Pauli group

In physics and mathematics, the Pauli group $G_{1}$ on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix $I$ and all of the Pauli matrices

X=\sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad Y=\sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad Z=\sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}

together with the products of these matrices with the factors $\pm 1$ and $\pm i$ :

G_{1}\ {\stackrel {\mathrm {def} }{=}}\ \{\pm I,\pm iI,\pm X,\pm iX,\pm Y,\pm iY,\pm Z,\pm iZ\}\equiv \langle X,Y,Z\rangle

The Pauli group is generated by the Pauli matrices, and like them it is named after Wolfgang Pauli.

The Pauli group on $n$ qubits, $G_{n}$ , is the group generated by the operators described above applied to each of $n$ qubits in the tensor product Hilbert space $(\mathbb {C} ^{2})^{\otimes n}$ . That is,

$G_{n}=\langle W_{1}\otimes \cdots \otimes W_{n}:W_{i}\in \{X,Y,Z\}\rangle .$

The order of $G_{n}$ is $4\cdot 4^{n}$ since a scalar $\pm 1$ or $\pm i$ factor in any tensor position can be moved to any other position.

As an abstract group, $G_{1}\cong C_{4}\circ D_{4}$ is the central product of a cyclic group of order 4 and the dihedral group of order 8.^[1]

The Pauli group is a representation of the gamma group in three-dimensional Euclidean space. It is not isomorphic to the gamma group; it is less free, in that its chiral element is $\sigma _{1}\sigma _{2}\sigma _{3}=iI$ whereas there is no such relationship for the gamma group.

References

Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge; New York: Cambridge University Press. ISBN 978-0-521-63235-5. OCLC 43641333.

External links

^ Pauli group on GroupNames

2. https://arxiv.org/abs/quant-ph/9807006

This quantum mechanics-related article is a stub. You can help Wikipedia by expanding it.

Read other articles:

Rincon South, San Jose

For the vast district immediately north of Rincon South, see Rincon de los Esteros, San Jose. Neighborhood of San Jose, California, United States Neighborhood of San Jose in California, United StatesRincon SouthNeighborhood of San JoseRincon SouthLocation within San JoseCoordinates: 37°22′01″N 121°54′50″W / 37.367067°N 121.913803°W / 37.367067; -121.913803CountryUnited StatesStateCaliforniaCitySan Jose Rincon South is a neighborhood of San Jose, California,...

Bernhard Rust

Minister of Science, Education and National Culture of Nazi Germany Bernhard Rustc.1934Reich Minister of Science, Education and CultureIn office1 May 1934 – 8 May 1945LeaderAdolf HitlerPreceded byOffice establishedSucceeded byGustav Adolf ScheelPrussian Minister for Science, Culture and Public EducationIn office2 February 1933 – 1 May 1934LeaderAdolf HitlerPreceded byWilhelm Kähler [de]Succeeded byOffice abolishedGauleiter of South Hanover-BrunswickIn offic...

Vanguard SLV-5

Failed rocket launch Vanguard SLV-5Vanguard rocket on LC-18A prior to its launchNamesVanguard Space Launch Vehicle-FiveMission typeMagnetic Field ExperimentAir Density ExperimentOperatorNaval Research LaboratoryMission durationFailed to orbit (500 seconds) Spacecraft propertiesSpacecraftVanguard 3ASpacecraft typeVanguardManufacturerNaval Research LaboratoryLaunch mass10.3 kg (23 lb) Start of missionLaunch date14 April 1959, 02:49:46 GMTRocketVanguard SLV-5Launch siteCape Canaveral, ...

Serviços Aéreos Cruzeiro do Sul

Serviços Aéreos Cruzeiro do SulLogo Stato Brasile Forma societariasussidiaria Fondazione1927 (come Syndicato Condor) a Rio de Janeiro Chiusura1993 (confluita nella Varig) Sede principaleRio de Janeiro Gruppo Deutsche Luft Hansa (1927-1975) Varig (1975-1993) SettoreTrasporto Prodotticompagnia aerea Compagnia aerea standardCodice IATASC Codice ICAOCRZ Indicativo di chiamataCRUZEIRO Primo volo1928 Ultimo volo1993 HubRio de Janeiro Voci di compagnie aeree presenti su Wikipedia Modifica...

