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the generation of nonclassical states of light is one of the primary research areas in quantum optics . in particular , the preparation of _ entangled states _ of light has attracted a considerable amount of attention recently , since these states have been identified as a key resource for quantum information processing @xcite . other interesting applications of the entangled states include , for instance , ultra - high precision measurements @xcite and quantum optical lithography @xcite . the range of interactions between light fields that are experimentally accessible and feasible is rather limited , which restricts the class of quantum states of the optical field that can be generated in the lab . however , this class can be significantly extended if one considers probabilistic generation schemes , whose success is conditioned on the detection of a particular outcome of a measurement performed on some ancilla system . schemes for probabilistic preparation of fock states @xcite , arbitrary superpositions of fock states of single - mode field @xcite , and superpositions of classically distinguishable states ( schrdinger - cat - like states ) @xcite , have been found . several setups for the generation of two - mode @xmath0-photon path - entangled states have been suggested @xcite . recently , the experimental conditional preparation of a single - photon fock state with negative wigner function has been reported @xcite .    however , all the above mentioned schemes are still quite restrictive since they are capable to prepare only single - mode states or only a particular class of two - mode states ( such as the @xmath0-photon states ) . in this paper , we design a truly _ universal _ scheme that can be used to probabilistically generate an _ arbitrary _ multimode entangled state of light , provided that each mode does not contain more than @xmath0 photons , where @xmath0 is an arbitrary but finite integer . the resources required for the present scheme comprise the passive linear optical elements ( beam splitters and phase shifters ) , single photon sources , strong coherent laser beams , and photodetectors . it was shown recently that these resources are sufficient for performing universal quantum computing @xcite . our proposal provides another illustration of the surprising versatility and power of such an approach , relying only on linear optics and single photons . in this context it is worth mentioning that , very recently , clausen _ et al . _ suggested a scheme for approximate conditional implementation of general single- or multi - mode operators acting on states of traveling optical field @xcite . the scheme described in @xcite is also based on linear optics and may be in principle used for ( approximate ) quantum state preparation . we emphasize that the approach proposed in the present paper conceptually differs from that of ref . we are primarily interested in exact state preparation and our setup is specifically tailored for that purpose . we will first explain all the essential features of the state - preparation procedure on the simplest yet nontrivial example of two - mode entangled state that is formed by a superposition of two product states , @xmath1 where @xmath2 is a normalization factor . then we will generalize the preparation procedure and design a scheme for the generation of arbitrary multimode entangled states of light . we note in passing that even the simple entangled states ( [ target ] ) may find applications in the investigations of the foundations of quantum mechanics . for instance , it was shown recently that the states of the form ( [ target ] ) yield a strong violation of bell - chsh inequalities when alice and bob perform balanced homodyne measurements followed by an appropriate binning @xcite . the paper is structured as follows . in section ii , we propose a method to probabilistically generate the two - mode entangled state ( [ target ] ) with the help of passive linear optics , single photon sources , strong coherent laser fields and photodetectors . in section iii we shall extend our scheme to allow for more than two modes and more than two terms in the superposition . we will thus design a universal device that enables one to conditionally prepare an arbitrary multimode state of light . finally , the conclusions are drawn in section iv . let us first note that , in order to simplify the notation , we shall omit the normalization factors in front of the quantum states in what follows . since the state @xmath3 is rather complicated , we divide its preparation into several steps . the main simplification stems from the observation that the entangled state ( [ target ] ) can be prepared by means of entanglement swapping @xcite if we possess the two three - mode states @xmath4 where @xmath5 and @xmath6 denote the state of a single photon in spatial mode @xmath7 ( or @xmath8 ) that is polarized vertically or horizontally , respectively . as shown in fig . 