qitd352 Dense Coding, Teleporttion, No Cloning Roert B. Griffiths Version of 8 Ferury 2012 Referenes: NLQI = R. B. Griffiths, Nture nd lotion of quntum informtion Phys. Rev. A 66 (2002) 012311; http://rxiv.org/rhive/qunt-ph/0203058 QCQI = Quntum Computtion nd Quntum Informtion y Nielsen nd Chung (Cmridge, 2000). Look up superdense oding, teleporttion, no-loning in the index. Add p. 187 to the teleporttion referenes. Contents 1 Fully Entngled Sttes 1 2 Dense Coding 2 3 Teleporttion 4 4 No Cloning 7 1 Fully Entngled Sttes As previously noted, entngled sttes on tensor produt re peulirly quntum in the sense tht there is no good lssil nlog for them. Dense oding nd teleporttion re two proesses whih mke use of entngled sttes, nd for this reson pper somewht strnge from n everydy lssil perspetive. We shll lter introdue mesure of entnglement for pure sttes, ut for the moment ll we need re fully entngled (or mximlly entngled) sttes. Let nd e two ilert spes of the sme dimension d = d = d. Any stte on Ψ on = n e written in the Shmidt form: = j λ j j j, (1) where { j } nd { k } re suitle orthonorml ses (whih depend upon ). A fully entngled stte is one for whih ll the λ re equl (or equl in mgnitude if one does not impose the ondition λ j > 0), nd thus equl to 1/ d if is normlized. In the se of two quits, d = 2, the Bell sttes B 0 = ( 00 + 11 ) / 2, B 1 = ( 01 + 10 ) / 2, B 2 = ( 00 11 ) / 2, B 3 = ( 01 10 ) / 2, (2) re exmples of fully-entngled sttes whih form n orthonorml sis. Fully entngled sttes n lso e hrterized in the following wy. Let the redued density opertors for normlized on nd e defined in the usul wy: ρ = Tr ([ψ]), ρ = Tr ([ψ]). (3) For fully entngled stte, ρ = I/d = ρ. (4) 1
Note tht we re ssuming tht the redued density mtries ome from pure stte on, nd not from mixed stte represented y density opertor. (Entnglement for mixed sttes is omplex prolem whih is fr from well understood t the present time.) Exerise. Show tht only one of the equlities in (4) is tully needed, s the seond is onsequene of the first (nd vie vers). Fully entngled sttes do not hve unique Shmidt deomposition. Given ny orthonorml sis { j } of, there is n orthonorml sis { k } of, one tht depends oth on { j } nd on, suh tht (1) holds (with λ j = 1/ d). Exerise. Prove this ssertion y expnding in the form j j β j, nd using (4). Given two normlized fully entngled sttes nd φ on one n lwys find unitry opertors U nd V on nd suh tht φ = (U V ). (5) Tht there is some unitry opertor W on mpping to φ is onsequene of the ft tht they hve the sme norm. Wht is speil out (5) is tht W is of the form U V. It will e onvenient to refer to ll suh n opertor lol unitry. The ide of lol is tht one thinks of the susystems nd s loted in two seprte lortories where Alie pplies U to the first nd Bo pplies V to the seond system. Exerise. Another lss of pure sttes mpped into eh other y lol unitries re the (normlized) produt sttes. Cn you think of other lsses? Wht is the most generl lss? [int: Shmidt.] For ny d = d = d 2 one n find n orthonorml sis for onsisting of fully entngled sttes, nlogous to the Bell sis in (2). These ses re not unique, there re lwys mny possiilities. If the dimensions d nd d re unequl, one nnot hve fully entngled stte of the form defined ove, euse the numer of nonzero Shmidt oeffiients n t most e the smller of d nd d. owever, one n hve mximlly entngled stte where, for exmple, if d < d, then d of the Shmidt oeffiients re equl to eh other (or equl in mgnitude). Sometimes it is helpful to define uniformly entngled stte s one in whih ll the nonzero Shmidt oeffiients re equl (or of the sme mgnitude). 