Lecture Notes PH 411/511 ECE 598 A. La Rosa Portland State University INTRODUCTION TO QUANTUM MECHANICS

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1 Lecture Notes PH 4/5 ECE 598. L Ros Portlnd Stte University INTRODUCTION TO QUNTUM MECHNICS CHPTER- QUNTUM ENTNGLEMENT The nnihiltion of the positronium This chpter is tken completely from the The Feynmn Lectures, Vol III, Chpter 8 nd from the book Quntum Mechnics by D. J. Griffiths. The nnihiltion of the positronium process with the consequent genertion of two entngled photons is described by Feynmn in gret detil, ccounting for the conservtion of energy, liner momentum, ngulr momentum nd prity. Feynmn rgues there does not exists prdox when stting tht mesurement on one side ffect the result of mesurement mde t nother fr wy loction. I. The QUNTUM THEORY nd the ELL s DISCOVERY I. The EPR pper on the (lck of) Completeness of the Quntum Theory I. The ohm experimentl version to settle the EPR prdox II. ILLUSTRTION of n ENTNGLEMENT PROCESS: The NNIHILTION of the POSITRONIUM III. ELL S THEOREM IV. QUNTUM TELEPORTTION V. ENTNGLEMENT from INDEPENDENT PRTICLE SOURCES PPENDIX-: Tensor Product of Stte-Spces PPENDIX-: THE EPR Pper on the (lck of) Completeness nd Loclity of the Quntum Theory. Einstein, Podolsky, nd Rosen, Phys. Rev. 47, 777 (935). L () Polrizer R Polrizer () x Polrizer F Polrizer y () () Polrizer R () Polrizer L ) () Fig. Positronium decy into directionl two-photon stte F. (ll direction re eqully probble). mesurement long one direction (using polriztion filters) mkes the stte F to collpse into correlted-sttes determined by the polrizers.

2 I. The QUNTUM THEORY nd the ELL s DISCOVERY Consider F ~ to be n observble-opertor ssocited to the observble f (quntity one mesures experimentlly), which hs complete orthonorml set of eigenfunctions {,, } nd corresponding eigenvlues { f, f, }. Let s consider system (n ensemble of systems) is in the stte described by, = m m c m = m m m Wht exctly does this wvefunction men? We hve so fr dopted the orn s sttisticl interprettion, postulting tht j c gives the probbility P( f j ) of obtining the vlue when mking mesurement of f on prticulr system of n ensemble. Under this interprettion, the wve function does not uniquely determine the outcome of mesurement; insted it provides sttisticl distribution of possible results. Such n interprettion hs cused deep controversil discussions. Suppose, for exmple, one mesurement renders the vlue f 4. Wht ws the vlue f of the system before we mde the mesurement? There re three min schools of thought for nswering tht question: i) The relistic viewpoint: The system hd the vlue f 4. Tht is, the physicl system hs the prticulr property being mesured prior to the ct of mesurement. This view ws dvocted by Einstein. ccordingly, quntum mechnics is n incomplete theory, for even when the system hd the vlue f 4, still quntum mechnics is unble to tell us so. (The theory is silent bout wht is likely to be true in the bsence of observtion.) Einstein hoped for progress in physics to yield more complete theory, nd one where the observer did not ply fundmentl role. Therefore, there is some other dditionl informtion (known s hidden vrible), which together with the wve function is required for complete description of the physicl relity of the system. [ut fter 97 Einstein regrded the hidden vribles project the project of developing more complete theory by strting with the existing quntum theory nd dding things, like trjectories or rel sttes n improbble route to tht gol.] In 935 Einstein co-uthored celebrted pper supporting the relistic view point nd questioning the completeness of the quntum theory. Fifteen yers lter hom proposed to nlyze the EPR pper but thought n experiment involving the dissocition of ditomic molecule where the two prts together should stisfy the conservtion of ngulr momentum. Different EPR-ohn type experimentl setup hve been suggested nd implemented since. ii) The orthodox viewpoint: the system hd no specific vlue of f.

3 It is the ct of mesurement tht forces the system to dopt specific vlue. mesurement forces system to dopt given vlue (corresponding to the the type of mesurement being done). Or equivlently, mesurement mkes the wvefunction to collpse into given sttionry stte, thus creting n ttribute on the system tht ws not there previously. Mesurements not only disturb wht is to be mesured, they produce it. We compel the system to ssume definite vlue of f. For exmple, two-electron system my be in the stte S / [ () () - () () ] (where one electron is flying in the opposite direction of the other). Upon using mgnetic field pprtus to mesure the spin of the prticles, one possible outcome is electron - in the stte () nd electron- in the stte (). Tht is, the mesurement hs creted these new sttes. Furthermore, ohr enuncited the principle of complementrity, which holds tht objects hve complementry properties tht cnnot ll be observed or mesured simultneously. iii) gnostic response: duck the question on the grounds tht it is methphysicl. There were so mny direct pplictions of the (mybe incomplete) quntum mechnics theory tht mny physicists left the conceptul foundtion interprettions side for the time being. These three views on the interprettion of the wvefunction were subject of controversil discussions. ut in 964 John ell stonished the physics community by showing tht it mkes n observble difference whether the prticle hd precise (though unknown) vlue of f prior to the mesurement, or not. 3 ell ws ble to ly down conditions tht ll deterministic locl theories must stisfy. It turns out those conditions re found to be violted by experiment. System with hidden vribles stisfy ell s inequlities Therefore, No ell s inequlities fulfilled non-existence of hidden vribles ell s discovery effectively eliminted gnosticism s vible option, nd mke it n experimentl question whether i) or ii) is the correct choice. Current experiments hve decisively confirmed the orthodox interprettion. system simply does not hve precise vle of f prior to mesurement. It is the mesurement process tht insists on one prticulr number, nd thereby in sense cretes the specific result, limited only by the sttisticl weighting imposed by the wvefunction. 4 Wht if we mde second mesurement, immeditely fter the first? Would we get f 4 gin? There is consensus tht the nswer is yes. Evidently the first mesurement rdiclly lters the wvefunction, so it is 4 right fter the mesurement. It is sid tht, upon mesurement, the wvefunction collpses to 4, nd then the ltter strts to evolve in ccordnce with the Schrodinger eqution.

4 = m c m Evolution with 4 m Hence, if the second mesurement is mde quickly, then it will render the sme vlue f 4. I. THE EPR Pper on the (lck of) Completeness nd Loclity of the Quntum Theory. Einstein, Podolsky, nd Rosen, Phys. Rev. 47, 777 (935). This pper is vilble online Einstein, Podolsky, nd Rosen questioned the completeness of the quntum mechnics theory. Here we highlight some of their sttements: It describes mesurements of two non-commuting vribles, position nd momentum The EPR pper emphsizes on the distinction between the objective relity (which should be independent of ny theory), nd the physicl concepts with which given theory opertes The concepts re intended to correspond with the objective relity, nd by mens of these concepts we picture this relity to ourselves. The EPR text is concerned with the logicl connections between two ssertions. Quntum mechnics is incomplete. Incomptible quntities (those whose opertors do not commute, like the x- coordinte of position nd liner momentum in direction x) cnnot hve simultneous relity (i.e., simultneously rel vlues). The uthors ssert tht one or nother of these must hold. Condition of completeness: Every element of the physicl relity must hve counter prt in the physicl theory. Criterion of relity If, without in ny wy disturbing system, we cn predict with certinty (i.e., with probbility equl to unity) the vlue of physicl quntity, then there exists n element of physicl relity corresponding to this physicl quntity. The Criterion of Relity implies tht quntity is definite if the stte of the system is n eigenstte for tht quntity. The elitity ssumption is tht physiclly mesurble quntities hve definite vlues before (nd whether or not) they re ctully mesured. In quntum mechnics, corresponding to ech physiclly observble quntity there is n opertor, which here will be designted by the sme letter. If is n eigenfunction of the opertor, =, where is number, then the physicl quntity hs with certinty the vlue whenever the prticle is in the stte given by. In ccordnce with our criterion of relity, for prticle in the stte given by for which = holds, there is n element of physicl relity corresponding to the physicl quntity.