1 , we easily obtain the state ( [ target ] ) from the state @xmath9 if we project the single photons in spatial modes @xmath7 and @xmath8 onto the maximally entangled bell state @xmath10 the generation of state ( [ target ] ) thus boils down to the preparation of the entangled state ( [ targetsp ] ) . [ state ( [ targetspprime ] ) can be prepared similarly ] . this latter task will be accomplished with the help of a quantum non - demolition ( qnd ) measurement of a single photon , which creates entanglement . the proposed setup is depicted in fig . the input polarization modes @xmath11 and @xmath12 are prepared in the pure single - mode states @xmath13 and @xmath14 that contain no more than @xmath15 photons , @xmath16 where @xmath17 denotes the @xmath18-photon fock state . the states ( [ fgtildeexpansion ] ) are , of course , closely related to the states @xmath19 and @xmath20 appearing in eq . ( [ targetsp ] ) . the exact relationship between them will be specified later . note that it was shown by dakna _ _ @xcite that the single - mode states ( [ fgtildeexpansion ] ) can be probabilistically generated with the help of passive linear optics , single photon sources , strong coherent laser pulses and photodetectors with single photon sensitivity .    as shown in fig . 2 , the input beam containing the states @xmath13 and @xmath14 in the @xmath21 and @xmath22 polarization modes impinges on a beam splitter @xmath23 with transmittance @xmath24 and reflectance @xmath25 . the purpose of this beam splitter is to separate a single photon from one of the input states and thus create entanglement . of course , @xmath23 may split more than one photon or no photon at all . therefore we must verify that there is exactly a single photon present in the output spatial mode @xmath26 , without disturbing its polarization state . such qnd measurements of a single photon have been recently thoroughly discussed in ref . @xcite where several schemes have been proposed and analyzed . one possible method relies on teleportation @xcite . with the help of the conditional c - not gate for photonic qubits @xcite we have to first prepare the singlet state @xmath27 in the auxiliary modes @xmath28 . then we teleport the polarization state of a photon in mode @xmath26 onto the photon in mode @xmath29 by performing a bell measurement on modes @xmath26 and @xmath30 . this measurement should be carried out with two detectors that are capable to resolve the number of photons . if exactly two photons are detected in coincidence at the two detectors , then we know that there was a single photon in mode @xmath26 and its polarization state has been coherently transfered onto the state of a single photon in mode @xmath29 .    kok _ et al . _ also proposed qnd measurement schemes that do not require a - priori entanglement @xcite . for our purposes it suffices to employ the simple interferometric scheme depicted in fig . 3 which performs a partial qnd photon number measurement . the coincident detection of a single photon in detectors @xmath31 and @xmath32 indicates that there was at least one photon in the input mode . moreover , if there was _ exactly _ a single photon at the input , then its polarization state is unperturbed by the measurement . the probability of successful qnd measurement of a single photon provided that there _ is _ a single photon in the spatial mode @xmath26 is equal to @xmath33 . in the preparation scheme shown in fig . 1 the two qnd measurements are followed by the entanglement swapping that succeeds only if each photodetector in fig . 1 detects exactly one photon . in this way we select the events when there is exactly a single photon in each mode @xmath7 and @xmath8 . the state after the qnd measurement can be written in the form @xmath34 where the non - unitary operators @xmath35 and @xmath36 describe removal of none or a single photon at @xmath37 , respectively , @xmath38      the qnd measurement has thus created the entanglement . still , @xmath39 and @xmath40 are states of two different polarization modes while @xmath41 and @xmath42 in eq . ( [ targetsp ] ) are states of a _ single _ mode . we achieve this by erasing the information about the polarization . this is done with the help of a polarizing beam splitter @xmath43 , see fig . the beam splitter is rotated such that it combines the @xmath21 and @xmath22 polarizations . if we associate the annihilation operators @xmath44 and @xmath45 with the modes @xmath11 and @xmath12 , then the transformation carried out by @xmath43 is given by @xmath46 where the polarization modes @xmath47 and @xmath48 are _ spatially separated _ at the output of @xmath43 as indicated in fig . 