On Bell measurements for teleportation N. Lu¨tkenhaus, J. Calsamiglia, and K-A. Suominen Helsinki Institute of Physics, PL 9, FIN-00014 Helsingin yliopisto, Finland (February 9, 2008) In this paper we investigate the possibility to make complete Bell measurements on a product Hilbert space of two two-level bosonic systems. We restrict our tools to linear elements, like beam splittersandphaseshifters,delaylinesandelectronically switchedlinearelements,photo-detectors, andauxiliary bosons. Asaresult weshowthatwiththesetools aneverfailing Bell measurementis impossible. 9 9 I. INTRODUCTION in the case of teleportation there have been two recent 9 experimentalrealizations[10,11]. Boschietal. presented 1 results in which Bell measurement is realized with 100 n Bell measurements project states of two two-levelsys- %efficiencyusinglinearopticalgates,buttheteleported a tems onto the complete set of orthogonal maximally en- state has to be prepared beforehand over one of the en- J tangledstates(Bell states). The motivationto dealwith tangled photons [10] . So, in some sense that scheme 8 Bellstatescomesfromthefactthattheyarekeyingredi- differs fromthe “genuine”teleportationsince itdoes not 2 ents in quantum information. Bell states provide quan- havesomeverycrucialproperties,liketheabilitytotele- 2 tum correlations which can be used in certain striking port entangled states or mixed states. This obstacle v applications such as: teleportation in which a quantum could,ofcourse,beovercomeifonehadthepossibilityto 3 stateistransferredfromoneparticletoanotherina“dis- swap the unknown state to the EPR photon. But, this 6 embodied” way [1], quantum dense coding in which two again requires quantum-quantum interaction (not linear 0 bits of informationcanbe communicatedby only encod- operator). On the other hand the Innsbruck experiment 9 ingasingletwo-levelsystem[2],andentanglementswap- canbeconsideredasa“genuine”teleportationbutithas 0 ping [4,5] which allows to entangle two particles that do the important drawback that it only succeeds in 50 % 8 9 not have any common past, and opens a source full of of the cases (in the remaining cases the original state / new applications since it provides a simple way of cre- is destroyed). For the same reason the Innsbruck dense h ating multiparticle entanglement [6,7]. But to take full coding experiment [12] can only reach a communication p advantage of these applications one needs to be able to rate of1.58 bits per photon insteadof 2 bits per photon. - t prepare and measure Bell states. The problem of creat- Recently, Kwiat and Weinfurter [13] have presented a n ingBellstateshasbeensolvedinopticalimplementations method which allows complete Bell measurements and a u byusingparametricdownconversioninanon-linearcrys- that operates on the product Hilbert spaces of two sys- q tal [8]. Particular Bell states can be prepared from any tems,butitaddsaveryrestrictiverequirementtoo. That : maximally entangled pair by simple local unitary trans- is, the particles need to be entangled in some other de- v i formations. The question arises whether it is possible to greeof freedombeforehand(so,half ofthe jobis already X performacompleteBellmeasurementwithlineardevices done). Notwithstanding, this method still represents an r (like beam-splitters and phase shifters). It is clear that importantprogresssinceitallows,inprinciple,torealize a thiscanbeachievedonceonehastheabilitytoperforma all applications which fulfill the condition that the Bell controlled NOT operation (CNOT) on the two systems, measurement is performed over photons which have al- which transforms the four