5 If the opertors corresponding to two physicl quntities, sy nd, do not commute, tht is, if, then the precise knowledge of one of them precludes such knowledge of the other. Furthermore, ny ttempt to determine the ltter experimentlly will lter the stte of the system in such wy s to destroy the knowledge of the first. From this follows tht either, () the quntum mechnicl description of relity given by the wve function is not complete, or () Quntum mechnics is complete, but when the opertors nd corresponding to two physicl quntities do not commute the two quntities cnnot hve simultneous relity. Seprted systems s described by EPR hve definite position nd momentum vlues simultneously. Since this cnnot be inferred from ny stte vector, the quntum mechnicl description of systems by mens stte vectors is incomplete. I. The ohm experimentl version to settle the EPR prdox: Experiment with spin prticles fter fifteen yers following the EPR publiction, in 95 Dvid ohm published textbook on the quntum theory in which he took close look t EPR in order to develop response. ohm showed how one could mirror the conceptul sitution in the EPR thought-experiment, which ddressed mesurement of two non-commuting vribles position nd momentum, by considering insted the dissocition of ditomic molecule nd ddressing tht totl spin ngulr momentum is (nd remins) zero, nd the non-commuttive reltion of the spin components (S x nd S y, for exmple). In the ohm experiment the tomic frgments seprte fter interction, flying off in different directions freely. Subsequently, mesurements re mde of their spin components (which here tke the plce of position nd momentum), whose mesured vlues would be nti-prllel fter dissocition. In the so-clled singlet stte of the tomic pir (the stte fter dissocition) if one tom's spin is found to be positive with respect to the orienttion of n xis t right ngles to its flight pth, the other tom would be found to hve negtive spin with respect to n xis with the sme orienttion. Like the opertors for position nd momentum, spin opertors for different orienttions do not commute. Moreover, in the experiment outlined by ohm, the tomic frgments cn move fr prt from one nother nd so become pproprite objects for ssumptions tht restrict the effects of purely locl ctions. Thus ohm's experiment mirrors the entngled correltions in EPR for sptilly seprted systems, llowing for similr rguments nd conclusions involving loclity, seprbility, nd completeness. Insted of ditomic molecule, let s consider here the decy of neutrl pi meson into n electron nd positron ( similr process involving insted the decy of positronium into two gmm photons is described in more detiled in Section III below).

6 e - e Decy of neutrl pi meson Grphiclly e - e + Fig. Decy of neutrl pi meson ssuming the pion ws t rest, the electron nd positron trvel in opposite direction. The pion hs spin zero, hence conservtion of ngulr momentum requires tht the electron nd positron hve opposite spin in the pion s single stte: [ ( ) ) / ( - ) ( ) ( ] If the observer on the left mkes mesurement nd finds the electron to hve spin up (down), the positron must then hve spin down (up). Tht is, the mesurements re correlted. This occurs even if the observers re rbitrrily fr wy. Relistic explntion: The electron hd spin up nd the positron spin down from the moment they were creted Orthodox explntion: Neither prticle hd spin up or spin down until the ct of mesurement intervened. The mesurement on the electron side collpsed the wve function into stte ( ) ( ) ; i.e. s soon s the electron ws found to hve spin up, instntneously the positron dopted the spin down. This occurs no mtter how fr wy the electron nd positron re seprted. Such n instntneous doption of stte, upon mesurement mde fr wy, constitutes the most problemtic issue rised by the relistic school ginst the orthodox. The fundmentl ssumption on which the EPR rgument rests is tht no influence cn propgte fster thn the speed of light. We cll this the principle of loclity. Interesting point: One my be tempted to propose tht the collpse of the wvefunction is not instntneous, but trvels t some finite velocity. However, this would led to violtions of ngulr momentum conservtion, for if we mesured the spin of the positron before the news of the collpse (t the other side) hs reched us, there would be fifty-fifty probbility of finding both prticles with spin up. It turns out, experiments indicte tht no such violtion occurs; the ntiprllelism of the fr-prt prticles is perfect. Tht is, the collpse of the wvefunction is instntneous. 5 Loclity ffirms tht the rel stte of system is not ffected by distnt mesurements. Loclity supposes tht no rel chnge cn tke plce in one system s direct consequence of mesurement mde on the other system. 6

7 Insted of the sketchy description of the decy of neutrl pi meson, more thorough description is offered by Feynmn bout the nnihiltion of the positronium. This is presented in the next section. II. Illustrtion of n entnglement process: The nnihiltion of the positronium Ref: The Feynmn Lectures, Vol III, pge Positronium, tom mde up of n electron nd positron. It is bound stte of n e + nd n e, like hydrogen tom, except tht positron replces the proton. This object hs like the hydrogen tom mny sttes. Like the hydrogen, the ground stte is split into hyperfine structure by the interction of the mgnetic moments. The electron nd positron hve ech spin ½, nd they cn be either prllel or ntiprllel to ny given xis. Sttes re indicted by: (electron s spin, positron s spin) (In the ground stte there is no other ngulr momentum due to orbitl motion.) The sttes of compound systems (i.e. systems composed of more thn one prticle) re subjected lso to the conditions of symmetry conditions: symmetric or ntisymmetric. So there re four sttes possible: Three re the sub-sttes of spin-one system, ll with the sme energy; (+ ½, + ½) m = [ (+ ½, - ½) + (- ½, + ½) ] m = () (- ½, - ½) m = - nd one is stte of spin zero with different energy. [ (+ ½, - ½) - (- ½, + ½) ] m = () However, the positronium does not lst forever. The positron is the ntiprticle of the electron; they cn nnihilte ech other. The two prticles dispper completely converting their rest energy into rdition, which ppers s -rys (photons). In the disintegrtion, two prticles with finite rest mss go into two or more objects which hve zero rest mss.

8 Cse: Disintegrtion of the spin-zero stte of the positronium. It disintegrtes into two γ-rys with lifetime of bout seconds. The initil nd finl sttes re illustrted in Fig. 7 below. Initil stte: we hve positron nd n electron close together nd with spins ntiprllel, mking the positronium system. [ (+ ½, - ½) - (- ½, + ½) ] spin zero stte Finl stte: fter the disintegrtion there re two photons going out with equl mgnitude but opposite moment, (becuse the totl momentum fter the disintegrtion must be zero; we re tking the cse of the positronium being t rest). ngulr distribution of the outgoing photons Since the initil stte () hs spin zero, it hs no specil xis; therefore tht stte is symmetric under ll rottions. The finl stte (b) (constituted by photons) must then lso be symmetric under ll rottions. Tht mens tht ll ngles for the disintegrtion re eqully likely (3) The mplitude is the sme for photon to go in ny direction. Of course, once we find one of the photons in some direction the other must be opposite. Fig. 3. nnihiltion of positronium nd emission of two photons. We re interested in the polriztion stte of the outgoing photons.