2 Dense Coding The phenomenon of dense oding is sed on the oservtion tht given some stte elonging to the Bell sis (2) there re lol unitries on whih will mp it onto ny of the other sttes elonging to the sis, prt from n overll phse. There re similr unitries on. Thus if we strt with B 0, it is mpped to B 1 y X I, to B 2 y Z I, nd to B 3 (up to phse) y Y I. As onsequene, given n entngled stte B j shred etween Alie s nd Bo s lortories, either one of them n onvert it into nother sis stte B k y pplying n pproprite unitry. On the other hnd, neither of them n determine y lol mesurements, i.e., entirely y mesurements on, or entirely y mesurements on, whih B j they jointly possess. Exerise. Disuss why this is so. [int. Suppose Alie rries out mesurements in some orthonorml sis of. Wht will e the proilities of the outomes?] If they oth rry out mesurements nd ompre them using lssil hnnel (telephone) they n mke some distintions. For exmple, mesurements in the stndrd sis when ompred with eh other will distinguish B 0 from B 1 nd B 3, ut not from B 2. To rry out mesurement in the Bell sis requires either ringing the quits together, or else doing something like teleporttion, whih requires nother quit pir in known Bell (or other fully entngled) stte. A onvenient wy of imgining how one of the B j might e shred y Alie nd Bo is to suppose tht Bo produes it in some system in his lortory (e.g., y photon down onversion), nd then ships the prt to Alie over perfet quntum hnnel. Putting inside refully onstruted ox 2
so tht it will not e distured, nd sending the ox to Alie y prel servie, is to e thought of s quntum hnnel though not very prtil one, given urrent tehnology. It is more relisti to imgine n optil fier etween the lortories, nd sending one of two down-onverted photons through the fier. The protool for dense oding is illustrted in the following figure, whih shows quntum iruit with prts in Alie s (upstirs) nd Bo s (sement) lortories. ā ā ā ā Alie Z Bo ā t 0 t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 Figure 1: Ciruit for dense oding. The prt of the iruit preeding t 2 is simply devie to turn the produt stte 00 of two quits nd in Bo s lortory into the entngled Bell stte B 0. The quit is then shipped upstirs to Alie. (Alterntively, Alie ould produe the entngled pir in her lortory nd ship downstirs to Bo.) Alie then rries out one of four unitry opertions on quit whih either leve the omintion in B 0 or mp it into one of the B j with j > 0. Wheres one ould imgine Alie doing this y hnd, the figure shows quntum iruit in whih the unitry is ontrolled y two quits nd ā, whih re initilly in one of the four stndrd sis sttes 00, 01, 10, 11 orresponding to the four possile messges whih Alie n trnsmit to Bo y this mens. At t 5 quit is shipped k to Bo. The iruit in his l following t 6 is just the mirror imge of the one used to produe B 0 in the first ple, nd its purpose is to mesure whih Bell stte the pir is in. The result of the mesurement will e two quits whose sttes, s indited in the digrm, re the sme s Alie s input, ssuming lwys tht oth nd ā re initilly either or 1. Exerise. Wht hppens if one hooses initil = x + = + nd initil ā =? Wht stte results t t 8 from unitry time evolution? Suppose t this time the four quits re mesured in the stndrd sis; wht will one find? Wht mkes this proess seem prdoxil, nd gives rise to the nme dense oding, is the ft tht only one quit,, psses from Alie s lortory to Bo s during the time intervl etween Alie s preprtion nd Bo s mesurement. On the other hnd, two its of informtion needed to identify one out of four possile messges hve somehow pssed etween them. To put it nother wy, if there were only one it lssil hnnel etween Alie nd Bo, it would hve to e used twie to get the messge through, wheres the quntum hnnel is only used one, t lest if we ignore wht hppened efore t 3. In summry, it looks s if single quit n rry two its of lssil informtion! But loser inspetion shows tht things re not quite so simple. Let us imgine, for exmple, tht quit ws ptured y some outsider (trditionlly know s n evesdropper, or Eve) who wnted to listen in on the messge etween Alie nd Bo. Wht ould she lern out the vlues of nd ā from mesuring or rrying out other opertions on? Asolutely nothing. From this point of view one might rgue tht rther 3
