
vegale karata yenyajogi sthitee

quantum yantrikimadhye, vegale karata yenyajogya avastha yaa sammishra jageshi sambandhit quantum avastha asatat jyaa vegalya upasthananshi sambandhit vaiyaktik avasthanmadhye ghatak kelya jaau shakatat. ekhade rajya vegale karata yet nasel tarr teey adakale aahe asey mhatale jaate. sarvasadharanpane, ekhade rajya vegale karata yenyajoge aahe qee nahee heey tharvane sope nahee aanee samasyeche aine. pii.-haard mhanoon vargikaran kele jaate.
dvidaleey pranalinchi pruthakkarankshamta.
done degree auph friedam asalelya pahilya sammishra avasthancha vichar qaraa, jyanna "dvidaleey rajye" mhanatat. quantum yantrikichya swapramananusar, tensor utpadan space sootra _ 1 madhye yaa vector mhanoon varnan kele jaau shakate. yaa charchet aapan hilbert space sootra _ 2 aanee sootra _ 3 maryadit-aayami asalyachya babatit laksh kendrit karoo.
shuddha rajye.
sootra _ 4 aanee sootra _ 5 heey anukrame sootra _ 2 aanee sootra _ 3 sathee orthonormal bases asoo dya. sootra _ 8 sathee aadhar mhanje sootra _ 9, kinwa adhik sankshipt sanketan sootra _ 10. tensor utpadanachya agadi vyakhyevarun, manak 1, i chaa konatahi vector.eae. sammishra pranalichi shuddha sthitee, asey lihile jaau shakate
jethe sootra _ 12 haah sthirank aahe. 
jarr sootra _ 13 heey "sadhe tensor" mhanoon lihile jaau shakate, mhanje sootra _ 14 madhye sootra _ 15 saha "i"-th jaaget shuddha sthitee, tarr tee "utpadan sthitee" aanee visheshtah, "vegale karnyayogya" asey mhatale jaate. anyatha tyala 'guntalele' mhanatat. lakshat ghya qee, jaree 'utpadan' aanee 'vegale karata yenyajogya' avasthanchya kalpana shuddha avasthansathi eksandh asalya taree, tyaa adhik samanya mishra avasthanchya babatit naheet.
jarr aanee keval tyanchya aanshik avastha shuddha nastil tarach shuddha avastha guntalya jaatat. heey pahanyasathi, sootra _ 16 chey schmitt vighatan asey liha
jethe sootra _ 18 hii sakaratmak vastavik sankhya aahet, sootra _ 19 hii sootra _ 16 chee schmitt shreni aahe aanee sootra _ 21 aanee sootra _ 22 heey anukrame sootra _ 2 aanee sootra _ 3 madheel orthonormal avasthanche sanch aahet.
jarr aanee keval sootra _ 26 asel tarach sthitee sootra _ 16 guntalele asate. tyach velee, aanshik sthitiche swaroop asel
teey asey darshavate qee sootra _ 28 shuddha aahe---- mhanje, ekak-shrenisah prakshepan aahe-----jarr aanee fakt sootra _ 29, jeey sootra _ 16 chya samatulya aahe, teey vegale karnyayogya asel.
bhautikdrushtya, yacha arth assa aahe qee uppranalila eka nishchit (shuddha) sthitee niyukt karane shakya nahee, jyache varnan tyaeivaji shuddha avasthanche sankhyikeey samooh mhanoon kele pahije, mhanje ghanata matricks mhanoon. aanshik sthitee sootra _ 32 chee wonn neuman entropy shunyetar asel tarach shuddha sthitee sootra _ 31 asha prakare guntalele asate.
aupcharikpane, utpadan jaaget sthitichya utpadanache embedding segger embedingdware dile jaate. mhanjech, jarr aanee keval jarr tee segger embeddingchya pratimet asel tarach quantum-yantrik shuddha sthitee vegali karata yete.
varchi charcha, jevha sthitichi jaaga aksharshah kahihi badallyashivay anant-aayami asate, tyaa prakaranaparyant vistarli jaau shakate.
mishra rajye.
mishra sthitiche prakaran vicharat ghya. sammishra pranalichya mishra sthitiche varnan sootra _ 8 varr karya karananya ghanata matricks sootra _ 33 dware kele jaate. sootra _ 35, sootra _ 36 aanee sootra _ 37 astitvat asalyas ρ vegale karata yete jeey sambandhit uppranalichya mishra avastha aahet jase qee
kuthe?
anyatha sootra _ 33 laa guntaleli sthitee mhanatat. vareel abhivyaktimadhye aapan samanyata naan gamavata asey gruheet dharoo shakato qee sootra 36 aanee sootra 37 heey serve shreni-1 chey andaj aahet, mhanje teey yogya uppranalichya "shuddha samoohanche" pratinidhitva karatat. yaa vyakhyevarun heey spasht hote qee, vibhakt avasthanche kutumb haah bahirmukh sanch aahe.
punha ekada tensor utpadanachya vyakhyevarun, konatehi ghanata matricks, kharokhar sammishra sthitee jagevar karya karnare konatehi matricks, ichchit swarupat kshullakpane lihile jaau shakatat, jarr aapan sootra _ 36 aanee sootra _ 37 heey swatach sthitee aahet aanee sootra _ 45 hii aavashyakta sodoon dilee tarr jarr yaa aavashyakta purn jhalya tarr aapan ekoon sthiticha arth asambandhit utpadan avasthanvaril sambhavyata vitaran mhanoon gheu shakato.
