Continuous-Variable Gaussian Analog of Cluster States

PHYSICAL REVIEW A 73, 032318 ͑2006͒

Continuous-variable Gaussian analog of cluster states

Jing Zhang1,* and Samuel L. Braunstein2 1State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, People’s Republic of China 2Computer Science, University of York, York YO10 5DD, United Kingdom ͑Received 21 October 2005; published 16 March 2006͒

We present a continuous-variable ͑CV͒ Gaussian analog of cluster states, a new class of CV multipartite entangled states that can be generated from squeezing and quantum nondemolition coupling HI = ប␹XAXB. The entanglement properties of these states are studied in terms of classical communication and local operations. The graph states as general forms of the cluster states are presented. A chain for a one-dimensional example of cluster states can be readily experimentally produced only with squeezed light and beam splitters.

DOI: 10.1103/PhysRevA.73.032318 PACS number͑s͒: 03.67.Hk, 03.65.Ta

I. INTRODUCTION ment. The study of entanglement properties of the harmonic chain has aroused great interest ͓12͔, which is in direct anal- Entanglement is one of the most fascinating features of ogy to a spin chain with an Ising interaction. To date, the quantum mechanics and plays a central role in quantum- study and application of CV multipartite entangled states has information processing. In recent years, there has been an mainly focused on CV GHZ-type states. However, a richer ongoing effort to characterize qualitatively and quantitatively understanding and more definite classification of CV multi- the entanglement properties of multiparticle systems and ap- partite entangled states are needed for developing a CV ply them in quantum communication and information. The quantum-information network. In this paper, we introduce a study of multipartite entangled states in the discrete-variable class of CV Gaussian multipartite entangled states, CV clus- regime has shown that there exist different types of entangle- terlike states, which are different from CV GHZ-like states. ment, inequivalent up to local operations and classical com- We propose an interaction model to realize CV clusterlike ͓ ͔͑ ͒ munication 1 LOCC . For example, it is now well known states and compare these states to GHZ-like states in terms of ͑ ͒ that the Greenberger-Horne-Zeilinger GHZ and W states LOCC. It is worth noting that a CV Gaussian cluster state in are not equivalent up to LOCC. Recently, Briegel and Raus- our protocol cannot be used in universal quantum computers sendorf introduced a special kind of multipartite entangled over continuous variables ͓13͔; however, it may be applied in states, the so-called cluster states, which can be created via quantum network communication as a different type of mul- ͓ ͔ an Ising Hamiltonian 2 . It has been shown that via cluster tipartite entanglement. states, one can implement a quantum computer on a lattice of qubits. In this proposal, which is known as a “one-way quantum computer,” information is written onto the cluster and II. CV GAUSSIAN CLUSTER STATE then is processed and read out from the cluster by one-qubit measurements ͓3͔. We consider an N-mode N-party system described by a set ͑ ͒ In recent years, quantum communication, or more gener- of quadrature operators R= X1 , P1 ;X2 , P2 ,...;X1 , Pn obey- ͓ ͔ ␦ ally quantum information with continuous variables ͑CVs͒, ing the canonical commutation relation Xl , Pk =i lk. The has attracted a lot of interest and appears to yield very prom- quantum nondemolition ͑QND͒ coupling Hamiltonian is em- ប␹ ising perspectives concerning both experimental realizations ployed in the form HI = XlXk, so in this process and general theoretical insights ͓4͔, due to its relative sim- quadrature-amplitude ͑“position”͒ and quadrature-phase plicity and high efficiency in the generation, manipulation, ͑“momentum”͒ operators are transformed in the Heisenberg and detection of the CV state. The investigation of CV mul- picture according to the following expressions: tipartite entangled states has made significant progress in XЈ = X , PЈ = P + gX , theory and experiment. The quantification and scaling of l l l l k multipartite entanglement has been developed by different Ј Ј ͑ ͒ methods, for example, by determining the necessary and suf- Xk = Xk, Pk = Pk + gXl, 1 ͓ ͔ ficient criteria for their separability 5–7 , and by the en- g ␹t ␹ t ͓ ͔ where =− is the gain of the interaction, and and are tanglement of formation 8 . Multipartite quantum protocols the coupling coefficient and the interaction time, respec- were proposed and performed experimentally, such as quan- ͓ ͔ ͓ ͔ tively. The important feature of this Hamiltonian is that the tum teleportation networks 9 , controlled dense coding 10 , momentum P and P pick up the information of the position and quantum secret sharing ͓11͔ based on tripartite entangle- l k Xk and Xl, respectively, while the position remains un- changed. The QND coupling of light was widely investigated in many previous experiments ͓14͔ and recently has been *Corresponding author. Email address: [email protected]; experimentally observed between light and the collective [email protected] spin of atomic samples ͓15͔.