9 Polriztion of the photons The only remining question is bout the polriztion of the (4) outgoing photons. In Fig. 3(b), let's cll the directions of motion of the two photons the plus nd minus Z-xes. See lso Fig. 5 below. Photon polriztion sttes: We cn use ny representtions we wnt for the polriztion sttes of the photons. We will choose for our description right nd left circulr polriztions. In the clssicl theory, right-hnd circulr polriztion hs equl components in x nd y which re 9 out of phse. Fig. 4. Clssicl picture of the electric field vector. In the quntum theory, right-hnd circulrly (RHC) polrized photon hs equl mplitudes to be x polrized or y polrized, nd the mplitudes re 9 out of phse. Similrly for left-hnd circulrly (LHC) polrized photon. R = L = [ x + i y ] RHC photon stte [ x - i y ] LHC photon stte (5) Cse-: Emitted photons in the RHC sttes If the photon going upwrd is RHC, then ngulr momentum will be conserved if the downwrd going photon is lso RHC. Ech photon will crry + unit of ngulr momentum with respect to its momentum direction, which mens plus nd minus one unit long the z-xis. The totl ngulr momentum will be zero. The ngulr momentum fter the disintegrtion will be the sme s before. See Fig. 5 below.

10 Fig. 5 Positronium nnihiltion long the z-xis. The finl stte is indicted s R R. Cse-: Emitted photons in the LHC sttes There is lso the possibility tht the two photons go in the LHC stte. Figure 6. nother possibility for positronium nnihiltion long the z-xis. The finl stte is indicted s L L.

11 Reltionship between the two decy modes mentioned bove Wht is the reltion between the mplitudes for these two possible decy modes? The nswer comes from using the principle of conservtion of prity. In tomic processes, prity is conserved, so the prity of the whole system must be the sme before nd fter the photon emission. The prity of stte, reltive to given opertor ction, is relted to whether, or F ~ = F ~ = - even prity odd prity efore the decy: Theoreticl physicists hve shown, in wy tht is not esy to explin, tht the spin-zero ground stte ( / ) [ (+ ½, - ½) - (- ½, + ½) ] of the positronium (e +, e ) is odd. fter the decy: Let's see then wht hppens if we mke n inversion of the process in Fig. 5. In the QM jrgon, we sy let s pply the opertor P ~ to the stte. Here P ~ stnds for the inversion opertor. 7 When we pply P ~ to the stte described in Fig. 5, we obtin Fig. 6 (this is illustrted in Fig. 7). (6) Direction of propgtion ngulr momentum pplying inversion procedure Fig. 7 In n inversion opertion ech prt of the system moves to n equivlent point on the opposite side of the origin. When we chnge x, y,z into x, y, z, polr vectors (like displcements nd velocities get reversed). xil vectors (like ngulr momentum derived from cross product of two polr vectors displcement velocity) hve the sme components fter n inversion. Let, R R stnd for the finl stte of Fig. 5 (7) in which both photons re RHC,

12 nd L L stnd for the finl stte of Fig. 6 (8) in which both photons re LHC. We notice tht n inversion of the photon stte in Fig. 5 results in n rrngement equl to the one in Fig. 6; nd vice vers. Tht is, P ~ R R = L L P ~ L L = R R (9) So neither the stte R R nor the stte L L conserve the prity condition stted in (6). So, how to build stte such tht P ~ = -? nswer, F = R R L L for n inversion chnges the R's into L's nd gives the stte P ~ F = P ~ ( R R L L ) = P ~ ( R R ) P ~ ( L L ) = L L R R = F () So the finl stte F of the emitted photons hs negtive prity, which is the sme prity the initil spin-zero stte of the positronium (e +, e ). ccordingly, Prity = - Prity = - [ (+ ½, - ½) - (- ½, + ½) ] F efore Figure 8. Decy of the spin-zero positronium stte. fter

13 Figure 9 shows more explicit picture of the ntisymmetric stte. () Prity = - Prity = - R L [ (+ ½, - ½) - (- ½, + ½) ] - R L R R L L F efore fter Figure 9. Decy of the spin-zero positronium stte. The stte F = R R L L is the only finl stte tht conserves both ngulr momentum nd prity. For normliztion purposes, let s define F = Exercise. For ech prticle, using R = [ R R L L ] () [ x + i y ] nd L = show tht R R = nd L R = Hint: ssume x x = y y = ; x y =. [ x - i y ] Exercise. Show tht R R F = nd L L F = -. Notice R R L L [ R L ] [ R L ] = - y y

14 nswer, < R R F = R R [ R R L L ] Similrly, = = R R F = L L F = - [ R R R R R R L L ] lthough we re working with two-prticle mplitudes for the two photons, we cn hndle them just s we did with the single prticle mplitudes. We men, the mplitude R R R R is just the product of the two independent mplitudes R R nd R R. [ R R R R R L R L ].. () (3) Exercise. Show tht nswer, R L F = (4) < R L F = R L [ R R L L ] = = [ R L R R R L L L ] [ R R L R R L L L ].. = The results in () nd (4), nmely R R F = / nd R L F =, suggests tht, If we set the polrizer- to detect the RHC stte, we will influence the results on other side to never detect LHC stte. Overll, 5% of the time polrizer- will detect RHC stte; from those 5% cses the other side will lwys detect RHC (the ltter is true becuse otherwise the probbility R R F wouldn t be equl to /.)

15 Prphrsing Feynmn If we observe the two photons in two detectors which cn be set to count seprtely the RHC or LHC photons, we will lwys see two RHC photons together, or two LHC photons together. Tht is, if you stnd on one side of the positronium nd someone else stnds on the opposite side, you cn mesure the polriztion (RHC or LHC) nd tell the other guy wht polriztion he/she will get. You hve 5-5 chnce of ctching RHC photon or LHC photon; whichever one you get, you cn predict tht the other will get the sme. Cse: Wht hppens if we observe the photon in counters tht ccept only linerly polrized light? Suppose tht i) you hve counter tht only ccepts light with x-polriztion, nd ii) tht there is guy on the other side tht lso looks for liner polrized light with, sy, y-polriztion. Wht is the chnce to pick up the two photons from n (e +, e ) nnihiltion? z x x y y efore fter Fig. Experimentl set up to observe the output stte x y. Clculte the mplitude tht F will collpse in the stte x y fter the mesurement. x y F =?

