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Hilbert–Schmidt volumes of quantum states of fixed rank
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Numbers
$\mathbb F$
$N$
$r$ 
$V_{N,r}^{(\beta)}$
real
2
1:
4.442882938158366247015880990060693698614621689375690223085395606956434793099473910575326934764765237
real
2
2:
1.570796326794896619231321691639751442098584699687552910487472296153908203143104499314017412671058534
real
3
1:
12.56637061435917295385057353311801153678867759750042328389977836923126562514483599451213930136846827
real
3
2:
2.467401100272339654708622749969037783828424851810197656603337344055011205604801310750443350929638058
real
3
3:
0.2014624562149674630583256873592397217473547768754365728574372205630677296937019279117545400480398343
real
4
1:
27.91545679855551813663024374467762115967946508379216946461398667007963105133230885866960534924324575
real
4
2:
2.583856390024985014623026255591782933518774047157092307845378175317199576455475503130558419383573849
real
4
3:
0.05021354011606970414515930779362672758620417445516107550781938877390597187821512738392369372066469708
real
4
4:
0.001610600050165384213565481691281499855319569868926676945956809844410893774087751259757306005865624877
real
5
1:
52.63789013914324596711728533267280605500639683861755000753786333984023905290242796267612481983227857
real
5
2:
2.029356063208384109092506931014689817702658034847612951905580403957726155350566587294205567390687345
real
5
3:
0.009013853763483505376274727598204419523512127279976364505244607698538597110414053612865715111654812206
real
5
4:
0.00003705357888929374780428751201161569521786922676401710233983131789060067682480907262102177949950958517
real
5
5:
5.918165800436569611102489828124165830747367227872864627728440678174602877436596840478384063016448165e-7
real
6
1:
87.69899399994526445902989806952582546859522718182378370825256577580866325065258393899582400855618061
real
6
2:
1.275082019938672721928088791847648360334586277616979127306218515359312467223391768690764585306271329
real
6
3:
0.001209739640976061844055508457548760595343813111447803336740972145527455919508246247678895667657165170
real
6
4:
6.722503038269950767341953264987700007572013554908776638404269040709673931329892630316040627385983155e-7
real
6
5:
8.369085188067079168247257754135081449176403721282970068996761643757139822168930067424591966537066631e-10
real
6
6:
7.639894571977841211999934387753513006731349201397618199589944800321482507396890977219152123101543806e-12
real
7
1:
132.2934471692792327486989442862992861961612312144431261616833625762406183145203457602845910724389811
real
7
2:
0.6676313844272947479376523914252915964355677278340601499590806857528429950294098704855704230221637057
real
7
3:
0.0001264642731883852414163471398883629150059975149550905127378039703729793055121393415992947474124871851
real
7
4:
9.312062536539746120143430345211648392902509633659776855890064920928824805309978514202060466502307588e-9
real
7
5:
1.026029143657959737620671471534874520747117274792890772622749359959648013987544771054719408270083803e-12
real
7
6:
4.801554371279329360080418467873417873139361092277732747747401822408994691831617354287719460685516933e-16
real
7
7:
2.744058979679727970448421723979543525206063093683910139512011645947038084079409524638926471861829558e-18
real
8
1:
183.6763435182955992718493306166415194236432305635143405413646470645936034094348654912366039648480803
real
8
2:
0.2996322646603960384438696917730200230076817931840676847970470726570570106442075278708782364176596960
real
8
3:
0.00001064511718812006642402363940670771404910924074019662826836470820634572898761714534200882257087109391
real
8
4:
9.989823194983914637560421030296849354237923185646185697602160024288886765228706273167253581008645502e-11
real
8
5:
9.944929676550440851194477542394546278499944409835134329748734492772826663400590943152862770819995564e-16
real
8
6:
2.875075765731019315011603136591462467696757349896704347692516704471156309922266670888542913109169153e-20
real
8
7:
5.855654007020638626404540372647594172151774065427041112951626120262150402782068428133908745563056656e-24
real
8
8:
2.235699088751142994552655764779460279479452135057722599661881508322271991474517010737740223498166907e-26
real
9
1:
237.4926409971868945955116682696048426660024945924520757430073524174527523518834497587733144009215818
real
9
2:
0.1176653151794466022709396763877327107725344676542792992573176018733495647548533766869477223092623404
real
9
3:
7.406257844680866845187584970194901820527831702712872617428504463775734254386298761732994638886936433e-7
real
9
4:
8.480347428919351162530072217868896921974444344981020804930339846189147263318184293226083743995913514e-13
real
9
5:
