
aanshik nishan

raikhik beejganit aur karyatmak vishleshan mein, aanshik nishan nishan kaa ack saamaanyikaran haye. jabaki trace prachaalak parr ack scalar moolya kaary haye, aanshik trace ack prachaalak-moolya kaary haye. aanshik nishan key quantum jaankaari aur decoherance mein anuprayog hain joe quantum maap key liye praasangik haye aur iss tarah quantum yantriki kii vyakhyaon key liye decoherant drishtikon key liye susangat haye, jisamen susangat itihas aur saapeksh sthiti vyakhya shamil haye.
vivaran.
maan leejiye kii sootra _ 1, sootra _ 2 ack kshetra parr parimit-ayami sadish sthaan hain, jinake ayam sootra _ 3 aur sootra _ 4 hain. kissy bhee sthaan sootra _ 5 key liye, sootra _ 6 koo sootra _ 5 parr raikhik prachaalak key sthaan koo darshaata haye. sootra _ 2 parr aanshik nishan koo sootra _ 9 key roop mein likhaa jaataa haye, jahaan sootra _ 10 kronkar utpaad koo darshaata haye.
ise nimnanusar paribhaashit kiya gayaa haih 
sootra _ 11 key liye, sootra _ 12, aur sootra _ 13, kramshah "v" aur "w" key liye aadhar houn; phir "t"
ack matricks pratinidhitva haye
sootra _ 16 key aadhar sootra _ 15 key saapeksh.
abb soochkaank key liye "k", "i" renz 1 mein,..., aim", yog parr vichaar karein
yeh ack matricks "b" "k", "i" detaa haye. v" parr sambaddh raikhik prachaalak aadharon kii pasand seey swatantra haye aur paribhasha key anusaar aanshik nishan haye.
bhautikavidon key beach, ise aksar "tracing out" yaa "tracing over" doubleu kahaa jaataa haye taaki kewal ack operator koo "vee" parr chhod diyaa jaa sakey, iss sandarbh mein jahaan "doubleu" aur "vee" quantum pranaliyon seey judey hilbert sthaan hain (neeche dekhein).
aparivartaniya paribhasha.
aanshik trace operator koo nirantar roop seey paribhaashit kiya jaa sakata haye (yaani, binaa kissy aadhar key) nimnanusarah yeh advitiya raikhik maanchitra haye. 
kii
yeh dekhane key liye kii uparokt sthitiyaan aanshik nishan koo vishisht roop seey nirdhaarit karati hain, sootra _ 20 koo sootra _ 1 key liye ack aadhar banaate hain, sootra _ 22 koo sootra _ 2 key liye ack aadhar banaate hain, sootra _ 24 koo vaha maanchitra honey dein joe sootra _ 25 koo sootra _ 26 (aur anya sabhi aadhar tatvon koo shoonya) parr bhejata haye, aur sootra _ 27 koo vaha maanchitra honey dein joe sootra _ 28 koo sootra _ 29 parr bhejata haye. kyonki vactor sootra _ 30 sootra _ 31 key liye ack aadhar banaata haye, maanchitra sootra _ 32 sootra _ 33 key liye ack aadhar banaata haye.
iss amoort paribhasha seey, nimnalikhit gunon kaa paalan hotaa haih
shreni saiddhaantik dhaarana.
yeh raikhik parivartanon kaa aanshik nishan haye joe wafadar, sadak aur satya kii anurekhit ekal shreni kii dhaarana kaa vishay haye. ack trace kii gaee monoidal shreni ack monoidal shreni sootra haye _ 36 key saath, shreni mein vastuon "x, y, u" key liye, home-sett kaa ack kaary,
kuchh swayamsiddhiyon koo santusht karna.
aanshik nishan kii iss amoort dhaarana kaa ack aur mamla unake beach seemit sett aur bijections kii shreni mein hotaa haye, jisamen monoidal utpaad asambaddh sangh haye. koee bhee yeh dikhaa sakata haye kii kissy bhee seemit sett key liye, "x, y, u" aur bijection sootra _ 38 key liye ack sambandhit "aanshik roop seey trace kiya gayaa" bijection sootra _ 39 maujood haye.
hilbert rikt sthaan parr prachaalak key liye aanshik nishan.
aanshik nishan anant ayami hilbert rikt sthaan parr prachaalak key liye saamaanyikrit hotaa haye. maan leejiye kii "v", "w" hilbert space hain, aur 
jaane dou.
doubleu" key liye ack orthonormal aadhar hoe. abb ack samamit samroopata haye
iss apghatan key tahat, kissy bhee prachaalak sootra _ 42 koo ack anant matricks key roop mein maanaa jaa sakata haye.
vee" parr prachaalak"
jahan sootra _ 44 haye.
pahale maan leejiye kii "tii" ack gair-nakaratmak prachaalak haye. iss mamle mein, uparokt matricks kii sabhi vikarn pravishtiyaan "v" parr gair-nakaratmak prachaalak hain. aggar raashi
l ("v") key majboot prachaalak topology mein abhisaran, yeh "w" key chune gaye aadhar seey swatantra haye. aanshik trace tra "doubleu" ("tii") koo iss prachaalak key roop mein paribhaashit kiya gayaa haye. ack swa-jodd prachaalak kaa aanshik nishan paribhaashit kiya jaataa haye yadi aur kewal tabhi jaba sakaratmak aur nakaratmak bhaagon key aanshik nishan paribhaashit kiye jaate hain.
