C/CS/Phys C9 Bell Inequlities, o Cloning, Teleporttion 9/3/7 Fll 7 Lecture 6 Redings Benenti, Csti, nd Strini: o Cloning Ch.4. Teleporttion Ch. 4.5 Bell inequlities See lecture notes from H. Muchi, Cltech, Ph 95 () on next pge C/CS/Phys C9, Fll 7, Lecture 6
Lecture otes from H. Muchi, Cltech, Ph 95 () onloclity nd Bell Inequlities (Bsed on the discussion in Chris Ishm s ook, Lectures on Quntum Theory: Mthemticl nd Structurl Foundtions (Imperil College Press, 995).) Sy we hve two experimenters, lice nd Bo, whose ls re locted mny kilometers prt. Their ls re siclly identicl, ctully, ech consisting of one prticle detector tht hs one meter, one switch, nd ell. The meter is for reding out the result of mesurement (which we ssume to e either ), while the switch is used to select which of two types of mesurements the experimenter would like to mke. On lice s side we ll lel the two possiilities nd Ë, nd on Bo s side B nd B Ë. The ell rings ech time prticle hits the detector, letting the experimenter know when he or she cn red out the result of his/her selected mesurement. So where do these prticles come from? Midwy etween lice s l nd Bo s there is pir source. This source lwys produces prticles in pirs, sending one to lice nd the other to Bo. We ssume tht the prticles hve some internl degree of freedom, which is wht lice s nd Bo s detectors re designed to mesure. The pir source prepres the internl sttes of the prticles in some unknown, possily rndom fshion. The experiment consists of the following procedure. The source prepres nd emits one pir of prticles per unit of time, so lice nd Bo know tht they my expect to receive prticles t regulr rte. Once per unit time, they ech (independently) select rndom setting for their switch, wit for their ell to ring, nd then red off nd write down the mesurement result. Hence fter ten rounds, e.g., lice s nd Bo s l ooks might look something like this: lice Bo Ë Ë Ë Ë Ý Ý Ý Ý Ý B Ë B Ë B B B B Ë B B Ë B Ë B Ý Ý Ý Ý
Lecture otes from H. Muchi, Cltech, Ph 95 () lthough this experimentl scenrio seems extremely generl, it turns out tht we hve lredy specified enough to derive some importnt predictions out the sttistics of lice s nd Bo s mesurement records! Let s strt y mking some resonle ssumptions out the overll ehvior of the experiment:. Locl determinism we might like to elieve tht the result of lice s mesurement (either or Ë )isloclly determined y the physicl stte of the prticle she receives from the pir source. It should not depend on the stte of Bo s prticle, since in this scenrio Bo could e relly fr wy! nd the result of lice s mesurement certinly should not depend on Bo s choice of mesurement tht is, whether lice s meter reds or Ý should not depend on whether Bo hs his switch set to B or B Ë.... Ojective relity Even though lice (nd Bo) must choose to mke one mesurement or the other ( or Ë ) on ny given prticle, ech prticle knows wht its vlue is for oth mesurements. Tht is, sufficient informtion to determine the outcome of either mesurement is encoded in the internl stte of ech prticle. Under these ssumptions, we cn write down the following model for this experiment. In ech round, the pir souce produces pir of prticles with the following informtion encoded in their internl sttes: Ë Ë n, n, B n, B n. Here the four possile mesurement lels re treted s rndom vriles, with the suscript lelling the round. s logicl consequence of locl determinism nd ojective relism, we cn ssume the existence of joint proility distriution P,, B, B. Hence, it should e meningful to consider correltion functions of ll four rndom vriles simultneously, nd these correltion functions should e mesurle y lice nd Bo. Consider the following function of the rndom vriles, g n n B n n B n n B n Ý n B n. 3 Were we to tulte the 6 possile vlues of g n, we would mgiclly find tht g n. However, n esier wy to see this is to note tht the lst term in the sum is equl to the Ë product of the first three, since n n Ë B n B n : n B n n B n n B n n B n n B n n B n. 4 Then if n B n, the set n B n, n B n, n B n hs either zero or two Ý s, hence g n n B n n B n n B n Ý n B n must e either or Ý. If on the other hnd n B n Ý, the set must hve either zero or two s, hence g n must e either Ý or. In ny cse, it follows tht