thn two its of informtion, quit ontins no informtion (of the relevnt sort) t ll! In the sme wy one n show tht quit y itself ontins no informtion out nd ā efore t 6 when the mesurement ours. Thus in some sense it is only the omined system of long with tht ontins or rries the informtion; one n sy tht the informtion resides in orreltions etween these quits. Exerise. Show tht no informtion is present in either quit or quit seprtely t the time t 5, y omputing the redued density opertor of eh quit nd showing tht it is independent of nd ā. Some insight into wht is going on is provided y the following nlogy. Imgine tht the quntum hnnel etween Alie nd Bo is repled y lssil hnnel, whih I like to think of s pipe whih n rry olored slips of pper. In the sement Bo hs mhine whih puts out two slips of pper of the sme olor, red(r) or green (G) in rndom fshion, suh tht RR or GG is produed with proility 1/2. The first slip is sent upstirs to Alie, who sends messge k to Bo in the following fshion. For 0, she returns slip with the sme olor, nd for 1 she returns slip with the opposite olor (G if she reeived R, R if she reeived G). Bo reds the messge y ompring the olor of the slip in his possession with the one sent k y Alie: if they re the sme tht signifies 0, if they re different tht mens 1. The min point of this nlogy is tht it shows how the messge is stored in orreltions rther thn in individul slips of pper. If Eve stels the slip whih Alie is sending k to Bo, she will lern nothing out the messge, for the proility tht it will e red or green is 1/2, independent of whtever messge Alie deided to send. There is tully loser onnetion etween this nlogy nd the quntum iruit in Fig. 1 thn one might t first suppose; there is wy of desriing the quntum proess using prtiulr onsistent fmily of histories whih mkes it look lmost like the lssil sitution just desried. (Detils re given in Se. V of NLQI.) owever, in the lssil se Alie n only send one it of informtion per slip of pper, wheres in the quntum se she n send two its per quit, so there is still quntum mystery. One wy of viewing the mystery is to rell tht, Se. 1, the Bell sis hs the property tht given n initil stte is one of the Bell sis sttes, Alie herself, with no help from Bo, n trnsform it to ny one of the other three sis sttes. Contrst this with wht hppens when one uses sis of produt sttes suh s the stndrd sis 00, 01, 10, 11. ere it is ovious tht there is no wy y whih Alie y herself n hnge 00 (for exmple) into eh of the other sis sttes. She n only reh one other sis stte, not ll three. Consequently, Alie n put two its of informtion into the two quit system if it strts off in fully-entngled stte, ut only one it of informtion if it strts off in produt stte. The ltter is wht orresponds est with our lssil experiene of the world, nd tht is one reson we find quntum entnglement peulir nd perplexing. Dense oding n e generlized in n ovious wy to systems with d > 2. A totl of d 2 different messges n e sent through d-dimensionl quntum hnnel, provided Alie nd Bo shre fullyentngled stte to egin with. 3 Teleporttion Teleporttion resemles dense oding in tht it requires the presene of fully entngled stte t the outset. owever, its gol is silly different: rther thn send lrger-thn-expeted mount of lssil informtion over quntum hnnel, the ide is to send quntum stte over lssil hnnel. Figure 10.2 shows the si rrngement for teleporttion in the form of iruit. As in Fig. 1, the elements preeding t 2 rete n entngled stte B 0 on quits nd, nd the quit is sent to Alie s l over quntum hnnel. Next Crol rings to Alie stte to e teleported to Crol s ssoite Chrlie, who is witing in Bo s lortory. The iruit in Alie s lortory etween t 3 nd t 5 followed y the stndrd-sis detetors D nd D serves to mesure the system onsisting of quits nd in the Bell sis. Note tht this is silly the sme rrngement s one finds in Fig. 1 following t 6 (where the detetors re not shown) nd its purpose is the sme. 4