aanshik nishan kii ganana karna.
maan leejiye kii "doubleu" kaa ack orthonormal aadhar haye, jise humm sootra _ 46 key roop mein kett vector sanketan dwara darshaate hain.
koshthak mein superscript matricks ghatakon kaa pratinidhitva naheen karate hain, balki isake bajaay matricks koo hii label karate hain.
aanshik nishan aur aparivartaniya ekikaran.
parimit ayami hilbert rikt sthaan key mamle mein, "doubleu" key ekaatmak samooh yuu ("doubleu") parr ack upayukt saamaanyikrit haar maap key sambandh mein ekikaran seey judey aanshik nishan koo dekhane kaa ack upayogi tarika haye. upayukt roop seey saamaanyikrit honey kaa arth haye kii kull dravyamaan mund ("doubleu") key saath ack maap key roop mein liya jaataa haye.
pramey. maan leejiye kii "v", "w" seemit ayami hilbert sthaan hain. taba
formula _ 49 key sabhi operators key saath awagaman kartaa haye aur isaliye formula _ 50 kaa vishisht roop haye. operator "r" "t" kaa aanshik nishan haye.
ack quantum operation key roop mein aanshik nishan.
aanshik nishan koo ack quantum operation key roop mein dekhaa jaa sakata haye. ack quantum yaantrik pranali parr vichaar karein jisaki avastha sthaan hilbert rikt sthaan kaa tensor utpaad sootra _ 51 haye. ack mishrit avastha koo ghanatva matricks ρ dwara varnit kiya jaataa haye, joe haye 
tensor utpaad sootra _ 52 parr trace 1 kaa ack gair-nakaratmak trace-class operator
pranali "b" key sambandh mein ρ kaa aanshik nishan, jise sootra _ 53 dwara darshaaya jaataa haye, pranali "a" parr ρ kii cumm hui sthiti kahalaati haye. prateekon mein,
yeh dikhaane key liye kii yeh vaastav mein ρ koo "a" upapranaali parr ack raajya nirdhaarit karane kaa ack samajhadaari bharaa tarika haye, humm nimnalikhit auchitya pradaan karate hain. maan leejiye kii "m" upapranaali "a" parr ack avalokan yogya haye, phir sanyukt pranali parr sambandhit avalokan yogya sootra _ 55 haye. haalaanki koee ack cumm avastha sootra _ 56 koo paribhaashit karane kaa vikalp chunata haye, maap saankhyiki kii sthirata honey chaahiye. upapranaali "a" key baad "m" kaa apekshit moolya sootra _ 53 mein taiyar kiya jaataa haye aur sootra _ 55 kaa jaba ρ mein samagr pranali taiyar kii jaati haye tou vahee honah chaahiye, i.ee. nimnalikhit samaantaa honey chaahieh
humm dekhate hain kii yeh santusht haye yadi sootra _ 53 aanshik nishan key madhyam seey oopar paribhaashit kiya gayaa haye. isake alawa, iss tarah kaa operation advitiya haye.
t (h)" koo hilbert space "h" parr trace-class operators kaa banach space honey dein. yeh aasaani seey jancha jaa sakata haye kii aanshik nishan, ack maanchitra key roop mein dekhaa jaataa haye 
yeh poori tarah seey sakaratmak aur trace-preserving haye.
ghanatva matricks ρ hermician haye, sakaratmak ardh-nishchit haye, aur iska 1 kaa nishan haye. iska ack varnakramiya apghatan haih
yeh dekhana aasaan haye kii aanshik trace sootra _ 53 bhee inn sthitiyon koo santusht kartaa haye. udaaharan key liye, sootra 65 mein kissy bhee shuddh avastha sootra key liye, hamaare paas haye
dhyaan dein kii sootra 67 shabd avastha sootra 64 koo khojane kii sambhavana kaa pratinidhitva kartaa haye jaba samagr pranali avastha sootra 69 mein hoti haye. yeh sootra 53 kii sakaratmak ardh-paribhasha koo saabit kartaa haye.
oopar diye gaye aanshik trace maanchitra mein sootra _ 72 parr bandhe operators key see *-algebra aur sootra _ 51 key beach ack dohra maanchitra sootra _ 71 shamil haye.
sootra _ 71 avalokan yogya maanchitron koo avalokan yogya banaata haye aur sootra _ 76 kaa heisenberg chitra pratinidhitva haye.
shaastriya mamle key saath tulanaa.
maan leejiye kii quantum yaantrik pranaliyon key bajaay, dou pranaaliyan "aeye" aur "bee" shaastriya hain. pratyek pranali key liye avalokan yogya sthaan taba abelian see *-algebra hotey hain. yeye compact space "x", "y" key liye kramshah "c" ("x") aur "c" ("y") key roop mein hain. samagr pranali kaa raajya sthaan saral haye
yaugik pranali parr ack avastha c ("x" × "y") key dohare kaa ack sakaratmak tatva ρ haye, joe ryz-markov pramey dwara "x" × "y" parr ack niyamit boral maap key anuroop haye. anuroop cumm avastha koo ρ koo "x" maapane key liye prakshepit karke praapt kiya jaataa haye. iss prakaar aanshik nishan iss operation key quantum yaantrik samkaksh haye.