Bell states into four disentan- ready quantum correlations (like in the case of quantum gled basis states. In principle we need to do less. As dense coding). we are not interested in the state of the system after the AtthisstagewechoosetocallaphysicalschemeaBell measurement, it can be vandalized by the measurement. analyzer only if it operates on product Hilbert spaces of Theonlyimportantthingisthemeasurementresultiden- two two-level systems. A generalization to systems with tifying unambiguously a Bell state. other structure than a two-level system is the measure- In an earlier papers Cerf, Adami, and Kwiat [9] have ment used in the teleportation of continuous variables shown that it is possible to implement quantum logic in [14] which successfully projects on singlet states. purelylinearopticalsystems. Theseoperations,however, In this paper we prove that all these turnabouts are donotoperateonaproductofHilbertspacesoftwosys- more than justified since we present a no-go-theorem tems, instead they operate on product of Hilbert spaces for Bell analyzer for experimentally accessible measure- of two degrees of freedom (polarization and momentum) ments involving only linear quantum elements. We now of the same system. Therefore these results can be used lay out the framework for this theorem in a language to implement quantum logic circuits but not to perform which clearly has the experimental situation of the tele- most of the applications mentioned above. For example, portation experiment performed in Innsbruck in mind. 1 Thismeansespeciallythatweconcentrateonbosonicin- 1 2 3 4 put states. Results concerning fermionic input or input of distinguishable particles can be found in the work of Vaidman [15]. The Hilbert space of the input states is spanned by states describing two photons coming into the measure- PB PB ment from two different spatial directions, each carry- ing two polarization modes. Therefore we can describe the input states in a sub-space of the excitations of four modeswithphotoncreationoperatorsa†,a†,b†,b†. Here 1 2 1 2 a and b refers to spatial modes, while “1” and “2” refer BS to polarization modes. The Hilbert space of interest is spanned by the orthonormalset of Bell states given by, a a b b 1 2 1 2 FIG. 1. The Innsbruck detection scheme uses an initial 50/50 beam-splitter (BS) mixing modes a† with b† and a† 1 1 2 Ψ = 1 a†b† a†b† 0 (1) withb†2. Theneachoftheresultingoutputsisseparatedfrom | 1i √2(cid:16) 1 2− 2 1(cid:17)| i each otherusing a polarizing beam-splitter (PB). 1 Ψ = a†b† +a†b† 0 (2) | 2i √2(cid:16) 1 2 2 1(cid:17)| i To illustrate the formalism we look at the Innsbruck detection scheme [17](fig. I), whichconsists of 8 POVM 1 Ψ = a†b† a†b† 0 (3) elements, corresponding to the events | 3i √2(cid:16) 1 1− 2 2(cid:17)| i Ψ = 1 a†b† +a†b† 0 (4) detectors going “click” could have been | 4i √2(cid:16) 1 1 2 2(cid:17)| i triggered by “1” and “4” Ψ 1 “2” and “3” Ψ 1 “1” and “2” Ψ 2 “3” and “4” Ψ 2 “1” sees 2 photons Ψ or Ψ 3 4 “2” sees 2 photons Ψ or Ψ 3 4 where 0 describes the vacuum state. Although we used “3” sees 2 photons Ψ or Ψ 3 4 | i spatial modes and polarization to motivate this form of “4” sees 2 photons Ψ or Ψ 3 4 Bell states, it should be noted that any two pairs of bosonic creation operators (all four commuting) can be Only the first four events allow assigning unambigu- chosen for the theorem to be valid. This includes all ously Bell states to the outcomes. The total fraction of possible degrees of freedom of the boson. In the pho- these events for teleportation, where all Bell states are ton case it includes especially polarization, time, spa- equally