncatlab.org\/nlab\/show\/quantum+mechanics+in+terms+of+dagger-compact+categories)\n  - [quantum information](https:\/\/ncatlab.org\/nlab\/show\/quantum+information)\n  - [Hamiltonian operator](https:\/\/ncatlab.org\/nlab\/show\/Hamiltonian+operator)\n  - [density matrix](https:\/\/ncatlab.org\/nlab\/show\/density+matrix)\n  - [Kochen-Specker theorem](https:\/\/ncatlab.org\/nlab\/show\/Kochen-Specker+theorem)\n  - [Bell's theorem](https:\/\/ncatlab.org\/nlab\/show\/Bell%27s+theorem)\n  - [Gleason's theorem](https:\/\/ncatlab.org\/nlab\/show\/Gleason%27s+theorem)\n- **[Quantization](https:\/\/ncatlab.org\/nlab\/show\/quantization)**\n  - [geometric quantization](https:\/\/ncatlab.org\/nlab\/show\/geometric+quantization)\n  - [deformation quantization](https:\/\/ncatlab.org\/nlab\/show\/deformation+quantization)\n  - [path integral quantization](https:\/\/ncatlab.org\/nlab\/show\/path+integral)\n  - [semiclassical approximation](https:\/\/ncatlab.org\/nlab\/show\/semiclassical+approximation)\n- **[Quantum Field Theory](https:\/\/ncatlab.org\/nlab\/show\/quantum+field+theory)**\n  - Axiomatizations\n    - [algebraic QFT](https:\/\/ncatlab.org\/nlab\/show\/AQFT)\n      - [Wightman axioms](https:\/\/ncatlab.org\/nlab\/show\/Wightman+axioms)\n      - [Haag-Kastler axioms](https:\/\/ncatlab.org\/nlab\/show\/Haag-Kastler+axioms)\n        - [operator algebra](https:\/\/ncatlab.org\/nlab\/show\/operator+algebra)\n        - [local net](https:\/\/ncatlab.org\/nlab\/show\/local+net)\n      - [conformal net](https:\/\/ncatlab.org\/nlab\/show\/conformal+net)\n      - [Reeh-Schlieder theorem](https:\/\/ncatlab.org\/nlab\/show\/Reeh-Schlieder+theorem)\n      - [Osterwalder-Schrader theorem](https:\/\/ncatlab.org\/nlab\/show\/Osterwalder-Schrader+theorem)\n      - [PCT theorem](https:\/\/ncatlab.org\/nlab\/show\/PCT+theorem)\n      - [Bisognano-Wichmann theorem](https:\/\/ncatlab.org\/nlab\/show\/Bisognano-Wichmann+theorem)\n        - [modular theory](https:\/\/ncatlab.org\/nlab\/show\/modular+theory)\n      - [spin-statistics theorem](https:\/\/ncatlab.org\/nlab\/show\/spin-statistics+theorem)\n        - [boson](https:\/\/ncatlab.org\/nlab\/show\/boson), [fermion](https:\/\/ncatlab.org\/nlab\/show\/fermion)\n    - [functorial QFT](https:\/\/ncatlab.org\/nlab\/show\/FQFT)\n      - [cobordism](https:\/\/ncatlab.org\/nlab\/show\/cobordism)\n      - [(∞,n)-category of cobordisms](https:\/\/ncatlab.org\/nlab\/show\/%28%E2%88%9E%2Cn%29-category+of+cobordisms)\n      - [cobordism hypothesis](https:\/\/ncatlab.org\/nlab\/show\/cobordism+hypothesis)\\-theorem\n      - [extended topological quantum field theory](https:\/\/ncatlab.org\/nlab\/show\/extended+topological+quantum+field+theory)\n  - Tools\n    - [perturbative quantum field theory](https:\/\/ncatlab.org\/nlab\/show\/perturbative+quantum+field+theory), [vacuum](https:\/\/ncatlab.org\/nlab\/show\/vacuum)\n    - [effective quantum field theory](https:\/\/ncatlab.org\/nlab\/show\/effective+quantum+field+theory)\n    - [renormalization](https:\/\/ncatlab.org\/nlab\/show\/renormalization)\n    - [BV-BRST formalism](https:\/\/ncatlab.org\/nlab\/show\/BV-BRST+formalism)\n    - [geometric ∞-function theory](https:\/\/ncatlab.org\/nlab\/show\/geometric+%E2%88%9E-function+theory)\n  - [particle physics](https:\/\/ncatlab.org\/nlab\/show\/particle+physics)\n    - [phenomenology](https:\/\/ncatlab.org\/nlab\/show\/phenomenology)\n    - [models](https:\/\/ncatlab.org\/nlab\/show\/model+%28in+particle+phyiscs%29)\n      - [standard model of particle physics](https:\/\/ncatlab.org\/nlab\/show\/standard+model+of+particle+physics)\n      - [fields and quanta](https:\/\/ncatlab.org\/nlab\/show\/fields+and+quanta+-+table)\n      - [Grand Unified Theories](https:\/\/ncatlab.org\/nlab\/show\/GUT), [MSSM](https:\/\/ncatlab.org\/nlab\/show\/MSSM)\n    - [scattering amplitude](https:\/\/ncatlab.org\/nlab\/show\/scattering+amplitude)\n      - [on-shell recursion](https:\/\/ncatlab.org\/nlab\/show\/on-shell+recursion), [KLT relations](https:\/\/ncatlab.org\/nlab\/show\/KLT+relations)\n  - Structural phenomena\n    - [universality class](https:\/\/ncatlab.org\/nlab\/show\/universality+class)\n    - [quantum anomaly](https:\/\/ncatlab.org\/nlab\/show\/quantum+anomaly)\n      - [Green-Schwarz mechanism](https:\/\/ncatlab.org\/nlab\/show\/Green-Schwarz+mechanism)\n    - [instanton](https:\/\/ncatlab.org\/nlab\/show\/instanton)\n    - [spontaneously broken symmetry](https:\/\/ncatlab.org\/nlab\/show\/spontaneously+broken+symmetry)\n    - [Kaluza-Klein mechanism](https:\/\/ncatlab.org\/nlab\/show\/Kaluza-Klein+mechanism)\n    - [integrable systems](https:\/\/ncatlab.org\/nlab\/show\/integrable+systems)\n    - [holonomic quantum fields](https:\/\/ncatlab.org\/nlab\/show\/holonomic+quantum+fields)\n  - Types of quantum field thories\n    - [TQFT](https:\/\/ncatlab.org\/nlab\/show\/TQFT)\n      - [2d TQFT](https:\/\/ncatlab.org\/nlab\/show\/2d+TQFT)\n      - [Dijkgraaf-Witten theory](https:\/\/ncatlab.org\/nlab\/show\/Dijkgraaf-Witten+theory)\n      - [Chern-Simons theory](https:\/\/ncatlab.org\/nlab\/show\/Chern-Simons+theory)\n    - [TCFT](https:\/\/ncatlab.org\/nlab\/show\/TCFT)\n      - [A-model](https:\/\/ncatlab.org\/nlab\/show\/A-model), [B-model](https:\/\/ncatlab.org\/nlab\/show\/B-model)\n      - [homological mirror symmetry](https:\/\/ncatlab.org\/nlab\/show\/homological+mirror+symmetry)\n    - [QFT with defects](https:\/\/ncatlab.org\/nlab\/show\/QFT+with+defects)\n    - [conformal field theory](https:\/\/ncatlab.org\/nlab\/show\/conformal+field+theory)\n    - [(1,1)-dimensional Euclidean field theories and K-theory](https:\/\/ncatlab.org\/nlab\/show\/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory)\n    - [(2,1)-dimensional Euclidean field theory and elliptic cohomology](https:\/\/ncatlab.org\/nlab\/show\/%282%2C1%29-dimensional+Euclidean+field+theory)\n    - [CFT](https:\/\/ncatlab.org\/nlab\/show\/CFT)\n      - [WZW model](https:\/\/ncatlab.org\/nlab\/show\/WZW+model)\n      - [6d (2,0)-supersymmetric QFT](https:\/\/ncatlab.org\/nlab\/show\/6d+%282%2C0%29-supersymmetric+QFT)\n    - [gauge theory](https:\/\/ncatlab.org\/nlab\/show\/gauge+theory)\n      - [field strength](https:\/\/ncatlab.org\/nlab\/show\/field+strength)\n      - [gauge group](https:\/\/ncatlab.org\/nlab\/show\/gauge+group), [gauge transformation](https:\/\/ncatlab.org\/nlab\/show\/gauge+transformation), [gauge fixing](https:\/\/ncatlab.org\/nlab\/show\/gauge+fixing)\n      - examples\n        - [electromagnetic field](https:\/\/ncatlab.org\/nlab\/show\/electromagnetic+field), [QED](https:\/\/ncatlab.org\/nlab\/show\/QED)\n      - [electric charge](https:\/\/ncatlab.org\/nlab\/show\/electric+charge)\n      - [magnetic charge](https:\/\/ncatlab.org\/nlab\/show\/magnetic+charge)\n        - [Yang-Mills field](https:\/\/ncatlab.org\/nlab\/show\/Yang-Mills+field), [QCD](https:\/\/ncatlab.org\/nlab\/show\/QCD)\n      - [Yang-Mills theory](https:\/\/ncatlab.org\/nlab\/show\/Yang-Mills+theory)\n      - [spinors in Yang-Mills theory](https:\/\/ncatlab.org\/nlab\/show\/spinors+in+Yang-Mills+theory)\n      - [topological Yang-Mills theory](https:\/\/ncatlab.org\/nlab\/show\/topological+Yang-Mills+theory)\n        - [Kalb-Ramond field](https:\/\/ncatlab.org\/nlab\/show\/Kalb-Ramond+field)\n        - [supergravity C-field](https:\/\/ncatlab.org\/nlab\/show\/supergravity+C-field)\n        - [RR field](https:\/\/ncatlab.org\/nlab\/show\/RR+field)\n        - [first-order formulation of gravity](https:\/\/ncatlab.org\/nlab\/show\/first-order+formulation+of+gravity)\n      - [general covariance](https:\/\/ncatlab.org\/nlab\/show\/general+covariance)\n      - [supergravity](https:\/\/ncatlab.org\/nlab\/show\/supergravity)\n      - [D'Auria-Fre formulation of supergravity](https:\/\/ncatlab.org\/nlab\/show\/D%27Auria-Fre+formulation+of+supergravity)\n      - [gravity as a BF-theory](https:\/\/ncatlab.org\/nlab\/show\/gravity+as+a+BF-theory)\n    - [sigma-model](https:\/\/ncatlab.org\/nlab\/show\/sigma-model)\n      - [particle](https:\/\/ncatlab.org\/nlab\/show\/particle), [relativistic particle](https:\/\/ncatlab.org\/nlab\/show\/relativistic+particle), [fundamental particle](https:\/\/ncatlab.org\/nlab\/show\/fundamental+particle), [spinning particle](https:\/\/ncatlab.org\/nlab\/show\/spinning+particle), [superparticle](https:\/\/ncatlab.org\/nlab\/show\/superparticle)\n      - [string](https:\/\/ncatlab.org\/nlab\/show\/string), [spinning string](https:\/\/ncatlab.org\/nlab\/show\/spinning+string), [superstring](https:\/\/ncatlab.org\/nlab\/show\/superstring)\n      - [membrane](https:\/\/ncatlab.org\/nlab\/show\/membrane)\n      - [AKSZ theory](https:\/\/ncatlab.org\/nlab\/show\/AKSZ+theory)\n- [String Theory](https:\/\/ncatlab.org\/nlab\/show\/string+theory)\n  - [string theory results applied elsewhere](https:\/\/ncatlab.org\/nlab\/show\/string+theory+results+applied+elsewhere)\n- [number theory and physics](https:\/\/ncatlab.org\/nlab\/show\/number+theory+and+physics)\n  - [Riemann hypothesis and physics](https:\/\/ncatlab.org\/nlab\/show\/Riemann+hypothesis+and+physics)\n## Contents\n- [Idea](https:\/\/ncatlab.org\/nlab\/show\/quantum+superposition#idea)\n- [Related concepts](https:\/\/ncatlab.org\/nlab\/show\/quantum+superposition#related_concepts)\n- [References](https:\/\/ncatlab.org\/nlab\/show\/quantum+superposition#references)\n## Idea\nIn [quantum theory](https:\/\/ncatlab.org\/nlab\/show\/quantum+theory) the [state](https:\/\/ncatlab.org\/nlab\/show\/state) of a [physical system](https:\/\/ncatlab.org\/nlab\/show\/physical+system) is given by a [vector](https:\/\/ncatlab.org\/nlab\/show\/vector) in a [complex vector space](https:\/\/ncatlab.org\/nlab\/show\/complex+vector+space) (usually a [Hilbert space](https:\/\/ncatlab.org\/nlab\/show\/Hilbert+space)). This means that given any two states ψ 1 \\\\psi\\_1 and ψ 2 \\\\psi\\_2, their sum, the vector ψ 1 \\+ ψ 2 \\\\psi\\_1 + \\\\psi\\_2, also represents a state. This is called the *quantum superposition* of the two states.