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Bures volumes of quantum states of fixed rank
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Numbers
$\mathbb F$
$N$
$r$ 
$V_{N,r}^{(\beta)}$
real
2
1:
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068
comment: The pure rebit states form a circle of radius $1/2$, with length $\pi$.
real
2
2:
1.570796326794896619231321691639751442098584699687552910487472296153908203143104499314017412671058534
comment: The full-rank rebit states form an open hemisphere of a sphere of radius $1/2$, with area $\pi/2$.
real
3
1:
6.283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234136
real
3
2:
2.467401100272339654708622749969037783828424851810197656603337344055011205604801310750443350929638058
real
3
3:
0.4844730731296846902418174229234593000347701338419548077210084078719749205854016568369797036344200968
real
4
1:
9.869604401089358618834490999876151135313699407240790626413349376220044822419205243001773403718552232
real
4
2:
2.583856390024985014623026255591782933518774047157092307845378175317199576455475503130558419383573849
real
4
3:
0.2319264072238153267534293635445359791660180611254414802177806175951687034686361814050520648446499823
real
4
4:
0.02490394570192720160015798421577438203778488823470662358019958037811157162545687048224149580676311189
real
5
1:
13.15947253478581149177932133316820151375159920965438750188446583496005976322560699066903120495806964
real
5
2:
2.029356063208384109092506931014689817702658034847612951905580403957726155350566587294205567390687345
real
5
3:
0.07823805286257360327394363962015135895729309310555392382332976786166128848000896919752778394790980926
real
5
4:
0.002560334292752407555062363088880786329411343447617765861302892466552000823375796746943539617826681973
real
5
5:
0.0001746108768417226331257981886789160973238238888019042452197244013153012882542001910669453592177504056
real
6
1:
15.50313834014991008773815753355069760111264428294255384707226905190319745873285301878335051630144310
real
6
2:
1.275082019938672721928088791847648360334586277616979127306218515359312467223391768690764585306271329
real
6
3:
0.01955641820627293491008939713203860290026827554581327546460913294731374428447042139949788023238804543
real
6
4:
0.0001974178640938216558198756800839613236577514659980181410357101495256331949360711853340252631940513794
real
6
5:
0.000002953364340593039704810900984142171820509312558915712660805020048290278168023806288400505628956204045
real
6
6:
1.396809380480961918021514551781747258935376445048966528383509738139323720796496777130309069975346138e-7
real
7
1:
16.53668089615990409358736803578741077452015390180539077021042032203007728931504322003557388405487264
real
7
2:
0.6676313844272947479376523914252915964355677278340601499590806857528429950294098704855704230221637057
real
7
3:
0.003790422990601375791741613057612057414228828147161948307885634870892157342772566758413285053325786485
real
7
4:
0.00001136587795183191796000285001844329231973573591961474692963413779091156159811433067029283149562454827
real
7
5:
3.995929277993565855024168412118739741695169274258474279040864009715554706091482509266901432403458201e-8
real
7
6:
3.138395516002238285237777052566104077454568674222591715953045990402231623953150489232356278170175850e-10
real
7
7:
1.091460487952709123233135191300027509439112723523963231290787228047916441235259212637167128892904803e-11
real
8
1:
16.23484850566707287274005544811751854162126427878090361524464323166180924280453269835364453912549876
real
8
2:
0.2996322646603960384438696917730200230076817931840676847970470726570570106442075278708782364176596960
real
8
3:
0.0005900122823151583141722877466726100421743030103886834615891945133900503396644402386598425511575862088
real
8
4:
5.007889243555571547586257327154491114793124342997643981577323545747945123141782914529446069904586402e-7
real
8
5:
4.114709100762914410606721280804079467453565157220405815825529153585494925999932857744999416697937984e-10
real
8
6:
6.013469268407967004430425555177761784497271206443295403063915372236874394236987679358072190907676650e-13
real
8
7:
2.726099693538016978642851530437757929105364122119321840211139276284003656057343804581645411968817133e-15
real
8
8:
7.270752228792055337120126207355517017418367700768815903519358902821812719962591421012685419520310495e-17
real
9
1:
14.84329006232418091221947926685030266662515591202825473393795952609079702199271560992333215005759886
real
9
2:
0.1176653151794466022709396763877327107725344676542792992573176018733495647548533766869477223092623404
real
9
3:
0.00007578619800423733406529558534130332894582590411088227582781956431596786136721857846143238792619528859
real
9
4:
1.733688593437517241397599796795933125617265533653847771864345721050888442481578853187379664946211549e-8
real
9
5:
3.238606435923164170627707618160056419777172577077754346170833460225396343396124439842994749418954754e-12
real
9
6:
8.998807526290423140297107395306021387940426919486721258343693807751195532213157639363243297685451934e-16
real
9
7:
6.179795923241037504871122546252849741271110859493673040543274126077213550766939120823588591973417444e-19
real
9
8:
1.733428702958767609286352050138669488099891084551909583190798016339090384330453995192158980595805823e-21
real
9
9:
3.659859761057376820648109750903260042516470882982518865037694576278067304766066818904879150550881902e-23
real
10
1:
12.75082019938672721928088791847648360334586277616979127306218515359312467223391768690764585306271329
real
10
2:
0.04107294330556411439940118276184917240391873039887656155846444558511713353438209905412914174185825778
real
10
3:
0.000008206455932129281367570324452881424147675489287567188395543507686903206806405495222129864187946699823
real
10
4:
4.827271428707062038440269914950568429593096324790233531771897458464993291762128419048725065463333784e-10
real
10
5:
1.980308982686319144778261465896902223348248299314793433599177648841380685662747177014590934225256065e-14
real
10
6:
1.036252692745785672998305794870749048438113565807670391446772842327371976671464159104727482686976702e-18
real
10
7:
1.127191625691117962143432284366552736235651795227324276996702921548022914577410675668142891997676225e-22
real
10
8:
3.982952953452015492362676922038128936799589083828725045002023089161862054856385082394990744467598579e-26
real
10
9:
7.291850663262599988292294728661139836584839199136995234274090451836690793760356948757689918693168788e-29
real
10
10:
1.249428281100024148586935170440977116847666952030492598672622048091609403763818825029372928777584320e-30
real
11
1:
10.36257133664445132741557875279358667446520911333163989175847428355504735267264421539161059804461778
real
11
2:
0.01290344569500703000629914712644942482859322052407334299590118625898344466797387792777436912661825256
real
11
3:
7.621225008727348519759953854197879760289231921678328791064275331326733840080637544516642729137285515e-7
real
11
4:
1.102853225417132767175430145371172316825294348980946632092446443438460851741341586079429547369665856e-11
real
11
5:
9.579580448494374221495449125916257870005672075022019618707944497959705384340646448612591839950717955e-17
real
11
6:
9.236419266703980112806467762599940050015507680454228602196804519098770232805639193273457897558471221e-22
real
11
7:
1.601525444392372539355581316689084162763433806184276435094215501768593835418781933901390646930990435e-26
real
11
8:
7.603920958297197740433818838700745364070439284432010197205071442223542142398844081947385102277650089e-31
real
11
9:
1.483180917386036071197157466905502874743422075444380045698135786850226100187219751099254418067177019e-34
real
11
10:
1.849027886509738477205190222446690914484393229029141984719269355109227315680089816771047865827103440e-37
real
11
11:
2.623157296588046870333626241689261131945694591038537898750560244821532828288298216991142220001366630e-39
real
12
1:
8.011576613127536975251828697103499157226812734008721799508968229034115940352918445826845076265964468
real
12
2:
0.003685215472857175388629544978645390750605819118010498724949921356908745087911048978643468175125543081
real
12
3:
6.156326389770984460581389874271432346356794952132363555953464234184566389370175699047508145888630065e-8
real
12
4:
2.102730975052874739002026457428030191686842200903429045313112559422435736329956246081009403263329980e-13
real
12
5:
3.729845877352905180357528919044178419519568365101829270632715104500011489311128480609728461898062681e-19
real
12
6:
6.450492276353210649276196649380874463417346377589165646032798236256144713128135515557417959309771553e-25
real
12
7:
1.764629328269210276067530067825491100254733848480705603169814033762925004958746571067848936439604530e-30
real
12
8:
1.148986324181007267827451182961245503070979264638774784614510797037242351483533192982396425302996432e-35
real
12
9:
2.591535372258410777949479281909094111675435829022903584726131237713300571003565989093534759160162928e-40
real
12
10:
2.951873452975756037437092352158139920299013854220641105900580871863581873204095247611192447410486254e-44
real
12
11:
2.594009497677026897257409480654049040972342793006851588172433577825063282244798325402365054559368629e-47
real
12
12:
3.097525562794428785444122302399981282268940279175685504197874293123958548093515326138961444433092237e-49
real
13
1:
5.919086906091340449248104318783537616883301760191147996174014045116114518702300496772503144369659174
real
13
2:
0.0009647871547019615239516727818429788200842359075001516761173238086657478171254927657436738490930719567
real
13
3:
4.376764830942922394090943889670158575472191115851731063851919692403538765370582008582673664736470761e-9
real
13
4:
3.394536284677054463514804821645496054786063481003209617615143857201698892498092586735095445506703054e-15
real
13
5:
1.187269843831725372806712844831383044009145477609783799012988042819587689364833105338298840139872173e-21
real
13
6:
3.578537935790938555015014171351904295723967792786880911109966908298263512186653593253871040621258244e-28
real
13
7:
1.516780959566497277344526042498642901586296205792106094694760640274903078461033014880388609248188793e-34
real
13
8:
1.354574696382236446230885967881345872065306341348354371304384765993719699514987940380713981284745390e-40
real
13
9:
3.651233955064297043936877270655783257143675831200566267335732633944409682642627639445943188214041224e-46
real
13
10:
4.187610942881420539342187713306028552566968662317538072510355593139924744851225516145599749718379806e-51
real
13
11:
2.917020549297180977741163016691293701715723762929307424754921994110558753584690667896568859375170227e-55
real
13
12:
1.859596251631998187848023035322510726577088114723022671438333263153264121859421785965988790503086246e-58
real
13
13:
1.895048556679410812293905974064576493175412433659450593220635991741948103667958508240737936161259705e-60
real
14
1:
4.194851705245544538214175684822280322107545104576947587158659017198798149018905390192295309847235744
real
14
2:
0.0002331514028838062822103144572351358719140999068079963385781517866269817873434795498450717479052623639
real
14
3:
2.765922929273496193676348440399269304258329613854693199261129233492138940097900010395856294364014051e-10
real
14
4:
4.697898873885080863293930772535736556050903831734762322547865697124267002596882231963986983563392124e-17
real
14
5:
3.133026048765781841909866743881158419953463161011722066822392405739649479423465731288321372300299110e-24
real
14
6:
1.598567828038878369966866929698045869114546316608052245750316230419088673934026122075326068262644534e-31
real
14
7:
1.027066171581794023718414332777755210663455442223660855441351142023050410493544834222319496945044625e-38
real
14
8:
1.245404025880604750191148008454433170029325003117419261292967612310080749641922880984064833058679831e-45
real
14
9:
4.050527924994779284814055422986645124372775481374934630382862993802124231895267908068253599742850388e-52
real
14
10:
4.889023198656825291854658464383923892510330785622374282246222042307921988612706153681351943468655629e-58
real
14
11:
3.015571156424664290333867960080067453353879671132737318884159673345617381020398712811751656753575831e-63
real
14
12:
1.335212746118269372216452163564388609749161157638822811203501215661921247004213967510393784106901521e-67
real
14
13:
6.326839448242245517948449184068896201010528012681700547884749849304301855655231216851749609108643692e-71
real
14
14:
5.567317737528482287859735620462632877188255450553397302684859257749468395647440027136916608613608708e-73
real
15
1:
2.860824606174783621133077523315361338553530594130399153679964591149896306756815930440832765423622645
real
15
2:
0.00005231905246242285355900836417611966380514866574545660659263856038571757852730071607543909299867458501
real
15
3:
1.567051226922928388221869035387303960845758551899426031554422902873689411196504430202826740045500354e-11
real
15
4:
5.634125518998761219686317724824490557629908055302891042349483201844056862967393018472005279491477125e-19
real
15
5:
6.938936209744526332214429168481411394431758868884922669002686700917037503305173007531802124991175032e-27
real
15
6:
5.823693157361027057346603105329442569626464392760881739835600362799794507590644105722445498282453065e-35
real
15
7:
5.539535669285392446681315036206571810832965350068486491543459167654435558170592747865000156856165566e-43
real
15
8:
8.979963968532068187857504513177897337546686321361482661379363176249019094183893320281570694970590922e-51
real
15
9:
3.514457401780600996520857087255436000371591715338256131165844407549880906275577490883939077780253119e-58
real
15
10:
4.546391473143470249205094849180338363371446492179200146482409848049442660445573859069390089039290236e-65
real
15
11:
2.621792048387253782412652299589038945155173631423256488635761745879412629981459341501335310921582611e-71
real
15
12:
9.116010248799351142394559972226604214287363932611078968479842029816334815732777312507143518435193990e-77
real
15
13:
2.651273066075916825434472880876938970761296861344792050816640903556663846679151866843445225868116672e-81
real
15
14:
9.534363538283638698161309998934978649973427613951187673828582006665628592734722319437238388908032164e-85
real
15
15:
7.318115821190032035414086876269312853684501412493172703693453608311136400618285750498938651185506556e-87
real
16
1:
1.882645042871145636335034822203723372360551482468468788117081629973593036077654026991163556948197446
real
16
2:
0.00001095767672391510791369232602854709442962435438297789015551763824190492426038172849839446478314528767
real
16
3:
8.018391318005588675964215822094921554417869118082062543594628614210448853039972141590089724231107048e-13
real
16
4:
5.910564415242351245969610792994502328392373482918696199396732387285723421378911348478218788806259742e-21
real
16
5:
1.304042949714033216329047840713355360057352337215281280969780917148888430495303575266842845051660917e-29
real
16
6:
1.750626567854513174061613333172830292527863982806290238841978549781465243684440164317046999857686379e-38
real
16
7:
2.406443583966069777978882945350423347045696632529300558256733840784079475250905970408187505451327399e-47
real
16
8:
5.119963385271823812382194428607874451259023014257736389164366806142467050364784083446429378802529227e-56
real
16
9:
2.388122512241369368799605516719100761601388208460561655298509522401137848747354281976199895278231364e-64
real
16
10:
3.324566659209915356591781497587577987430115954966308292389937188891866690495187231004125800296540285e-72
real
16
11:
1.839879460532835385408669545950577124622286576966501574556511790630481603464434759584222009641202035e-79
real
16
12:
5.353430652008341415430903591212196729727440540611903883685340013098656163342202193984538907436618587e-86
real
16
13:
1.092278971927596044104770096677940591751218640655636670964952232265594980650232540121407242132876904e-91
real
16
14:
2.146526809345679583030493724207834140872078073738711712195297909133286899481180293487426594592473053e-96
real
16
15:
5.965071079145445124875343659350576188669992904978144001070265848691868273498375060347889541061953664e-100
real
16
16:
4.028959820578095274413319768969623168143908605423529211724388650764328049490393702032907081136860906e-102
complex
2
1:
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068
comment: The pure qubit states form a sphere of radius $1/2$, with area $\pi$.