7.525542281117282029065055518092565282553102343977416880505362164768962694243726257375396070507785727e-19
real
9
6:
1.420807896833990430133569526100491961802311674914768686239643968962776875752554033648731180620413269e-24
real
9
7:
1.305263477204899151136835059635342749567162794104518181279206622592795654249533946018461991690947110e-29
real
9
8:
1.287278607039710532158617282812316189050664985274892640540322446307032304953467440121088351605563760e-33
real
9
9:
3.447891789068758677742053749247527875507141765864446626447523425760077073399771294519667091867944111e-36
real
10
1:
288.5181257176563456136532627588079410609451490661543939114219032884620370245874862832923852529826073
real
10
2:
0.04107294330556411439940118276184917240391873039887656155846444558511713353438209905412914174185825778
real
10
3:
4.348688390369776146331531729235236446915215131346721270390688050310719016589526811925765428482849044e-8
real
10
4:
5.816599738787468084642524718549918392742896256232354184511227666598167196806610106521944827514865807e-15
real
10
5:
4.482159396817723193190994326009725029783934089801828481732817903022217311596628774522816340671049951e-22
real
10
6:
5.577693607536563864625608647966886506877200896016623412398260955792980864638217292090403199060553748e-29
real
10
7:
2.518415070145058638396165270787483469846085564514760699572681233224540255003575119828250622591043208e-35
real
10
8:
8.461491221669683688082087148051450710263418465440147057431821681330709363940380914339729736379175648e-41
real
10
9:
4.384245721799050497619148317792082329011505188473305933912454451214609971712321861635326838670763522e-45
real
10
10:
8.558149569588844915918794383652251696332303320288799268107952470410725521442485455214380352621221034e-48
real
11
1:
331.6022827726224424772985200893947735828866916266124765362711770737615152855246148925315391374277691
real
11
2:
0.01290344569500703000629914712644942482859322052407334299590118625898344466797387792777436912661825256
real
11
3:
2.191564233513183431952384127713210476445476619147206995828490758339669913373066412813815350171998818e-9
real
11
4:
3.283714479125526024928128800206746042930047739751228865496970151634946655825007693062929245288374897e-17
real
11
5:
2.130148222134627267228701533795821311308920868489183441870654906016388619168332788980876106775007787e-25
real
11
6:
1.729238003384944391620452215923754623258448541967667975632863422915490926962147725575199644634519964e-33
real
11
7:
3.952719581212075804525728258090995612556964804551671151866563664468219705553518468976504926317058400e-41
real
11
8:
5.062151171548370320549611149265209509897458678776638702175465379112333852877153616280424101997074747e-48
real
11
9:
6.930924508822504656993998906819595734013590750066589530308414910241605079000397793453113454335976168e-54
real
11
10:
2.012585176476006189453166494637749676477352434690817960515705394750255040345993740767468344913774244e-58
real
11
11:
2.952191805292477248154832490953196108419176001776383085855977232694431435705193565129740229317163525e-61
real
12
1:
362.5625696728342269923481913771002705497074697875558716269447230722667491741462739361088837429232507
real
12
2:
0.003685215472857175388629544978645390750605819118010498724949921356908745087911048978643468175125543081
real
12
3:
9.612142513576103793406521979141974334050855271111040551342638889813026612494046675932056796508966233e-11
real
12
4:
1.550522614099442088514240910097823684541201675941928687029939401706679569632026297444012518393980323e-19
real
12
5:
8.197089218047005951148648368701446447581941773725940918844360926591048717638652824149560941274819922e-29
real
12
6:
4.257768882142400641208722578705969490651233811669865130499910238531063111388836038597039310871652895e-38
real
12
7:
4.947791915008927158299976666466391277862054442288463206779857751808727224368686753733667422707827458e-47
real
12
8:
2.533827550390673780197859915515926373012639429136636490023357533433814639912108046099364905616476750e-55
real
12
9:
1.049915640358829706877099101903033784855141356059586388084482100482205084805360868467880398960747441e-62
real
12
10:
6.385103058528580643030123259944158202002121754811526742218500706526516876575947272829198286833311579e-69
real
12
11:
1.095063960165268724845247469937670657332004233685601230883200220603480599086848097917845688688330556e-73
real
12
12:
1.237832255241127606894231338013162857928979622309820372533366914014346200376837680708365297162987553e-76
real
13
1:
378.8215619898457887518786764021464074805313126522334717551368988874313291969472317934402012396581871
real
13
2:
0.0009647871547019615239516727818429788200842359075001516761173238086657478171254927657436738490930719567
real
13
3:
3.711956450900524445815580344088463293844612145555811733603515591426940949899967957949114635151455838e-12