D Alie D Bo Z t 0 t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 Figure 2: Teleporttion. The outomes of the mesurements re lssil signls, shown s hevy lines, whih n e trnsmitted to Bo over lssil hnnel, for exmple, n ordinry phone line. At this point Bo rries out unitry trnsformtion on quit, the nture of whih is determined y the messge reeived from Alie. The figure shows n utomted version: if D = 1, n X or not opertion is performed on, nd if D = 1 Z or phse opertion is performed on, wheres if D or D is 0 the orresponding opertion is not rried out. The net result is tht fter these opertions quit, whih Bo hnds to Chrlie, is in the sme stte s the quit whih Crol rought to Alie. This is the sense in whih the quntum stte hs een teleported with the help of two lssil its of informtion, the outomes of the D nd D mesurements, nd without the need for rel quntum hnnel from Alie to Bo. Things re, of ourse, not quite tht simple, for quntum hnnel ws needed to set up the entngled stte essentil for teleporttion. owever, this ws used erlier, possily even efore Crol reted the stte to ring to Alie. One n well imgine tht even if good quntum hnnel were ville to Bo nd Alie, it might e dvntgeous to employ it to produe olletion of pirs of entngled quits on weekends when the rtes re low, nd then teleport using lssil hnnel during the week. Or, teleporttion might e reserved for use in n emergeny when one needs to trnsmit quntum stte, ut the quntum hnnel hs roken down. (To e sure, using urrent tehnology it is not possile to preserve pir of quits in fully entngled stte for ny signifint length of time, ut we must leve few prolems to future genertions of engineers.) Crol nd Chrlie n hek whether the lim of Alie nd Bo to e operting good teleporting servie is orret. Crol retes vrious different 1-quit sttes whih she rings to Alie, nd Chrlie mesures the sttes reeived from Bo in the, ψ sis, where ψ is the stte orthogonl to. The outome of one suessful mesurement ould e n ident, ut if things work in, sy, 20 ses, this is some evidene tht teleporttion is tking ple. Teleporttion Ltd., the firm tht employs Alie nd Bo, will not gurntee perfet quntum hnnel, ut insted promises to hieve ertin fidelity F or error rte ǫ, suh s ǫ < 0.01, or F > 0.99. Alie nd Bo oth know the outomes of the Bell-stte mesurements, i.e., the vlues of D nd D in ny prtiulr se in whih teleporttion is employed, nd one might suppose tht this would provide them some informtion out the stte supplied y Crol. owever, this is not t ll the se: for given, the D nd D outomes re ompletely rndom: eh of the four possiilities for (D, D ) ours with proility of 1/4. This is n exmple of very generl priniple in quntum informtion whih goes under the nme of no loning or no opying, see Se. 4. The si ide is tht quntum hnnel nnot e split into two or more hnnels without introduing some noise. Consequently, if the hnnel from t t 1 to t t 8 in Fig. 2 is perfet hnnel, it is not possile for informtion out t t 8 to e ville nywhere exept in quit itself. The outome of the D nd D mesurements ould e in Alie s noteook (for exmple), 5