complex
2
2:
1.233700550136169827354311374984518891914212425905098828301668672027505602802400655375221675464819029
comment: The full-rank qubit states form an open three-dimensional hemisphere of radius $1/2$, with volume $\pi^2/8$.
complex
3
1:
4.934802200544679309417245499938075567656849703620395313206674688110022411209602621500886701859276116
complex
3
2:
0.3805042618515720204548450495652543408192483815339274284822963257420736541282312351176635438857538772
complex
3
3:
0.05798160180595383168835734088613399479150451528136037005444515439879217586715904535126301621116249557
complex
4
1:
5.167712780049970029246052511183565867037548094314184615690756350634399152910951006261116838767147699
complex
4
2:
0.07823805286257360327394363962015135895729309310555392382332976786166128848000896919752778394790980926
complex
4
3:
0.0006984435073668905325031927547156643892952955552076169808788976052612051530168007642677814368710016224
complex
4
4:
0.00005745376717746416126510726386119761268190159553431606409048711028972146716545575033542369253381950213
complex
5
1:
4.058712126416768218185013862029379635405316069695225903811160807915452310701133174588411134781374690
complex
5
2:
0.01005440925605622822139377117570958221933277921850531121583524430070125675395475630869914619341841287
complex
5
3:
0.000006984046902404809590107572758908736294676882225244832641917548690696618603982483885651545349876730691
complex
5
4:
1.380139095356886182715901539577930669905046330690600885559068406103727079094161175361166738166435180e-8
complex
5
5:
7.135587996417081883971729307354892395884230846890132641144400017349204832306218766548038272149032502e-10
complex
6
1:
2.550164039877345443856177583695296720669172555233958254612437030718624934446783537381529170612542658
complex
6
2:
0.0008682891161343536732144048893377985152297378923212195222766758941973417813989990487897486549131239894
complex
6
3:
4.393753722403133599332887873592545708436396143911628402334264386563124273534410684925298789438246190e-8
complex
6
4:
3.366338511097874013828273788627437849298786121145548777430192983589261428703537496490166505946722566e-12
complex
6
5:
2.044574770153512733982138647828318446829023091589491380158354457213002742043007853436234058976612850e-15
complex
6
6:
7.270752228792055337120126207355517017418367700768815903519358902821812719962591421012685419520310495e-17
complex
7
1:
1.335262768854589495875304782850583192871135455668120299918161371505685990058819740971140846044327411
complex
7
2:
0.00005356043801261003897883870694794033343767503800144083916677632670407344048548620689341615877476301645
complex
7
3:
1.767959964911942299302545113222261964642197734694328848500791119437641071905674377731273224126856935e-10
complex
7
4:
5.397695335834452214390842997478047223525501960147601301053066859127412390196458055594442867444873095e-16
complex
7
5:
7.173325131672458568470295111770389683332282930645736975473881369505011917341490965053563135079728529e-21
complex
7
6:
1.643579667460958761394557135151150284470148955756437238994610162516674337725790918235129241000497208e-24
complex
7
7:
4.269665248829992280846334410479366425135044678891379837108723187681904979459888922985389742415440245e-26
complex
8
1:
0.5992645293207920768877393835460400460153635863681353695940941453141140212884150557417564728353193920
complex
8
2:
0.000002472288980471440489217839184517518312180723012308599594461387965995896489370810347728700006997896528
complex
8
3:
4.751370373881343418076872704721579292832545413488988824405470330492631241008547233583792461177918621e-13
complex
8
4:
5.467203405842133223902854854366786306261503486428212831791671244595465517011070910417333907264053911e-20
complex
8
5:
1.763193190288354078485882912526306907295427602705431155246804234266879190591501125881451077090926992e-26
complex
8
6:
5.414107738902009834256755943639437258587979655793929701760762883210659376660333993503541840146832867e-32
complex
8
7:
5.393385154131040258898754425110919544521534819797745620720493770364458546135344549451834247517007342e-36
complex
8
8:
1.068770432664475173565011593582225996156886674920210538118943366215351514575712198878066806121239961e-37
complex
9
1:
0.2353306303588932045418793527754654215450689353085585985146352037466991295097067533738954446185246807
complex
9
2:
