Key words: Qubits, Geometric algebra, Clifford translation, Geometric phase, Topological quantum computing

Transcription

1 omtc phas n th + quantum stat voluton Alxand OUNE OUNE upcomputng Al Vjo A 9656 UA Emal addss alx@gun.com Ky wods Qubts omtc algba lffod tanslaton omtc phas opologcal quantum computng Abstact Whn quantum mchancal qubts as lmnts of two dmnnal complx Hlbt spac a gnald to lmnts of vn subalgba of gomtc algba ov th dmnnal Eucldan spac gomtcally fomal complx plan bcoms xplctly dfnd as an abtay vaabl plan n D []. h sult s that th quantum stat dfnton and voluton cv mo dtald dscpton ncludng cla calculatons of gomtc phas wth mpotant consquncs fo topologcal quantum computng.. ntoducton Qubts unt valu lmnts of th Hlbt spac of two dmnnal complx vctos 0 0 k k k k can b gnald to unt valu lmnts ov Eucldan spac E g-qubts [] of vn subalgba b b b b of gomtc algba b b b b scalas - unt valu bvctos satsfyng. b Multplcaton uls. assum th ght scw spac ontaton that can b sn though th od of vctos dual to th bvctos usd to cat th spac ontd unt volum s Fg...

2 Fg... Rght scw unt valu ontd volum h always a two optons to cat ontd unt volum dpndng on th od of vctos n th poduct. hy cospond to th two typs of th th dmnnal spac handdnss lft and ght scw handdnss. On can al thnk about as a ght lft sngl thad scw hlx of th hght on s th abov pctu. n ths way s lft ght scw hlx. Mappngs btwn g-qubts and qubts a not on-to-on and a dfnd by. that actually dfns pncpal fb bundl wh g g s total spac and } ; { s bas spac. wll dnot thm as and spctvly. h pojcton dpnds on whch patcula s takn fom an abtaly slctd tpl n D satsfyng.. vcto dfns complx plan fo th complx vctos of w should wt. Fo any y x y x th fb n conssts of all lmnts y x y x F f s optonally chosn as complx plan. hat patculaly mans that standad fb s quvalnt to th goup of otatons of th tpl y x y as a whol. All such otatons n a al dntfd by lmnts of snc fo any bvcto th sult of ts otaton s s fo xampl t s convnnt to wt lmnts as xponnts.

3 [] [] wh. o standad fb s dntfd as and th composton of otatons s Multplcatons of by bass bvctos gv bass bvctos of tangnt spacs to ognal bvctos [4] hs lmnts a othogonal to and to ach oth and a th tangnt spac bass lmnts at ponts. Pojctons of onto a hs lmnts of a mutually othogonal n th sns of Eucldan scala poduct n R w w w and othogonal to th pojcton of th ognal stat n. hy a th tangnt spac bass lmnts n at ponts.. lffod tanslatons Lt s tak lffod tanslaton n l and lft t to usng l F F sn cos sn cos sn cos sn cos

4 anslatonal vlocty s F l F Fl. and s othogonal to Fl F F F 0 l F 0 l l l 0 0 ndx 0 mans scala pat of lmnt wo oth componnts of th tangnt spac othogonal to Fl and F l at any pont of th obt a F l and F l. h vlocts whl movng along lffod obt a Fl F Fl Fl. dvatv of F l s othogonal to F l and lookng n th dcton F l Fl F Fl Fl. dvatv of F l s othogonal to F l and lookng n th dcton F l hs two quatons xplctly show that th two tangnts othogonal to lffod tanslaton vlocty otat n movng plan F F wth th sam by valu otatonal vlocty as tanslaton vlocty s s Fg.. l l Fg... angnts otat n th plan wth th sam by valu spd as tanslatonal vlocty laly th th F l a dntcal to al consdd tangnts 4

5 f a fb g-qubt maks full ccl n lffod tanslaton F l and F l g-qubt gomtc phas ncmntng n th F Fl F 0 both al mak full otaton n th common plan by. hs s spcal cas of th sph bg ccl closd cuv quantum stat path. h dmonstatd otaton of tangnts n th plan othogonal to th obt of g-qubt lffod tanslaton that s what actually s not ntutvly obvous and s mo mpotant than all wdly accptd mysts of quantum mchancs. hs otaton phnomna has nothng to do wth th s of physcal systm. hs s topologcal popty of th spac of dmnn 4 not ou magnaton cannot asly dal wth. At th sam tm w should mmb that g-qubts stats a opatos actng on obsvabls. hough obsvabls a lmnts of th sam spac as stats s nxt scton acton of a stat on obsvabl s and th sult of ths acton changs dffntly compad to th stat modfcaton subjctd to lffod tanslaton.. Masumnt of obsvabls n bass stats Lt s consd th cas whn fo an abtay g-qubt of complx plan. hn du to. th s. 0 th plan s takn as playng th ol lmnt gvn by pojcton Lt s call th dfntons of stats obsvabls and masumnts appopat fo th cas of th fomalsm of th two stat systms []. tats and obsvabls a lmnts of Dfnton. stat unt valu lmnt of masumnt b b b Dfnton. obsvabl lmnt of dfns opaton actng on obsvabl n a b b b 0 b 5

6 6 Dfnton. masumnt Masumnt of obsvabl masud n stat s gnald Hopf fbaton gnatd by th obsvabl Explct fomulas can b found n []. Du to dfnton. th stat cosponds to th stabl stat s [5] 0 n famla tms of quantum mchancal notatons 0 f s slctd as complx plan. h stat s stabl n th sns that th masumnt of any obsvabl wth th bvcto pat paalll to dos not chang th obsvabl omt scala pat whch dos not chang n otatons. h g-qubt stat cospondng to s wh s any unt bvcto othogonal to and f w kp ght scw spac ontaton wth multplcaton uls.. Masumnt of any obsvabl wth th bvcto pat paalll to n ths stat gvs. h last fomula mans that masumnt of any obsvabl wth th bvcto pat paalll to n th stat cospondng to flps bvcto pat of th obsvabl. Fomulas.. tv th actual sns of th two bass stats. onsd th sults of masumnts n stats and of an abtay obsvabl 0 0 cos sn sn cos 0. though paamtaton cos sn 0 cos sn sn cos 0.4