2 . the erasing succeeds if and only if the detector @xmath49 placed on the @xmath48 mode does not detect any photon , which ensures that all photons have been transmitted to the linearly polarized output mode @xmath47 . since the input states do not contain more than @xmath15 photons each and since a single photon was subtracted on @xmath23 , the state after erasure of the polarization information reads @xmath50 where the output complex amplitudes @xmath51 and @xmath52 are given by the following formula , @xmath53    it follows that we can fix the values of output coefficients @xmath51 by properly choosing the complex amplitudes of the input states . we have the recurrence equations @xmath54 and a similar formula holds for @xmath55 . the amplitudes @xmath56 and @xmath55 are calculated as follows . first , some nonzero values for @xmath57 and @xmath58 are chosen . then one uses repeatedly eq . ( [ inoutsolution ] ) to calculate @xmath56 and @xmath55 from @xmath59 and @xmath60 , @xmath61 , and the iterations stop at @xmath62 . we have thus almost obtained the state ( [ targetsp ] ) . however , while the effective states @xmath41 and @xmath42 in ( [ psierasure ] ) have the correct structure in the subspace spanned by the first @xmath15 fock states @xmath63 , they also contain an unwanted tail in the subspace spanned by the fock states @xmath64 . the last step in our scheme is to get rid of that tail . we do so with the help of the quantum scissors @xcite that will project the state in the mode @xmath65 onto the subspace spanned by the first @xmath15 fock states . this action is formally described by the projector @xmath66 ) in the output modes 1 and 2 of our device . the quantum scissors have been discussed in detail in several recent papers @xcite . the schemes proposed there can be implemented with the use of linear optics , single photon sources , and photodetectors . the setup that we shall consider in the present paper is depicted in fig . the modes @xmath26 and @xmath30 are prepared in a pure @xmath0-photon two - mode entangled state @xmath67 it was shown recently that these states may be conditionally generated with the help of linear optics , single photon sources and photodetectors @xcite . we then combine the modes @xmath65 and @xmath26 on a balanced beam splitter and measure the number of photons in output modes @xmath65 and @xmath26 by means of two photodetectors @xmath31 and @xmath32 with single - photon resolution . the truncation is a conditional operation that succeeds when certain particular measurement outcome has been detected , say , no photons at @xmath31 and @xmath0 photons at @xmath32 , which we denote as @xmath68 . note , that one could also consider other detection events , but the total number of photons detected by the photodetectors should be equal to @xmath0 . the quantum scissors is essentially a quantum state teleportation in the subspace of the first @xmath15 fock states , where the state ( [ psiscissors ] ) serves as the quantum channel , and the bell - type measurement is carried out with the use of the beam splitter in fig . 4 that couples the modes @xmath65 and @xmath26 .    for our particular choice of the detection event @xmath68 , and assuming a balanced beam splitter , we find that @xmath69 should hold , where @xmath70 is the probability of the successful quantum - state truncation that projects onto the subspace spanned by @xmath71 . so far we have considered the generation of a two - mode state that consists of a superposition of two product states . in this section we shall generalize our scheme in several ways . first , we will discuss the preparation of a @xmath72-mode entangled state which has a form limited to the superposition of two product states , namely @xmath73 to prepare this state , we first generate @xmath72 states @xmath74 as before , and then project the @xmath72 single photons onto the maximally entangled state @xmath75 this projection can be performed with the greenberger - horne - zeilinger ( ghz ) state analyzer described by pan and zeilinger @xcite . this analyzer consists of polarizing beam splitters and half - wave plates and allows one to distinguish two ghz states of m qubits . our second generalization extends the preparation procedure to two - mode states that are superpositions of @xmath76 product states , @xmath77 obviously , a natural strategy is to prepare this state via @xmath76-dimensional entanglement swapping from two states of the form @xmath78 where @xmath79 denotes a state of a single photon in the @xmath80th auxiliary spatial mode . to prepare the states ( [ psiquditsp ] ) , we proceed essentially as in sec . the extended scheme is depicted in fig . 