8.850632941728054733985276233325461274611398521739256322100430669896446608571824919879239056013439539e-8
complex
9
3:
8.891048103758841074325463719991026835599014394629811958613310987168851559841162485118561801454260102e-16
complex
9
4:
3.567397927384009831414557382924934972555548803275475759172365030189542768395336757765347666038781194e-24
complex
9
5:
2.767772862758697331758905140176002696073398536669142054086579912275900184295212844928246129399274598e-32
complex
9
6:
1.345604904826482519319921934837414250293014757377276861300106604197290681098005417413950740333161520e-39
complex
9
7:
1.167304273954662103765834266294322068437554256853083214677595110021278477010159246431064274551715883e-45
complex
9
8:
5.614132818681673416106336243353855242330812604792557308255339483942282690207913124182418172830124834e-50
complex
9
9:
8.768236463210580390611497326365421254571572252748612606950099501326581309658196705808478032097808979e-52
complex
10
1:
0.08214588661112822879880236552369834480783746079775312311692889117023426706876419810825828348371651556
complex
10
2:
2.527553409551668573009080833841620447316199418921365778988145293464655221278755543460902870444104098e-9
complex
10
3:
1.200734504671517414664840809137992206502015998459049718285584587482840130264733095125549526682601259e-18
complex
10
4:
1.547026460392291042163185819231978665104250850939160028035384606924794731159535322762594374959049508e-28
complex
10
5:
2.758118383269333362651796484347183932210427182857306664991478544596200086210724843748488997547356758e-38
complex
10
6:
2.192255941433984884827394016255019330440562116806850546533986862535161791958741551357810796150940381e-47
complex
10
7:
2.063707878223878374588063605756323801068024140255141696017272595006981484894406815474163612479611819e-55
complex
10
8:
5.876661815326835611443545696877183101844259971325550750449437393092823439919598119831635603163906537e-62
complex
10
9:
1.479381454289964820611198960007882384887107373985840521430612084647575245791011927399799674624006981e-66
complex
10
10:
1.868182053837595224801643450309907976744270010466045531610444959104779393425844402136284303468334002e-68
complex
11
1:
0.02580689139001406001259829425289884965718644104814668599180237251796688933594775585554873825323650511
complex
11
2:
5.890016504629063312987776592504940475386167014582829152942087421135178228527661739034689360741727237e-11
complex
11
3:
1.206242055678050391414648773434223524659043772609173975808133270179899061354961341409237158390927280e-21
complex
11
4:
4.601041424899859296351440012954653150190769644562600820974327957859937781719807847663465927041102266e-33
complex
11
5:
1.776261805146658697432574168424592969561910000559455713385681539043140032114797939452404402820730683e-44
complex
11
6:
2.268363981445510077733889528974914079603700125172036671398173732164579745766559810214203010093187529e-55
complex
11
7:
2.479567631432600169150784520047863469988461005728220659019886522026237189468108234283018752349374328e-65
complex
11
8:
5.452495400324302984619140427595840269942326789808939898415256947039463158673484369743261071489265531e-74
complex
11
9:
5.728643589199748854956628903323385990394945003977139966943099095184934383003167426572444148750752096e-81
complex
11
10:
8.049927403309035238955495563896244139052951553742489974062798969142250040680114325548832516304057211e-86
complex
11
11:
8.390220293704836614054484744114427213127083177221697014241689410352510125581894241422872888924236687e-88
complex
12
1:
0.007370430945714350777259089957290781501211638236020997449899842713817490175822097957286936350251086161
complex
12
2:
1.141023268093949704376671886160208563175257035860990586198801079838806701419512101037805111045460980e-12
complex
12
3:
9.246942026083785226158796618773320700150732776561300274308154491042712915490312884968002768965331302e-25
complex
12
4:
9.659186588175795829312068976146762359095046859058638498596407618694006222817475732048321205955796571e-38
complex
12
5:
7.575296715786063739502755308802044339025909606620828653205021186527035879088986771952647701301683123e-51
complex
12
6:
1.495600955126021278973251820921812609856133926497531557528999119972941917895985163319542612950344535e-63
complex
12
7:
1.913714266408661352061770789797740304243379407283570515299020312704204123524477657625428331387732818e-75
complex
12
8:
3.588344588079528862872188748085220235969737808126767756697510632465327154853366664982317855108543149e-86
complex
12
9:
2.134645423705291128775078293576048818436971985993216166863002660948250250683444753158444881440578083e-95
complex
12
10:
9.089332256249258204974190324607377176023470586027699688692143334443241939151854510735469419557179350e-103
complex
12
11:
7.513947321368119323768053894752481719181651338386009538251426042539600931918372229938986969320066043e-108
complex
12
12:
6.574126697777072451414844725301271424779604125745232491562530703546382577987915254681497042441210574e-110
complex
13
1:
0.001929574309403923047903345563685957640168471815000303352234647617331495634250985531487347698186143913
complex
13
2:
1.866243794499949432518579452627517279038832084221091554919452725320921760369993301455605238672532327e-14
complex
13
3:
5.527847549358441403345425842895842615398101162830644665804593524789208634463862330136477544052224798e-28
complex
13
4:
1.468540828066737938785752860203919960264677838946876791042148348061979259763853658148156697995899652e-42
complex
13
5:
2.193857022685226276761289250784711031333597203070300225820952108359198350310764055765983843627027587e-57
complex
13
6:
6.383591834424907453285493535349664902194725005149465252503998779352704399953497926271440489043477340e-72
complex
13
7:
9.377420674486131622677289142867747981346229426140934967638121077361025780940539844470813846790451202e-86
complex
13
8:
1.546464088327113786422414803260364141163342125716376617022761297072155754739522872871910322650232537e-98
complex
13
9:
5.914479704391580467579077751867421017823630967939541704809122157521992809850444187180984954887560816e-110
complex
13
10:
1.071523232037430929618023113596241349670209076517600605835807398576466693622634366980271516254136021e-119
complex
13
11:
1.997571399253212730849683196225969560781599778488685760749979210741620860233198039999475999710463471e-127
complex
13
12:
1.014889353363788171074202035416703334635624044996162674272653164223659591362326676785382143777072281e-132
complex
13
13:
7.560089025652581988502749522434852579781203492079653890316228557615074291440179043234520807034303979e-135
complex
14
1:
0.0004663028057676125644206289144702717438281998136159926771563035732539635746869590996901434958105247278
complex
14
2:
2.611187273527831151818287268764816688406674129857776688086796719834109405999888229975570412535170218e-16
complex
14
3:
2.625320875338350997451848679882512710387610148256605488317888643550147364021862362859678430625729842e-31
complex
14
4:
1.653923748251248340584574752615186069907114837362219952644506075317183685786878415404767492720963859e-47
complex
14
5:
4.418852628956181442690387001519515759856600269512762917640191903923920393797046231062491073487814953e-64
complex
14
6:
1.799528582824728134289359968282414126206854851628363966578375914475327948469891793389961276593468830e-80
complex
14
7:
2.936195216090068136121003339962423590990897232471084608491402790850829005819709935342268054471561313e-96
complex
14
8:
4.250140519385494921392746264326177060950611368948910951239852697227691756795237061100097515849906709e-111
complex
14
9:
1.100430864266450588913340127315960247181756707981553430448311059923049539799224951423998257536575020e-124
complex
14
10:
9.874424742496158836509041049681645667061996850911673800380109925621360232175527121184098270815675991e-137
complex
14
11:
6.008304876663799132765045364595218574691237843308703557828094816640264927389835752014182241161741252e-147
complex
14
12:
5.232010734622043534396553861181192836196284585221590278652770781228918853173096077977729403163816347e-155
complex
14
13:
1.695214282029491365953001168856228061880494686199529114071306639957975817416681147648316865466058289e-160
complex
14
14:
1.088168087095727858275468380366180261089800796362950108880105833349690262747429136240535310149193041e-162
complex
15
1:
0.0001046381049248457071180167283522393276102973314909132131852771207714351570546014321508781859973491700
complex
15
2:
3.160737024980230232075563485466230343127309542183989312117181864466776846002814859816804505126513734e-18
complex
15
3:
1.006610276516345075085427666574377418203521790113616887334137666124342515118553094482302024918169668e-34
complex
15
4:
1.407633820680385036487524377228576826975633988487589386415527145904026970144079783173574626119617527e-52
complex
15