\n\nIf these quantum states are thought of as [wavefunctions](https:\/\/ncatlab.org\/nlab\/show\/wavefunctions) solving a [linear differential equation](https:\/\/ncatlab.org\/nlab\/show\/linear+differential+equation) such as the [Schroedinger equation](https:\/\/ncatlab.org\/nlab\/show\/Schroedinger+equation), [Dirac equation](https:\/\/ncatlab.org\/nlab\/show\/Dirac+equation), [Klein-Gordon equation](https:\/\/ncatlab.org\/nlab\/show\/Klein-Gordon+equation) or [Tomonaga-Schwinger equation](https:\/\/ncatlab.org\/nlab\/show\/Tomonaga-Schwinger+equation), their quantum superposition is an example of the general principle of *[superposition](https:\/\/ncatlab.org\/nlab\/show\/superposition)* of solutions to [linear differential equations](https:\/\/ncatlab.org\/nlab\/show\/linear+differential+equations).\n\nThe concept of quantum superposition is at the heart of what makes [quantum physics](https:\/\/ncatlab.org\/nlab\/show\/quantum+physics) peculiar, notably via the phenomenon of [entanglement](https:\/\/ncatlab.org\/nlab\/show\/entanglement).\n\nThe various fundamental quantum physics phenomena at play here may be identified with four [modal](https:\/\/ncatlab.org\/nlab\/show\/modality)\\-[units](https:\/\/ncatlab.org\/nlab\/show\/unit+of+a+monad) of [quantum modal logic](https:\/\/ncatlab.org\/nlab\/show\/necessity+and+possibility#ModalQuantumLogic) (as discussed at *[quantum circuits via dependent linear types](https:\/\/ncatlab.org\/nlab\/show\/quantum+circuits+via+dependent+linear+types)*):\n\n![](https:\/\/ncatlab.org\/nlab\/files\/QCwthLHT-QuantumModalUnits-221110.jpg)\n\n|   |   |\n|---|---|\n| [quantum measurement](https:\/\/ncatlab.org\/nlab\/show\/quantum+measurement) | [quantum state preparation](https:\/\/ncatlab.org\/nlab\/show\/quantum+state+preparation) |\n| quantum superposition | [quantum parallelism](https:\/\/ncatlab.org\/nlab\/show\/quantum+parallelism) |\n\n- [quantum parallelism](https:\/\/ncatlab.org\/nlab\/show\/quantum+parallelism)\n- [superposition](https:\/\/ncatlab.org\/nlab\/show\/superposition)\n\n**[quantum probability theory](https:\/\/ncatlab.org\/nlab\/show\/quantum+probability+theory) – [observables](https:\/\/ncatlab.org\/nlab\/show\/observables) and [states](https:\/\/ncatlab.org\/nlab\/show\/states)**\n\n- **[states](https:\/\/ncatlab.org\/nlab\/show\/states)**\n  - [classical state](https:\/\/ncatlab.org\/nlab\/show\/classical+state)\n  - [quantum state](https:\/\/ncatlab.org\/nlab\/show\/quantum+state)\n    - [space of states (in geometric quantization)](https:\/\/ncatlab.org\/nlab\/show\/space+of+states+%28in+geometric+quantization%29)\n    - [state on a star-algebra](https:\/\/ncatlab.org\/nlab\/show\/state+on+a+star-algebra), [quasi-state](https:\/\/ncatlab.org\/nlab\/show\/quasi-state)\n    - [qbit](https:\/\/ncatlab.org\/nlab\/show\/qbit), [Bell state](https:\/\/ncatlab.org\/nlab\/show\/Bell+state)\n      [dimer](https:\/\/ncatlab.org\/nlab\/show\/dimer), [tensor network state](https:\/\/ncatlab.org\/nlab\/show\/tensor+network+state)\n    - [quantum state preparation](https:\/\/ncatlab.org\/nlab\/show\/quantum+state+preparation)\n    - [probability amplitude](https:\/\/ncatlab.org\/nlab\/show\/probability+amplitude), [quantum fluctuation](https:\/\/ncatlab.org\/nlab\/show\/quantum+fluctuation)\n    - [pure state](https:\/\/ncatlab.org\/nlab\/show\/pure+state)\n      - [wave function](https:\/\/ncatlab.org\/nlab\/show\/wave+function)\n      - [bra-ket](https:\/\/ncatlab.org\/nlab\/show\/bra-ket)\n      - [Bell state](https:\/\/ncatlab.org\/nlab\/show\/Bell+state)\n    - [quantum superposition](https:\/\/ncatlab.org\/nlab\/show\/quantum+superposition), [quantum interference](https:\/\/ncatlab.org\/nlab\/show\/quantum+interference)\n    - [quantum entanglement](https:\/\/ncatlab.org\/nlab\/show\/quantum+entanglement)\n    - [quantum measurement](https:\/\/ncatlab.org\/nlab\/show\/quantum+measurement)\n      [wave function collapse](https:\/\/ncatlab.org\/nlab\/show\/wave+function+collapse)\n      [Born rule](https:\/\/ncatlab.org\/nlab\/show\/Born+rule)\n      [deferred measurement principle](https:\/\/ncatlab.org\/nlab\/show\/deferred+measurement+principle)\n      [quantum reader monad](https:\/\/ncatlab.org\/nlab\/show\/quantum+reader+monad)\n      [measurement problem](https:\/\/ncatlab.org\/nlab\/show\/measurement+problem)\n    - [superselection sector](https:\/\/ncatlab.org\/nlab\/show\/superselection+sector)\n    - [mixed state](https:\/\/ncatlab.org\/nlab\/show\/mixed+state), [density matrix](https:\/\/ncatlab.org\/nlab\/show\/density+matrix)\n      [entanglement entropy](https:\/\/ncatlab.org\/nlab\/show\/entanglement+entropy)\n      [holographic entanglement entropy](https:\/\/ncatlab.org\/nlab\/show\/holographic+entanglement+entropy)\n    - [coherent quantum state](https:\/\/ncatlab.org\/nlab\/show\/coherent+quantum+state)\n  - [ground state](https:\/\/ncatlab.org\/nlab\/show\/ground+state), [excited state](https:\/\/ncatlab.org\/nlab\/show\/excited+state)\n  - [quasi-free state](https:\/\/ncatlab.org\/nlab\/show\/quasi-free+state)\n  - [Fock space](https:\/\/ncatlab.org\/nlab\/show\/Fock+space), [second quantization](https:\/\/ncatlab.org\/nlab\/show\/second+quantization)\n  - [vacuum](https:\/\/ncatlab.org\/nlab\/show\/vacuum), [vacuum state](https:\/\/ncatlab.org\/nlab\/show\/vacuum+state)\n    - [Hadamard state](https:\/\/ncatlab.org\/nlab\/show\/Hadamard+state)\n    - [vacuum diagram](https:\/\/ncatlab.org\/nlab\/show\/vacuum+diagram)\n    - [vacuum expectation value](https:\/\/ncatlab.org\/nlab\/show\/vacuum+expectation+value), [vacuum amplitude](https:\/\/ncatlab.org\/nlab\/show\/vacuum+amplitude), [vacuum fluctuation](https:\/\/ncatlab.org\/nlab\/show\/vacuum+fluctuation)\n    - [vacuum energy](https:\/\/ncatlab.org\/nlab\/show\/vacuum+energy)\n    - [vacuum polarization](https:\/\/ncatlab.org\/nlab\/show\/vacuum+polarization)\n    - [interacting vacuum](https:\/\/ncatlab.org\/nlab\/show\/interacting+vacuum)\n    - [thermal vacuum](https:\/\/ncatlab.org\/nlab\/show\/thermal+vacuum), [KMS state](https:\/\/ncatlab.org\/nlab\/show\/KMS+state)\n    - [vacuum stability](https:\/\/ncatlab.org\/nlab\/show\/vacuum+stability)\n      - [false vacuum](https:\/\/ncatlab.org\/nlab\/show\/false+vacuum), [tachyon](https:\/\/ncatlab.org\/nlab\/show\/tachyon), [Coleman-De Luccia instanton](https:\/\/ncatlab.org\/nlab\/show\/Coleman-De+Luccia+instanton)\n    - [theta vacuum](https:\/\/ncatlab.org\/nlab\/show\/theta+vacuum)\n    - [perturbative string theory vacuum](https:\/\/ncatlab.org\/nlab\/show\/perturbative+string+theory+vacuum)\n      - [non-geometric string theory vacuum](https:\/\/ncatlab.org\/nlab\/show\/non-geometric+string+theory+vacuum)\n      - [landscape of string theory vacua](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua)\n  - [entangled state](https:\/\/ncatlab.org\/nlab\/show\/entangled+state)\n  - [tensor network state](https:\/\/ncatlab.org\/nlab\/show\/tensor+network+state)\n    - [matrix product state](https:\/\/ncatlab.org\/nlab\/show\/matrix+product+state)\n    - [tree tensor network state](https:\/\/ncatlab.org\/nlab\/show\/tree+tensor+network+state)\n- **[observables](https:\/\/ncatlab.org\/nlab\/show\/observables)**\n  - [quantum observable](https:\/\/ncatlab.org\/nlab\/show\/quantum+observable), [beable](https:\/\/ncatlab.org\/nlab\/show\/beable)\n  - [algebra of observables](https:\/\/ncatlab.org\/nlab\/show\/algebra+of+observables), [star-algebra](https:\/\/ncatlab.org\/nlab\/show\/star-algebra)\n  - [Bohr topos](https:\/\/ncatlab.org\/nlab\/show\/Bohr+topos)\n  - [quantum operator (in geometric quantization)](https:\/\/ncatlab.org\/nlab\/show\/quantum+operator+%28in+geometric+quantization%29)\n  - [quantum operation](https:\/\/ncatlab.org\/nlab\/show\/quantum+operation), [quantum effect](https:\/\/ncatlab.org\/nlab\/show\/quantum+effect), [effect algebra](https:\/\/ncatlab.org\/nlab\/show\/effect+algebra)\n  - in [quantum