nd therefore nnot possily tell one nything out. In understnding how the iruit in Fig. 2 funtions, it n e helpful to reple it with the one shown in Fig. 3, whih hs no lssil elements up to time t 8. By onsistent histories rguments, or y the priniple of deferred mesurements in QCQI (p. 186), one n show tht the finl results (D, D, nd ) re the sme. D Alie D Bo Z t 0 t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 Figure 3: Alterntive teleporttion iruit. ere detetion tkes ple fter t 8, nd so fr s teleporttion is onerned, nothing would hnge if there were no detetors: the outomes oviously do not influene quit. owever, rrying out the two opertions etween t 6 nd t 8 requires pproprite onnetions etween Alie s nd Bo s lortory, nd these, in turn, neessitte using quntum hnnel. This undermines the originl ide of teleporttion, whih ws to onstrut quntum hnnel without hving to use quntum hnnel! We introdue the iruit in Fig. 3 not s prtil replement for tht in Fig. 2, ut s n id to understnding. Exerise. Assume generl initil stte = α+β 1, nd work out the unitry time development of the three quits in Fig. 3, strting with Ψ 2 = B 0 t t 2. In prtiulr otin Ψ 5 nd Ψ 8 t times t 5 nd t 8. ) Show tht Ψ 8 is produt stte. Then rgue tht it hs to e of the form if the output is to e the sme s the input. Cn you see from Ψ 8 why the outomes of D nd D provide no informtion out? ) Expnd Ψ 5 in the form j,k j k jk, i.e., find the kets jk. Use these to explin why the iruit in Fig. 2 is suessful. ) Show tht t time t 6 in Fig. 3 the informtion out is ontined in quits nd in the sense tht it ould e reovered from them y modifying the iruit t lter times. (This is in ontrst with the sitution t t 6 in the iruit in Fig. 2, where informtion out is not present in the lssil its representing the mesurement outomes.) If the two lssil its representing the mesurement outomes in Fig. 2 ontin no informtion out, where is tht informtion t, sy, t 6? Is it in quit? But how n tht e, given tht the results of the mesurements on nd hve yet to reh Bo s lortory? See NLQI Se. VII for disussion of these questions. The rief nswer is tht the informtion is present in orreltions etween the D nd D outomes nd the quit, the nture of whih will e evident if you did prt () of the preeding exerise. There is nothing prtiulrly weird or osure going on in prtiulr, there re no mgil long-rnge influenes t work for one n onstrut lssil nlogy for suh orreltions, s explined in NLQI. In wht sense is quntum informtion, in ontrst to lssil informtion, teleported? If one were simply onerned with the question of whether quit is in the stte rther thn the orthogonl ψ, this type of informtion n e trnsmitted in lssil fshion y the simple devie of 6
mesuring the quit in the, ψ sis, nd sending the informtion over lssil hnnel. Wht the teleporttion iruit llows one to do is to trnsmit n unknown quntum stte without tully determining wht it is. Think of the simple perfet quntum hnnel produed y sending spin-hlf prtile through good pipe, or polrized photon through good optil fier. Wht hppens is tht whtever goes in omes out gin, independent of wht it is tht goes in. In some sense the issue is not quntum informtion, ut the ility to use one iruit or piee of pprtus to trnsmit the lssil informtion distinguishing from ψ for mny different possile hoies of, when the hoie is not speified in dvne. Tht is wht distinguishes quntum hnnel from its lssil ounterprt. 4 No Cloning There is sense in whih quntum informtion nnot e perfetly opied or loned, nd this is expressed in vrious no-loning theorems. The simplest exmple is illustrted in the following figure, where nd hve the sme dimension, ut re otherwise ritrry. T Figure 4: ypothetil loning mhine. Suppose tht T nd re fixed, while n e vried. For prtiulr one n lwys find unitry T suh tht T ( ) is equl to. But there is no fixed T tht will omplish this for ll possile inputs. The rgument is strightforwrd. Suppose the opying iruit n e mde to work for two nonorthogonl normlized sttes ψ nd ψ, nd ssume tht is normlized. Then we hve T ( ψ ) = e iφ ψ ψ, T ( ψ ) = e iφ ψ ψ, (6) where we hve llowed for phses φ nd φ, whih do not ffet the physil interprettion of the finl sttes. Tke the inner produt of the first of these equtions with the seond, nd use