Considering first a one-dimensional example of a chain of product state. From this point of view, it is impossible to N modes which are numbered from 1 to N with next- destroy all the entanglement of the cluster if fewer than neighbor interaction. Initially, all modes are prepared in the int ͑N/2͒ parties are traced out ͑discarded͒. But only if one +r ͑0͒ −r ͑0͒ quadrature-phase squeezed state Xl =e Xl , Pl =e Pl , party for the GHZ state is traced, will the remaining state be where r is the squeezing parameter and the superscript ͑0͒ completely unentangled. Note that this is not true in the case denotes initial vacuum modes. Applying QND coupling to of finite squeezing. For example, if three weakly squeezed next-neighbor modes of a chain at different or the same time vacuum states are used to generate a tripartite GHZ state, the yields a CV clusterlike state in the form state is a fully inseparable tripartite entangled state, but the remaining bipartite state after tracing out one of the three C +r ͑0͒ C −r ͑0͒ +r ͑0͒ X1 = e X1 , P1 = e P1 + e X2 , subsystems is still entangled ͓9,10͔. ӇӇ Next, we investigate the entanglement properties of the ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ quantum teleportation network. In other words, we will an- XC = e+rX 0 , PC = e−rP 0 + e+rX 0 + e+rX 0 , ͑2͒ i i i i i−1 i+1 swer the question of how many parties need to be measured ӇӇ from an N-partite entangled state to make bipartite entangle- C +r ͑0͒ C −r ͑0͒ +r ͑0͒ ment between any two of the N parties “distilled,” which XN = e XN , PN = e PN + e XN−1. enables quantum teleportation. We first show that the parties Here, without loss of generality, we put the gain of the inter- at the ends of the chain, i.e., parties 1 and N, can be brought action g=1. For simplicity we discuss the properties of CV into bipartite entanglement by measuring the parties clusterlike states in the ideal case r→ϱ corresponding to 2,...,N−1. After measurement of the momentum of party 2, infinite squeezing. When N=2, one obtains a state with total the remaining parties are identical to an entangled chain of C C → C C → position X1 +X2 0 and relative momentum P1 − P2 0by length N−1 when one displaces the position of party 1 as ؠ C C C applying a local −90 rotation transformation X → P , P XЈC =XC − PC and measuring the result PC on party 2, then C 1 1 2 2 →−X on mode 2, which corresponds to Einstein-Podolsky- applies a local rotation transformation XC →−PC, PC →XC Rosen or two-mode squeezed states. The entanglement of a on parties 1. We can repeat this procedure and measure party state is not affected by local unitary transformation. It obvi- 3, and so on. At the end, party 1 and N are brought into a ously exhibits maximum bipartite entanglement. Similarly, bipartite entanglement. To bring any two parties j,k ͑jϽk͒ C C one obtains the state for N=3 with total position X1 +X2 from the chain ͕1,2,...,N͖ into bipartite entanglement, we C → C C → ͑ ͒ +X3 0 and relative momentum Pi − Pj 0 i, j=1,2,3 , first measure the position of the “outer” parties j−1 and k which corresponds to a CV three-mode GHZ-like state ͓9͔. ЈC +1, then displace the momentum of parties j and k as Pj However, quantum correlation of position and momentum of = PC −XC and PЈC = PC −XC , which projects the parties C C C → C j j−1 k k k+1 the state for the N=4 case become X1 +X2 +X3 0, X3 j, j+1,...,k into an entangled chain of length k− j+1. This C → C C → C C C → +X4 0 and P1 − P2 0, P2 − P3 + P4 0 by applying a lo- ;”ؠ process that breaks chain into parts is called “disconnection cal −90 rotation transformation on modes 2 and 4. Clearly, a for example, one breaks a chain into two independent chains CV four-partite clusterlike state is not equivalent to a GHZ- when measuring the position of party j and displacing the → like state with total position X1 +X2 +X3 +X4 0 and relative momentum of parties j−1 and j+1 as PЈ C = PC −XC and → ͑ ͒ j−1 j−1 j momentum Pi − Pj 0 i, j=1,2,3,4 . Note that when the Ј C C C Pj+1 = Pj+1−Xj . A subsequent measurement of the “inner” CV clusterlike state is generated by finite squeezing, quan- parties j+1,...,k−1 will then project parties j, k into a bi- tum correlation is expressed by the variance, such as partite entanglement, as shown previously. Note that the as- ͗␦2͑ C C C͒͘ X1 +X2 +X3 . More generally, CV N-partite clusterlike sistance of all the “inner” parties is necessary to bring any states and GHZ-like states are not equivalent for NϾ3, as we two parties from the chain into a bipartite entanglement; shall see below. however, for the “outer” parties it depends on which strategy We now compare the entanglement properties of CV clus- is chosen. The optimum strategy as shown previously is to ter and GHZ-like states by LOCC in the limit of infinite measure next-neighbor parties j−1 and k+1. The other strat- squeezing. First, we discuss the persistence of entanglement egies may be with the help of parties j−2, j−3 and k of an entangled N-partite state, which means the minimum +2, k+3, or j−2, j−4, j−5 and k+2, k+4, k+5, and so number of local measurements such that, for all measure- on. For the N-partite GHZ state, in contrast, bipartite en- ment outcomes, the state is completely disentangled ͓2͔.An tanglement between any two of the N parties is generated explicit strategy to disentangle the CV cluster-type state ͑2͒ with the help of a local position measurement of N−2 is to measure the positions of all even-number parties, j modes ͓9͔. =2,4,6,..., and then displace momentum of the remaining In the following we will generalize the one-dimensional ͑unmeasured͒ parties with the measured results; such as the CV cluster to graph states that correspond to mathematical ЈC C C −r ͑0͒ momentum of party 1 displaced as P1 = P1 −X2 =e P1 by graphs, where the vertices of the graph play the role of quan- C ЈC C the measured result X2 of party 2; and party 3 as P3 = P3 tum physical systems, i.e., the individual modes and the C C ͑ ͒ −X2 −X4 by parties 2 and 4. It is obvious that the remaining edges represent interactions. A graph G= V,E is a pair of a parties become the originally prepared quadrature-phase finite set of n vertices V and a set of edges E, the elements of squeezed states that are product states and completely unen- which are subsets of V with two elements each ͓16͔. We will tangled. Thus the minimum number of measurements to dis- consider only simple graphs, which are graphs that contain entangle the cluster state is int ͑N/2͒. For the GHZ state, in neither loops ͑edges connecting a vertex with itself͒ nor mul- contrast, a single local measurement suffices to bring it into a tiple edges. When the vertices a, b෈V are the end points of