real
13
4:
6.209192378456535462586429431936480371121883328340847157290213949814789742510187706093570728550768933e-22
real
13
5:
2.590019305338504559047197731708652550752568064071817823434269222806467217974303126167551547600193010e-32
real
13
6:
8.407853505327599099733244572901987471877245169548819282164350730279766022218040348523567382351135471e-43
real
13
7:
4.925113193363874270147374604842099832360833877792081897360950423533394560683843581516687300105041331e-53
real
13
8:
1.026398612284579543126487911441049920405530561979434029964678760148858191194135415928827801186028209e-62
real
13
9:
1.372774221400691145723401702533951272989636655530009302829663482023943251181594246291426977680742205e-71
real
13
10:
2.044455371356993278739689446886946850964162692998344607175088446448104661278662500384615443284629272e-79
real
13
11:
5.925720592901642735135897755345725445395427128409769698606841058500004165425058327136545112545540391e-86
real
13
12:
6.269596843719449238163195284332077344576714155404029235304022334527122972645695767000802727755319284e-91
real
13
13:
5.577438709372204487167087220230989663715498931504986121513522686182636669558841406407496848436160951e-94
real
14
1:
379.6746351169378577729246795023769057728137117903952501743502469135919657902627198714982484077828576
real
14
2:
0.0002331514028838062822103144572351358719140999068079963385781517866269817873434795498450717479052623639
real
14
3:
1.274620488499800970319049091928218882936958239653692350350762427728592124980105127611850958949288546e-13
real
14
4:
2.134324330932992569913542641619388049897863405555345706668991507747180834375246855668529943857432570e-24
real
14
5:
6.806235157350276840466579285306523721894707020331163941532003653956052032208268159708529514279721696e-36
real
14
6:
1.346313778952350485595431108097861476258583033651602564420998358894813414075087613621654235544479745e-47
real
14
7:
3.915664535655349257532125681461564114320091110096306788382567249041097638123606525759339086902344883e-59
real
14
8:
3.326128355502197206496203647682361168330526694020594918187369395482651422345051504629777378446970924e-70
real
14
9:
1.479294676778094599701355030576705196537405343206741300917900095513260022284532836204104720699570758e-80
real
14
10:
5.841046485238697089630448740793238519248144452047599501493932487251781261601274983851955546437503318e-90
real
14
11:
3.408030142584609295442640986214093308260456222868079568589288345290754844450932798972135392081721040e-98
real
14
12:
4.992786481627694054145761005556283238883718374654806498133143940401835819236070392278866164309905329e-105
real
14
13:
3.380945454299542202304556248795741602861462081063983277586297904134418347853497306671853187238852768e-110
real
14
14:
2.409734141988453792291585573160849167106608527720525343931095535970599836460475110951008019932501551e-113
real
15
1:
366.1855495903723035050339229843662513348519160486910916710354676671867272648724390964265939742236986
real
15
2:
0.00005231905246242285355900836417611966380514866574545660659263856038571757852730071607543909299867458501
real
15
3:
3.924870097305429095303775250622676595567375595552831494111275876885825189310394761625531687164949934e-15
real
15
4:
6.363653095881005012247384628621419856246550532436215365120097272807706950114419550751844722765018778e-27
real
15
5:
1.504883298206943995807539859055329374052246267272455290856981413462977321284398367148160274994568847e-39
real
15
6:
1.767464580251087496756979055184377776379400740411409413136640029788885581832489478419535686598281338e-52
real
15
7:
2.505134779973581587870299547116197249958358324011800349063053875889892447703357968353878642519601576e-65
real
15
8:
8.610988466580496659778527105946117580763504154892666113036726718437416435990005084769794108477308062e-78
real
15
9:
1.287612082716818577011684993102982963248282674911334956445058074558581261363564501735036877828539769e-89
real
15
10:
1.404446081242258384920677736888321447179366023226195947638686411354849754765160782328925944979469993e-100
real
15
11:
1.810439108269072158497962935899719700265863430466185290488763178938734409726327171895806619947988308e-110
real
15
12:
4.447844460048410329890370380836640599596180587433364227721920462143639468868392698019784433081259677e-119
real
15
13:
3.461924162296789915127902618130097203977936002198812997635771311932426487886862608286859711136854182e-126
real
15
14:
1.548389185655705554564878544988100924038944637399936268875456250368957201272802804652073673119048866e-131
real
15
15:
8.978907978816443727305536548139204333554157280424492519116441604678150237575226512468543421867824431e-135
real
16
1:
340.7951555536449435556343888467527386740270559542821875632357236071511891396913700562992524145520895