quantum vahinyanchya babatit, sthanik kriya aanee shastriy sanvad vaparoon itar konatyahi avasthepasun vegale karata yenyajogi sthitee tayar kelee jaau shakate, tarr guntaleli sthitee tayar houu shakat nahee.
jevha sthitichi jaaga anant-aayami asate, tevha ghanata matrixchi jaaga trace 1 saha sakaratmak trace class operatordware ghetali jaate aanee jarr sthiticha andaj, trace normmadhye, vareel swarupachya avasthandvare lavala jaau shakato tarr tee vegali karata yete.
jarr keval ekach shoonya nasalele sootra _ 46 asel, tarr sthitee fakt sootra _ 47 mhanoon vyakt kelee jaau shakate aanee tyala fakt vegale karata yenyajogi kinwa utpadan sthitee mhanatat. utpadanachya sthiticha eka gunadharm mhanje entropychya babatit,
bahupakshiy prakaranaparyant vistarane.
vareel charcha donpeksha jaast upapranali asalelya quantum pranalichya babatit sahajpane samanyikrut hote. ekhadya pranalimadhye "aine" upapranali asoo dya aanee tyache sthitee avkash sootra _ 49 asoo dya. jarr tee rachana karat asel tarr shuddha sthitee sootra _ 50 vegale karata yeil.
tyachpramane, "h" varr karya karnari mishra sthitee ρ jarr uttal berries asel tarr tee vegale karata yete.
kinwa, anant-aayami prakaranat, jarr vareel swarupachya avasthandvare trace normmadhye andaje andaj lavala jaau shakato tarr ρ vegale karata yete.
vibhajaniyata nikash.
ekhadi sthitee sarvasadharanpane vegale karata yete qee nahee heey tharvanyachya samasyela kadhikadhi quantum mahitee siddhantat pruthakkaran samasya mhanatat. hii eka katheen samasya manali jaate. heey anek prakarananmadhye aine. pii.-haard asalyache disun aale aahe aanee sarvasadharanpane asey manale jaate. jarr ekhadyane eka nishchit parimanasathi thet broot forcecha vapar karoon samasya sodavanyacha prayatna keela tarr yaa adchanibaddal kahee prashansa miloo shakate. aapan pahato qee qamee parimanansathihi samasya patkan nirakaran karnyayogya hott jaate. tyamule adhik atyadhunik formulation aavashyakta aahe. pruthakkaran samasya haah sadhyachya sanshodhanacha vishay aahe.
pruthakkarankshamatecha nikash' hii eka aavashyak aut aahe jee rajyane vegale karata yenyajogi asali pahije. qamee-aayami ("2 x 2" aanee "2 x 3") prakarananmadhye, peres-horodeki nikash haah pratyakshat pruthakkarankshamatesathi aavashyak aanee pureshi aut aahe. itar pruthakkarankshamatechya nikashanmadhye shreni nikash, qamee karnyache nikash aanee anishchitata sambandhanvar aadharit nikash yancha samavesh (parantu itakech maryadit nahee) aahe. ref pahaa. swatantra chull pranalinmadhil pruthakkarankshamatechya nikshambabatchya punaravalokanasathi.
satat badalatya pranalinmadhye, peres-horodeki nikash dekhil lagoo hoto. visheshtah, saymanne canonical operatorschya dusya kramankachya kshananchya sandarbhat peres-horodeki nikashachi eka vishisht aavrutti tayar kelee aanee heey darshavile qee sootra _ 53-mode gossian avasthansathi tee aavashyak aanee pureshi aahe (sandarbh pahaa. eka bhinn parantu moolatah samatulya drushtikon mhanoon). nantar asey aadhalun aale qee saymanchi sthitee dekhil aavashyak aahe aanee formula _ 54-mode gossian avasthansathi pureshi aahe, parantu formula _ 55-mode gossian avasthansathi aata pureshi nahee. canonical operatorsche uchch kramankache kshan vicharat gheuun kinwa entropic upay vaparoon saymanchi sthitee samanya kelee jaau shakate.
bijganitiy bhoomitidware vaishishtyikaran.
quantum yantriki prakshepak hilbert jagevar tayar kelee jaau shakate aanee asha done jaganche vargikaranatmak utpadan mhanje segger embedding. dvidaleey prakaranat, jarr aanee keval jarr tee segger embeddingchya pratimet asel tarach quantum sthitee vegali karata yete.
johnn magney linus, zaan mirhem aanee erich ovaram yanni tyanchya 'guntaguntichya bhaumitik pailun' yaa peparmadhye samasyeche varnan kele aahe aanee samanya sthitee matrixcha upasanch mhanoon vibhakt avasthanchya bhoomiticha abhyas keela aahe. yaa upasanchala peres-horodeki nikash dharan karananya avasthanchya upasanchasah kahee aantarchhed aahe. yaa peparmadhye, leenas aanee itar. sarvasadharan babatit pruthakkarankshamatesathi chachanisathi sankhyatmak drushtikon dekhil dya.
pruthakkarankshamatechi chachani.
sarvasadharanpane, pruthakkarankshamatechi chachani hii eka aine. pii.-katheen samasya aahe. leena aanee itar. dileli sthitee vegali karata yenyajogi asalyas chachanisathi punaravrutti, sambhavya algoridam tayar keela. jevha algoridam yashasvi hoto, tevha too dilelya sthitiche vegale karata yenyajogi sthitee mhanoon spasht, yadruchchik pratinidhitva deto. anyatha teey dilelya sthitiche, tyala sapadu shakananya sarvat jawalchya vegale karata yenyajogya sthitipasoonche antar dete.