probable, is 50%. The state demolishing pro- tial mode and frequency. For example, all wave packets jection on entangled states is indeed possible using only containing one photon can be modeled. The Bell mea- linear elements, but not 100 % efficient. surement we are looking for is described by a positive operator valued measure (POVM) [16] given by a col- olepcetrioantoorfFposciotirvreesoppoenrdastotros FonkewciltahssPicakllFykd=ist11in.gEuiaschh- II. DESCRIPMTEIOANSUORFETMHEENTCSONSIDERED k able measurement outcome, for example that detectors “1” and “2”out of four detectors go “click”and the rest Before we continue we shall describe our tools more do not. The probability p for the outcome k to occur precisely. We restrict our measurement apparatus to k whilethe inputisbeingdescribedbydensitymatrixρ,is linear elements only. This means that the vector of givenbyp =Tr(ρF ). ABellmeasurementwith100% creation operators of the input modes is mapped by a k k efficiencyischaracterizedbythe propertythatallF are unitary matrix onto the vector of creation operators of k triggeredwithprobabilityTr(ρ F )=0foronlyoneof the output modes. Reck et al. [18] have shown that all Ψi k 6 the four Bell state inputs ρ (i=1,...,4). This allows these unitary mappings can be realized using only beam Ψi us to rephrase the problem as one of distinguishing be- splitters and phase shifters. The number of modes is tween four orthogonal equally probable Bell states with not necessarily four: we can couple to more modes us- 100 % efficiency. ing beam-splitters so that the input states are described 2 by the direct product of the Hilbert space of the Bell surements are allowedneither. Both tools might be very states and the initial state of the additional modes. All usefulandwedonotseeanyessentialreasontodisregard thosemodesaremappedintooutputmodes,whereplace them. For instance the apparatus proposed by Vaidman detectors. We assume these detectors to be ideal, so and Yoran can not distinguish between the four disen- thattheyaredescribedasperformingaPOVMmeasure- tangled basis-states of the from, mentonthemonitoredmodewhereeachPOVMelement F(detector) = k k is the projectionontoa Fock stateof k | ih | , , , that mode. For experimental reasons, one would like to |↑i|←i |↑i|→i |↓i|↑i |↓i|↓i reducethistoasimplerdetectorthatcannotdistinguish thenumberofphotonsbywhichitistriggered. Thesim- for which a conditional measurement is needed. ple“click”or“noclick”detectorisdescribedbyaPOVM withtwoelements, 0 0 and ∞ k k. However,we | ih | Pk=1| ih | will show that even a more fancier detector does not al- lowustoimplementaBellmeasurementthatneverfails. III. CRITICISM TO A PRIORI ARGUMENTS The last tool introduced here is the ability to perform AGAINST LINEAR BELL MEASUREMENTS conditional measurements. With that we mean that we monitoroneselectedmodewhilekeepingtheothermodes inawaitingloop. Thenwecanperformsomelinearoper- ationontheremainingmodesdependingontheoutcome Intuitively, one needs to operate a “non-linear” mea- of the measurement with all the tools described above. suring device to perform Bell measurements in the sense The general strategy is shown schematically in figure 2. that one two-level has to interact with the other. In the case of photons there is no direct interaction between them. Onecantrytocouplethemthroughathirdsystem suchasanatom[19]ormapthestateofthephotonsinto atomorionstatesandperformtherethedesiredmeasure- ment [20]. These schemes are closest to the simple idea ofperformingaCNOToperation,aHadamardtransform and than projecting on the disentangled base, but they bring up a whole new