real
16
2:
0.00001095767672391510791369232602854709442962435438297789015551763824190492426038172849839446478314528767
real
16
3:
1.091738795537521195493199519065898197417757994795817324192019414783565376277728291946597295963504507e-16
real
16
4:
1.660986072786410345676294779899141031312677893701139387091686772008376781031716141777137747765556410e-29
real
16
5:
2.828875027009125621717010255834617849622532245152007826782408944333908094589519655240454200201180410e-43
real
16
6:
1.922493373391471771759142564919392745048080439217400571581879785248944395718620967042994598414564131e-57
real
16
7:
1.300979440963215435408941228140586166911141373589485357864602737783978327374666978606981900835126263e-71
real
16
8:
1.787399263088893728833279194398827381306560271382492877583689822731874210010571293511340902794008709e-85
real
16
9:
8.987733611911856629104882194414809186042265922354258114729708149486264861729186380915242952488514924e-99
real
16
10:
2.762141555364946616656264036779160054340378256685850319790579972089560089651841209377492779101688427e-111
real
16
11:
8.282121663025456991120807305446951720873703370808129456999022613298336060180683849196798146782572842e-123
real
16
12:
3.791959010677061063180428682417759279555238496378169359771313447178508016177898028351024669569237232e-133
real
16
13:
4.170795496414967024486180390497393053940573494764511353218722433579148505931155028168924378597520952e-142
real
16
14:
1.800251799654907791172970588505326969407088637047105923121916455247375486666094257339667318069196021e-149
real
16
15:
5.465178748653330651475870633124267853836022135594481614874921848053896953916997737626304056762616024e-155
real
16
16:
2.613153861429758011554440311704262326386439656799993867427431024153617424121438314287432778687824267e-158
complex
2
1:
6.283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234136
complex
2
2:
1.480960979386122082338626996686897899538207229791896741028465202318811597699824636858442311588255079
complex
3
1:
19.73920880217871723766898199975230227062739881448158125282669875244008964483841048600354680743710446
complex
3
2:
0.4176142571425964974239518955691705974695010818182084938488217417895650630983456378047420190883627648
complex
3
3:
0.02131128831897513940362728906742617750820941698676499217779451803413046697312475540128724123630517387
complex
4
1:
41.34170224039976023396842008946852693630038475451347692552605080507519322328760805008893471013718159
complex
4
2:
0.08326641434852996987407020570239767418786295788345004153287789416694431082960677814813633052034130082
complex
4
3:
0.00005867765141807391014264485650041247534887001392596328377361383722195523041654055405342463692445656649
complex
4
4:
0.000001129251928054666599749714651782573568544585958522425976071049688094496817517990784640663756175530215
complex
5
1:
64.93939402266829149096022179247007416648505711512361446097857292664723697121813079341457815650199503
complex
5
2:
0.01053598166047668602988468974088070001589559639777986597220419155604373309441288488931896571228168658
complex
5
3:
1.564650408341061880217586562611919463778580316446232207276020695105839617514883272134482354811196171e-7
complex
5
4:
1.068308645171401238093767384749976243617650536517303861645674410367104563899578975036849183684489412e-11
complex
5
5:
9.953378131474815159004733469539537950846926742111657963028967322183709470497562714811434667755656750e-14
complex
6
1:
81.60524927607505420339768267824949506141352176748666414759798498299599790229707319620893345960136504
complex
6
2:
0.0009014263736993960158189921065157382406383235413908222828326105345550367942602479701062755890979372014
complex
6
3:
2.753403338666418628244515166217332745173035224755743066868270101042213916083922655642749847535502830e-10
complex
6
4:
1.319135526589060101037564752130334163212975078897944829792574668027395816597104318812382757396024375e-16
complex
6
5:
1.474834905688501911970881177364444219783578397277790342151340761916885905551697497919133895821622314e-21
complex
6
6:
7.693336632777111714489686039925127055438623584018148649055492909713946767957645994854627884451216725e-24
complex
7
1:
85.45681720669372773601950610243732434375266916275969919476232777636390336376446342215301414683695432
complex
7
2:
0.00005525914102559987443791150239231940519263973696348070772106151114769607236571413240171831621114798428
complex
7
3:
3.165472218433527774720487109082255815953974786671550990990329815702924558774703231528306012339178161e-13
complex
7
4:
1.169591356066153168106057077941675729547180026633681180281489834009867416914989710038767493952659718e-21
complex
7
5:
3.772639639306958277821182248443668169021136727572917414044962417876832304389546255707654340722502898e-29
complex
7
6:
9.532121737975924238355387290356403503501634678571255057025302354329201557227697682373343771412642755e-35
complex
7
7:
3.064246492533672506414731666180265496630559513293357400514096563417389860409044355155037965749987738e-37
complex
8
1:
76.70585975306138584163064109389312588996653905512132730804405060020659472491712713494482852292088217
complex
8
2:
0.000002539362741220511578544911751779802382458302509598668669922509341748228744155027790518962104948898845
complex
8
3:
2.458211351030487276291767369010576607909054265278457008404994508758535996694031662338965711642880265e-16
complex
8
4:
6.831995654512950942772677333410875728749135738320167996869869388934530419923022940190055810555275339e-27
complex
8
5:
7.718964228288718666860289250602328595226630840888447287642802248166659498467884572771301619516749365e-37
complex
8
6:
2.645967563448066444177849141618037362327391147541700283315663096049983742235074772575015806690598862e-45
complex
8
7:
1.884151037507058403343176019544921537897587095761761708679082061715697788441679409648348406349564972e-51
complex
8
8:
3.996512271704121636405687162032482306835392574150148738459925352061836129695851061968938238673954617e-54
complex
9
1:
60.24464137187666036272111431051914791553764743899100121974661215915497715448492886371723382234231827
complex
9
2:
9.060416405096214926212351260601996935532075940746233184168848531348625359482103739915849336726573049e-8
complex
9
3:
1.338167837227080285921045914483240345559109274182461357300616639739785664873600507110001739767537675e-19
complex
9
4:
2.631954256346913294539084804482246490045873685713888136964567189897449094399739225691460301028917192e-32
complex
9
5:
1.078251002418875891920143595407302633938949898568047486344951608372035425716804702205405895859637364e-44
complex
9
6:
6.623075046026896329693916346861141259480051594503480896161212506993268347866758336055869649313372909e-56
complex
9
7:
3.308797173833865555363912246206085487828077072072242574013060364761787020306246213543452105428193315e-65
complex
9
8:
7.806966547581354509037948180721269973564446577482391652112501741986572204155941489739271755493640416e-72
complex
9
9:
1.150074788810688584382743196433548324563930284816279545888385000501507990713875881897560238223042396e-74
complex
10
1:
42.05869394489765314498681114813355254161277992844959903586759227915994473920726943142824114366285597
complex
10
2:
2.580742442350355785287505746903771940898338075402354977591622895324505984002081523019982981503888051e-9
complex
10
3:
5.280147508672025663529294865650127693863821932452746777112502114744962467038520071290860459970498015e-23
complex
10
4:
6.836739362471459427321524484329005162784321479979186169135258115287121305205032546821563018081638956e-38
complex
10
5:
9.938914852761570594796829749508696348091243042685554104050448638049981566234760514351041701875312438e-53
complex
10
6:
1.193823296581665161285696291039728480712078380631790235736731481955413512317485673324367616547131715e-66
complex
10
7:
5.939992657257079696680717338307095936326113475454455573850534115175797620259205910832597963248219854e-79
complex
10
8:
5.463035915319529804756244369216585831185158399221778974929011802263031769998279599187390308260062176e-89
complex
10
9:
4.835787337407218145110128116254971529455099918372670991550429868861876599873485460410980847974024022e-96
complex
10
10:
5.148855981955547105922468619933076880553468690743971715732473413804923060991852065011347827477821369e-99
complex
11
1:
26.42625678337439745290065331496842204895891563330220645560562945839809468001050199608190797131418123
complex
11
2:
6.001461101049007913850187966595781274798385511495631967948092246674462376656416488472861753428451559e-11
complex
11
3:
1.554393138731887579358275937765708621797270291869576980159370361461250387199799753648840592822710747e-26
complex
11
4:
1.229652304335418377195111782838310633919071169134478815266389590545093152380034946340375323715466849e-43
complex
11
5:
6.067691159249710348324046762895302149573630557992796944994513122255570758164676371785170128980363247e-61
complex
11
6:
1.445763127749154575134821653892698270982360879696732609726871840013202313682091367711664171345983229e-77
complex
11
7:
8.194054825971834456941579852237307016208620465200905625140436946022451473146313935335584898155262005e-93
complex
11
8:
4.438370486519549974324419736365716982088374600038244524986565313086708151615644977341740185017570392e-106
complex
11
9:
9.003370069754663442551708755324610555346381285037242158313170170058814425849330975070708663239282411e-117
complex
11
10:
3.299561287558662115993842508774415147183652581296011412006265779031552461059823663951052439528920652e-124
complex
11
11:
2.621673540508501972551487040607153331898972483612098859980694532680397349937964339487912576370591261e-127