032318-2 CONTINUOUS-VARIABLE GAUSSIAN ANALOG OF¼ PHYSICAL REVIEW A 73, 032318 ͑2006͒

an edge, they are referred to as being adjacent. An ͕a,c͖ path is an ordered list of vertices a=a1 ,a2 ,...,an−1,an =c, such that for all i, ai and ai+1 are adjacent. A connected graph is a graph that has an ͕a,c͖ path for any two a,c෈V. Otherwise ʚ it is referred to as disconnected. The neighborhood Na V is deﬁned as the set of vertices b for which ͕a,b͖෈E. In other words, the neighborhood is the set of vertices adjacent to a given vertex. The CV graph states with a given graph G are conveniently deﬁned as FIG. 1. A schematic diagram for generating a CV four-partite G +r ͑0͒ cluster and GHZ-like state by means of squeezed light and beam Xa = e Xa , splitters.

G −r ͑0͒ +r ͑0͒ ෈ ͑ ͒ Pa = e Pa + e ͚ Xb for a V. 3 squeezing in mode 1 and position squeezing in modes 2,3,4 ෈ b Na yields a GHZ-like state͒. The four-partite clusterlike ͑GHZ- Equation ͑3͒ can be used to generalize some of the entangle- like͒ states are expressed by the Heisenberg operators ment properties from the one-dimensional case to graph 1 ͱ3 states. To bring any two parties on sites a, d෈V into C͑Z͒ +r ͑0͒ −r ͑0͒ X1 = e X1 + e X2 , bipartite entanglement, we ﬁrst select a one-dimensional path 2 2 PʚV that connects sites a and d. Then we measure all ͱ neighboring modes surrounding this path in the position ͑ ͒ 1 ͑ ͒ 3 ͑ ͒ PC Z = e−rP 0 + e+rP 0 , component. By this procedure, we project the parties on the 1 2 1 2 2 path P into a state that up to local displacement on the momentum with measured results is identical to the linear chain 1 1 2 Eq. ͑2͒. We have thereby reduced the graph states to the C͑Z͒ +r ͑0͒ −r ͑0͒ ͱ −r ͑0͒ X2 = e X − e X + e X , one-dimensional problem. 2 1 ͱ12 2 3 3