5:
6.327928691502045102177386779123082023976819699212517708306030033358554854284258524091103191128731913e-71
complex
15
6:
3.421071583845799031677200792987016239955989503456020170011994036656884968206605001242931239693978200e-89
complex
15
7:
5.957588731501211230584611120300277893859531389905201640601828286638145513006155078068892085823387075e-107
complex
15
8:
7.424017902168821841012988335123420685218490605741856042760766865614654201046143005691173595620319432e-124
complex
15
9:
1.319588044154505116436360627385975608725394954416369686718326090322114863064187293393154874179712352e-139
complex
15
10:
6.293005454416156391901381078104364655993920470561995281231811963831887264284162872003582018088355845e-154
complex
15
11:
1.487779620647682220602812804559821565601793828437289056890509359875771494548181781798389352969440319e-166
complex
15
12:
3.301496353777244490774670754174284501399262955059425142698710175162351158545824023637174954948364355e-177
complex
15
13:
1.419003247874803445439604929227816297839603494609615093810651519442814796859013670345471938748824254e-185
complex
15
14:
3.026546452987328050182035530484892954629994119132722056142022209976824949039860267019808751686789259e-191
complex
15
15:
1.691518361444211680313847900471429516465150336774393365916943017079202372664609335321287657336717500e-193
complex
16
1:
0.00002191535344783021582738465205709418885924870876595578031103527648380984852076345699678892956629057534
complex
16
2:
3.342345434189032106950446877702909475412939858000984792579763112648187650902014432073510638709795657e-20
complex
16
3:
3.159991570608373393762836638463933580448364951223359457389371717354650070414699755972525759155617269e-38
complex
16
4:
9.214389505068048192863423470366223022019862751290874570665022189734159980251424121155704761101855433e-58
complex
16
5:
6.572908724511163823359506992074711590716629947228815870350941065251628780254221828049733006433831294e-78
complex
16
6:
4.476013046563258955795874426027434698739090205445881362990149112571999655618817334367447159817300737e-98
complex
16
7:
7.967474074197495542615347715699175687904524130931987722164940369045292812428625241063062601441986001e-118
complex
16
8:
8.304130287013477613760173538321082056678000041900036302975164423552898901119622832108076507940183757e-137
complex
16
9:
1.006523476014322284871639630627620765651119612741667290077299940020642661168976984147790456269073814e-154
complex
16
10:
2.625116998941059446232490618944204127777402022752948655805166835651986651264433407233472369095778204e-171
complex
16
11:
2.632234324041404305523374765763034492453837690546478318080416842712651525484584501343310449159505251e-186
complex
16
12:
1.808120585178009167810550258225434224575035971730558914903886070162537998024729514517675530559204410e-199
complex
16
13:
1.570027294471639323192689852759797882743408174008485234922253611603640031694992497305594443202662961e-210
complex
16
14:
3.492835156335936618076462139860944713014729141106627261949504382098255080292471348146358113435789393e-219
complex
16
15:
5.040679018138304917390106355336861092511474702415023513114460282054790463531807803977443811504598610e-225
complex
16
16:
2.475054045350604009230811310582221595465332416616917752217224821482090919336149091471142133801905931e-227
Definition
$V_{N,r}^{(\beta)}$ is the intrinsic Bures volume of the manifold of positive semidefinite, trace-one $N\times N$ density matrices of rank $r$, with metric (1). The field $\mathbb R$ ($\beta=1$) or $\mathbb C$ ($\beta=2$) specifies the space whose volume is taken; see [1].
Parameters
$\mathbb F$
—   scalar field ($\mathbb F\in\{\mathbb R,\mathbb C\}$, identified by real and complex, respectively)
$N$
—   matrix size ($N\geq2$)
$r$
—   rank ($1\leq r\leq N$)
Formulas
(1)
$\displaystyle ds_B^2=\frac12\sum_{\lambda_i+\lambda_j>0}\frac{|d\rho_{ij}|^2}{\lambda_i+\lambda_j}$, where $\lambda_i$ are the eigenvalues of $\rho$ and the matrix entries are taken in an eigenbasis. The metric is restricted to the tangent space of the rank stratum; see [1], equation (3.5).
(2)
$\displaystyle V_{N,r}^{(\beta)}=\frac{\pi^{(D+1)/2}}{2^D\Gamma((D+1)/2)}\prod_{j=1}^{r}\frac{\Gamma(j\beta/2)\Gamma(1+(2d+j-1)\beta/2)}{\Gamma((d+j)\beta/2)\Gamma(1+(d+j-1)\beta/2)}$, where $d=N-r$ and $D=r-1+\beta r(2N-r-1)/2$. This is [1], equation (5.15).