complex
12
1:
15.09464257682299039182661623253152051448143510737100277739487787789821988008365661652364564531422446
complex
12
2:
1.160634137082161247321990236334081340093745720719681086907933031083789291533621481021056469317787689e-12
complex
12
3:
3.499200070405782890266717460388143476095593861016278312164846992663292732408377940509553171488449035e-30
complex
12
4:
1.571556311012566458752308663545498887483432709429413519222751910781221744389785995371454721196435528e-49
complex
12
5:
2.494020408223694874213676381081627719186835511137285447303169189185305175916917785946793799980777265e-69
complex
12
6:
1.157166764193672230285371049698006058067604391421542357927565502353144597793669960284729319210335558e-88
complex
12
7:
7.830315133429225049241797636200858538385972614865259522455273983113785370016918290805427240255487252e-107
complex
12
8:
2.979157479440653430329106152713310153184201467925934169203697061209477156111681359154802638136636534e-123
complex
12
9:
2.209516878351259336797340685589326412846558542502470377793359370038508502425563654968346217570715893e-137
complex
12
10:
1.142262311572632454713257696674522954948480697559764312569565100775439050812619108800036206308160407e-148
complex
12
11:
1.877891639492169563759790032459339549000851277507056358208761735417622571488682215469126398611604560e-156
complex
12
12:
1.143003408187409960543891156313184041085678293333594366027526035532207280371054591985784563551515522e-159
complex
13
1:
7.903536371318468804212103428857682494130060554241242530753116640589806117892036736972176171770445469
complex
13
2:
1.895626442359445396225960110298314563871936588152018590900742079689130618400525437881121199829403045e-14
complex
13
3:
6.152445564402807660551669820864642579785303821239482902397308079258759553792982261725429158864760218e-34
complex
13
4:
1.461606726638905547617275432088329676381940507796861804405039345002569680877059111300208549752943123e-55
complex
13
5:
7.043430596110040191801789903718989475366583198857269185676667265405266068289196189515703650424402606e-78
complex
13
6:
6.148244926585167299925970551803421075510534529982906120913843066226249486210282176494203893349006293e-100
complex
13
7:
4.994599190823741781799269577745202369364427233216052311798925106038044589314433198320820652969856310e-121
complex
13
8:
1.441477535609939064276398889466334031664197376613398101193186213290692910532696659735432890775290505e-140
complex
13
9:
4.840246428599136406664414464494097794382462887200045655706368808324179501137298165761463518917461218e-158
complex
13
10:
5.907704081650240044102791147240698275933574066231812932470206319868190897079315796270932864961058383e-173
complex
13
11:
8.762103941230327164467391892506576196145728779467245244167802909462438522866770220063517310066815993e-185
complex
13
12:
6.907209969424431278448802948230284463165284578542855265969304230940525097854075026691760026130881597e-193
complex
13
13:
3.291782084369126572453283109712057484338787045396144399207515524183716868091701561531529669066298574e-196
complex
14
1:
3.819952584848282127733792067340466125440612873142212011264438872096469603835568944661655517679818570
complex
14
2:
2.649114481494993999730384497208991447889247949006749687192210291122068773277870636692904744420848844e-16
complex
14
3:
8.604172639826388493354975859994588587005188061494483036808444699885535163822217958652545454014881099e-38
complex
14
4:
1.010599984068109316682986053857535836689578287184390508379139058197293062258324780345690755187490855e-61
complex
14
5:
1.395404138291368466623152505997606938984988986527129848772037619452362512951719044135653676831384242e-86
complex
14
6:
2.196793854485489852514161271392088944206170450440763349065844210588160047873841329170231517766797921e-111
complex
14
7:
2.108217472379379666829530150193046650355592725486588775896309257402231860693353749390628534519118255e-135
complex
14
8:
4.745498832309586681241363737091409118294862069661711360237430641548704017081000511602263687522751209e-158
complex
14
9:
8.007098004406873409965610387339175664059365320750900373242364940570314067708748976425757293551788144e-179
complex
14
10:
2.955634758396683954884218423988068951303492751875820613949477730084699875424047761560863535772883661e-197
complex
14
11:
6.905847119596441720849804925001760273387558088450667525184830680045333455561656225324165019342140261e-213
complex
14
12:
3.244302678648996924439699993652115060710215252989111844294462270217064776249905090104615788436897213e-225
complex
14
13:
1.297300597099406784616002284720864277052011193812090936938559550582712362040602205246915873501883392e-233
complex
14
14:
4.931400928738327271357110581203478718467736106580056629220715473937388781593046266143269122779231120e-237