1 1 2 C͑Z͒ −r ͑0͒ +r ͑0͒ ͱ +r ͑0͒ III. PHYSICALLY REALIZABLE MODELS P2 = e P − e P + e P , 2 1 ͱ12 2 3 3 We briefly mention that, to embody Eqs. ͑2͒ and ͑3͒,we can use the off-resonant interaction of linearly polarized op- ͑ ͒ 1 ͑ ͒ 1 ͑ ͒ 1 ͑ ͒ 1 ͑ ͒ tical buses with N ensembles of atoms ͑confined in a vapor XC Z = e+rX 0 − e−rX 0 − ͱ e−rX 0 + e+͑−͒rX 0 , ͒ ប␹ 3 2 1 ͱ 2 6 3 ͱ 4 cell , providing the Hamiltonian HOA= XOXA. Here, XO 12 2 ͑ ͒ ͑ ͒ XA is the position operator of the bus atomic ensemble . First, we prepare the spin-squeezed atomic ensembles by 1 1 1 1 C͑Z͒ −r ͑0͒ +r ͑0͒ ͱ +r ͑0͒ −͑+͒r ͑0͒ means of QND coupling H and projection measurements P3 = e P1 − e P2 − e P3 + e P4 , OA 2 ͱ12 6 ͱ2 on light ͓15,17͔. In the second step, the QND coupling ប␹ HA = ЈXA XA between ensembles can be implemented by lk l k 1 1 1 1 first letting an optical bus interact with ensemble l, then ap- C͑Z͒ +r ͑0͒ −r ͑0͒ ͱ −r ͑0͒ +͑−͒r ͑0͒ , ؠ X4 = e X − e X − e X − e X plying a 90 rotation transformation on the optical bus, then 2 1 ͱ12 2 6 3 ͱ2 4 letting the optical bus interact with ensemble k, then apply- ؠ ing −90 rotation transformation on the optical bus, and fi- 1 1 1 1 C͑Z͒ −r ͑0͒ +r ͑0͒ ͱ +r ͑0͒ −͑+͒r ͑0͒ nally letting the optical bus interact with ensemble l. P4 = e P1 − ͱ e P2 − e P3 − ͱ e P4 . Stimulating opportunities also come from the experimen- 2 12 6 2 tal demonstration of CV entanglement swapping ͓18͔,in ͑4͒ which a four-partite entangled state may be generated. Here →ϱ we show that a chain for a one-dimensional example of clus- In the ideal case r corresponding to infinite squeezing, ter states can be experimentally produced with only squeezed the four-partite clusterlike state is a simultaneous eigenstate C C → C C C → light and beam splitters as shown in Fig. 1 and give an ex- of position X1 −X2 0, 2X2 −X3 −X4 0 and momentum C C C → C C → P1 + P2 +2P3 0, P3 − P4 0; however, a GHZ-like state is plicit example where the quantum teleportation network is Z Z Z implemented with four-partite clusterlike states compared a simultaneous eigenstate of total momentum P1 + P2 + P3 Z → Z Z → ͑ ͒ with GHZ-like states. To make this example simple and easy + P4 0 and relative position Xi −Xj 0 i, j=1,2,3,4 . to compare quadripartite clusterlike states with GHZ-like ͓ ͔ states, we employ the scheme in Ref. 9 to generate the IV. APPLICATION TO THE TELEPORTATION cluster and GHZ-like states and implement the quantum tele- NETWORK portation network. Applying the beam splitter operations ͑beam splitters 1, 2, and 3 give 1:3,1:2, and 1:1, respectively͒ The teleportation network protocol involving four partici- to momentum squeezing in modes 1,4 and position squeez- pants Alice, Bob, Claire, and Doris is shown as follows. Let ing in modes 2,3 yields a clusterlike state ͑momentum us send the four modes of Eq. ͑4͒ to Alice, Bob, Claire, and