field theory](https:\/\/ncatlab.org\/nlab\/show\/quantum+field+theory)\n    - [local observable](https:\/\/ncatlab.org\/nlab\/show\/local+observable)\n    - [polynomial observable](https:\/\/ncatlab.org\/nlab\/show\/polynomial+observable)\n      - [linear observable](https:\/\/ncatlab.org\/nlab\/show\/linear+observable)\n        - [field observable](https:\/\/ncatlab.org\/nlab\/show\/field+observable)\n    - [regular observable](https:\/\/ncatlab.org\/nlab\/show\/regular+observable)\n    - [microcausal observable](https:\/\/ncatlab.org\/nlab\/show\/microcausal+observable)\n    - [normal-ordered product](https:\/\/ncatlab.org\/nlab\/show\/normal-ordered+product), [time-ordered products](https:\/\/ncatlab.org\/nlab\/show\/time-ordered+products), [retarded product](https:\/\/ncatlab.org\/nlab\/show\/retarded+product)\n    - [Wick algebra](https:\/\/ncatlab.org\/nlab\/show\/Wick+algebra)\n    - [scattering amplitude](https:\/\/ncatlab.org\/nlab\/show\/scattering+amplitude)\n    - [interacting field algebra of observables](https:\/\/ncatlab.org\/nlab\/show\/interacting+field+algebra+of+observables), [Bogoliubov's formula](https:\/\/ncatlab.org\/nlab\/show\/Bogoliubov%27s+formula)\n- [GNS construction](https:\/\/ncatlab.org\/nlab\/show\/GNS+construction)\n- [theorems](https:\/\/ncatlab.org\/nlab\/show\/theorems)\n  - [order-theoretic structure in quantum mechanics](https:\/\/ncatlab.org\/nlab\/show\/order-theoretic+structure+in+quantum+mechanics)\n    - [Gleason's theorem](https:\/\/ncatlab.org\/nlab\/show\/Gleason%27s+theorem)\n    - [Alfsen-Shultz theorem](https:\/\/ncatlab.org\/nlab\/show\/Alfsen-Shultz+theorem)\n    - [Harding-Döring-Hamhalter theorem](https:\/\/ncatlab.org\/nlab\/show\/Harding-D%C3%B6ring-Hamhalter+theorem)\n    - [Kochen-Specker theorem](https:\/\/ncatlab.org\/nlab\/show\/Kochen-Specker+theorem)\n    - [Nuiten's lemma](https:\/\/ncatlab.org\/nlab\/show\/Nuiten%27s+lemma)\n  - [Wigner's theorem](https:\/\/ncatlab.org\/nlab\/show\/Wigner%27s+theorem)\n  - [no-cloning theorem](https:\/\/ncatlab.org\/nlab\/show\/no-cloning+theorem)\n  - [Bell's theorem](https:\/\/ncatlab.org\/nlab\/show\/Bell%27s+theorem)\n## References\nSee also\n\n- Wikipedia, *[Quantum superposition](https:\/\/en.wikipedia.org\/wiki\/Quantum_superposition)*\n\nLast revised on January 13, 2023 at 22:36:23. See the [history](https:\/\/ncatlab.org\/nlab\/history\/quantum+superposition) of this page for a list of all contributions to it.","meta_canonical":null}

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Boilerpipe Text	Contents Context Physics physics , mathematical physics , philosophy of physics Surveys, textbooks and lecture notes (higher) category theory and physics geometry of physics books and reviews , physics resources theory (physics) , model (physics) experiment , measurement , computable physics mechanics mass , charge , momentum , angular momentum , moment of inertia dynamics on Lie groups rigid body dynamics field (physics) Lagrangian mechanics configuration space , state action functional , Lagrangian covariant phase space , Euler-Lagrange equations Hamiltonian mechanics phase space symplectic geometry Poisson manifold symplectic manifold symplectic groupoid multisymplectic geometry n-symplectic manifold spacetime smooth Lorentzian manifold special relativity general relativity gravity supergravity , dilaton gravity black hole Classical field theory classical physics classical mechanics waves and optics thermodynamics Quantum Mechanics in terms of ∞-compact categories quantum information Hamiltonian operator density matrix Kochen-Specker theorem Bell's theorem Gleason's theorem Quantization geometric quantization deformation quantization path integral quantization semiclassical approximation Quantum Field Theory Axiomatizations algebraic QFT Wightman axioms Haag-Kastler axioms operator algebra local net conformal net Reeh-Schlieder theorem Osterwalder-Schrader theorem PCT theorem Bisognano-Wichmann theorem modular theory spin-statistics theorem boson , fermion functorial QFT cobordism (∞,n)-category of cobordisms cobordism hypothesis -theorem extended topological quantum field theory Tools perturbative quantum field theory , vacuum effective quantum field theory renormalization BV-BRST formalism geometric ∞-function theory particle physics phenomenology models standard model of particle physics fields and quanta Grand Unified Theories , MSSM scattering amplitude on-shell recursion , KLT relations Structural phenomena universality class quantum anomaly Green-Schwarz mechanism instanton spontaneously broken symmetry Kaluza-Klein mechanism integrable systems holonomic quantum fields Types of quantum field thories TQFT 2d TQFT Dijkgraaf-Witten theory Chern-Simons theory TCFT A-model , B-model homological mirror symmetry QFT with defects conformal field theory (1,1)-dimensional Euclidean field theories and K-theory (2,1)-dimensional Euclidean field theory and elliptic cohomology CFT WZW model 6d (2,0)-supersymmetric QFT gauge theory field strength gauge group , gauge transformation , gauge fixing examples electromagnetic field , QED electric charge magnetic charge Yang-Mills field , QCD Yang-Mills theory spinors in Yang-Mills theory topological Yang-Mills theory Kalb-Ramond field supergravity C-field RR field first-order formulation of gravity general covariance supergravity D'Auria-Fre formulation of supergravity gravity as a BF-theory sigma-model particle , relativistic particle , fundamental particle , spinning particle , superparticle string , spinning string , superstring membrane AKSZ theory String Theory string theory results applied elsewhere number theory and physics Riemann hypothesis and physics Contents Idea Related concepts References Idea In quantum theory the state of a physical system is given by a vector in a complex vector space (usually a Hilbert space ). This means that given any two states ψ 1 \psi_1 and ψ 2 \psi_2 , their sum, the vector ψ 1 + ψ 2 \psi_1 + \psi_2 , also represents a state. This is called the quantum superposition of the two states. If these quantum states are thought of as wavefunctions solving a linear differential equation such as the Schroedinger equation , Dirac equation , Klein-Gordon equation or Tomonaga-Schwinger equation , their quantum superposition is an example of the general principle of superposition of solutions to linear differential equations . The concept of quantum superposition is at the heart of what makes quantum physics peculiar, notably via the phenomenon of entanglement . The various fundamental quantum physics phenomena at play here may be identified with four modal - units of quantum modal logic (as discussed at quantum circuits via dependent linear types ): quantum measurement quantum state preparation quantum superposition quantum parallelism quantum parallelism superposition quantum probability theory – observables and states states classical state quantum state space of states (in geometric quantization) state on a star-algebra , quasi-state qbit , Bell state dimer , tensor network state quantum state preparation probability amplitude , quantum fluctuation pure state wave function bra-ket Bell state quantum superposition , quantum interference quantum entanglement quantum measurement wave function collapse Born rule deferred measurement principle quantum reader monad measurement problem superselection sector mixed state , density matrix entanglement entropy holographic entanglement entropy coherent quantum state ground state , excited state quasi-free state Fock space , second quantization vacuum , vacuum state Hadamard state vacuum diagram vacuum expectation value , vacuum amplitude , vacuum fluctuation vacuum energy vacuum polarization interacting vacuum thermal vacuum , KMS state vacuum stability false vacuum , tachyon , Coleman-De Luccia instanton theta vacuum perturbative string theory vacuum non-geometric string theory vacuum landscape of string theory vacua entangled state tensor network state matrix product state tree tensor network state observables quantum observable , beable algebra of observables , star-algebra Bohr topos quantum operator (in geometric quantization) quantum operation , quantum effect , effect algebra in quantum field theory local observable polynomial observable linear observable field observable regular observable microcausal observable normal-ordered product , time-ordered products , retarded product Wick algebra scattering amplitude interacting field algebra of observables , Bogoliubov's formula GNS construction theorems order-theoretic structure in quantum mechanics Gleason's theorem Alfsen-Shultz theorem Harding-Döring-Hamhalter theorem Kochen-Specker theorem Nuiten's lemma Wigner's theorem no-cloning theorem Bell's theorem References See also Wikipedia, Quantum superposition Last revised on January 13, 2023 at 22:36:23. See the history of this page for a list of all contributions to it.