16 x y F = x y = = [ R R L L ] [ x y R R x y L L ] lthough we re working with two-prticle mplitudes for the two photons, we cn hndle them just s we did with the single prticle mplitudes. We men, the mplitude x y R R is just the product of the two independent mplitudes x R nd y R. [ x R y R x L y L ] Using R = [ x + i y ], L = x y F = [ x - i y ] [ x R y R x L y L ] i -i Similrly, x y F = i (5) x x F = (6) Similr to the conclusions rrived from () nd (4), this time (5) nd (6) suggests tht, If we set the polrizer- to detect x-polriztion, we will influence the results on other side to never detect x-polriztion. Overll, 5% of the time polrizer- will detect x-polriztion; from those 5% cses the other side will lwys detect y-polriztion (the ltter is true becuse otherwise x y F wouldn t be equl to /.) Exercise. Express the stte F in terms of the x nd y polriztions nswer: F = i [ y x + x y ] Exercise: Clculte the probbility to find photon- in the stte x regrdless of the polriztion of the photon- Hint: Clculte F x x F (Notice tht expression resembles x F ) nswer:

17 nswer: Exercise: Knowing tht photon- is in the stte x clculte the probbility tht photon- will be found in the stte y Hint: photon t x Prob regrdless of photon photon- t Prob knowing photon- y is t x = photon t y = Prob nd photon is t x nswer: Since F = i then Therefore, x y F = i [ y x + x y ], nd F x y x y F = - i i = photon t y F x x F Prob knowing photon is t x = F x y x y F Prob photon t knowing photon y is t x = Prob photon t knowing photon y is t x =. nswer (7) Tht is, knowing tht photon- hs collpsed into stte x, we cn mke deterministic prediction tht the photon- will be found in the stte y. Using polrized bem splitters ( let s mke the photons decide ) Now ll this leds to n interesting sitution. Suppose you were to set up something like piece of clcite which seprtes the photons into x -polrized nd y -polrized bems

18 You put counter in ech bem. Let's cll one the x -counter nd the other the y - counter. The guy on the other side does the sme thing. The results (5) nd (6) bove indicte tht, You cn lwys tell him which bem his photon is going to go into. Whenever you nd he get simultneous counts, you cn see which of your detectors cught the photon nd then tell him which of his counters hd photon. Let's sy tht in certin disintegrtion you find tht photon went into your x - counter; you cn tell him tht he must hve hd count in his y -counter. z y x x y fter y x Figure. Photon detection with polrized bem splitters t ech side. Now mny people who lern quntum mechnics in the usul (old-fshioned) wy find this disturbing. They would like to think tht, Once the photons re emitted it goes long s wve with definite chrcter. Since ny given photon hs some mplitude to be x-polrized or to be y-polrized, there should be some chnce of picking it up in either the x- or y-counter nd tht this chnce shouldn't depend on wht some other person finds out bout completely different photon.

19 Someone else mking mesurement shouldn't be ble to chnge the probbility tht I will find something. Our quntum mechnics sys, however, tht, by mking mesurement on photon number one, you cn predict precisely wht the polriztion of photon number two is going to be when it is detected. This point ws never ccepted by Einstein, nd he worried bout it gret del it becme known s the Einstein-Podolsky-Rosen prdox. ut when the sitution is described s we hve done it here, there doesn't seem to be ny prdox t ll; it comes out quite nturlly tht wht is mesured in one plce is correlted with wht is mesured somewhere else. The rgument tht the result is prdoxicl runs something like this (let s enuncite some sttements tht my be right or wrong; let s stte them just for the ske of helping to contrst wht quntum mechnics stnds): () If you hve counter which tells you whether your photon is RHC or LHC, you cn predict exctly wht kind of photon (RHC or LHC) he (on the other end) will find. () The photons he receives must, therefore, ech be purely RHC or purely LHC, some of one kind nd some of the other. (3) Surely you cnnot lter the physicl nture of his photons by chnging the kind of observtion you mke on your photons. No mtter wht mesurements you mke on yours, his must still be either RHC or LHC. (4) Now suppose he chnges his pprtus to split his photons into two linerly polrized bems with piece of clcite so tht ll of his photons go either into n x- polrized bem or into y-polrized bem. There is bsolutely no wy, ccording to quntum mechnics, to tell into which bem ny prticulr RHC photon will go. There is 5% probbility it will go into the x-bem nd 5% probbility it will go into the y-bem. nd the sme goes for LHC photon. (5) Since ech photon is RHC or LHC ccording to () nd (3) ech one must hve 5-5 chnce of going into the x-bem or the y-bem nd there is no wy to predict which wy it will go. (6) Yet the theory predicts tht if you see your photon go through n x-polrizer you cn predict with certinty tht his photon will go into his y-polrized bem. This is in contrdiction to (5) so there is prdox. My own comment-: The photons re lredy in globl stte, the stte F, which by the wy it is the only one possible. In mking one mesurement (on either side) the observer is just mking the stte F to collpse into the stte tht we re mesuring. Comment-: In Fig the photon tht rrives first then decides the outcome. Wht bout if both rrive simultneously? ut simultneity is reltive concept. Would different observers (trveling different speed on the side-) see different results?

20 Nture pprently doesn't see the prdox, however, becuse experiment shows tht the prediction in (6) is, in fct, true. In the rgument bove, Steps (), (), (4), nd (6) re ll correct, but Step (3), nd its consequence (5), re wrong; they re not true description of nture. rgument (3) sys tht by your mesurement (seeing RHC or LHC photon) you cnnot determine which of two lterntive events occurs for him (seeing RHC or LHC photon), nd tht even if you do not mke your mesurement you cn still sy tht his event will occur either by one lterntive or the other. ut this is not the wy Nture works. Her wy requires description in terms of interfering mplitudes, one mplitude for ech lterntive. mesurement of which lterntive ctully occurs destroys the interference, but if mesurement is not mde you cnnot still sy tht one lterntive or the other is still occurring. If you could determine for ech one of your photons whether it ws RHC nd LHC, nd lso whether it ws x-polrized (ll for the sme photon) there would indeed be prdox. ut you cnnot do tht it is n exmple of the uncertinty principle. Do you still think there is prdox? Mke sure tht it is, in fct, prdox bout the behvior of Nture, by setting up n imginry experiment for which the theory of quntum mechnics would predict inconsistent results vi two different rguments. Otherwise the prdox is only conflict between relity nd your feeling of wht relity ought to be. Do you think tht it is not prdox, but tht it is still very peculir? On tht we cn ll gree. It is wht mkes physics fscinting.

21 dditionl own comments efore fter () () F More precisely - R R L L F efore fter Fig.. The output stte F of the twin photons fter positronium decy. lthough the stte F is the only one tht fulfills the conservtion of prity, quntum mechnics llows us to clculte the mplitude probbility to obtin nother stte upon mking the system (the two outgoing photons) to interct with some (polrizers) pprtus. For exmple, we cn clculte the mplitude probbility x y F of detecting photon () in stte x nd photon () in stte y.

22 L () Polrizer R Polrizer () x Polrizer F Polrizer y () () Polrizer R () Polrizer L ) () Figure 3. Pictoril representtion tht fter the positronium decy the globl stte of the photons is F. Every ngulr outcome direction of the photons hs equl probbility to occur (see expression (3) bove). fterwrds different quntum sttes cn be obtined fter mking mesurement (in this cse using corresponding polrizers). Three different mesurements re indicted (with different color lines) in the figure. Notice, those output sttes re dopted only right fter the mesurement. So it does not mke sense to sk whether those sttes were creted t the time of the positronium decy, since t the time of the decy the output stte is F. The sttes shown in Fig. 3 re different possible sttes fter mesurement is performed on the two photons. Notice ech possible stte is composed of one photon going to one side nd nother going to the other side. different color is used in the figure to represent ech different entngled stte. One stte (depicted in green) shows RHC polriztion (on the left side) nd RHC polriztion (on the right side). nother stte shows y-polriztion (on the photon going left) nd x-polriztion (on the photon going right). Etc. If this experiment (the positroniun decy ) is repeted N times, with the detectors set to detect polriztion S on () nd polriztion T on (8) () then frction S T F of N will give such n expected result. (Correspondingly, similr for ny other specific stte) The following is not correct: Fig. 3 shows the pths vilble for given twin of photons, nd tht in prticulr single experiment ll they hve to do is to choose one vilble (9) pth.

