Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics
aa r X i v : . [ qu a n t - ph ] A ug Nonlocal entanglement concentration scheme for partially entangled multipartitesystems with nonlinear optics ∗ Yu-Bo Sheng, , , Fu-Guo Deng, , † and Hong-Yu Zhou , , The Key Laboratory of Beam Technology and Material Modification of Ministry of Education,Beijing Normal University, Beijing 100875, China Institute of Low Energy Nuclear Physics, and Department of Material Science and Engineering,Beijing Normal University, Beijing 100875, China Beijing Radiation Center, Beijing 100875, China Department of Physics, Applied Optics Beijing Area Major Laboratory,Beijing Normal University, Beijing 100875, China (Dated: October 27, 2018)We present a nonlocal entanglement concentration scheme for reconstructing some maximallyentangled multipartite states from partially entangled ones by exploiting cross-Kerr nonlinearities todistinguish the parity of two polarization photons. Compared with the entanglement concentrationschemes based on two-particle collective unitary evolution, this scheme does not require the parties toknow accurately information about the partially entangled states—i.e., their coefficients. Moreover,it does not require the parties to possess sophisticated single-photon detectors, which makes thisprotocol feasible with present techniques. By iteration of entanglement concentration processes, thisscheme has a higher efficiency and yield than those with linear optical elements. All these advantagesmake this scheme more efficient and more convenient than others in practical applications.
PACS numbers: 03.67.Pp, 03.67.Mn, 03.67.Hk, 42.50.-p
I. INTRODUCTION
Entanglement is a unique phenomenon in quantummechanics and it plays an important role in quantum-information processing and transmission. For instance,quantum computers exploit entanglement to speedup thecomputation of problems in mathematics [1, 2]. Thetwo legitimate users in quantum communication—say,the sender Alice and the receiver Bob—use an entangledquantum system to transmit a private key [3, 4, 5, 6, 7] ora secret message [8]. Also quantum dense coding [9, 10],quantum teleportation [11], controlled teleportation [12],and quantum-state sharing [13] need entanglements toset up the quantum channel. However, in a practicaltransmission or the process for storing quantum systems,we can not avoid channel noise, which will make the en-tangled quantum system less entangled. For example,the Bell state | φ + i AB = √ ( | H i A | H i B + | V i A | V i B ) maybecome a mixed one such as a Werner state [14]: W F = F | φ + ih φ + | + 1 − F | φ − ih φ − | + | ψ + ih ψ + | + | ψ − ih ψ − | ) , (1)where | φ ± i AB = 1 √ | H i A | H i B ± | V i A | V i B ) , (2) | ψ ± i AB = 1 √ | H i A | V i B ± | V i A | H i B ) . (3) ∗ Published in Phys. Rev. A 77, 062325 (2008) † Email address: [email protected]
Here H and V represent the horizontal and vertical po-larizations of photons, respectively. The Bell state | φ + i can also be degraded as a less pure entangled state like | Ψ i = α | H i A | V i B + β | V i A | H i B , where | α | + | β | = 1.Multipartite entanglement states also suffer from channelnoise. For instance, | Φ ± i = √ ( | HH · · · H i±| V V · · · V i )will become | Φ ′± i = α | HH · · · H i ± β | V V · · · V i . Forthree-particle quantum systems, their states with theform | Φ ± i are called Greenberg-Horne-Zeilinger (GHZ)states. Now, the multipartite entangled states like | Φ ± i = √ ( | · · · i ± | · · · i ) are also called mul-tipartite GHZ states.The method of distilling a mixed state into a max-imally entangled state is named entanglement purifi-cation, which has been widely studied in recent years[15, 16, 17, 18, 19, 20, 21, 22]. Another way of distill-ing less entangled pure states into maximally entangledstates that will be detailed here is called entanglementconcentration. Several entanglement concentration pro-tocols of pure nonmaximally entangled states have beenproposed recently. The first entanglement concentrationprotocol was proposed by Bennett et al [23] in 1996,which is called Schmidt projection method. In their pro-tocol [23], the two parties of quantum communicationneed some collective and nondestructive measurements ofphotons, which, however, are not easy to manipulate inexperiment. Also the two parties should know accuratelythe coefficients α and β of the partially entangled state α | i + β | i before entanglement