Markdown	# nLab quantum superposition [Skip the Navigation Links](https://ncatlab.org/nlab/show/quantum+superposition#navEnd) \\| [Home Page](https://ncatlab.org/nlab/show/HomePage "Home page") \\| [All Pages](https://ncatlab.org/nlab/all_pages "List of all pages") \\| [Latest Revisions](https://ncatlab.org/nlab/latest_revisions "Latest edits and page creations") \\| [Discuss this page](https://nforum.ncatlab.org/discussion/14313/#Item_5 "Discuss this page in its dedicated thread on the nForum") \\| ### Context #### Physics [physics](https://ncatlab.org/nlab/show/physics), [mathematical physics](https://ncatlab.org/nlab/show/mathematical+physics), [philosophy of physics](https://ncatlab.org/nlab/show/philosophy+of+physics) ## Surveys, textbooks and lecture notes - [(higher) category theory and physics](https://ncatlab.org/nlab/show/higher+category+theory+and+physics) - [geometry of physics](https://ncatlab.org/nlab/show/geometry+of+physics) - [books and reviews](https://ncatlab.org/nlab/show/books+and+reviews+in+mathematical+physics), [physics resources](https://ncatlab.org/nlab/show/physics+resources) * [theory (physics)](https://ncatlab.org/nlab/show/theory+%28physics%29), [model (physics)](https://ncatlab.org/nlab/show/model+%28physics%29) [experiment](https://ncatlab.org/nlab/show/experiment), [measurement](https://ncatlab.org/nlab/show/measurement), [computable physics](https://ncatlab.org/nlab/show/computable+physics) - [mechanics](https://ncatlab.org/nlab/show/mechanics) - [mass](https://ncatlab.org/nlab/show/mass), [charge](https://ncatlab.org/nlab/show/charge), [momentum](https://ncatlab.org/nlab/show/momentum), [angular momentum](https://ncatlab.org/nlab/show/angular+momentum), [moment of inertia](https://ncatlab.org/nlab/show/moment+of+inertia) - [dynamics on Lie groups](https://ncatlab.org/nlab/show/dynamics+on+Lie+groups) - [rigid body dynamics](https://ncatlab.org/nlab/show/rigid+body+dynamics) - [field (physics)](https://ncatlab.org/nlab/show/field+%28physics%29) - [Lagrangian mechanics](https://ncatlab.org/nlab/show/Lagrangian+mechanics) - [configuration space](https://ncatlab.org/nlab/show/configuration+space), [state](https://ncatlab.org/nlab/show/state) - [action functional](https://ncatlab.org/nlab/show/action+functional), [Lagrangian](https://ncatlab.org/nlab/show/Lagrangian) - [covariant phase space](https://ncatlab.org/nlab/show/covariant+phase+space), [Euler-Lagrange equations](https://ncatlab.org/nlab/show/Euler-Lagrange+equations) - [Hamiltonian mechanics](https://ncatlab.org/nlab/show/Hamiltonian+mechanics) - [phase space](https://ncatlab.org/nlab/show/phase+space) - [symplectic geometry](https://ncatlab.org/nlab/show/symplectic+geometry) - [Poisson manifold](https://ncatlab.org/nlab/show/Poisson+manifold) - [symplectic manifold](https://ncatlab.org/nlab/show/symplectic+manifold) - [symplectic groupoid](https://ncatlab.org/nlab/show/symplectic+groupoid) - [multisymplectic geometry](https://ncatlab.org/nlab/show/multisymplectic+geometry) - [n-symplectic manifold](https://ncatlab.org/nlab/show/n-symplectic+manifold) - [spacetime](https://ncatlab.org/nlab/show/spacetime) - [smooth Lorentzian manifold](https://ncatlab.org/nlab/show/smooth+Lorentzian+manifold) - [special relativity](https://ncatlab.org/nlab/show/special+relativity) - [general relativity](https://ncatlab.org/nlab/show/general+relativity) - [gravity](https://ncatlab.org/nlab/show/gravity) - [supergravity](https://ncatlab.org/nlab/show/supergravity), [dilaton gravity](https://ncatlab.org/nlab/show/dilaton+gravity) - [black hole](https://ncatlab.org/nlab/show/black+hole) - [Classical field theory](https://ncatlab.org/nlab/show/classical+field+theory) - [classical physics](https://ncatlab.org/nlab/show/classical+physics) - [classical mechanics](https://ncatlab.org/nlab/show/classical+mechanics) - [waves](https://ncatlab.org/nlab/show/waves) and [optics](https://ncatlab.org/nlab/show/optics) - [thermodynamics](https://ncatlab.org/nlab/show/thermodynamics) - [Quantum Mechanics](https://ncatlab.org/nlab/show/quantum+mechanics) - [in terms of ∞-compact categories](https://ncatlab.org/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories) - [quantum information](https://ncatlab.org/nlab/show/quantum+information) - [Hamiltonian operator](https://ncatlab.org/nlab/show/Hamiltonian+operator) - [density matrix](https://ncatlab.org/nlab/show/density+matrix) - [Kochen-Specker theorem](https://ncatlab.org/nlab/show/Kochen-Specker+theorem) - [Bell's theorem](https://ncatlab.org/nlab/show/Bell%27s+theorem) - [Gleason's theorem](https://ncatlab.org/nlab/show/Gleason%27s+theorem) - [Quantization](https://ncatlab.org/nlab/show/quantization) - [geometric quantization](https://ncatlab.org/nlab/show/geometric+quantization) - [deformation quantization](https://ncatlab.org/nlab/show/deformation+quantization) - [path integral quantization](https://ncatlab.org/nlab/show/path+integral) - [semiclassical approximation](https://ncatlab.org/nlab/show/semiclassical+approximation) - [Quantum Field Theory](https://ncatlab.org/nlab/show/quantum+field+theory)** - Axiomatizations - [algebraic QFT](https://ncatlab.org/nlab/show/AQFT) - [Wightman axioms](https://ncatlab.org/nlab/show/Wightman+axioms) - [Haag-Kastler axioms](https://ncatlab.org/nlab/show/Haag-Kastler+axioms) - [operator algebra](https://ncatlab.org/nlab/show/operator+algebra) - [local net](https://ncatlab.org/nlab/show/local+net) - [conformal net](https://ncatlab.org/nlab/show/conformal+net) - [Reeh-Schlieder theorem](https://ncatlab.org/nlab/show/Reeh-Schlieder+theorem) - [Osterwalder-Schrader theorem](https://ncatlab.org/nlab/show/Osterwalder-Schrader+theorem) - [PCT theorem](https://ncatlab.org/nlab/show/PCT+theorem) - [Bisognano-Wichmann theorem](https://ncatlab.org/nlab/show/Bisognano-Wichmann+theorem) - [modular theory](https://ncatlab.org/nlab/show/modular+theory) - [spin-statistics theorem](https://ncatlab.org/nlab/show/spin-statistics+theorem) - [boson](https://ncatlab.org/nlab/show/boson), [fermion](https://ncatlab.org/nlab/show/fermion) - [functorial QFT](https://ncatlab.org/nlab/show/FQFT) - [cobordism](https://ncatlab.org/nlab/show/cobordism) - [(∞,n)-category of cobordisms](https://ncatlab.org/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms) - [cobordism hypothesis](https://ncatlab.org/nlab/show/cobordism+hypothesis)\-theorem - [extended topological quantum field theory](https://ncatlab.org/nlab/show/extended+topological+quantum+field+theory) - Tools - [perturbative quantum field theory](https://ncatlab.org/nlab/show/perturbative+quantum+field+theory), [vacuum](https://ncatlab.org/nlab/show/vacuum) - [effective quantum field theory](https://ncatlab.org/nlab/show/effective+quantum+field+theory) - [renormalization](https://ncatlab.org/nlab/show/renormalization) - [BV-BRST formalism](https://ncatlab.org/nlab/show/BV-BRST+formalism) - [geometric ∞-function theory](https://ncatlab.org/nlab/show/geometric+%E2%88%9E-function+theory) - [particle physics](https://ncatlab.org/nlab/show/particle+physics) - [phenomenology](https://ncatlab.org/nlab/show/phenomenology) - [models](https://ncatlab.org/nlab/show/model+%28in+particle+phyiscs%29) - [standard model of particle physics](https://ncatlab.org/nlab/show/standard+model+of+particle+physics) - [fields and quanta](https://ncatlab.org/nlab/show/fields+and+quanta+-+table) - [Grand Unified Theories](https://ncatlab.org/nlab/show/GUT), [MSSM](https://ncatlab.org/nlab/show/MSSM) - [scattering amplitude](https://ncatlab.org/nlab/show/scattering+amplitude) - [on-shell recursion](https://ncatlab.org/nlab/show/on-shell+recursion), [KLT relations](https://ncatlab.org/nlab/show/KLT+relations) - Structural phenomena - [universality class](https://ncatlab.org/nlab/show/universality+class) - [quantum anomaly](https://ncatlab.org/nlab/show/quantum+anomaly) - [Green-Schwarz mechanism](https://ncatlab.org/nlab/show/Green-Schwarz+mechanism) - [instanton](https://ncatlab.org/nlab/show/instanton) - [spontaneously broken symmetry](https://ncatlab.org/nlab/show/spontaneously+broken+symmetry) - [Kaluza-Klein mechanism](https://ncatlab.org/nlab/show/Kaluza-Klein+mechanism) - [integrable systems](https://ncatlab.org/nlab/show/integrable+systems) - [holonomic quantum fields](https://ncatlab.org/nlab/show/holonomic+quantum+fields) - Types of quantum field thories - [TQFT](https://ncatlab.org/nlab/show/TQFT) - [2d TQFT](https://ncatlab.org/nlab/show/2d+TQFT) - [Dijkgraaf-Witten theory](https://ncatlab.org/nlab/show/Dijkgraaf-Witten+theory) - [Chern-Simons theory](https://ncatlab.org/nlab/show/Chern-Simons+theory) - [TCFT](https://ncatlab.org/nlab/show/TCFT) - [A-model](https://ncatlab.org/nlab/show/A-model), [B-model](https://ncatlab.org/nlab/show/B-model) - [homological mirror symmetry](https://ncatlab.org/nlab/show/homological+mirror+symmetry) - [QFT with defects](https://ncatlab.org/nlab/show/QFT+with+defects) - [conformal field theory](https://ncatlab.org/nlab/show/conformal+field+theory) - [(1,1)-dimensional Euclidean