23 Indeed, the bove cnnot be true. We know tht the output stte of the combined photons is F ; it is the only one stisfying ll the conservtion lws in positronium decy. Fig. 3 shows output sttes tht do not stisfy prity; i.e. they re not () llowed outcomes from decy. Insted, wht Fig. 3 shows re sttes obtined fter mking selected mesurement. Sttement () prevents us from ffirming, for exmple, the following: Once decy occurs ( single event) twin of photons cn go either pth red, or blue or green, ech with its own weight of mplitude probbilities. Tht ltter is not true, since ech pth shown with different colors in Fig. 3 violtes the prity conservtion. Insted, wht occurs is the following: The mesurement mde on one side forces the stte F to collpse on prticulr stte. The collpsing implies tht the other side collpses too () in the corresponding twin stte. In single observtion (i.e. just one positronium decy), if one of the observers detects the photon in the red stte, then he/she cn predict with certinty tht the observer on the other side will detect the photon on the red stte (even though they re fr prt in spce). This is becuse the the quntum stte is composed of entngled photon. If observer on the right side is set to observe only blue sttes, the the observer on the left will detect only the corresponding blue stte.

24 III. ELL S THEOREM Determinism of clssicl mechnics nd the imposition of hidden vribles in QM Clssicl mechnics is deterministic theory. In principle, nd prticulrly when deling with just few prticles, we expect to obtin explicitly (fter pplying Newton s lw) the position nd velocity of ech prticipting prticle t ny time t provided we know their corresponding vlues t t=. It is of course true tht in system composed of huge number of prticles (like when describing gs) the motion of the prticles cn be described only in sttisticl mnner. This clssicl indeterminism rises merely from our lck of detiled knowledge bout the position nd velocities of ech molecule t given time. If we knew those vlues (lthough this is prcticlly impossible) clssicl mechnics conceives, t lest in principle, tht the motion of ech prticle could be determined. Some schools ssume tht mybe such type of clssicl indeterminism occurs lso in quntum mechnics. Tht is, quntum mechnics is n incomplete theory mybe becuse there re other vribles, clled hidden vribles, of which we re not directly wre, but which re required to determine the system completely. These hidden vribles re postulted to behve in clssicl deterministic mnner. The pprent indeterminism exhibited by quntum system would rise from our lck of knowledge of the hidden sub-structure of the system. Thus, pprently identicl systems re perhps chrcterized by different vlues of one or more hidden vribles, which determine in some wy which prticulr eigenvlues re obtined in prticulr mesurement. 3 In the cse of the positronium decy (describe bove), for exmple, there might be clssicl hidden vrible, the vlue of which ws determined when the stte F ws creted nd which subsequently determined the experimentl results when the photons re nlyzed fr wy. Over the yers, number of hidden vrible theories hve been propossed; 4 they tend to be cumbersome, but until 964 they pper worth pursuing. ut in 964, J. S. ell proved tht ny locl hidden vrible theory is incomptible with quntum mechnics. 5 ell ws ble to ly down conditions tht ll deterministic locl theories must stisfy. It turns out those conditions re found to be violted by experiment. Therefore, System with hidden vribles stisfy ell s inequlities No ell s inequlities fulfilled non-existence of hidden vribles Following Griffith s book For the cse of, for exmple, the decy of neutrl pi meson e - e insted of orienting the electron nd positron detectors long the sme direction let them to be rotted independently. 3. H. rnsden & C. J. Jochin, Quntum mechnics, Prentice Hll (). 4 D. ohm, Phys. Rev. 85, 66, 8 (95) 5 J. S. ell, Physics, 95 (964).

25 The first detector mesures the electron spin in the direction of unit vector â. The second detector mesures the positron spin in the direction of unit vector bˆ. Fig. 4 ell s version of the EPR-ohm experiment: detectors independently oriented in directions â nd bˆ. Recording the dt in units of /, ech detector registers the vlue + (for spin up) or - (spin down), long the selected direction. tble of results might look like this, For given set â nd bˆ of detector orienttions, clculte the product verge P( â, bˆ ); P( â, bˆ ) the verge of the product of () the mesured spins When the detectors re prllel â = bˆ ; in this cse if one is spin up the other is spin down, so the product is lwys -; hence, P( â, â ) = () If â nd bˆ re ntiprllel ( bˆ =- â ), then every product is +; hence, P( â, - â ) = + (3) On one hnd, quntum mechnics predicts, P( â, bˆ ) = ˆ b ˆ (4) On the other hnd, ell discovered tht such quntum mechnics result is incomptible with ny locl hidden vrible theory. Demonstrtion: Suppose tht complete stte of the electron/positron system is chrcterized by the hidden vrible.

26 vries (in some wy tht we neither understnd nor control) from one pion decy to the next. Loclity ssumption: Suppose the outcome of the electron mesurement is independent of the positron detector, which my be chosen by the observer t the positron end just before the electron mesurement is mde, nd hence fr too lte for ny subliminl messge to get bck to the electron detector Upon one pion decy, let s ssume the hidden vrible hs cquired specific vlue. There exists then some function ( â, ) tht gives the result of n electron mesurement, nd some function ( bˆ, ) tht gives the result of n positron mesurement. These functions cn tke only the vlues + or -. ( â, ) (5) ( bˆ, ) When the detectors re ligned bˆ = â, the results re perfectly correlted, ( â, ) ( â, ), for ll (6) The verge of the product of series of mesurements is given by, P( â, bˆ ) () ( â, ) ( bˆ, ) d (7) where () is the probbility density for the hidden vrible, Using (6), is non negtive ( s ny other probbility density), nd stisfies () d. Different theories would give different expressions for (). ( bˆ, ) ( bˆ, ), expression (7) is then given by, P( â, bˆ ) () ( â, ) ( bˆ, ) d (8) If ĉ is nother unit vector P( â, ĉ ) () ( â, ) ( ĉ, ) d (9) In (9) we re using the sme () s in (8) since we re using the sme â nd we re ssuming the outcome of the electron mesurement is independent of the positron detector s orienttion (whether the ltter is bˆ or ĉ ). From (8) nd (9), P( â, bˆ ) P( â, ĉ ) () ( â, ) [ ( bˆ, ) ( ĉ, ) ] d Using ( bˆ, ) ( bˆ, )

27 P( â, bˆ ) P( â, ĉ ) () ( â, ) [ ( bˆ, ) ( bˆ, ) ( bˆ, ) ( ĉ, ) ] d () ( â, ) ( bˆ, ) [ ( bˆ, ) ( ĉ, ) ] d () - ( â, ) ( bˆ, ) [ ( bˆ, ) ( ĉ, ) ] P ( â, bˆ ) P ( â, ĉ ) () [ ( bˆ, ) ( ĉ, ) ] d () d () [ ( bˆ, ) ( ĉ, ) ] d () [ ( bˆ, ) ( ĉ, ) ] d using ( ĉ, ) ( ĉ, ) + () [ ( bˆ, ) ( ĉ, ) ] d + P ( bˆ, ĉ ) P ( â, bˆ ) P ( â, ĉ ) + P ( bˆ, ĉ ) (3) This is the fmous ell inequlity. It holds for ny hidden vrible theory since no ssumptions re mde bout the nture of the hidden vrible.