range of problems (e.g. weak cou- pling, decoherence, pulse shape design) that breaks with the idea of having simple and controlled “table-top” op- tical implementations of Quantum Information applica- tions. Therefore it is worth checking the possibility of performing it by linear means. U It is true that linear operations can not make the two input photons interact, they can only make them inter- fere. Therefore the unitary transformation U is sepa- L rable in the sense that it can be written in terms of a a a c c unitary operation U over each photon, and of course a 1 2 1 ..... D-4 CNOTcannotbeperformedbythesemeans(U =U U b b c L ⊗ 1 2 2 acts on the symmetric subspace of the single photon Hilbert space product , dim(U) > 2 ). Even 1 1 H ⊗ H if this kind of operation preserves the entanglement, the Hilbert space might be large enough to span outputs FIG.2. The general scheme mixes the modes of the Bell which trigger different combination of detectors for dif- state with auxiliary modes (not necessarily in the vacuum ferent input Bell state. state). Then one selected mode is measured and, depending on the measurement outcome, the other output modes are mixedwithnewmodesandinputslinearlyandagain amode isselectedtobemeasured. Thisprocesscanberepeatedover and over again. IV. NO-GO THEOREM VaidmanandYoran[15]havearrivedtotheconclusion thataBellstateanalyzercannotbebuildusingonlylin- WenowshowthatitisnotpossibletoconstructaBell ear devices, but their measurement apparatus does only measurementusingonlythetoolsmentionedabovetore- a very restrictive type of measurement. It is not allowed alize a measurement, for which all POVM elements are tomakeuseofauxiliaryphotonsandnoconditionalmea- projections on one of the four orthogonal Bell states. 3 N is defined as the maximal order in d† among the Bell four polynomials P˜ and it is independent of the index Ψi i. As a consequence, the polynomials Q˜ can be zero C Ψi for some i. Similarly N is defined as the order in d† aux B of the polynomia P˜aux. Inthe mode d wewillfind arangeofphotonnumbers. To prove the theorem it suffices to see that for any of U these events the conditional states Φ(total) that arise (cid:12) i E (cid:12) for each of the Bell states, are not p(cid:12)erfectly distinguish- A able. We concentrate on the measurement outcomes in this mode which leads to the maximum photon number a a c c detected in that mode, N = Naux +NBell. The states 1 2 1 ..... D-4 Φ(total) of the remainingmodes conditionedon the oc- b b c (cid:12) i E (cid:12) 1 2 2 c(cid:12)urrence of this event is then given by Φ(total) =Q˜ e† Q˜ e† 0 . (7) (cid:12)(cid:12) i E aux(cid:16) j(cid:17) Ψi(cid:16) j(cid:17)| i FIG.3. The initial step takes the input state at stage A (cid:12) from the input mode description via the linear transforma- The reasonof starting out from the event of detecting tionU totheoutputmodedescriptionatstageB.Depending the N photons in the selected mode d, is that the prob- on the detected photon number in mode d we find different lem reduces to a much simpler form in which the mea- conditional state for the four Bell state inputsat stage C. suring apparatus is not allowed to make use of auxiliary photons. That is, by imposing the orthogonality condi- To do so we concentrate on the first step of our mea- tionoftheconditionalstatesonthis particularevent,we surement set-up: We measure the photon number in one prove that the contribution Q˜ e† of the auxiliary selected mode d (see figure 3). For each result we will aux(cid:16) j(cid:17) find the remaining modes in four conditional states cor- photons can not make non-orthogonal states