5 . initially , @xmath76 spatial modes are prepared in states @xmath81 . the array of @xmath76 beam splitters @xmath82 extracts , with a certain probability , a single photon . we must verify the presence of exactly one single photon in total in the modes @xmath83 while preserving the coherence among these modes by performing a qnd measurement . this can be accomplished by @xmath76-dimensional teleportation similarly as discussed in section ii . however , this time we need to teleport a qudit encoded as a state of a single photon in @xmath76 spatial modes . fortunately , it was shown recently that the probabilistic gates for quantum computing with linear optics @xcite , that were introduced for qubits , can be easily extended to qudits @xcite . this means that all the necessary manipulations , such as the generation of maximally entangled state of two qudits and the bell measurement , can be probabilistically carried out with the resources that we assume here . after the qnd measurement , we must erase the spatial information contained in the resulting state @xmath84 where @xmath85 are given by eq . ( [ b01operators ] ) . this is done by the second array of @xmath86 beam splitters indicated by the shaded zone in fig . the preparation is finished by the application of the quantum scissors , that truncate the fock state expansion of the output state at @xmath0 photons . suppose that the output mode after erasing is a balanced linear superposition of all @xmath76 modes @xmath87 the complex amplitudes of the states @xmath88 can then be expressed in terms of the complex amplitudes of the input states @xmath89 as follows @xmath90 where the prime indicates summation over all @xmath91 satisfying the constraint @xmath92 . note that the formula ( [ finoutglobal ] ) may be easily inverted and we can determine the complex amplitudes @xmath93 of the input states for any given prescribed output states @xmath94 .      finally , we point out that those two generalizations can be combined and we can thus generate an arbitrary multimode entangled state of light where each mode contains no more than @xmath0 photons . any @xmath72-mode state of this kind can be written as a superposition of no more than @xmath95 product states , @xmath96 note that the norm of each product state in the superposition ( [ superposition ] ) may be arbitrary . we can conditionally generate the state ( [ superposition ] ) if we first prepare @xmath72 entangled states ( @xmath97 ) @xmath98 where @xmath99 denotes a single photon state of the @xmath100-th auxiliary spatial mode . as discussed above , the states ( [ superpositionsp ] ) can be prepared with the help of the scheme depicted in fig . , we carry out a joint measurement on all auxiliary photons . the state @xmath101 is obtained if we project the auxiliary modes onto an @xmath72-photon ghz entangled state : @xmath102 if we denote @xmath103 then we can write @xmath104      the above described procedure for the preparation of the generic @xmath72-mode entangled state clearly shows that our approach is indeed general since it allows us , in principle , to generate an arbitrary state . but the success probability of the scheme will be exponentially small in general . the preparation procedure will therefore have to be repeated exponentially many times to produce the desired state . in the rest of this section we shall argue that the number of operations required to produce the desired state can be significantly decreased if a quantum memory is available . the situation is similar to that analyzed in ref . @xcite where the operations required to produce `` cat states '' was drastically decreased by using a recursive procedure that uses a quantum memory . the advantage of the quantum memory is that it allows one to wait until successive steps in the state generation protocol have succeeded before proceeding with the next steps . for instance in the scheme described in fig . 5 , one can first produce the states @xmath81 and store them in a memory before proceeding with the production of the entangled state ( [ psiquditsp ] ) . suppose that the state @xmath109 can be prepared with the probability @xmath110 . without quantum memory the probability of simultaneous preparation of the @xmath76 states @xmath109 is @xmath111 and the number of required operations thus scales as @xmath112 . if , after a successful preparation , we store each state @xmath13 in a memory , we reduce the number of required operations to @xmath113 . a more elaborate use of a quantum memory is to build the state using a _ recursive procedure _ based on the schmidt decomposition . the method is best explained on an explicit example . suppose we would like to prepare a four - mode state @xmath114 . the schmidt decomposition of this four - mode state with respect to a bi - partite splitting into the @xmath115 and @xmath116 modes can be written as follows , @xmath117 where @xmath118 and @xmath119 form orthogonal bases in some subspaces of the hilbert space of modes @xmath115 and @xmath116 , respectively . the number of the terms in the schmidt decomposition ( [ schmidt ] ) is bounded by @xmath120 while , in general , @xmath121 terms were necessary in the general procedure of section iiic . the decomposition ( [ schmidt ] ) suggests that we could obtain the state @xmath114 via entanglement swapping if we first prepare two entangled states @xmath122 now the entangled states @xmath123 and @xmath124 can be prepared using the general procedure of sec . however , for states of the specific form ( [ abx ] ) , a simpler procedure can be devised . one first generates @xmath76 states @xmath125 where the tilde indicates that @xmath126 is related to @xmath127 via relations similar to those described by eqs . ( [ inoutsolution ] ) . the state @xmath123 is then prepared with the help of the device shown in fig . 5 where the mode @xmath128 of @xmath125 is sent to the @xmath80th input port of the interferometer . the only modification is that we must also erase the spatial information carried by the @xmath129 modes of the states @xmath125 and perform the quantum state truncation after the erasing . briefly , all @xmath76 modes @xmath130 are combined on an array of @xmath86 beam splitters and the detectors monitor the first @xmath86 output ports . if these detectors do not register any photon , then the erasing procedure succeeded and we apply the quantum scissors to the state in the @xmath76th output port .    clearly , this recursive procedure can be extended to any number of modes . the resulting scheme resembles a tree structure involving repeated applications of the entangling scheme shown in fig . 5 and the entanglement swapping that produces an @xmath131-mode state from two @xmath132-mode states entangled with auxiliary single photons . this scheme relying on quantum memory may possibly lead to a substantial improvement of the generation probability , similarly to the case of the two - mode @xmath0-photon cat states @xcite . in summary , the scheme proposed in the present paper is universal in that arbitrary multimode entangled states of light can be probabilistically generated . the resources required are passive linear optics , single photon sources , strong coherent states , and detectors with single - photon resolution . this result clearly illustrates the versatility and power of the approach relying only on linear optics and single photons . it is fair to say , though , that the suggested setup is rather complicated and involves several nontrivial operations such as the quantum non - demolition measurement of a single photon , quantum scissors , and transformations of photonic qudits encoded as a state of a single photon in @xmath76 spatial modes . we have shown that all these operations can be ( probabilistically ) implemented with the resources that we consider here . the resulting procedure may become very complex for general states . however , it should be emphasized that building a state like ( [ superposition ] ) must necessarily be complicated because of the large number of parameters [ @xmath133 complex numbers ] that characterize this state and that must be fixed by the preparation scheme . we have also argued that the use of a quantum memory can decrease the number of operations required to produce the desired state . we hope that this paper will stimulate the efforts towards experimental demonstrations of the basic building blocks of our scheme , such as the conditional generation of single - mode finite superpositions of fock states and the preparation of two - mode entangled @xmath0-photon states required for the quantum state truncation . we are grateful to m. duek , r. filip and p. grangier for stimulating discussions . we acknowledge financial support from the communaut franaise de belgique under grant arc 00/05 - 251 , from the iuap programme of the belgian government under grant v-18 , from the eu under project resq ( ist-2001 - 37559 ) and chic ( ist-2001 - 32150 ) . j.f . was also partially supported by the grant ln00a015 of the czech ministry of education .                                        n.j . cerf , j. fiurek , s. iblisdir , and s. massar , quant - ph/0210059 ; to appear in the proceedings of the sixth international conference on quantum communication , measurement and computing ( qcmc02 ) , boston ( 2002 ) .
we propose a universal scheme for the probabilistic generation of an arbitrary multimode entangled state of light with finite expansion in fock basis . 
 the suggested setup involves passive linear optics , single photon sources , strong coherent laser beams , and photodetectors with single - photon resolution . 
 the efficiency of this setup may be greatly enhanced if , in addition , a quantum memory is available .