28 IV. QUNTUM TELEPORTTION References:. Chrles H. ennett, Gilles rssrd, Clude Crépeu, Richrd Jozs, sher Peres, nd Willim K. Wootters. Teleporting n unknown quntum stte vi dul clssicl nd Einstein-Podolsky-Rosen chnnels. Phys. Rev. Lett. 7, 895 (993).. D. ouwmeester, J. Pn, K. Mttle, M. Eibl, H. Weinfurter nd. Zeilinger. Experimentl quntum teleporttion. Phil. Trns. R. Soc. Lond. 356, (998). 3. Chrles H. ennett. Quntum Informtion nd Computtion. Physics Tody 48, 4-3 (October 995). 4. Mrk eck, Quntum Mechnics, Theory nd Experiments, Oxford University Press (). 5. EPR Prdox Timeline 6. Yoon-Ho Kim, S. P. Kulik, nd Y. Shih. Quntum Teleporttion with Complete ell Stte Mesurement. Journl of Modern Optics 49, -36 (). 7. C. K. Hong, Z. Y. Ou, nd L. Mndel. Mesurement of subpicosecond time intervls between two photons by interference. Phys. Rev. Lett. 59, 44 (987). Hong-Ou- Mndel(HOM) interference [] between independent photon sources (HOMI-IPS) is t the hert of quntum informtion processing involving the quntum interference of single photons. Fig. 5 Principle of quntum teleporttion proposed by ennett [Ref nd 3]. Fundmentls. Entnglement between quntum systems is pure quntum effect describing correltions between systems tht re much stronger nd richer thn ny clssicl correltions cn be. Originlly this property ws introduced by Einstein, Podolsky nd Rosen, nd by Schrodinger nd ohr, in the discussion on the completeness of quntum mechnics nd by von Neumnn in his description of the mesurement process. Since then entnglement hs been seen s just one of the fetures which mkes quntum mechnics so counterintuitive. pplictions: However, recently the new field of quntum informtion theory hs shown the tremendous importnce of quntum entnglement lso for the formultion of new methods of informtion trnsfer nd for lgorithms exploiting the cpbility of quntum computers. (See Fig. 5).

29 Opticl implementtion: While quntum computers need entnglement between number of quntum systems, bsic quntum communiction schemes only rely on entnglement between the members of pir of prticles, directly pointing t possible reliztion of such schemes by mens of correlted photon pirs s produced by opticl prmetric down-conversion processes (see Fig. 6). Fig. 6 Implementtion of quntum teleporttion vi opticl mens [Ref ]. More specific ppers, ddressing the different spects of the teleporttion strtegy shown in the figure bove, re provided in Projects section of this course. Those ppers re intended to serve the seed topics for your project ssignment. You cn choose the one tht fits better your quntum curiosity nd interest.

30 Fundmentl Properties of Quntum Systems Superposition: quntum computer cn exist in n rbitrry complex liner combintion of sttes, which evolve in prllel ccording to unitry trnsformtion. Interference: prllel computtion pths in the superposition (like pths of prticle through n interferometer) cn reinforce or cncel one nother, depending on their reltive phse. Nonclonbility nd uncertinty: n unknown quntum stte cnnot be ccurtely copied (cloned) nor cn it be observed without being disturbed. QUITS 8 Clssicl informtion cn be represented s binry bits: s nd s. ll computer informtion is stored nd processed s bits. In quntum mechnics ny two orthogonl sttes cn be used to encode bits. For exmple, the polriztion stte H could signify, while V could signify. bit of informtion stored in this mnner is known s qubit (i.e. quntum-bit) t first, it my seem tht there is little difference between clssicl bit nd qbit. ut clssicl bit, t ny instnt in time, cn represent either or, but no both. However, quntum systems cn exist in superposition sttes. For exmple, the linerly polriztion stte 45 (/ ) H V superposition of the sttes H nd V. is qubit in this stte signifies both nd ; property referred to s quntum prllelism. (Quntum prllelism cn give quntum informtion processing n dvntge over clssicl informtion processing) Pir of Qubits ( two-photon system) Consider the spontneous prmetric down-conversion process, shown in Fig. 3. In this process single photon from pump lser incident in crystl is split into two photons, clled the signl nd the idler inside the crystl. The signl nd the idler emerge from the crystl t essentilly the sme time. For Type-I down-conversion the two down-converted photons hve the sme polriztion, nd orthogonl to tht of the pump. s p i Fig. 7 Type-I spontneous prmetric down-conversion. Polriztions of the signl nd idler photons re orthogonl to tht of the pump.

31 In order to describe the polriztion stte of the two-photons system, the polriztion of ech photon must be specified (when possible). For the cse shown in the Fig. 7 the polriztion stte is, H, H H s H i Product stte (T) If one plces hlf-wve polrizers long the photons propgtion direction, different twophoton polriztion sttes cn result, H s H i, s V i H, V s H i, V s V i Product sttes (T) Entngled sttes Fig. 8 shows vrition with respect to the one in Fig. 7. This time, there re two crystls sndwiched together, with their orienttion rotted by 9 o with respect to ech other. One crystl converts verticlly (horizontlly) polrized pump photons into horizontlly (verticlly) polrized signl nd idler photons. If the pump is polrized t 45 o, ech of these processes is eqully likely. s p polrized light Fig. 8 Setting to produce entngle sttes, s the one in Eq. (3). i If the crystls re thin enough, observers detecting the signl nd idler photons hve no informtion bout which crystl given photon ws produced in. (ctully photons with given polriztion re emitted in conicl region of some thickness. It turn out the cones corresponding to ech polriztion intersect in some regions. Hence, photon contined in tht intersected regions hve polriztion tht cnnot be distinguished. ) If the photons re indistinguishble, their polriztion stte is superposition of the two possible sttes generted by the down-conversion process. One possible stte of the two photon system in Fig. 8 is, si ( H s H i V s V i ) (T3) Here the two photons re in the the sme time. (Not s mening tht they re in one stte H s H i stte nd in the s V i V stte t H s H i or the other V s V i ).

32 Notice si cnnot be expressed s the product of (stte of the signl photon) (stte of the idler photon) Sttes of the combined system tht cnnot be written s single product the product of sttes of the individul prticles re known s entngled sttes. EITS Let s cll H, nd V Under certin experimentl conditions (see prgrph below Fig. 8) lice cn hve qubit, nd ob cn hve qubit in such wy tht neither qubit by itself hs definite stte of the type,,. (?) (?) Qubit is not in individul Qubit is not in individul,,, or else,,, or else? p polrized light Fig. 9 lice nd ob hve photons in such quntum sttes tht the globl stte is not product stte like (stte-) (stte-) Wht we men is tht experimentl conditions hve been crete so tht tht pir cn exists in n entngled stte. n exmple of one possible such entngled stte is, ( ) Entngled two-qubit?