concentration. Thatis, their protocol works under the condition that the twousers obtain information about the coefficients and pos-sess the collective and nondestructive measurement tech-nique. Another similar scheme is called entanglementswapping [24, 25]. In these schemes [24, 25], two pairs ofless entangled pairs belong to Alice and Bob. Then Alicesends one of her particles to Bob, and Bob performs aBell-state measurement on one of his particle and Alice’sone. So Bob has to own three photons of two pairs, andthey have to perform collective Bell-state measurements.Moreover, the parties should exploit a two-particle col-lective unitary evaluation of the quantum system and anauxiliary particle to project the partially entangled stateinto a maximally entangled one probabilistically.Recently, two protocols of entanglement concentrationbased on a polarization beam splitter (PBS) were pro-posed independently by Yamamoto et al [26] and Zhao et al [27]. The experimental demonstration of the lat-ter has been reported [28]. In their protocol, the partiesexploit two PBSs to complete the parity-check measure-ments of polarization photons. However, each of the twousers Alice and Bob has to choose the instances in whicheach of the spatial modes contains exactly one photon.With current technology, sophisticated single-photon de-tectors are not likely to be available, which makes it suchthat these schemes can not be accomplished simply withlinear optical elements.Cross-Kerr nonlinearity is a powerful tool to con-struct a nondestructive quantum nondemolition detec-tor (QND). It also has the function of constructing acontrolled-not (CNOT) gate and a Bell-state analyzer[33]. Cross-Kerr nonlinearity was widely studied in thegeneration of qubits [29, 30, 31] and the discrimina-tion of unknown optical qubits [32]. Cross-Kerr nonlin-earities can be described with the Hamiltonian H ck =¯ hχa + s a s a + p a p [33, 34], where a + s and a + p are the creationoperations and a s and a p are the destruction operations.If we consider a coherent beam in the state | α i p witha signal pulse in the Fock state | Ψ i s = c | i s + c | i s ( | i s and | i s denote that there are no photons and onephoton, respectively, in this state), after the interactionwith the cross-Kerr nonlinear medium the whole systemevolves as U ck | Ψ i s | α i p = e iH ck t/ ¯ h [ c | i s + c | i s ] | α i p = c | i s | α i p + c | i s | αe iθ i p , (4)where θ = χt and t is the interaction time. From thisequation, the coherent beam picks up a phase shift di-rectly proportional to the number of photons in the Fockstate | Ψ i s . This good feature can be used to construct aparity-check measurement device [33].In this paper, we present a different scheme for non-local entanglement concentration of partially entangledmultipartite states with cross-Kerr nonlinearities. By ex-ploiting a new nondestructive QND, the parties of quan-tum communication can accomplish entanglement con-centration efficiently without sophisticated single-photondetectors. Compared with the entanglement concentra-tion schemes based on linear optical elements [26, 27],the present scheme has a higher efficiency and yield.Moreover, it does not require that the parties knowaccurately information about the partially entangledstates—i.e., the coefficients of the states—different from schemes based on two-particle collective unitary evalua-tion [23, 24, 25]. These good features give this schemethe advantage of high efficiency and feasibility in practi-cal applications. b b b b Homodyne (cid:84)(cid:14) (cid:68)
X X (cid:84)(cid:14)
PBS
FIG. 1: The principle of our nondestructive quantum nonde-molition detector (QND). Two cress-Kerr nonlinearities areused to distinguish superpositions and mixtures of | HH i and | V V i from | HV i and | V H i . Each polarization beam splitter(PBS) is used to pass through | H i polarization photons andreflect | V i polarization photons. Cross-Kerr nonlinearity willcause the coherent beam to pick up a phase shift θ if thereis a photon in the mode. So the probe beam | α i will pick upa phase shift of θ if the state is | HH i or | V V i . Here b and b represent the up spatial mode and the down spatial mode,respectively. II. ENTANGLEMENT CONCENTRATION OFPURE ENTANGLED PHOTON PAIRSA. Primary entanglement concentration of lessentangled photon pairs