the ft tht T T = I. The result is ψ ψ = e i(φ φ ) ( ψ ψ ) 2, (7) nd upon tking the solute vlue of oth sides, ψ ψ = ψ ψ 2. (8) This lst eqution hs two solutions: either ψ ψ is equl to 0 or it is equl to 1. In the first se the two sttes re orthogonl, nd in the seond, sine we hve ssumed tht oth of them re normlized, they re identil prt from phse ftor. (See the exerise following (12).) Both re exluded y our definition of the term nonorthogonl. Conlusion: There is no quntum opying mhine tht n mke two perfet opies (or one perfet opy plus remining perfet originl) of two (or more) nonorthogonl sttes. Notie the qulifition. There is no rule ginst mking s mny perfet opies s one wnts of mutully orthogonl sttes using quntum opying mhine. Sine ny two mrosopilly distint sttes of the world orrespond to orthogonl quntum sttes, there is no similr restrition on opying lssil ojets. Quntum physis is no thret to ordinry photoopying. 7
A seond no-loning result n e otined y slight modifition of the ove rgument. One gin, imgine quntum system with time development given y unitry opertor T ting on tensor produt spe, where is of dimension 2 (one quit), wheres is of ritrry size, nd suppose tht Ψ := T ( ψ ) = ψ, (9) tht is, the result is perfet opy or, if one prefers, perfet preservtion of the initil stte ψ. The sttes ψ nd re ssumed to e normlized, nd sine T is unitry, must lso e normlized, ut otherwise we know nothing out it. Unlike (6), there is no need to introdue phse ftor on the right side of this eqution, s it n lwys e inorported into the definition of. Note tht in order to produe perfet opy (or preservtion) on, Ψ in (9) must e produt stte on, not n entngled stte. If we hve n entngled stte, then the proility of otining ψ t the lter time nnot e 1. Exerise. Justify the preeding sttement. [One pproh: Form n orthonorml sis { j } of, with ψ = 0. Expnd Ψ in the { j } with oeffiients in E. Use this to lulte the proility of ψ.] Next suppose there is seond stte ψ nonorthogonl to ψ, whih is lso perfetly opied (or preserved): T ( ψ ) = ψ. (10) Tke the inner produt of (9) with (10). The result is ψ ψ = ψ ψ. (11) Sine ψ nd ψ re nonorthogonl, ψ ψ nnot e 0, so (11) tells us tht = 1. Sine oth nd re normlized, this mens tht =. (12) Exerise. Complete the rgument tht for normlized sttes, = 1 implies tht they re identil. [int: Wht is the norm of?] Also show tht if = 1, is multiplied y some phse ftor e iφ. Sine T is liner opertor, (12) omined with (9) nd (10) tells us tht T ( ) = (13) for ny whih is liner omintion of ψ nd ψ. Beuse ψ nd ψ re nonorthogonl nd thus not multiples of eh other, their liner omintions form two-dimensionl ilert spe, in effet the input of one-quit quntum hnnel. Wht (13) tells us is tht this is perfet hnnel, nd tht no informtion distinguishing different in the hnnel resides in the -independent stte (nd thus in ). It my not e ovious t first how this result pplies to the teleporttion iruit in Fig. 3, sine the input nd output of the hnnel re different: the quit nd the quit, respetively. All one needs to do is to dd t the end of the iruit unitry opertion whih exhnges the nd the quits, so the output is lso the quit, nd let T in (13) e the unitry orresponding to this ugmented iruit. The result n e summrized s follows: if one-quit hnnel is perfet for two nonorthogonl sttes, it is perfet for ll sttes, nd no informtion distinguishing these sttes n lek out of it into the environment. In the sme wy, to hek whether d > 2 hnnel is perfet, it suffies to show tht it is perfet for d nonorthogonl sttes whih together spn the d-dimensionl ilert spe of the hnnel. No informtion n lek out of suh perfet hnnel. Exerise. Wht is ment y d nonorthogonl sttes? Suppose we lel them s φ k, 1 k d. Is it neessry tht φ k φ j e nonzero for ll j nd k, or will the rgument go through with some of these inner produts equl to 0? 8