33 p polrized light ( ) Fig. Production of two-qubit entngled stte. There re four two-qubit entngled sttes tht we will find useful, ( ( ) ) (T4) The no-cloning theorem TELEPORTTION It is possible to trnsmit n unknown quntum stte with perfect fidelity if the sender nd the receiver hve t their disposl two resources: - The bility to send clssicl messges, nd - entnglement of qubits between the sender nd the receiver X Following Ref. 4 For ech of the three qubits we will hve,

34 The digrm bove indictes tht lice nd ob hve qubit, nd tht these qubits re entngled. There re four two-qubit entngled sttes tht re possible: They re known s the ell sttes. (T5) The ell sttes form bsis, i.e. ny two-qubit stte cn be expressed s liner combintion of the four ell sttes. pir of entngled qubits, shred by seprte prties, is known s ebit. t the beginning: lice hs qubit hs qubit in stte, The stte lice wnts to teleport (T6) lice nd ob hve shred ebit in stte, (T7)

35 The totl stte of the system is then, (T8) which cn be written s, - (T9) Proof:

36 dding nd subtrcting terms (b) From, one obtins + = nd - = (c) (c) in (b) - (d) From, one obtins + = nd - = (e)

37 (e) in (d) - -

38 - Proven If lice performs mesurement nd let s ssume she finds her qubits re in the stte,, then, ccording to (5), ob s qubit is projected into the stte, which is the desired teleported stte. Thus, by simply performing ell mesurement, lice successfully teleport the stte to ob, with no further ction on ob s prt. If lice s msurement projects her qubits into the stte, then, ccording to (5), ob s qubit is projected into the stte. Subsequently, ob cn trnsform this stte into the desired stte by pplying shift to the component of his qubit. For tht purpose, he uses device ( polrizer) whose mtrix representtion is,

39 ~ T For, ~ T Thus, upon receiving informtion from lice (tht her qubits hve collpsed to stte ), ob knows which trnsformtion to pply to his entngled qubit in order to obtined the desired stte. In short, is is possible to teleport n unknown quntum stte by mking it to become prt of sttionry stte of compound system. The quntum teleporttion is ctully done through the lice-ob entngled system. mzingly, mesurement on mde by lice ffects the entngled system own by ob. Once lice tell ob wht to do, then ob cn perform the pproprite mesurement to ttin (construct)

40 Nture, Vol 55, pp. 4 5 (3 September 5) Quntum spookiness psses toughest test yet Experiment plugs loopholes in previous demonstrtions of 'ction t distnce', ginst Einstein's objections nd could mke dt encryption sfer. y Zeey Merli John ell devised test to show tht nture does not 'hide vribles' s Einstein hd proposed. Physicists hve now conducted virtully unssilble version of ell's test. efore investing too much ngst or money, one wnts to be sure tht ell correltions relly exist. s of now, there re no loophole-free ell experiments. Experiments in 98 by tem led by French physicist lin spect, using well-seprted detectors with settings chnged just before the photons were detected, suffered from n efficiency loophole in tht most of the photons were not detected. This llows the experimentl correltions to be reproduced by (dmittedly, very contrived) locl hidden vrible theories. In 3, this loophole ws closed in photon-pir experiments using high-efficiency detectors 7, 8. ut they lcked lrge seprtions nd fst switching of the settings, opening the seprtion loophole : informtion bout the detector setting for one photon could hve propgted, t light speed, to the other detector, nd ffected its outcome.

41 Entnglement independent of polriztion It is now well known tht the photons produced vi the down-conversion process shre nonclssicl correltions. In prticulr, when pump photon splits into two dughter photons, conservtion of energy nd momentum led to entnglements in these two continuous degrees of freedom. 6 IV. ENTNGLEMENT from INDEPENDENT PRTICLE SOURCES Ref: ernrd Yurke nd Dvid Stoler. Einstein-Podolsky-Rosen Effects from Independent Prticle Sources. Physicl Rev. Lett. 68, 5 (99). lthough considerble discussion bout the completeness of quntum mechnics followed the publiction in 935 of the EPR pper, progress in reveling the true extent of the incomptibility of our ingrined notions of locl relism with quntum mechnics ws slow. It ws not until ell's work published in 965 tht it ws relized tht the issue could be rigorously formulted nd put to experimentl test. n even more provoctive demonstrtion of the incomptibility of quntum mechnics with locl relism ws recently discovered by Greenberger, Horne, nd Zeilinger (GHZ) in gednken experiment employing more thn two prticles. 7 In typicl of EPR experiments proposed to dte, one hs metstble system which decys into n prticles which re in n entngled stte. Ech prticle is delivered to its respective detector, consisting usully of polriztion nlyzer nd pir of prticle counters. Upon performing the pproprite set of experiments stronger correltions re found mong the firing ptterns of the prticle counters thn llowed by locl relism. Indeed, entnglement between two or more prticles ws generlly viewed s consequence of the fct tht the prticles involved did originte from the sme source or t lest hd intercted t some erlier time. However, it hs been suggested in seminl ppers by Yurke nd Stoler tht, the correltion of prticle detection events required for ell test cn even rise for photons, or ny kind of prticle for tht mtter, originting from independent sources. This triggered the interesting possibility: If we cn observe violtions of ell s inequlity for registrtion of prticles coming from independent sources, cn we lso entngle them in nondestructive mnner? Is this possible for prticles tht do not interct t ll nd tht shre no common pst? 6 J. G. Rrity nd P. R. Tpster. Experimentl Violtion of ell's Inequlity sed on Phse nd Momentum. Phys. Rev. Lett. 64, 495 (99). 7 Dniel M. Greenberger, Michel. Horne, bner Shimony, nd nton Zeilinger. ell s theorem without inequlities. mericn Journl of Physics 58, 3 (99).

42 This possibilities nd reliztion were ddressed in the following ppers: ernrd Yurke nd Dvid Stoler. Einstein-Podolsky-Rosen Effects from Independent Prticle Sources. Physicl Rev. Lett. 68, 5 (99). M. Zukowski,. Zeilinger, M.. Horne, nd. K. Ekert. Event-Redy-Detectors" ell Experiment vi Entnglement Swpping. Physicl Review Letters 7, 487 (993). H. WEINFURTER, Experimentl ell-stte nlysis. Europhys. Lett., 5 (8), pp (994). M. Zukowski,. Zeilinger, H. Weinfurter, Entngling Photons Rdited by Independent Pulse Sources. nnls of the New York cdemy of Sciences 755, 9 (995). C. K. Hong, Z. Y. Ou, nd L. Mndel. Mesurement of subpicosecond time intervls between two photons by interference. Phys. Rev. Lett. 59, 44 (987). This pper is considered to be t the hert of quntum informtion processing involving the quntum interference of single photons. y Dik ouwmeester, Jin-Wei Pn, Klus Mttle, Mnfred Eibl, Hrld Weinfurter nd nton Zeilinger. Experimentl quntum teleporttion. Phil. Trns. R. Soc. Lond. 356, 733{737 (998). The positive nswers cn be summrized s follows: The conditions for obtining entngled sttes require specific (not immeditely intuitive) choices of coincidence timing tht enble one to monitor the emission events of the independent sources s well s to erse the Welcher-Weg informtion. Complementry informtion: Multiple prticle interference