(3)
$\displaystyle V_{N,1}^{(1)}=\frac{\pi^{N/2}}{\Gamma(N/2)},\qquad V_{N,1}^{(2)}=\frac{\pi^{N-1}}{\Gamma(N)}$. These are the volumes of $\mathbb{RP}^{N-1}$ and $\mathbb{CP}^{N-1}$ in the Bures metric; see [1], equation (5.16). The real value is half the unit $(N-1)$-sphere volume, and the complex value equals the unit $2(N-1)$-ball volume.
(4)
$\displaystyle V_{N,N}^{(\beta)}=\frac{\pi^{(D+1)/2}}{2^D\Gamma((D+1)/2)}$, with $D=N-1+\beta N(N-1)/2$. This equals the volume of a $D$-dimensional hemisphere of radius $1/2$, as in [1], equation (5.15) at nullity zero. For $N>2$ this equality of total volumes does not assert an isometry.
(5)
$\displaystyle \frac{V_{N,N-1}^{(2)}}{V_{N,N}^{(2)}}=\frac{2N}{\sqrt\pi}\frac{\Gamma(N^2/2)}{\Gamma((N^2-1)/2)}$; see [1], equation (4.20).
Comments
(6)
The real dimension is $D=r-1+\beta r(2N-r-1)/2$, and the volume includes variation of the support subspace. Lower-rank strata have zero volume in the ambient full-rank measure, but their intrinsic volumes here are positive. The cases $r=1$, $r=N-1$, and $r=N$ give pure states, the generic boundary, and the interior, respectively.
(7)
The scalar field is $\mathbb R$ for real symmetric matrices and $\mathbb C$ for complex Hermitian matrices. The nullity is $d=N-r$; it is the second subscript of $S_{N,d}^{(\beta)}$ in [1]. The corrected version [3], equations (4.19) and (5.15), is used; note [54] corrects the printed complex rank-volume formula.
(8)
At $N=2$, the real and complex full-rank spaces are open hemispheres of dimensions two and three, both of radius $1/2$. Multiplying the metric by four would multiply a $D$-dimensional volume by $2^D$. No volume is normalized to a probability.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

def bures_volume(field, N, r, digits=100):
    beta = {"real": 1, "complex": 2}[field]
    if N < 2 or not 1 <= r <= N:
        raise ValueError("need N >= 2 and 1 <= r <= N")
    R = RealBallField(numberdb.bits(digits, losing=128))
    d = N-r
    D = r-1 + beta*r*(2*N-r-1)//2
    v = R.pi()**(QQ(D+1)/2) / (R(2)**D * R(QQ(D+1)/2).gamma())
    for j in range(1, r+1):
        v *= R(QQ(j*beta)/2).gamma() * R(1+QQ((2*d+j-1)*beta)/2).gamma()
        v /= R(QQ((d+j)*beta)/2).gamma() * R(1+QQ((d+j-1)*beta)/2).gamma()
    return v

# One more value:
bures_volume("complex", 3, 2)
# The table's selection:
# for field in ("real", "complex"):
#     for N in range(2, 17):
#         for r in range(1, N+1):
#             bures_volume(field, N, r)
References
[1]
H.-J. Sommers and K. Życzkowski, Bures volume of the set of mixed quantum states, Journal of Physics A: Mathematical and General 36 (2003), 10083–10100. The formulas used here are from the corrected version dated 17 February 2005. (arXiv) (doi)
[2]
A. Sarkar and S. Kumar, Bures–Hall ensemble: spectral densities and average entropies, Journal of Physics A: Mathematical and Theoretical 52 (2019), 295203. (arXiv) (doi)
Links
Similar tables
Hilbert–Schmidt volumes of quantum states of fixed rank —   Volumes of the same real and complex rank strata under the Hilbert–Schmidt metric.
Volume of the unit sphere —   Unit-sphere volumes determine the real pure-state volumes and the hemispherical total volumes of full-rank state spaces after scaling.
Volume of the unit ball —   Even-dimensional unit-ball volumes equal the complex pure-state Bures volumes.
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds both real and complex state spaces for every integer matrix size $2\leq N\leq16$ and every rank $1\leq r\leq N$, covering pure states, all boundary depths, and interiors before extending to larger dimensions; the cutoff 16 is a noncanonical practical bound for this discrete geometric family)
How they were obtained:

Integer and half-integer gamma factors in [1], equation (5.15), are reduced exactly to a rational times an integer power of $\pi$. Arb ball arithmetic at 397 bits encloses the result before serialization to 100 significant digits; every serialized interval contains both the generating ball and a separate 560-bit gamma evaluation.

more

All complex entries also agree exactly with Pfaffian integration of the Bures eigenvalue weight using [2], equations (21)–(23) and (37). Independent geometric formulas check all pure-state and full-rank cases, and a beta integral checks every rank-two case in both fields. The complex boundary-to-interior ratio is checked for every matrix size. The attached checks repeat the enclosure and identity tests on values read from the database.



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