complex
15
1:
1.714390711088672065421586077323089143567111479147122084827580346719193613182589864359988199380568801
complex
15
2:
3.203346231996146416369570825442217956261960239282581057114878879523806797734651582230822763320192072e-18
complex
15
3:
9.723497564484660269838948107397018648151395480697301998015757069747074363242550673838068705289715048e-42
complex
15
4:
5.295088875906372310296338194778004502319758395709691744308216820026549487581940980626291941473056834e-68
complex
15
5:
1.978311076887698329923259529518230929436645882451629653745884264342574937588205590696282924649305353e-95
complex
15
6:
5.364936265977378617834445369485844010169625917172587398988705087552364966548560862417147434863012106e-123
complex
15
7:
5.915341263811421131042408257176766999388920852684639742152133309948388719698105561212973595899447302e-150
complex
15
8:
1.039662826534084933541494807707075328826440409914043873683652570382578442578923288504961904112243294e-175
complex
15
9:
9.252836892696393796835816181689522275041606020009053400955940526193683894194495449710781482083498252e-200
complex
15
10:
1.175559725898617137953683994195940269602574882033978614071285496354048890351541916393119475700012665e-221
complex
15
11:
5.715350268098279850974171637938128842885408843743041993367793465737159420502561113363287652398078507e-241
complex
15
12:
2.901933683642251267683699065571774219837780343816956147338470690852784203234279220662553090603348793e-257
complex
15
13:
4.697020331933559385919167152832958381090875927188382371614731336350462728827119667925274058747066997e-270
complex
15
14:
9.997387083035979185443588010272261808011280744258667333326827833180918625114674410911732237225097609e-279
complex
15
15:
3.079844870275558048452401470096556424151079179404894872145444265278599470494433459446237826601446671e-282
complex
16
1:
0.7181223017785005122317402786068623805398616888428390092320039398214811163283769588707796440282095726
complex
16
2:
3.384381095756165596536807274195837997893134920152062737243802316273736513487389554618063114031181475e-20
complex
16
3:
9.003225034228960957037456488004886904248997570997847726671675960407677716954720507370076163034868063e-46
complex
16
4:
2.138510724206215336624843259671760199205298526853016348013281413432524394001153122837537110692987318e-74
complex
16
5:
2.044676218917447080871236202352416975542501820210689022238745164798985395151761278026072798658866667e-104
complex
16
6:
9.110004545486482951656774529998541926436119478806760930791274663634606095722194125949960946417408936e-135
complex
16
7:
1.115131598856225608906772844596401510913497552440234975003399264324927410724901670183537471079920926e-164
complex
16
8:
1.508801811580894538705097724295324999732569320401998666032824429377239630903020738043107465102432525e-193
complex
16
9:
7.206276236491195575233917463620798196555222730742645085336362999208936333677864702493936491633379797e-221
complex
16
10:
3.371992329912778775325376747740520165824516695286949045956244530190663880959579448569966049156434552e-246
complex
16
11:
3.972858511254346142386349885759586773315288475229253712269883304149522284031555620216915106326846494e-269
complex
16
12:
2.961048054756492339448952481229825502540591597845264338256716189568399211471106084682727730128997443e-289
complex
16
13:
3.639814891577090859032005059771242028047616022708697887833697850354013813713732685714468178175388078e-306
complex
16
14:
2.182083939246938626440340000917664433194903449100020687953427624323748181726503112673966716783365901e-319
complex
16
15:
2.577503342871813000673542770408444916064102428452316194881435663192113154462324011249848591873777356e-328
complex
16
16:
6.524593151461749855785969612230852873744848079057771926083951073714729093513058083156525606841394823e-332
Definition
$V_{N,r}^{(\beta)}$ is the intrinsic $D$-dimensional Riemannian volume of the manifold of positive semidefinite, trace-one $N\times N$ density matrices of rank exactly $r$ over the real ($\beta=1$) or complex ($\beta=2$) field, with metric $ds^2=\operatorname{Tr}(d\rho^2)$ and $D=r-1+\beta r(2N-r-1)/2$. 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)
$\displaystyle V_{N,r}^{(\beta)}=\frac{\sqrt r}{r!\,\Gamma(A)}\left(\prod_{k=d+1}^{N}P_\beta(k)\right)\prod_{j=1}^{r}\frac{\Gamma(1+j\beta/2)\Gamma(\alpha+(j-1)\beta/2)}{\Gamma(1+\beta/2)}$, where $d=N-r$, $\alpha=1+\beta d$, $A=\alpha r+\beta r(r-1)/2$, $\displaystyle P_1(k)=\frac{2^{(k-1)/2}\pi^{k/2}}{\Gamma(k/2)}$, and $\displaystyle P_2(k)=\frac{(2\pi)^{k-1}}{\Gamma(k)}$. The factors $P_\beta(k)$ are the successive projective-space volumes in the unit-trace metric. This is [1], equations (4.3), (5.3), and (7.8), with the flag-volume ratio expanded.