field theories and K-theory](https://ncatlab.org/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory) - [(2,1)-dimensional Euclidean field theory and elliptic cohomology](https://ncatlab.org/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory) - [CFT](https://ncatlab.org/nlab/show/CFT) - [WZW model](https://ncatlab.org/nlab/show/WZW+model) - [6d (2,0)-supersymmetric QFT](https://ncatlab.org/nlab/show/6d+%282%2C0%29-supersymmetric+QFT) - [gauge theory](https://ncatlab.org/nlab/show/gauge+theory) - [field strength](https://ncatlab.org/nlab/show/field+strength) - [gauge group](https://ncatlab.org/nlab/show/gauge+group), [gauge transformation](https://ncatlab.org/nlab/show/gauge+transformation), [gauge fixing](https://ncatlab.org/nlab/show/gauge+fixing) - examples - [electromagnetic field](https://ncatlab.org/nlab/show/electromagnetic+field), [QED](https://ncatlab.org/nlab/show/QED) - [electric charge](https://ncatlab.org/nlab/show/electric+charge) - [magnetic charge](https://ncatlab.org/nlab/show/magnetic+charge) - [Yang-Mills field](https://ncatlab.org/nlab/show/Yang-Mills+field), [QCD](https://ncatlab.org/nlab/show/QCD) - [Yang-Mills theory](https://ncatlab.org/nlab/show/Yang-Mills+theory) - [spinors in Yang-Mills theory](https://ncatlab.org/nlab/show/spinors+in+Yang-Mills+theory) - [topological Yang-Mills theory](https://ncatlab.org/nlab/show/topological+Yang-Mills+theory) - [Kalb-Ramond field](https://ncatlab.org/nlab/show/Kalb-Ramond+field) - [supergravity C-field](https://ncatlab.org/nlab/show/supergravity+C-field) - [RR field](https://ncatlab.org/nlab/show/RR+field) - [first-order formulation of gravity](https://ncatlab.org/nlab/show/first-order+formulation+of+gravity) - [general covariance](https://ncatlab.org/nlab/show/general+covariance) - [supergravity](https://ncatlab.org/nlab/show/supergravity) - [D'Auria-Fre formulation of supergravity](https://ncatlab.org/nlab/show/D%27Auria-Fre+formulation+of+supergravity) - [gravity as a BF-theory](https://ncatlab.org/nlab/show/gravity+as+a+BF-theory) - [sigma-model](https://ncatlab.org/nlab/show/sigma-model) - [particle](https://ncatlab.org/nlab/show/particle), [relativistic particle](https://ncatlab.org/nlab/show/relativistic+particle), [fundamental particle](https://ncatlab.org/nlab/show/fundamental+particle), [spinning particle](https://ncatlab.org/nlab/show/spinning+particle), [superparticle](https://ncatlab.org/nlab/show/superparticle) - [string](https://ncatlab.org/nlab/show/string), [spinning string](https://ncatlab.org/nlab/show/spinning+string), [superstring](https://ncatlab.org/nlab/show/superstring) - [membrane](https://ncatlab.org/nlab/show/membrane) - [AKSZ theory](https://ncatlab.org/nlab/show/AKSZ+theory) - [String Theory](https://ncatlab.org/nlab/show/string+theory) - [string theory results applied elsewhere](https://ncatlab.org/nlab/show/string+theory+results+applied+elsewhere) - [number theory and physics](https://ncatlab.org/nlab/show/number+theory+and+physics) - [Riemann hypothesis and physics](https://ncatlab.org/nlab/show/Riemann+hypothesis+and+physics) [Edit this sidebar](https://ncatlab.org/nlab/edit/physicscontents) # Contents - [Idea](https://ncatlab.org/nlab/show/quantum+superposition#idea) - [Related concepts](https://ncatlab.org/nlab/show/quantum+superposition#related_concepts) - [References](https://ncatlab.org/nlab/show/quantum+superposition#references) ## Idea In [quantum theory](https://ncatlab.org/nlab/show/quantum+theory) the [state](https://ncatlab.org/nlab/show/state) of a [physical system](https://ncatlab.org/nlab/show/physical+system) is given by a [vector](https://ncatlab.org/nlab/show/vector) in a [complex vector space](https://ncatlab.org/nlab/show/complex+vector+space) (usually a [Hilbert space](https://ncatlab.org/nlab/show/Hilbert+space)). This means that given any two states ψ 1 \\psi\_1 and ψ 2 \\psi\_2, their sum, the vector ψ 1 \+ ψ 2 \\psi\_1 + \\psi\_2, also represents a state. This is called the quantum superposition of the two states. If these quantum states are thought of as [wavefunctions](https://ncatlab.org/nlab/show/wavefunctions) solving a [linear differential equation](https://ncatlab.org/nlab/show/linear+differential+equation) such as the [Schroedinger equation](https://ncatlab.org/nlab/show/Schroedinger+equation), [Dirac equation](https://ncatlab.org/nlab/show/Dirac+equation), [Klein-Gordon equation](https://ncatlab.org/nlab/show/Klein-Gordon+equation) or [Tomonaga-Schwinger equation](https://ncatlab.org/nlab/show/Tomonaga-Schwinger+equation), their quantum superposition is an example of the general principle of [superposition](https://ncatlab.org/nlab/show/superposition) of solutions to [linear differential equations](https://ncatlab.org/nlab/show/linear+differential+equations). The concept of quantum superposition is at the heart of what makes [quantum physics](https://ncatlab.org/nlab/show/quantum+physics) peculiar, notably via the phenomenon of [entanglement](https://ncatlab.org/nlab/show/entanglement). The various fundamental quantum physics phenomena at play here may be identified with four [modal](https://ncatlab.org/nlab/show/modality)\-[units](https://ncatlab.org/nlab/show/unit+of+a+monad) of [quantum modal logic](https://ncatlab.org/nlab/show/necessity+and+possibility#ModalQuantumLogic) (as discussed at [quantum circuits via dependent linear types](https://ncatlab.org/nlab/show/quantum+circuits+via+dependent+linear+types)): ![](https://ncatlab.org/nlab/files/QCwthLHT-QuantumModalUnits-221110.jpg) \| \| \| \|---\|---\| \| [quantum measurement](https://ncatlab.org/nlab/show/quantum+measurement) \| [quantum state preparation](https://ncatlab.org/nlab/show/quantum+state+preparation) \| \| quantum superposition \| [quantum parallelism](https://ncatlab.org/nlab/show/quantum+parallelism) \| ## Related concepts - [quantum parallelism](https://ncatlab.org/nlab/show/quantum+parallelism) - [superposition](https://ncatlab.org/nlab/show/superposition) [quantum probability theory](https://ncatlab.org/nlab/show/quantum+probability+theory) – [observables](https://ncatlab.org/nlab/show/observables) and [states](https://ncatlab.org/nlab/show/states) - [states](https://ncatlab.org/nlab/show/states) - [classical state](https://ncatlab.org/nlab/show/classical+state) - [quantum state](https://ncatlab.org/nlab/show/quantum+state) - [space of states (in geometric quantization)](https://ncatlab.org/nlab/show/space+of+states+%28in+geometric+quantization%29) - [state on a star-algebra](https://ncatlab.org/nlab/show/state+on+a+star-algebra), [quasi-state](https://ncatlab.org/nlab/show/quasi-state) - [qbit](https://ncatlab.org/nlab/show/qbit), [Bell state](https://ncatlab.org/nlab/show/Bell+state) [dimer](https://ncatlab.org/nlab/show/dimer), [tensor network state](https://ncatlab.org/nlab/show/tensor+network+state) - [quantum state preparation](https://ncatlab.org/nlab/show/quantum+state+preparation) - [probability amplitude](https://ncatlab.org/nlab/show/probability+amplitude), [quantum fluctuation](https://ncatlab.org/nlab/show/quantum+fluctuation) - [pure state](https://ncatlab.org/nlab/show/pure+state) - [wave function](https://ncatlab.org/nlab/show/wave+function) - [bra-ket](https://ncatlab.org/nlab/show/bra-ket) - [Bell state](https://ncatlab.org/nlab/show/Bell+state) - [quantum superposition](https://ncatlab.org/nlab/show/quantum+superposition), [quantum interference](https://ncatlab.org/nlab/show/quantum+interference) - [quantum entanglement](https://ncatlab.org/nlab/show/quantum+entanglement) - [quantum measurement](https://ncatlab.org/nlab/show/quantum+measurement) [wave function collapse](https://ncatlab.org/nlab/show/wave+function+collapse) [Born rule](https://ncatlab.org/nlab/show/Born+rule) [deferred measurement principle](https://ncatlab.org/nlab/show/deferred+measurement+principle) [quantum reader monad](https://ncatlab.org/nlab/show/quantum+reader+monad) [measurement problem](https://ncatlab.org/nlab/show/measurement+problem) - [superselection sector](https://ncatlab.org/nlab/show/superselection+sector) - [mixed state](https://ncatlab.org/nlab/show/mixed+state), [density matrix](https://ncatlab.org/nlab/show/density+matrix) [entanglement entropy](https://ncatlab.org/nlab/show/entanglement+entropy) [holographic entanglement entropy](https://ncatlab.org/nlab/show/holographic+entanglement+entropy) - [coherent quantum state](https://ncatlab.org/nlab/show/coherent+quantum+state) - [ground state](https://ncatlab.org/nlab/show/ground+state), [excited state](https://ncatlab.org/nlab/show/excited+state) - [quasi-free state](https://ncatlab.org/nlab/show/quasi-free+state) - [Fock space](https://ncatlab.org/nlab/show/Fock+space), [second quantization](https://ncatlab.org/nlab/show/second+quantization) - [vacuum](https://ncatlab.org/nlab/show/vacuum), [vacuum state](https://ncatlab.org/nlab/show/vacuum+state) - [Hadamard state](https://ncatlab.org/nlab/show/Hadamard+state) - [vacuum diagram](https://ncatlab.org/nlab/show/vacuum+diagram) - [vacuum