Cinnamomum burmanni

Species of flowering plant Indonesian cinnamon Conservation status Least Concern (IUCN 3.1)[1] Scientific classification Kingdom: Plantae Clade: Tracheophytes Clade: Angiosperms Clade: Magnoliids Order: Laurales Family: Lauraceae Genus: Cinnamomum Species: C. burmanni Binomial name Cinnamomum burmanni(Nees & T.Nees) Blume[2] Synonyms[2] Synonymy Cinnamomum ammannii Lukman. Cinnamomum burmanni var. angustifolium Meisn. Cinnamomum burmanni var. chinense (B...

コルク

この記事は検証可能な参考文献や出典が全く示されていないか、不十分です。出典を追加して記事の信頼性向上にご協力ください。（このテンプレートの使い方）出典検索?: コルク – ニュース · 書籍 · スカラー · CiNii · J-STAGE · NDL · dlib.jp · ジャパンサーチ · TWL（2017年4月）コルクを打ち抜いて作った瓶の栓コルク（木栓、�...

Nokia 7280

Nokia 7280 adalah produk telepon genggam yang dirilis oleh perusahaan Nokia. Telepon genggam ini memiliki dimensi 115 x 32 x 19 mm dengan berat 84 gram. Fitur Kamera digital VGA, 640x480 pixels Bluetooth v1.1 SMS MMS Polifonik baterai Li-Ion 700 mAh (BL-8N) Inframerah Lihat pula Daftar produk Nokia Pranala luar informasi di GSM Arena lbsNokiaAnak usaha Nokia Bell Labs Nokia Networks Nokia Technologies Unit lain NGP Capital Nuage Networks Akuisisi Alcatel-Lucent Dopplr earthmine Enpocket ...

Santo Georgius dan Naga

Untuk kegunaan lain, lihat Santo Georgius dan Naga (disambiguasi). Santo Gregorius Membunuh Naga, cukil kayu karya Albrecht Dürer (1501/4). Legenda Santo Gregorius dan Naga mengisahkan Santo Georgius yang menjinakkan dan membunuh seekor naga yang menuntut pengurbanan manusia. Sehingga, orang kudus tersebut menyelamatkan putri raja yang dipilih sebagai pengurbanan berikutnya. Kisah tersebut berlatar Cappadocia dalam sumur-sumur terawal dari abad ke-11 dan ke-12, namun diganti menjadi Libya da...

University of Lynchburg

Christian university in Lynchburg, Virginia, U.S. This article is about the private institution the University of Lynchburg formerly known as Lynchburg College. For the historically black college in Lynchburg, Virginia, see Virginia University of Lynchburg. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: University of Lynchburg – n...

SEPTA Key

Public transit smart card used in Philadelphia SEPTA KeyLocationPhiladelphia areaLaunched2016–2017TechnologyMastercard Contactless card issued by MetaBankOperatorConduentManagerSEPTACurrencyUSD ($1[1] minimum load, $250 maximum load)Auto rechargeAvailableValiditySEPTAWebsitewww.septakey.org The SEPTA Key card is a smart card that is used for automated fare collection on the SEPTA public transportation network in the Philadelphia metropolitan area. It can be used throughout SEPTA's t...

قائمة البحيرات حسب العمق

لا يزال النص الموجود في هذه الصفحة في مرحلة الترجمة من الإنجليزية إلى العربية. إذا كنت تعرف اللغة الإنجليزية، لا تتردد في الترجمة من النص الأصلي باللغة الإنجليزية. (أبريل 2019) هذه القائمة تسرد أعمق البحيرات في العالم. البحيرات مرتبة حسب العمق الأقصى تحتوي هذه القائمة على جمي...

Katie Woodencloak

Norwegian fairy tale An illustration by Theodor Kittelsen of Katie dressed in her wooden clothing. Katie Woodencloak or Kari Woodengown (originally Kari Trestakk) is a Norwegian fairy tale collected by Peter Christen Asbjørnsen and Jørgen Moe in Norske Folkeeventyr.[1] Andrew Lang included it in The Red Fairy Book.[2] It is Aarne–Thompson type 510A, the persecuted heroine. Others of this type include Cinderella, The Sharp Grey Sheep, The Golden Slipper, The Story of Tam an...