032318-3 J. ZHANG AND S. L. BRAUNSTEIN PHYSICAL REVIEW A 73, 032318 ͑2006͒

Doris, respectively. Alice wants to teleport an unknown 3 g ͑ ͒ 1 ͑ ͒ PClaire = P + ͩ + 2 ͪe−rP 0 + ͑1−g ͒e+rP 0 quantum state and couples her mode 1 with the unknown tel in 2 2 1 ͱ 2 2 ͑ ͒ ͱ ͑ ͒ ͱ 2 3 input mode Xu = Xin−X1 / 2, Pv = Pin+ P1 / 2. Alice mea- sures classical values x and p for X and P . Alice sends 2 u v u v ͱ ͑ ͒ +r ͑0͒ ͱ −r ͑0͒ ͑ ͒ − 1−g2 e P + 2g2e P . 7 her classical results xu and pv to Bob or Claire or Doris via 3 3 4 classical channels. For clusterlike states, Bob, Claire, or Doris want to reconstitute the input state provided that additional classical information is received: Bob needs the result We may achieve the optimum teleportation by the optimum of a momentum detection by Claire reducing PC to pC; Claire 3 3 g and g . Similarly, if Alice teleports the unknown state to needs the results of a momentum detection by Bob reducing 2 4 Doris, Doris needs Alice’s classical values x and p and the PC to pC and a position detection by Doris reducing XC to xC; u v 2 2 4 4 additional classical information of the momentum detection Doris needs the results of a momentum detection by Bob by Bob and the position detection by Claire. Thus we see that reducing PC to pC and a position detection by Claire reduc- 2 2 the quantum teleportation network using a CV clusterlike ing XC to xC. Assuming that Claire detects her mode 3 and 3 3 state is quite asymmetric among the parties, and the required sends the result to Bob, a displacement of Bob’s mode 2, additional classical information and fidelity for the finite XC →X XC gͱ X PC → P PC gͱ P g PC 2 tel= 2 + 2 u, 2 tel= 2 + 2 v + 3 3 , ac- squeezing depend on the partners. In contrast, the quantum g g complishes the teleportation. Here, the parameters and 3 teleportation network using a CV GHZ-like state is symmet- g describe the normalized gain. For =1, the teleported mode ric among the parties, and the required additional classical becomes information and fidelity for the finite squeezing are indepen- ͓ ͔ 2 2 dent of the partners 9 . Bob −r ͑0͒ ͱ −r ͑0͒ Xtel = Xin − e X + e X , ͱ3 2 3 3 V. CONCLUSION g 1 g Bob ͩ 3 ͪ −r ͑0͒ ͩ 3 ͪ +r ͑0͒ Ptel = Pin + 1+ e P1 + 1− e P2 2 ͱ3 2 We have introduced a continuous-variable Gaussian analog of cluster states. We have proposed an interaction model 2 g g ͱ ͩ 3 ͪ +r ͑0͒ 3 −r ͑0͒ ͑ ͒ to realize CV clusterlike states and compared these states to + 1− e P3 + ͱ e P4 . 5 3 2 2 GHZ-like states in terms of LOCC. Those states are different from GHZ states and the entanglement of the states is harder Now we assume arbitrary coherent-state input a =X +iP in in in to destroy than for GHZ states. We have generalized the and calculate teleportation fidelity, in this case defined by one-dimensional CV cluster to graph states. To embody CV Fϵ͗a ͉ ͉a ͘. It describes the overlap between the input in tel in cluster states, we have described experimental setups of and the teleported states. In the case of unity gain g=1, the F atomic ensembles and squeezed light that offer nice perspec- fidelity for the Gaussian states is simply given by tives in the study of CV multipartite entangled states. Cluster ͱ͑ ͗␦2 ˆ ͒͑͘ ͗␦2 ˆ ͒͘ =2/ 1+ Xtel 1+ Ptel . The optimum teleportation states for CV multipartite entanglement may be applied to ͑ +4r ͒ ͑ +4r ͒ fidelity for Bob is achieved with g3 =2 e −1 / e +3 and multipartite quantum protocols and are of practical impor- becomes tance in realizing more complicated quantum communication among many parties. 3e+6r +10e+4r +3e−2r −1/2 F = ͫ͑1+e−2r͒ͩ1+ ͪͬ . ͑6͒ opt ͑e+4r +3͒2 F ACKNOWLEDGMENTS For r=0, we obtain class=1/2, which corresponds to the classical limit. When rϾ0, the fidelity of Bob’s output is larger than 1/2; thus the quantum teleportation is successful. J.Z. thanks K. Peng, C. Xie, T. Zhang, and J. Gao for Assuming that Alice teleports the unknown state to Claire, helpful discussions. This research was supported in part by National Natural Science Foundation of China ͑Approval Claire needs Alice’s classical values xu and pv and the additional classical information of the momentum detection by No. 60178012͒, Program for New Century Excellent Talents Bob and the position detection by Doris. Claire performs a in University ͑Approval No. NCET-04-0256͒, Doctoral Pro- Ј 1 C C gram Foundation of Ministry of Education China ͑Approval local unitary squeezed transform a3 = 2 X3 +2P3 and dis- C → 1 C ͱ C C No. 20050108007͒, the Cultivation Fund of the Key Scien- places her mode 3, X3 Xtel= 2 X3 +g 2Xu +g4X4 , P3 → P =2PC +gͱ2P +g PC, to accomplish the teleportation. tific and Technical Innovation Project of the Ministry of Edu- tel 3 v 2 2 ͑ ͒ For g=1, the teleported mode becomes cation of China Approval No. 705010 , Program for Changjiang Scholars and Innovative Research team in 1 1 University, the Natural Science Foundation of Shanxi Prov- Claire ͑ ͒ +r ͑0͒ ͑ ͒ −r ͑0͒ Xtel = Xin − 1−2g4 e X − 7+2g4 e X 4 1 4ͱ3 2 ince, and the Research Fund for the Returned Abroad Schol- ars of Shanxi Province. SLB was funded in part by EPSRC 1 g 1 ͩ 4 ͪ −r ͑0͒ ͑ ͒ +r ͑0͒ Grant GR/S56252 and by a Wolfson Royal Society Research − + e X + 1−2g4 e X , 2ͱ6 ͱ6 3 2ͱ2 4 Merit Award.