(2)
$\displaystyle V_{N,1}^{(1)}=\frac{2^{(N-1)/2}\pi^{N/2}}{\Gamma(N/2)},\qquad V_{N,1}^{(2)}=\frac{(2\pi)^{N-1}}{\Gamma(N)}$. These are the volumes of $\mathbb{RP}^{N-1}$ and $\mathbb{CP}^{N-1}$ in the metric induced by $ds^2=\operatorname{Tr}(d\rho^2)$.
(3)
$\displaystyle \frac{V_{N,N-1}^{(\beta)}}{V_{N,N}^{(\beta)}}=\left(N-1+\frac{\beta N(N-1)}2\right)\sqrt{N(N-1)}$; see [1], equations (6.5) and (7.9).
(4)
$\displaystyle V_{N,N}^{(2)}=\frac{\sqrt N\,(2\pi)^{N(N-1)/2}\prod_{j=1}^{N}\Gamma(j)}{\Gamma(N^2)}$; see [1], equation (4.5).
Comments
(5)
The dimension $D$ is the real dimension of the rank stratum, and its 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 case $r=N$ is the interior volume, $r=N-1$ is the generic boundary volume, and $r=1$ is the pure-state volume.
(6)
In [1], the second subscript of $S_{N,n}^{(\beta)}$ is the nullity $n=N-r$. Here $r$ always denotes the actual rank. Real matrices are symmetric; complex matrices are Hermitian.
(7)
For $N=2$, the real state space is a disk and the complex state space is a ball, both of Hilbert–Schmidt radius $1/\sqrt2$. Their interior volumes follow by scaling the unit-ball volumes, and their pure-state volumes follow by scaling the unit-sphere volumes.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField
from math import factorial

def hs_volume(field, N, r, digits=100):
    beta = {'real': 1, 'complex': 2}[field]
    assert N >= 2 and 1 <= r <= N
    R = RealBallField(numberdb.bits(digits, losing=96))
    d = N-r
    alpha = 1+beta*d
    A = alpha*r+beta*r*(r-1)//2
    v = R(r).sqrt()/R(factorial(r))/R(A).gamma()
    for k in range(d+1, N+1):
        if beta == 1:
            v *= R(2)**(QQ(k-1)/2)*R.pi()**(QQ(k)/2)/R(QQ(k)/2).gamma()
        else:
            v *= (2*R.pi())**(k-1)/R(k).gamma()
    for j in range(1, r+1):
        v *= R(1+QQ(j*beta)/2).gamma()*R(alpha+QQ((j-1)*beta)/2).gamma()/R(1+QQ(beta)/2).gamma()
    return v

hs_volume('complex', 2, 2)
References
[1]
K. Życzkowski and H.-J. Sommers, Hilbert–Schmidt volume of the set of mixed quantum states, Journal of Physics A: Mathematical and General 36 (2003), 10115–10130. (arXiv) (doi)
[2]
N. G. de Bruijn, On some multiple integrals involving determinants, Journal of the Indian Mathematical Society (N.S.) 19 (1955), 133–151. https://pure.tue.nl/ws/files/1920642/597510.pdf
Links
Similar tables
Volume_of_the_unit_ball —   Euclidean ball volumes give the real and complex $N=2$ interior cases after scaling the radius to $1/\sqrt2$.
Volume_of_the_unit_sphere —   Sphere volumes give the $N=2$ pure-state cases after scaling, and determine the real projective-space factors.
Data properties
Entries are of type: real number
digits: 100 (Unknown key)
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 are reduced exactly to a rational coefficient times a square root and an integer power of $\pi$. Arb ball arithmetic at 397 bits encloses the final expression, which is serialized to 100 significant digits. Each serialized interval contains both its generating ball and an independently computed 560-bit ball.

more

The independent calculation uses exact factorial-moment determinants for complex matrices and exact skew-moment Pfaffians for real matrices, following [2], with projective-space orbit factors. Elementary disk and ball controls, pure-state volumes, boundary-to-interior ratios, and the complex full-rank formula are also checked. The attached checks repeat these tests on values read from the database.



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