43 PPENDIX- Tensor Product of Stte Spces Consider the stte spce comprised of wvefunctions describing the sttes of given system (n electron, for exmple). We use the index- to differentite it from the stte spce comprised of wvefunctions describing the stte of nother system (nother electron, for exmple) which is initilly locted fr wy from the system. The systems (the electrons) my eventully get closer, interct, nd then go fr wy gin. How to describe the spce stte of the globl system? The concept of tensor product is introduced to llows such description Let { u (), i,, 3,...} be bsis in the spce ε, nd i { vi (), i,, 3,...} be bsis in the spce The tensor product of of ε nd ε, denoted by ε ε ε, is defined s spce whose bsis is formed by elements of the type, { u i () v () j Tht is, n element of ε ε is liner combintion of the form, } ε i,j c ij u i () v j () The tensor product is defined with the following properties, [ () ] () [ () () ]

44 () [ () ] [ () () ] [ () () ] () [ () () ] + [ () () ] ] [ () () [ () () () ] + [ () () ] Two importnt cses will rise. Cse : The wvefunction is the tensor product of the type, () () Tht is cn be expressed s the tensor product of stte from ε nd stte from ε. i i u i () b j v j () j i,j b i j ui () v j () Cse : The wvefunction cnnot be expressed s the tensor product between stte purely from the spce ε nd stte purely from the spce ε. In this cse, the wvefuntion tkes the form, i, j c i j ui () v j () Let s consider, for exmple the cse in which ech spce ε nd ε hs dimension. i, j c i j ui () v j () i c i u v() ci u v() i() i () i c u () v() c u () v() c u() v() c u() v()

45 Notice, cnnot be expressed in the form () () To mke it even simpler, let s ssume c = c =, c u () v () + c u () v () cnnot be expressed in the form of single term of the form () ().

46 PPENDIX- THE EPR Pper on the (lck of) Completeness nd Loclity of the Quntum Theory. Einstein, Podolsky, nd Rosen, Phys. Rev. 47, 777 (935). On the completeness Einstein, Podolsky, nd Rosen questioned the completeness of the quntum mechnics theory. Here we bullet the min points in the pper: The EPR pper emphsizes on the distinction between the objective relity (which should be independent of ny theory) nd the physicl concepts with which given theory opertes The concepts re intended to correspond with the objective relity, nd by mens of these concepts we picture this relity to ourselves. Condition of completeness: Every element of the physicl relity must hve counter prt in the physicl theory. Criterion of relity If, without in ny wy disturbing system, we cn predict with certinty (i.e., with probbility equl to unity) the vlue of physicl quntity, then there exists n element of physicl relity corresponding lo this physicl quntity. In quntum mechnics, corresponding to ech physiclly observble quntity there is n opertor, which my be designted by the sme letter. If is n eigenfunction of the opertor, =, where is number, then the physicl quntity hs with certinty the vlue whenever the prticle is in the stte given by. In ccordnce with our criterion of relity, for prticle in the stte given by for which = holds, there is n element of physicl relity corresponding to the physicl quntity. If the opertors corresponding to two physicl quntities, sy nd, do not commute, tht is, if, then the precise knowledge of one of them precludes such knowledge of the other. Furthermore, ny ttempt to determine the ltter experimentlly will lter the stte of the system in such wy s to destroy the knowledge of the first. From this follows tht either

47 () the quntum mechnicl description of relity given by the wve function is not complete, or () when the opertors nd corresponding to two physicl quntities do not commute the two quntities cnnot hve simultneous relity. (For if both of them hd simultneous relity nd thus definite vlues these vlues would enter into the complete description, ccording to the condition of completeness. If the wve function provided such complete description of relity, it would contin these vlues; these would be predictble. Our comment: Nowdys, we tend to ccept the uncertinty principle. We hve therefore to disgree with the criterion of relity estblished in the EPR pper. It s not tht given system hs to hve some definite vlues of given property; insted severl outcomes re possible depending on how we mke the mesurement. On the loclity Let us suppose tht we hve two systems, I nd II, which we permit to interct from the time t = to t =T, fter which time we suppose tht there is no longer ny interction between the two prts. We suppose further tht the sttes of the two systems before t= were known. We cn then clculte with the help of Schrodinger's eqution the stte of the combined system I + II t ny subsequent time. Let us designte the corresponding wve function by. ccording to QM, to find out the stte of the individul systems t t>t we hve to mke some mesurements. Let,, 3, be the eigenvlues of some physicl quntity pertining to system nd u (x ), u (x ), u 3 (x ), the corresponding eigenfunctions, where x stnds for the vribles used to describe the first system. Then cn be expressed s, where x stnds for the vribles used to describe the second system. Here ( x ) re to be regrded merely s the coefficients of the expnsion of into series of orthogonl functions u ( x ). Suppose now tht the quntity is mesured nd it is found tht it hs the vlue k. It is then concluded tht fter the mesurement the first system is left in the stte given by the

48 wve function u k (x ), nd tht the second system is left in the stte given by the wve function k ( x ) So the wve pcket hs been reduced to the term k ( x ) u k (x ) If, insted we hd chosen nother quntity, sy, hving the eigenvlues b, b, b 3, nd eigenfunctions v (x ), v (x ), v 3 (x ), we should hve obtined, insted of Eq. (7), the expnsion, If the quntity is now mesured nd is found to hve the vlue b r we conclude tht fter the mesurement the first system is left in the stte given by v r (x ) nd the second system is left in the stte given by r (x ), r ( x ) u r (x ) Thus, by mesuring either or we re in position to predict with certinty, nd without in ny wy disturbing the second system, either the vlue of the quntity (tht is k ) or the vlue of the quntity (tht is b r ). We see therefore tht, s consequence of two different mesurements performed upon the first system, the second system my be left in sttes with two different wve functions. Thus, it is possible to ssign two different wve functions (in our exmple k nd r ) to the sme relity (the second system fter the interction with the first). This mkes the relity of nd depend upon the process of mesurement crried out on the first system, which does not disturb the second system in ny wy (since no interction ws ssumed). No resonble definition of relity could be expected to permit this. Our comments. y ssuming tht quntum mechnics is complete theory ble to describe system composed of two seprted sub-systems () nd () (which initilly intercted but, now, being fr wy, they re ssumed tht cnnot interct), the EPR pper rrives to the following conclusions: i) two different wve functions describe the sme relity, ii) quntum mechnics predict different outcomes from mesurements on system () depending on wht type of mesurement is mde on system (), even though the two subsystems re not intercting. The ltter constitutes then prdox, ccording to EPR. It is reveling tht ll the objections of the EPR pper to the quntum theory could be surpssed if the interction t distnce were ccepted. Quntum theory is indeed non-locl theory. On the other hnd, tht there re different relities ssocited to system (i.e. different outcomes from mesurement) is n rgument tht we hve lerned to ccept s prt of quntum behvior.

49 ppendix

50 ppendix Mch Zender interferometer (Simultion experiment online) %Mch%Zehnder-PhseShifter%V7%REV%-%Copy.swf