Lecture otes from H. Muchi, Cltech, Ph 95 () Ý n g n Ý n n B n Ý n n B n Ý n B n n Ë ÝÝ B n n Ë n â. 5 This is one form (due to Cluser, Horne, Shimony, nd Holt) of Bell s fmous inequlity. It should e noted tht t this point, ll we hve relied on in our derivtion is sic proility theory! Hence the Bell Inequlity is model-independent prediction out mesurement sttistics in world tht is loclly deterministic nd llows ojective relism. Hence experimentl violtions of the Inequlity ctully tell us something out ture, not just quntum theory! s it turns out, one cn ctully go to the l nd perform experiments of precisely the type descried ove, nd find tht this inequlity is strongly violted! For exmple, see Ê G. Weihs et l, Violtion of Bell s Inequlity under Strict Einstein Loclity Conditions, Phys. Rev. Lett. 8, 539-543 (998); Ê W. Tittel et l, Violtion of Bell Inequlities y Photons More Thn km prt, Phys. Rev. Lett. 8, 3563-3566 (998); Ê. spect, Bell s inequlity test: more idel thn ever, ture 398, 89-9 (999). In experiments of this type, the key is to construct source tht produces pirs of photons n entngled stte such s è è Ý è. 6 In ech round of the experiment, lice s two mesurements correspond to the oservles z nd Ë cos z sin x, where z èè Ý èè, x èè èè. 7 On Bo s side we choose B z nd B Ë cos z Ý sin x. The eigenvlues of nd B re clerly, nd it turns out tht those of Ë nd B Ë re lso. For exmple, the eigensttes of cos z sin x re simply cos è sin è, sin è Ý cos è. 8 Hence Ë corresponds to projectors on sis tht is rotted from tht of y n ngle / (nd similrly rottion of Ý / for B,B Ë ). ow we cn compute the necessry correltion functions using the stndrd quntum proility rules: 3
Lecture otes from H. Muchi, Cltech, Ph 95 () Similrly, Ý n B n è å B è n P P P P Ý P P Ý P P Ý. 9 Ý Ë n B n èp cos z è Ý èp sin x è Ý èp cos z è èp sin x è n Ý cos Ý cos Ý cos. Ý Ë n B n P cos z P sin x Ý P cos z Ý P sin x n Ý n B n n Ý cos. ècos z z è ècos sin z x è Ý ècos sin x z è Ý sin x x cos Ý Ý Ý sin Ý Ý sin Ý cos Ý cos. Finlly, we cn construct the overll quntity Ý g n Ý Ý cos cos n cos Ý cos. Plotting this, we find tht the Bell Inequlity is violted (èg n è ) for 9 Ý :.5 <g n >= +cosφ-cos φ.5.5 4 6 8 4 6 8 φ [d eg] 4
Lecture otes from H. Muchi, Cltech, Ph 95 () So wht s going on here? From the grph we see tht our Bell Inequlity cn e violted when the two possile mesurements tht lice nd Bo cn perform correspond to projections on nonorthogonl ses. Hence wht is eing exploited here is the extr-strong quntum correltion etween two prticles tht hve een prepred in n entngled stte such s è è Ý è. èè èè p i, i è å i è 5
3 o Cloning Theorem quntum opertion which copied sttes would e very useful. For exmple, we considered the following prolem in Homework : Given n unknown quntum stte, either φ or ψ, use mesurement to guess which one. If φ nd ψ re not orthogonl, then no mesurement perfectly distinguishes them, nd we lwys hve some constnt proility of error. However, if we could mke mny copies of the unknown stte, then we could repet the optiml mesurement mny times, nd mke the proility of error ritrrily smll. The no cloning theorem sys tht this isn t physiclly possile. Only sets of mutully orthogonl sttes cn e copied y single unitry opertor. There re two wys to prove the no cloning theorem. The first follows from the norm preserving property of the inner product, the second from the linerity of quntum mechnics. o Cloning ssume we hve unitry opertor U cl nd two quntum sttes φ nd ψ which U cl copies, i.e., φ ψ Then φ ψ is or. U cl φ φ U cl ψ ψ. Proof : φ ψ =( φ )( ψ )=( φ φ )( ψ ψ )= φ ψ. In the second equlity we used the fct tht U, eing unitry, preserves inner products. Proof : Suppose there exists unitry opertor U cl tht cn indeed clone n unknown quntum stte φ = α + β. Then φ Ucl φ φ =(α + β )(α + β ) = α + βα + αβ + β But now if we use U cl to clone the expnsion of φ, we rrive t different stte: (α + β ) U cl α + β. Here there re no cross terms. Thus we hve contrdiction nd therefore there cnnot exist such unitry opertor U cl. ote tht it is however possile to clone known stte such s nd. 4 Teleporttion Contrry to its sci-fi counterprt, quntum teleporttion is rther mundne. Quntum teleporttion is mens to replce the stte of one quit with tht of nother. It gets its out-of-this-world nme from the fct tht the stte is trnsmitted y setting up n entngled stte-spce of three quits nd then removing two quits from the entnglement (vi mesurement). Since the informtion of the source quit is preserved y these mesurements tht informtion (i.e. stte) ends up in the finl third, destintion quit. This occurs, however, without the source (first) nd destintion (third) quit ever directly intercting. The interction occurs vi entnglement. Figure (see elow) shows the set up for quntum teleporttion, nd Figure (see elow) presents quntum circuit implementing teleporttion of one-quit stte. C/CS/Phys C9, Fll 7, Lecture 6