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͓1͔ W. Dür et al., Phys. Rev. A 62, 062314 ͑2000͒; F. Verstraete et ͓10͔ J. Zhang et al., Phys. Rev. A 66, 032318 ͑2002͒; J. Jing et al., al., ibid. 65, 052112 ͑2002͒. Phys. Rev. Lett. 90, 167903 ͑2003͒. ͓2͔ H. J. Briegel and R. Raussendorf, Phys. Rev. Lett. 86, 910 ͓11͔ A. M. Lance et al., Phys. Rev. Lett. 92, 177903 ͑2004͒. ͑2001͒. ͓12͔ K. Audenaert et al., Phys. Rev. A 66, 042327 ͑2002͒; A. Bot- ͓3͔ R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 ero and B. Reznik, ibid. 70, 052329 ͑2004͒. ͑2001͒. ͓13͔ S. Lloyd and S. L. Braunstein, Phys. Rev. Lett. 82, 1784 ͓4͔ S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513 ͑1999͒; S. D. Bartlett et al., ibid. 88, 097904 ͑2002͒. ͑2005͒. ͓14͔ S. F. Pereira et al., Phys. Rev. Lett. 72, 214 ͑1994͒; P. Grangier ͓5͔ G. Giedke et al., Phys. Rev. A 64, 052303 ͑2001͒. et al., Nature ͑London͒ 396, 537 ͑1998͒; J. Zhang et al., Phys. ͓6͔ P. van Loock and A. Furusawa, Phys. Rev. A 67, 052315 Rev. A 68, 035802 ͑2003͒ ͑2003͒. ͓15͔ B. Julsgaard et al., Nature ͑London͒ 413, 400 ͑2001͒. ͓7͔ G. Adesso et al., Phys. Rev. Lett. 93, 220504 ͑2004͒. ͓16͔ M. Hein et al., Phys. Rev. A 69, 062311 ͑2004͒. ͓8͔ M. M. Wolf et al., Phys. Rev. Lett. 92, 087903 ͑2004͒. ͓17͔ J. M. Geremia et al., Science 304, 270 ͑2004͒. ͓9͔ P. van Loock and S. L. Braunstein, Phys. Rev. Lett. 84, 3482 ͓18͔ J. Zhang et al., Phys. Lett. A 299, 427 ͑2002͒;X.Jiaet al., ͑2000͒; H. Yonezawa et al., Nature ͑London͒ 431, 430 ͑2004͒. Phys. Rev. Lett. 93, 250503 ͑2004͒.

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