orthogonal responding to each Bell state input. We then show that in the sense that two conditionalstates Φ(total) are or- (cid:12) i E there is always at least one photon number detection (cid:12) thogonal if and only if the the states, (cid:12) eventin the firstmode that leads to non-orthogonal(i.e. not distinguishable) conditional states in the remaining Φ =Q˜ d†,e† 0 modes. | ii Ψi(cid:16) j(cid:17)| i In stage A (fig. 3) the input state can be described as are orthogonal. a product of two polynomials in the creation operators To prove this statement we observe that the overlap of the auxiliary and the Bell states modes respectively oftwoconditionalstatesbelongingtodifferentBellstate acting onto the vacuum (denoted by 0 ): | i input i and j is given by Ψ(total) =P c† P a†,a†,b†,b† 0 (cid:12)(cid:12)(cid:12) i E aux(cid:16) j(cid:17) Ψi(cid:16) 1 2 1 2(cid:17)| i DΦ(itotal) (cid:12)(cid:12)Φ(jtotal)E =h0|Q˜†auxQ˜†ΨiQ˜auxQ˜Ψj|0i Since we use detectors with photon number resolution (cid:12) = 0 Q˜† Q˜ n n Q˜† Q˜ 0 it is enough to assume that the auxiliary input is in a Xh | aux aux| ih | Ψi Ψj| i n state of definite photon number. Then P c† con- aux(cid:16) j(cid:17) = 0 Q˜† Q˜ 0 0 Q˜† Q˜ 0 . (8) tains only products of a fixed number of creation opera- h | aux aux| ih | Ψi Ψj| i tors, and PBell(cid:16)a†1,a†2,b†1,b†2(cid:17) contains only products of The first step makes use of the commutativity of Q˜† Ψi two creation operators. Now the modes of the Bell state and Q˜ following the commutativity of the two set of aux inputa1,a2,b1,b2 andtheauxiliarymodescj arelinearly creation operators for the auxiliary modes and the Bell mappedbytheunitarytransformationU intotheoutput modes. Furthermore, the first step inserts the identity modes d and ek. At stage B the state is described by operator of the Fock space for all involved modes. We denote by n the vector of photon numbers in each in- (cid:12)(cid:12)Ψ(itotal)E=P˜aux(cid:16)d†,e†k(cid:17)P˜Ψi(cid:16)d†,e†k(cid:17)|0i volved mode. The second step then uses the fact that (cid:12) onlyoneofthesetermsisnonzero. Thisisaconsequence We expand the two polynomials in powers of d† as ofQ˜ 0 beingastatewithtotalphotonnumber2while Ψj| i the conjugate state n Q˜† is a two photon state if and P˜aux(cid:16)d†,e†k(cid:17) =(cid:0)d†(cid:1)NauxQ˜aux(cid:16)e†k(cid:17)+... (5) only if hn|=h0|. h | Ψi Nowthatisclearthattheuseofauxiliaryphotonsdoes P˜Ψi(cid:16)d†,e†k(cid:17) =(cid:0)d†(cid:1)NBellQ˜Ψi(cid:16)e†k(cid:17)+... (6) notprovideanyhelpinbuildingaBellstateanalyzer,itis 4 mucheasiertocheckiftheorthogonalityconditionofthe a=0,b=0 c,d i.e. v1 =(0,0,c,d) (11) ∀ conditional states is fulfilled when only one or two pho- c=0,d=0 a,b i.e. v1 =(a,b,0,0). tons are detected in the selected mode d. To do this, we ∀ introduce a formalism for the linear mapping of modes. Butforboth solutionsM˜ =0. ThereforeaperfectBell Consider the unnormalized input state 11 analyzer can never detect two photons in the selected µ µ mode. Now we have left only the case where only one Ψ = 1 a†b†+a†b† + 2 a†b†+a†b† (9) | i √2(cid:16) 1 1 2 2(cid:17) √2(cid:16) 1 1 2 2(cid:17) photon is detected. µ µ Afterasinglephotondetectionatmoded,thefirstline +√32(cid:16)a†1b†2−a†2b†1(cid:17)+ √42(cid:16)a†1b†2−a†2b†1(cid:17)|0i . of M˜, denoted by M˜1,i tells us the state of the remain- ing modes. Their state is derivedfromthe unnormalized By choosing one of the weights µ as one and the others state i as zero, we recover the four Bell states. This state can be written with the help of a symmetric real matrix M |Φi=M˜1,i(d†,e†1,...,e†D−1)T as by choosing,as before, one of the µ to one, and the rest i Ψ =(a†,a†,b†,b†,...)M(a†,a†,b†,b†,...)T 0 tozero. Wehaveshownabovethatthe firstcolumnofU | i 1 2 1 2 1 2 1 2 | i is of the form v1 =(a,b,0,0) or v1 =(0,0,c,d) in order with toavoidtwophotonsenteringthe selectedmode. Due to thesymmetryoftheproblemwecanrestrictourselvesto 0 0 µ +µ µ +µ 0 ... 