7 though paamtaton cos sn. Fomulas..4 man th followng Masumnt of obsvabl n pu qubt stat has bvcto 0 pat wth th componnt qual to unchangd valu. h a qual to and wh plan of otaton s. Masumnt of obsvabl componnts of otatd by angl 0 and masumnt componnts dfnd by cos and sn n pu qubt stat has bvcto pat wth th componnt qual to flppd valu flppng n plan. h masumnt componnts a qual to and and componnts of otatd by angl dfnd by cos sn wh plan of otaton s. h ablut valu of angl of otaton s th sam as fo but th otaton dcton s oppost to th cas of. h abov two sults a gomtcally ptty cla. h two stats and only dff by addtonal facto n. hat mans that masumnts of an obsvabl f t s pu bvcto n stats and a quvalnt up to addtonal wapp hat smply mans that th masumnt n stat cospondng to s cvd fom th masumnt masumnt n stat cospondng to 0 just by mong th sult latv to th plan s Fg... Fg... Rsults of masumnt of n th two stabl stats. 0 any masumnt n a stat cospondng to 0 and oth masumnt n a stat cospondng to can b mad mod of ach oth by otatng n a plan paalll to. h componnts a th sam n ablut valu fo all th th 0 and. 7

8 4. qunc of nfntsmal lffod tanslatons nc any bvcto n abtay unt bvcto can b gnally takn as playng th ol of complx plan lt s tak m and mak nfntsmal lffod tanslaton of an abtay g-qubt stat d l d nstant tanslatonal vlocty tangnt of t s l d plans to cat otatonal tangnt componnts. h fst on dfnd up to abtay angl of otaton aound nomal to. W al nd two bvctos s any unt bvcto othogonal to. h bvcto fo th scond tangnt can b takn n th cas of th ght scw spac ontaton as. h tanslatonal vlocty tangnt wll otat by th valu l d bcaus fom. F F h otatonal vlocty tangnts wll otat by th sam l l l d n th dcton oppost to F d valu n th plan othogonal to tanslatonal vlocty tangnt as nomal s Fg.. placng to. All that mans that whl movng n a squnc of nfntsmal lffod tanslatons th two otatonal tangnts otat n ach nfntsmal stp by th sam angl n th plan as tanslatonal vlocty tangnt otats n plan movng along obt lyng on. W saw abov that t dos not matt do w look at th angl accumulatd by tanslatonal vlocty n th vayng plan spannd by ths vlocty togth wth nstant stat g-qubt o accumulatd angl of otaton of any of th two tangnts of otatonal spd - thy a qual! Lt w mov along m abtay path on. h sph s cnt symmtcal sufac at any nstant valu of stat on nfntsmal ncmntng of tanslatonal vlocty angl dos not dpnd n what dcton th dsplacmnt happns. f th path s appoxmatd wth nfntsmal pcs of godscs thn th accumulatd angl btwn tanslatonal vlocty and nstant godsc s obvously qual to th total lngth of th path s Fg.4.. ndx h s takn to stss that ths bvcto wll play th sam ol as th pvously usd n c.. Hopfully ad mmbs that s a plan n D bang ndx and dmnnal spac. 8 s unt -dmnnal sph n 4-

9 Fg.4.. Accumulatng of angl whl movng along path. Al quals by valu to otaton angl of two vctos n plan manng othogonal to tanslatonal vlocty hs s puly gomtcal phas spaatd fom g-qubt xponnt phas modfcatons causd by xtnal factos dfnng th g-qubt stat path on th tanslatons along any path L wth vayng ak a squnc of nfntsmal lffod tansfomatons y takng th logathm appoachng stat l. nfnt composton of nfntsmal lffod gvs th fnal stat g-qubt. l l l N N l l... N and gttng back to xponnt w cv th fnal l dl 4. L 5. oncluns Evoluton of a quantum stat dscbd n tms of gvs mo dtald nfomaton about two stat systm compad to th Hlbt spac modl. t confms th da that dstnctons btwn quantum and classcal stats bcom lss dp f a mo appopat mathmatcal fomalsm s usd. h paadgm spads fom tval phnomna lk tossd con xpmnt [6] to cnt sults on ntanglmnt and ll thom [7] wh th fom was dmonstatd as not xclusvly quantum popty. 9

10 Woks td [] A. ogun "omtc Algba Qubts omtc Evoluton and All hat" Januay 05. [Onln]. Avalabl http//axv.og/abs/ [] D. Hstns Nw Foundatons of lasscal Mchancs Dodcht/oston/London Kluw Acadmc Publshs 999. []. Doan and A. Lasnby omtc Algba fo Physcsts ambdg ambdg Unvsty Pss 00. [4] A. ogun "Quantum stat voluton n and +" cnc Rsach vol. no. 5 August 05. [5] A. ogun "What quantum "stat" ally s?" Jun 04. [Onln]. Avalabl http//axv.og/abs/ [6] A. ogun "A tossd con as quantum mchancal objct" ptmb 0. [Onln]. Avalabl http//axv.og/abs/ [7] X.-F. Qan. Lttl J.. Howll and J. H. Ebly "hftng th quantum-classcal bounday thoy and xpmnt fo statstcally classcal optcal flds" Optca pp July 05. 0