expectation value](https://ncatlab.org/nlab/show/vacuum+expectation+value), [vacuum amplitude](https://ncatlab.org/nlab/show/vacuum+amplitude), [vacuum fluctuation](https://ncatlab.org/nlab/show/vacuum+fluctuation) - [vacuum energy](https://ncatlab.org/nlab/show/vacuum+energy) - [vacuum polarization](https://ncatlab.org/nlab/show/vacuum+polarization) - [interacting vacuum](https://ncatlab.org/nlab/show/interacting+vacuum) - [thermal vacuum](https://ncatlab.org/nlab/show/thermal+vacuum), [KMS state](https://ncatlab.org/nlab/show/KMS+state) - [vacuum stability](https://ncatlab.org/nlab/show/vacuum+stability) - [false vacuum](https://ncatlab.org/nlab/show/false+vacuum), [tachyon](https://ncatlab.org/nlab/show/tachyon), [Coleman-De Luccia instanton](https://ncatlab.org/nlab/show/Coleman-De+Luccia+instanton) - [theta vacuum](https://ncatlab.org/nlab/show/theta+vacuum) - [perturbative string theory vacuum](https://ncatlab.org/nlab/show/perturbative+string+theory+vacuum) - [non-geometric string theory vacuum](https://ncatlab.org/nlab/show/non-geometric+string+theory+vacuum) - [landscape of string theory vacua](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua) - [entangled state](https://ncatlab.org/nlab/show/entangled+state) - [tensor network state](https://ncatlab.org/nlab/show/tensor+network+state) - [matrix product state](https://ncatlab.org/nlab/show/matrix+product+state) - [tree tensor network state](https://ncatlab.org/nlab/show/tree+tensor+network+state) - [observables](https://ncatlab.org/nlab/show/observables) - [quantum observable](https://ncatlab.org/nlab/show/quantum+observable), [beable](https://ncatlab.org/nlab/show/beable) - [algebra of observables](https://ncatlab.org/nlab/show/algebra+of+observables), [star-algebra](https://ncatlab.org/nlab/show/star-algebra) - [Bohr topos](https://ncatlab.org/nlab/show/Bohr+topos) - [quantum operator (in geometric quantization)](https://ncatlab.org/nlab/show/quantum+operator+%28in+geometric+quantization%29) - [quantum operation](https://ncatlab.org/nlab/show/quantum+operation), [quantum effect](https://ncatlab.org/nlab/show/quantum+effect), [effect algebra](https://ncatlab.org/nlab/show/effect+algebra) - in [quantum field theory](https://ncatlab.org/nlab/show/quantum+field+theory) - [local observable](https://ncatlab.org/nlab/show/local+observable) - [polynomial observable](https://ncatlab.org/nlab/show/polynomial+observable) - [linear observable](https://ncatlab.org/nlab/show/linear+observable) - [field observable](https://ncatlab.org/nlab/show/field+observable) - [regular observable](https://ncatlab.org/nlab/show/regular+observable) - [microcausal observable](https://ncatlab.org/nlab/show/microcausal+observable) - [normal-ordered product](https://ncatlab.org/nlab/show/normal-ordered+product), [time-ordered products](https://ncatlab.org/nlab/show/time-ordered+products), [retarded product](https://ncatlab.org/nlab/show/retarded+product) - [Wick algebra](https://ncatlab.org/nlab/show/Wick+algebra) - [scattering amplitude](https://ncatlab.org/nlab/show/scattering+amplitude) - [interacting field algebra of observables](https://ncatlab.org/nlab/show/interacting+field+algebra+of+observables), [Bogoliubov's formula](https://ncatlab.org/nlab/show/Bogoliubov%27s+formula) - [GNS construction](https://ncatlab.org/nlab/show/GNS+construction) - [theorems](https://ncatlab.org/nlab/show/theorems) - [order-theoretic structure in quantum mechanics](https://ncatlab.org/nlab/show/order-theoretic+structure+in+quantum+mechanics) - [Gleason's theorem](https://ncatlab.org/nlab/show/Gleason%27s+theorem) - [Alfsen-Shultz theorem](https://ncatlab.org/nlab/show/Alfsen-Shultz+theorem) - [Harding-Döring-Hamhalter theorem](https://ncatlab.org/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem) - [Kochen-Specker theorem](https://ncatlab.org/nlab/show/Kochen-Specker+theorem) - [Nuiten's lemma](https://ncatlab.org/nlab/show/Nuiten%27s+lemma) - [Wigner's theorem](https://ncatlab.org/nlab/show/Wigner%27s+theorem) - [no-cloning theorem](https://ncatlab.org/nlab/show/no-cloning+theorem) - [Bell's theorem](https://ncatlab.org/nlab/show/Bell%27s+theorem) ## References See also - Wikipedia, [Quantum superposition](https://en.wikipedia.org/wiki/Quantum_superposition) Last revised on January 13, 2023 at 22:36:23. See the [history](https://ncatlab.org/nlab/history/quantum+superposition) of this page for a list of all contributions to it. [Edit](https://ncatlab.org/nlab/edit/quantum+superposition)[Discuss](https://nforum.ncatlab.org/discussion/14313/#Item_5)[Previous revision](https://ncatlab.org/nlab/revision/quantum+superposition/12)[Changes from previous revision](https://ncatlab.org/nlab/show/diff/quantum+superposition)[History (12 revisions)](https://ncatlab.org/nlab/history/quantum+superposition) [Cite](https://ncatlab.org/nlab/show/quantum+superposition/cite) [Print](https://ncatlab.org/nlab/print/quantum+superposition) [Source](https://ncatlab.org/nlab/source/quantum+superposition)
Readable Markdown	### Context #### Physics [physics](https://ncatlab.org/nlab/show/physics), [mathematical physics](https://ncatlab.org/nlab/show/mathematical+physics), [philosophy of physics](https://ncatlab.org/nlab/show/philosophy+of+physics) ## Surveys, textbooks and lecture notes - [(higher) category theory and physics](https://ncatlab.org/nlab/show/higher+category+theory+and+physics) - [geometry of physics](https://ncatlab.org/nlab/show/geometry+of+physics) - [books and reviews](https://ncatlab.org/nlab/show/books+and+reviews+in+mathematical+physics), [physics resources](https://ncatlab.org/nlab/show/physics+resources) * [theory (physics)](https://ncatlab.org/nlab/show/theory+%28physics%29), [model (physics)](https://ncatlab.org/nlab/show/model+%28physics%29) [experiment](https://ncatlab.org/nlab/show/experiment), [measurement](https://ncatlab.org/nlab/show/measurement), [computable physics](https://ncatlab.org/nlab/show/computable+physics) - [mechanics](https://ncatlab.org/nlab/show/mechanics) - [mass](https://ncatlab.org/nlab/show/mass), [charge](https://ncatlab.org/nlab/show/charge), [momentum](https://ncatlab.org/nlab/show/momentum), [angular momentum](https://ncatlab.org/nlab/show/angular+momentum), [moment of inertia](https://ncatlab.org/nlab/show/moment+of+inertia) - [dynamics on Lie groups](https://ncatlab.org/nlab/show/dynamics+on+Lie+groups) - [rigid body dynamics](https://ncatlab.org/nlab/show/rigid+body+dynamics) - [field (physics)](https://ncatlab.org/nlab/show/field+%28physics%29) - [Lagrangian mechanics](https://ncatlab.org/nlab/show/Lagrangian+mechanics) - [configuration space](https://ncatlab.org/nlab/show/configuration+space), [state](https://ncatlab.org/nlab/show/state) - [action functional](https://ncatlab.org/nlab/show/action+functional), [Lagrangian](https://ncatlab.org/nlab/show/Lagrangian) - [covariant phase space](https://ncatlab.org/nlab/show/covariant+phase+space), [Euler-Lagrange equations](https://ncatlab.org/nlab/show/Euler-Lagrange+equations) - [Hamiltonian mechanics](https://ncatlab.org/nlab/show/Hamiltonian+mechanics) - [phase space](https://ncatlab.org/nlab/show/phase+space) - [symplectic geometry](https://ncatlab.org/nlab/show/symplectic+geometry) - [Poisson manifold](https://ncatlab.org/nlab/show/Poisson+manifold) - [symplectic manifold](https://ncatlab.org/nlab/show/symplectic+manifold) - [symplectic groupoid](https://ncatlab.org/nlab/show/symplectic+groupoid) - [multisymplectic geometry](https://ncatlab.org/nlab/show/multisymplectic+geometry) - [n-symplectic manifold](https://ncatlab.org/nlab/show/n-symplectic+manifold) - [spacetime](https://ncatlab.org/nlab/show/spacetime) - [smooth Lorentzian manifold](https://ncatlab.org/nlab/show/smooth+Lorentzian+manifold) - [special relativity](https://ncatlab.org/nlab/show/special+relativity) - [general relativity](https://ncatlab.org/nlab/show/general+relativity) - [gravity](https://ncatlab.org/nlab/show/gravity) - [supergravity](https://ncatlab.org/nlab/show/supergravity), [dilaton gravity](https://ncatlab.org/nlab/show/dilaton+gravity) - [black hole](https://ncatlab.org/nlab/show/black+hole) - [Classical field theory](https://ncatlab.org/nlab/show/classical+field+theory) - [classical physics](https://ncatlab.org/nlab/show/classical+physics) - [classical mechanics](https://ncatlab.org/nlab/show/classical+mechanics) - [waves](https://ncatlab.org/nlab/show/waves) and [optics](https://ncatlab.org/nlab/show/optics) - [thermodynamics](https://ncatlab.org/nlab/show/thermodynamics) - [Quantum Mechanics](https://ncatlab.org/nlab/show/quantum+mechanics) - [in terms of ∞-compact categories](https://ncatlab.org/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories) - [quantum information](https://ncatlab.org/nlab/show/quantum+information) - [Hamiltonian operator](https://ncatlab.org/nlab/show/Hamiltonian+operator) - [density matrix](https://ncatlab.org/nlab/show/density+matrix) - [Kochen-Specker