Edmund A. Walsh

American Jesuit priest and author The ReverendEdmund A. WalshSJWalsh with General Douglas MacArthur in Tokyo in 1948Born(1885-10-10)October 10, 1885South Boston, Massachusetts, U.S.DiedOctober 31, 1956(1956-10-31) (aged 71)Washington, D.C., U.S.Alma materGeorgetown University (AB)Woodstock College (MA)OrdersOrdinationJune 28, 1916 Edmund Aloysius Walsh SJ (October 10, 1885 – October 31, 1956)[1] was an American Roman Catholic priest of the Society of Jesus and career diplo...

53e cérémonie des Grammy Awards

Si ce bandeau n'est plus pertinent, retirez-le. Cliquez ici pour en savoir plus. Cet article contient une ou plusieurs listes (janvier 2020). Ces listes gagneraient à être rédigées sous la forme de paragraphes synthétiques, plus agréables à la lecture, les listes pouvant être aussi introduites par une partie rédigée et sourcée, de façon à bien resituer les différents items.D'autre part, Wikipédia n'a pas pour rôle de constituer une base de données et privilégie un contenu e...

Minima naturalia

Part of a series onScholasticism Schools Thomism Scotism Occamism Major works Summa Theologica Cur Deus Homo Summa Grammatica Summa logicae Opus Oxoniense Libri Quattuor Sententiarum Precursors Augustine of Hippo Boethius Pope Gregory I Alcuin of York John Scotus Eriugena Philosophers Thomas Aquinas (Doctor Angelicus) Duns Scotus (Doctor Subtilis) William of Ockham (Doctor Invincibilis) Francisco Suárez (Doctor Eximius) Averroes (The Commentator) Maimonides (Rabbi Moses) Albertus Magnus (Doc...

Ordnance Survey National Grid

System of geographic grid references used in Great Britain For the electricity network, see National Grid (Great Britain). This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Ordnance Survey National Grid – news · newspapers · books · scholar · JSTOR (November 2008) (Learn how and when to remove this message) Ge...

Form of the Good

Superlative concept in the philosophy of Plato The Good redirects here. For other uses, see Good (disambiguation). Part of a series onPlatonism Life Works Theory of forms Form of the Good Theory of soul Epistemology Political philosophy Euthyphro dilemma Demiurge Atlantis The Republic Allegory of the cave Analogy of the Sun Analogy of the divided line Philosopher king Ship of State Ring of Gyges Myth of Er The works of Plato Euthyphro Apology Crito Phaedo Cratylus Theaetetus Sophist Statesman...

Grace Potter

American singer-songwriter and musician Grace PotterPotter in 2019Background informationBirth nameGrace Evelyn PotterBorn (1983-06-20) June 20, 1983 (age 41)Waitsfield, VermontGenres Rock pop Occupations Singer-songwriter musician Instruments Vocals keyboards guitar Years active2002–presentLabelsFantasyFormerly ofGrace Potter and the NocturnalsWebsitegracepotter.comMusical artist Grace Potter in concert in 2007 Grace Evelyn Potter (born June 20, 1983) is an American singer-songwriter a...

جغتاي خان

خان خانية جغتاي جغتاي خان (بالمنغولية: ᠴᠠᠭᠠᠲᠠᠢ)‏ تمثال جغتاي خان في منغوليا خان خانية جغتاي فترة الحكم1226 – 1241-42 جنكيز خان قره هولاكو [لغات أخرى]‏ معلومات شخصية الميلاد 22 ديسمبر 1183(1183-12-22)منغوليا الوفاة 1 جولاي 1242 (عن عمر 59)المالق، شينجيانغ الديانة التنغرية...

USS Milius

US Navy Arleigh Burke-class destroyer USS MiliusUSS Milius on 20 May 2003 History United States NameMilius NamesakePaul L. Milius Ordered8 April 1992 BuilderIngalls Shipbuilding Laid down8 August 1994 Launched1 August 1995 Sponsored byAnnette Milius Christened28 October 1995 Commissioned23 November 1996 HomeportYokosuka[1] Identification MMSI number: 367199000 Callsign: NPLM Hull number: DDG-69 Motto Alii Prae Me (Others Before Me) Honours andawardsSee Awards Statusin active serv...

/profillengkap.com/index.php/article/Electoral history of Dan Quayle

/profillengkap.com/index.php/article/2014 Texas Tech Red Raiders baseball team

Hotel Tjampuhan

2010 OT83

Kanibalisme pada unggas

The Milkmaid (Vermeer)

Hitoshi Imamura

2010 NG1

Christo and Jeanne-Claude

Chang Wanquan

Yazīd III le Réducteur

Wittenberge