0  0 0 µ1 µ2 µ3 µ4 0 ... 0  the first situation, v1 = (a,b,0,0). We now write U in 3− 4 1− 2 the form M=232  µµ13++µµ24 µµ31−−µµ42 00 00 00 ...... 00 . a aR  0 0 0 0 0 ... 0   b bR   : : : : : : :  U = 0 cR  A linear transformation of the modes is now equivalent  0 dR   : :  to the transformation M˜ = UTMU HThereenaM˜R,1b,iRis,cgRiv,ednRbyare D−1 dimensional row vectors. forsomematrixU ofdimensionD D (withD 4)sat- iesnfylairnggemUeUn†t=of11th.eTnhuemcbheoricoefDmo≥×de4scdoureretsopoand≥ddsittioonaanl M˜1,i = 2√12(0,µ1(acR+bdR)+µ2(acR−bdR)+ unexcited input modes of beam-splitters. The output µ3(bcR+adR)+µ4(bcR adR)) (12) − modes are now d,e ,...,e . The entries of the matrix 1 D−1 M˜ reveal the distinguishability of the Bell states in the From this it follows that the conditional states are (up following way: if two photons are detected in the mode to normalization) d then the presence of µ in the matrix element M˜ re- i 11 vealswhich Bellstates Ψi couldhavecontributed to this Ψ1 =(acR+bdR)e† 0 (13) | i | i event. ForallBellstatesthatcontribute,theconditional Ψ2 =(acR bdR)e† 0 (14) stateoftheremainingmodesisvacuum. Itturnsoutthat | i − | i thiseventcannotbeattributedtoasingleBellstate. To |Ψ3i =(adR+bcR)e†|0i (15) provethisstatementwecalculateM˜11withageneralfirst |Ψ4i =(adR−bcR)e†|0i (16) column of the matrix U given as v1 =(a,b,c,d,...)T: with the vector of creation operators e† = M˜11 =v1TMv1 (10) (e†1,...,e†D−1)T. Thesixdifferentoverlapsbetweenthese 1 1 states are (up to the missing normalization factors): = µ (ac+bd)+ µ (ac bd) 1 2 √2 √2 − 1 1 hΨ1 |Ψ2i =|a|2|cR|2−|b|2|dR|2 (17) +√2µ3(ad+bc)+ √2µ4(ad−bc). hΨ1 |Ψ3i =a∗b|cR|2+b∗a|dR|2 (18) Ψ1 Ψ4 =b∗adR 2 a∗bcR 2 (19) TmoodbeeuanbalemtboigautoturisblyutteotohneeeBveelnltstoafttew,oonpehaontodnosnilny oonnee hhΨ2 ||Ψ3ii =a∗b||cR||2−−b∗a||dR||2 (20) of the coefficients of the µ ’s should be non zero. It is Ψ2 Ψ4 = a∗bcR 2 b∗adR 2 (21) i h | i − | | − | | easily verified that this condition can not be satisfied. Ψ3 Ψ4 = a2 dR 2 b2 cR 2 . (22) h | i | | | | −| | | | If we impose that three of the coefficients vanish we obtain two possible solutions, These overlaps are zero if, 5 (a2 b2)(cR 2+ dR 2)=0 (23) VI. ACKNOWLEDGMENTS | | −| | | | | | (a2+ b2)(cR 2 dR 2)=0 (24) | | | | | | −| | a∗b cR 2 =0 (25) | | b∗a dR 2 =0. (26) The authors thank the organizers of the ISI (Italy) | | and Benasque Center for Physics (Spain) workshops on Since the column vector v1 can not be a zero vector quantum computation and quantum information held in (a2+ b2 =0) this simplifies to summer1998whichbroughtusincontactwiththeworks | | | | 6 by Vaidman and Yoran [15], and Kwiat and Weinfurter cR 2 = dR 2 (27) [13]. We also thank L. Vaidman and M. Plenio for use- | | | | 2(a2 b2)cR 2 =0 (28) ful discussions and the Academy of Finland for financial | | −| | | | b∗acR 2 =0. (29) support. | | fromwhichwecanconcludethat cR 2 = dR 2 =0. But | | | | for this choice the matrix U does not haverank 4 andso the restriction on U given by UU† =11 can no longer be satisfied. Obviouslynowwecandiscardtheonlyremain- ing case; the zero photon case represents a bad choice of [1] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. the mode d since it would be disconnected from the in- Peres, and W. K. Wootters, Phys. Rev. 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