theorem](https://ncatlab.org/nlab/show/Kochen-Specker+theorem) - [Bell's theorem](https://ncatlab.org/nlab/show/Bell%27s+theorem) - [Gleason's theorem](https://ncatlab.org/nlab/show/Gleason%27s+theorem) - [Quantization](https://ncatlab.org/nlab/show/quantization) - [geometric quantization](https://ncatlab.org/nlab/show/geometric+quantization) - [deformation quantization](https://ncatlab.org/nlab/show/deformation+quantization) - [path integral quantization](https://ncatlab.org/nlab/show/path+integral) - [semiclassical approximation](https://ncatlab.org/nlab/show/semiclassical+approximation) - [Quantum Field Theory](https://ncatlab.org/nlab/show/quantum+field+theory)** - Axiomatizations - [algebraic QFT](https://ncatlab.org/nlab/show/AQFT) - [Wightman axioms](https://ncatlab.org/nlab/show/Wightman+axioms) - [Haag-Kastler axioms](https://ncatlab.org/nlab/show/Haag-Kastler+axioms) - [operator algebra](https://ncatlab.org/nlab/show/operator+algebra) - [local net](https://ncatlab.org/nlab/show/local+net) - [conformal net](https://ncatlab.org/nlab/show/conformal+net) - [Reeh-Schlieder theorem](https://ncatlab.org/nlab/show/Reeh-Schlieder+theorem) - [Osterwalder-Schrader theorem](https://ncatlab.org/nlab/show/Osterwalder-Schrader+theorem) - [PCT theorem](https://ncatlab.org/nlab/show/PCT+theorem) - [Bisognano-Wichmann theorem](https://ncatlab.org/nlab/show/Bisognano-Wichmann+theorem) - [modular theory](https://ncatlab.org/nlab/show/modular+theory) - [spin-statistics theorem](https://ncatlab.org/nlab/show/spin-statistics+theorem) - [boson](https://ncatlab.org/nlab/show/boson), [fermion](https://ncatlab.org/nlab/show/fermion) - [functorial QFT](https://ncatlab.org/nlab/show/FQFT) - [cobordism](https://ncatlab.org/nlab/show/cobordism) - [(∞,n)-category of cobordisms](https://ncatlab.org/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms) - [cobordism hypothesis](https://ncatlab.org/nlab/show/cobordism+hypothesis)\-theorem - [extended topological quantum field theory](https://ncatlab.org/nlab/show/extended+topological+quantum+field+theory) - Tools - [perturbative quantum field theory](https://ncatlab.org/nlab/show/perturbative+quantum+field+theory), [vacuum](https://ncatlab.org/nlab/show/vacuum) - [effective quantum field theory](https://ncatlab.org/nlab/show/effective+quantum+field+theory) - [renormalization](https://ncatlab.org/nlab/show/renormalization) - [BV-BRST formalism](https://ncatlab.org/nlab/show/BV-BRST+formalism) - [geometric ∞-function theory](https://ncatlab.org/nlab/show/geometric+%E2%88%9E-function+theory) - [particle physics](https://ncatlab.org/nlab/show/particle+physics) - [phenomenology](https://ncatlab.org/nlab/show/phenomenology) - [models](https://ncatlab.org/nlab/show/model+%28in+particle+phyiscs%29) - [standard model of particle physics](https://ncatlab.org/nlab/show/standard+model+of+particle+physics) - [fields and quanta](https://ncatlab.org/nlab/show/fields+and+quanta+-+table) - [Grand Unified Theories](https://ncatlab.org/nlab/show/GUT), [MSSM](https://ncatlab.org/nlab/show/MSSM) - [scattering amplitude](https://ncatlab.org/nlab/show/scattering+amplitude) - [on-shell recursion](https://ncatlab.org/nlab/show/on-shell+recursion), [KLT relations](https://ncatlab.org/nlab/show/KLT+relations) - Structural phenomena - [universality class](https://ncatlab.org/nlab/show/universality+class) - [quantum anomaly](https://ncatlab.org/nlab/show/quantum+anomaly) - [Green-Schwarz mechanism](https://ncatlab.org/nlab/show/Green-Schwarz+mechanism) - [instanton](https://ncatlab.org/nlab/show/instanton) - [spontaneously broken symmetry](https://ncatlab.org/nlab/show/spontaneously+broken+symmetry) - [Kaluza-Klein mechanism](https://ncatlab.org/nlab/show/Kaluza-Klein+mechanism) - [integrable systems](https://ncatlab.org/nlab/show/integrable+systems) - [holonomic quantum fields](https://ncatlab.org/nlab/show/holonomic+quantum+fields) - Types of quantum field thories - [TQFT](https://ncatlab.org/nlab/show/TQFT) - [2d TQFT](https://ncatlab.org/nlab/show/2d+TQFT) - [Dijkgraaf-Witten theory](https://ncatlab.org/nlab/show/Dijkgraaf-Witten+theory) - [Chern-Simons theory](https://ncatlab.org/nlab/show/Chern-Simons+theory) - [TCFT](https://ncatlab.org/nlab/show/TCFT) - [A-model](https://ncatlab.org/nlab/show/A-model), [B-model](https://ncatlab.org/nlab/show/B-model) - [homological mirror symmetry](https://ncatlab.org/nlab/show/homological+mirror+symmetry) - [QFT with defects](https://ncatlab.org/nlab/show/QFT+with+defects) - [conformal field theory](https://ncatlab.org/nlab/show/conformal+field+theory) - [(1,1)-dimensional Euclidean field theories and K-theory](https://ncatlab.org/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory) - [(2,1)-dimensional Euclidean field theory and elliptic cohomology](https://ncatlab.org/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory) - [CFT](https://ncatlab.org/nlab/show/CFT) - [WZW model](https://ncatlab.org/nlab/show/WZW+model) - [6d (2,0)-supersymmetric QFT](https://ncatlab.org/nlab/show/6d+%282%2C0%29-supersymmetric+QFT) - [gauge theory](https://ncatlab.org/nlab/show/gauge+theory) - [field strength](https://ncatlab.org/nlab/show/field+strength) - [gauge group](https://ncatlab.org/nlab/show/gauge+group), [gauge transformation](https://ncatlab.org/nlab/show/gauge+transformation), [gauge fixing](https://ncatlab.org/nlab/show/gauge+fixing) - examples - [electromagnetic field](https://ncatlab.org/nlab/show/electromagnetic+field), [QED](https://ncatlab.org/nlab/show/QED) - [electric charge](https://ncatlab.org/nlab/show/electric+charge) - [magnetic charge](https://ncatlab.org/nlab/show/magnetic+charge) - [Yang-Mills field](https://ncatlab.org/nlab/show/Yang-Mills+field), [QCD](https://ncatlab.org/nlab/show/QCD) - [Yang-Mills theory](https://ncatlab.org/nlab/show/Yang-Mills+theory) - [spinors in Yang-Mills theory](https://ncatlab.org/nlab/show/spinors+in+Yang-Mills+theory) - [topological Yang-Mills theory](https://ncatlab.org/nlab/show/topological+Yang-Mills+theory) - [Kalb-Ramond field](https://ncatlab.org/nlab/show/Kalb-Ramond+field) - [supergravity C-field](https://ncatlab.org/nlab/show/supergravity+C-field) - [RR field](https://ncatlab.org/nlab/show/RR+field) - [first-order formulation of gravity](https://ncatlab.org/nlab/show/first-order+formulation+of+gravity) - [general covariance](https://ncatlab.org/nlab/show/general+covariance) - [supergravity](https://ncatlab.org/nlab/show/supergravity) - [D'Auria-Fre formulation of supergravity](https://ncatlab.org/nlab/show/D%27Auria-Fre+formulation+of+supergravity) - [gravity as a BF-theory](https://ncatlab.org/nlab/show/gravity+as+a+BF-theory) - [sigma-model](https://ncatlab.org/nlab/show/sigma-model) - [particle](https://ncatlab.org/nlab/show/particle), [relativistic particle](https://ncatlab.org/nlab/show/relativistic+particle), [fundamental particle](https://ncatlab.org/nlab/show/fundamental+particle), [spinning particle](https://ncatlab.org/nlab/show/spinning+particle), [superparticle](https://ncatlab.org/nlab/show/superparticle) - [string](https://ncatlab.org/nlab/show/string), [spinning string](https://ncatlab.org/nlab/show/spinning+string), [superstring](https://ncatlab.org/nlab/show/superstring) - [membrane](https://ncatlab.org/nlab/show/membrane) - [AKSZ theory](https://ncatlab.org/nlab/show/AKSZ+theory) - [String Theory](https://ncatlab.org/nlab/show/string+theory) - [string theory results applied elsewhere](https://ncatlab.org/nlab/show/string+theory+results+applied+elsewhere) - [number theory and physics](https://ncatlab.org/nlab/show/number+theory+and+physics) - [Riemann hypothesis and physics](https://ncatlab.org/nlab/show/Riemann+hypothesis+and+physics) ## Contents - [Idea](https://ncatlab.org/nlab/show/quantum+superposition#idea) - [Related concepts](https://ncatlab.org/nlab/show/quantum+superposition#related_concepts) - [References](https://ncatlab.org/nlab/show/quantum+superposition#references) ## Idea In [quantum theory](https://ncatlab.org/nlab/show/quantum+theory) the [state](https://ncatlab.org/nlab/show/state) of a [physical system](https://ncatlab.org/nlab/show/physical+system) is given by a [vector](https://ncatlab.org/nlab/show/vector) in a [complex vector space](https://ncatlab.org/nlab/show/complex+vector+space) (usually a [Hilbert space](https://ncatlab.org/nlab/show/Hilbert+space)). This means that given any two states ψ 1 \\psi\_1 and ψ 2 \\psi\_2, their sum, the vector ψ 1 \+ ψ 2 \\psi\_1 + \\psi\_2, also represents a state. This is called the quantum superposition of the two states. If these quantum states are thought of as [wavefunctions](https://ncatlab.org/nlab/show/wavefunctions) solving a [linear differential equation](https://ncatlab.org/nlab/show/linear+differential+equation) such as the [Schroedinger equation](https://ncatlab.org/nlab/show/Schroedinger+equation), [Dirac equation](https://ncatlab.org/nlab/show/Dirac+equation), [Klein-Gordon equation](https://ncatlab.org/nlab/show/Klein-Gordon+equation) or [Tomonaga-Schwinger