Suppose ψ = + nd given n EPR pir ( + ), the stte of the entire system is: [ ( ) ( )] + + + = Perform the COT opertion nd you otin [ ( + ) + ( + )] = ext we pply the H gte. However, s n side, lets exmine wht hppens when we pply the H gte to nd to. Recll tht: [ ] H = H = [ H = [ ][ ][ Thus, pplying H to our system we hve: ] = [ ] [ = ] ϕ [ = ( )( ) + + + ( )( ) ] + = We cn rewrite this expression s: ] C/CS/Phys C9, Fll 7, Lecture 6 3
[ ] [ ] [ ] [ ] = [ ( + ) + ( + ) + ( ) + ( )], which we cn shorten to: [ [ ] ψ + [ ] ψ + [ ] ψ + i [ i i ] ψ ]. We recognise tht the third quit is now in stte given y the ction of one of the well-known Puli opertors I,X,Y,Z on the unknown initil stte ψ of quit. The stte of quit 3 cn lso e written s: [ I ψ + X ψ + Z ψ + iy ψ ] nd lterntively s: ϕ = [ I ψ + X ψ + Z ψ + XZ ψ ]. otice tht the two-quit stte of quits nd is different in ech term. This result implies tht we cn mesure the first nd second quit nd otin two clssicl its which will tell us wht trnsform ws pplied to the third quit. Thus we cn susequently fixup the third quit once we know the clssicl outcome of the mesurement of the first two quits. This fixup is firly strightforwrd, either pplying nothing, X, Z or oth X nd Z. (Recll tht X = Y = Z = I.) Lets work through n exmple. Suppose the result of mesuring quits nd is. Then from the ove, quit 3 must e in the stte Z ψ. The mtrix representing the mesurement opertor is M = P()= ϕ M M ϕ = ϕ M ϕ, since here M M. Thus: M ϕ = C/CS/Phys C9, Fll 7, Lecture 6 4
Hence: ϕ M ϕ = [,,,,,,,] = 4 [ + ] Recll tht y definition of quit we know tht + =, hence the proility of mesuring is /4. The sme is true for the other outcomes. Wht hve we done? We hve inserted n unknown single quit quntum stte into system of 3 quits where the other two quits shred some entnglement. We crried out some unitry opertions on quits nd, nd then mesured out these two quits. The result is tht the unknown quntum stte hs een migrted through entnglement to quit 3, where it is cn e recovered y mking single quit unitry opertion dependent on the two mesured vlues from quits nd. Quntum teleporttion hs een termed disemodied trnsfer of quntum informtion from one plce to nother (S. Brunstein). It does not violte reltivity: the source sends only clssicl informtion (the result of the mesurements of quits nd ) nd this must e done y conventionl mens, e.g., opticl fier. The source sends no informtion out the quntum stte. either does it violte the no-cloning theorem since the quntum stte is destroyed t the source nd creted t the destintion. ie., ψ x ψ. Here x is the stte of quit fter mesurement. Teleporttion illustrtes n equivlence etween quntum its (quits), entnglement its (e-its), nd clssicl its (c-its): quit e-it + c-its ote the difference etween mking FX copy nd creting copy y quntum teleporttion. With FX, i) the originl is preserved, nd ii) only prtil copy is otined. With quntum teleporttion, i) the originl stte is destroyed (ut not the quit), nd ii) n exct copy of the quntum stte results. ccessile sources on quntum teleporttion: IBM we pge: http : //wwww.reserch.im.com/quntum i nfo/teleporttion C. Cves, Science 8 (3 Octoer) 998, p. 637 C/CS/Phys C9, Fll 7, Lecture 6 5
Generte EPR pir nd distriute to ech end Destintion in stte Source in stte (destroyed in process) Trnsmit clssicl informtion Fixup result Figure : Teleporttion requires pre-trnsmitting n EPR pir to the source nd destintion. The quit contining the stte to e teleported then intercts with one hlf of this EPR pir, creting joint stte spce. Unitries re performed in this joint stte spce nd then these quits re mesured. The resulting clssicl informtion of the mesurement outcome is trnsmitted to the destintion. This clssicl informtion is used to fixup the destintion quit with single quit unitries. y> H > > H X Z y> Figure : Quntum circuit implementing teleporttion. The first two opertions on quits nd 3 t the ottom right form the EPR pir. ote tht in this digrm single lines represent quntum dt while doule lines represent clssicl informtion. C/CS/Phys C9, Fll 7, Lecture 6 6