equation](https://ncatlab.org/nlab/show/Tomonaga-Schwinger+equation), their quantum superposition is an example of the general principle of [superposition](https://ncatlab.org/nlab/show/superposition) of solutions to [linear differential equations](https://ncatlab.org/nlab/show/linear+differential+equations). The concept of quantum superposition is at the heart of what makes [quantum physics](https://ncatlab.org/nlab/show/quantum+physics) peculiar, notably via the phenomenon of [entanglement](https://ncatlab.org/nlab/show/entanglement). The various fundamental quantum physics phenomena at play here may be identified with four [modal](https://ncatlab.org/nlab/show/modality)\-[units](https://ncatlab.org/nlab/show/unit+of+a+monad) of [quantum modal logic](https://ncatlab.org/nlab/show/necessity+and+possibility#ModalQuantumLogic) (as discussed at [quantum circuits via dependent linear types](https://ncatlab.org/nlab/show/quantum+circuits+via+dependent+linear+types)): ![](https://ncatlab.org/nlab/files/QCwthLHT-QuantumModalUnits-221110.jpg) \| \| \| \|---\|---\| \| [quantum measurement](https://ncatlab.org/nlab/show/quantum+measurement) \| [quantum state preparation](https://ncatlab.org/nlab/show/quantum+state+preparation) \| \| quantum superposition \| [quantum parallelism](https://ncatlab.org/nlab/show/quantum+parallelism) \| - [quantum parallelism](https://ncatlab.org/nlab/show/quantum+parallelism) - [superposition](https://ncatlab.org/nlab/show/superposition) [quantum probability theory](https://ncatlab.org/nlab/show/quantum+probability+theory) – [observables](https://ncatlab.org/nlab/show/observables) and [states](https://ncatlab.org/nlab/show/states) - [states](https://ncatlab.org/nlab/show/states) - [classical state](https://ncatlab.org/nlab/show/classical+state) - [quantum state](https://ncatlab.org/nlab/show/quantum+state) - [space of states (in geometric quantization)](https://ncatlab.org/nlab/show/space+of+states+%28in+geometric+quantization%29) - [state on a star-algebra](https://ncatlab.org/nlab/show/state+on+a+star-algebra), [quasi-state](https://ncatlab.org/nlab/show/quasi-state) - [qbit](https://ncatlab.org/nlab/show/qbit), [Bell state](https://ncatlab.org/nlab/show/Bell+state) [dimer](https://ncatlab.org/nlab/show/dimer), [tensor network state](https://ncatlab.org/nlab/show/tensor+network+state) - [quantum state preparation](https://ncatlab.org/nlab/show/quantum+state+preparation) - [probability amplitude](https://ncatlab.org/nlab/show/probability+amplitude), [quantum fluctuation](https://ncatlab.org/nlab/show/quantum+fluctuation) - [pure state](https://ncatlab.org/nlab/show/pure+state) - [wave function](https://ncatlab.org/nlab/show/wave+function) - [bra-ket](https://ncatlab.org/nlab/show/bra-ket) - [Bell state](https://ncatlab.org/nlab/show/Bell+state) - [quantum superposition](https://ncatlab.org/nlab/show/quantum+superposition), [quantum interference](https://ncatlab.org/nlab/show/quantum+interference) - [quantum entanglement](https://ncatlab.org/nlab/show/quantum+entanglement) - [quantum measurement](https://ncatlab.org/nlab/show/quantum+measurement) [wave function collapse](https://ncatlab.org/nlab/show/wave+function+collapse) [Born rule](https://ncatlab.org/nlab/show/Born+rule) [deferred measurement principle](https://ncatlab.org/nlab/show/deferred+measurement+principle) [quantum reader monad](https://ncatlab.org/nlab/show/quantum+reader+monad) [measurement problem](https://ncatlab.org/nlab/show/measurement+problem) - [superselection sector](https://ncatlab.org/nlab/show/superselection+sector) - [mixed state](https://ncatlab.org/nlab/show/mixed+state), [density matrix](https://ncatlab.org/nlab/show/density+matrix) [entanglement entropy](https://ncatlab.org/nlab/show/entanglement+entropy) [holographic entanglement entropy](https://ncatlab.org/nlab/show/holographic+entanglement+entropy) - [coherent quantum state](https://ncatlab.org/nlab/show/coherent+quantum+state) - [ground state](https://ncatlab.org/nlab/show/ground+state), [excited state](https://ncatlab.org/nlab/show/excited+state) - [quasi-free state](https://ncatlab.org/nlab/show/quasi-free+state) - [Fock space](https://ncatlab.org/nlab/show/Fock+space), [second quantization](https://ncatlab.org/nlab/show/second+quantization) - [vacuum](https://ncatlab.org/nlab/show/vacuum), [vacuum state](https://ncatlab.org/nlab/show/vacuum+state) - [Hadamard state](https://ncatlab.org/nlab/show/Hadamard+state) - [vacuum diagram](https://ncatlab.org/nlab/show/vacuum+diagram) - [vacuum expectation value](https://ncatlab.org/nlab/show/vacuum+expectation+value), [vacuum amplitude](https://ncatlab.org/nlab/show/vacuum+amplitude), [vacuum fluctuation](https://ncatlab.org/nlab/show/vacuum+fluctuation) - [vacuum energy](https://ncatlab.org/nlab/show/vacuum+energy) - [vacuum polarization](https://ncatlab.org/nlab/show/vacuum+polarization) - [interacting vacuum](https://ncatlab.org/nlab/show/interacting+vacuum) - [thermal vacuum](https://ncatlab.org/nlab/show/thermal+vacuum), [KMS state](https://ncatlab.org/nlab/show/KMS+state) - [vacuum stability](https://ncatlab.org/nlab/show/vacuum+stability) - [false vacuum](https://ncatlab.org/nlab/show/false+vacuum), [tachyon](https://ncatlab.org/nlab/show/tachyon), [Coleman-De Luccia instanton](https://ncatlab.org/nlab/show/Coleman-De+Luccia+instanton) - [theta vacuum](https://ncatlab.org/nlab/show/theta+vacuum) - [perturbative string theory vacuum](https://ncatlab.org/nlab/show/perturbative+string+theory+vacuum) - [non-geometric string theory vacuum](https://ncatlab.org/nlab/show/non-geometric+string+theory+vacuum) - [landscape of string theory vacua](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua) - [entangled state](https://ncatlab.org/nlab/show/entangled+state) - [tensor network state](https://ncatlab.org/nlab/show/tensor+network+state) - [matrix product state](https://ncatlab.org/nlab/show/matrix+product+state) - [tree tensor network state](https://ncatlab.org/nlab/show/tree+tensor+network+state) - [observables](https://ncatlab.org/nlab/show/observables) - [quantum observable](https://ncatlab.org/nlab/show/quantum+observable), [beable](https://ncatlab.org/nlab/show/beable) - [algebra of observables](https://ncatlab.org/nlab/show/algebra+of+observables), [star-algebra](https://ncatlab.org/nlab/show/star-algebra) - [Bohr topos](https://ncatlab.org/nlab/show/Bohr+topos) - [quantum operator (in geometric quantization)](https://ncatlab.org/nlab/show/quantum+operator+%28in+geometric+quantization%29) - [quantum operation](https://ncatlab.org/nlab/show/quantum+operation), [quantum effect](https://ncatlab.org/nlab/show/quantum+effect), [effect algebra](https://ncatlab.org/nlab/show/effect+algebra) - in [quantum field theory](https://ncatlab.org/nlab/show/quantum+field+theory) - [local observable](https://ncatlab.org/nlab/show/local+observable) - [polynomial observable](https://ncatlab.org/nlab/show/polynomial+observable) - [linear observable](https://ncatlab.org/nlab/show/linear+observable) - [field observable](https://ncatlab.org/nlab/show/field+observable) - [regular observable](https://ncatlab.org/nlab/show/regular+observable) - [microcausal observable](https://ncatlab.org/nlab/show/microcausal+observable) - [normal-ordered product](https://ncatlab.org/nlab/show/normal-ordered+product), [time-ordered products](https://ncatlab.org/nlab/show/time-ordered+products), [retarded product](https://ncatlab.org/nlab/show/retarded+product) - [Wick algebra](https://ncatlab.org/nlab/show/Wick+algebra) - [scattering amplitude](https://ncatlab.org/nlab/show/scattering+amplitude) - [interacting field algebra of observables](https://ncatlab.org/nlab/show/interacting+field+algebra+of+observables), [Bogoliubov's formula](https://ncatlab.org/nlab/show/Bogoliubov%27s+formula) - [GNS construction](https://ncatlab.org/nlab/show/GNS+construction) - [theorems](https://ncatlab.org/nlab/show/theorems) - [order-theoretic structure in quantum mechanics](https://ncatlab.org/nlab/show/order-theoretic+structure+in+quantum+mechanics) - [Gleason's theorem](https://ncatlab.org/nlab/show/Gleason%27s+theorem) - [Alfsen-Shultz theorem](https://ncatlab.org/nlab/show/Alfsen-Shultz+theorem) - [Harding-Döring-Hamhalter theorem](https://ncatlab.org/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem) - [Kochen-Specker theorem](https://ncatlab.org/nlab/show/Kochen-Specker+theorem) - [Nuiten's lemma](https://ncatlab.org/nlab/show/Nuiten%27s+lemma) - [Wigner's theorem](https://ncatlab.org/nlab/show/Wigner%27s+theorem) - [no-cloning theorem](https://ncatlab.org/nlab/show/no-cloning+theorem) - [Bell's theorem](https://ncatlab.org/nlab/show/Bell%27s+theorem) ## References See also - Wikipedia, [Quantum superposition](https://en.wikipedia.org/wiki/Quantum_superposition) Last revised on January 13, 2023 at 22:36:23. See the [history](https://ncatlab.org/nlab/history/quantum+superposition) of this page for a list of all contributions to it.
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