The principle of our nondestructive QND is shownin Fig.1. It is made up of four PBSs, two identi-cal cross-Kerr nonlinear media, and an X homodynemeasurement. If two polarization photons are initiallyprepared in the states | ϕ i b = c | H i b + c | V i b and | ϕ i b = d | H i b + d | V i b , the two photons combinedwith a coherent beam whose initial state is | α i p interactwith cross-Kerr nonlinearities, which will evolve the stateof the composite quantum system from the original one | Ψ i O = | ϕ i b ⊗ | ϕ i b ⊗ | α i p to | Ψ i T = [ c d | HH i + c d | V V i ] | αe iθ i p + c d | HV i| αe i θ i p + c d | V H i| α i p . (5)One can find immediately that | HH i and | V V i cause thecoherent beam | α i p to pick up a phase shift θ , | HV i topick up a phase shift 2 θ , and | V H i to pick up no phaseshift. The different phase shifts can be distinguished bya general homodyne-heterodyne measurement ( X homo-dyne measurement). In this way, one can distinguish | HH i and | V V i from | HV i and | V H i . This device isalso called a two-qubit polarization parity QND detec-tor. Our QND shown in Fig.1 is a little different fromthe one proposed by Nemoto and Munro [33]. With theQND in [33], the | HH i and | V V i pick up no phase shift.However, it is well known that a vacuum state (zero-photon state) can also cause there to be no phase shift onthe coherent beam. So one can not distinguish whethertwo photons or no photons pass through the two spa-tial modes. This modified QND can exactly check thenumber of photons if they have the same parity. a b PBS D R R PBS a b b S Alice Bob D D QND R R a S D FIG. 2: Schematic diagram of the proposed entanglement con-centration protocol. Two pairs of identical less entanglementphotons are sent to Alice and Bob from source 1 ( S ) andsource 2 ( S ). The QND is a parity-checking device. Thewave plates R and R rotate the horizontal and verticalpolarizations by 45 ◦ and 90 ◦ respectively. With the QND shown in Fig.1, the principle of ourentanglement concentration protocol is shown in Fig.2.Suppose there are two identical photon pairs with lessentanglement a b and a b . The photons a belong toAlice and photons b to Bob. The photon pairs a b and a b are initially in the following unknown polarizationentangled states: | Φ i a b = α | H i a | H i b + β | V i a | V i b , | Φ i a b = α | H i a | H i b + β | V i a | V i b , (6)where | α | + | β | = 1. The original state of the fourphotons can be written as | Ψ i ≡ | Φ i a b ⊗ | Φ i a b = α | H i a | H i b | H i a | H i b + αβ | H i a | H i b | V i a | V i b + αβ | V i a | V i b | H i a | H i b + β | V i a | V i b | V i a | V i b . (7)After the two parties Alice and Bob rotate the polariza-tion states of their second photons a and b by 90 ◦ withhalf-wave plates (i.e., R shown in Fig.2), the state ofthe four photons can be written as | Ψ i ′ = α | H i a | V i a | H i b | V i b + αβ | H i a | H i a | H i b | H i b + αβ | V i a | V i a | V i b | V i b + β | V i a | H i a | V i b | H i b . (8) Here a ( b ) is used to label the photon a ( b ) after thehalf-wave plate R .From Eq.(8), one can see that the terms | H i a | H i a | H i b | H i b and | V i a | V i a | V i b | V i b havethe same coefficient of αβ , but the other two termsare different. Now Bob lets the two photons b and b enter into the QND. With his homodyne measurement,Bob may get one of three different results: | HH i and | V V i lead to a phase shift of θ on the coherent beam, | HV i leads to 2 θ , and the other is | V H i , which leadsto no phase shift. If the phase shift of homodynemeasurement is θ , Bob asks Alice to keep these twopairs; otherwise, both pairs are removed. After onlythis parity-check measurement, the state of the photonsremaining becomes | Ψ i ′′ = 1 √ | H i a | H i a | H i b | H i b + | V i a | V i a | V i b | V i b ) . (9)The probability that Alice and Bob get the above stateis P s = 2 | αβ | .Now both pairs a b and a b are in the same polar-izations. Alice and Bob use their λ/ R to rotate the photons a and b by 45 ◦ . The unitarytransformation of 45 ◦ rotations can be described as | H i a → √ | H i a + | V i a ) , | H i b → √ | H i b + | V i b ) , | V i a → √ | H i a − | V i a ) , | V i b → √ | H i b − | V i b ) . (10)After the rotations, Eq. (9) will evolve into | Ψ i ′′′ = 12 √ | H i a | H i b + | V i a | V i b )( | H i a | H i b + | V i a | V i b ) + 12 √ | H i a | H i b − | V i a | V i b )( | H i a | V i b + | V i a | H i b ) . (11)The last step is to distinguish the photons a and b in different polarizations. Two PBSs are used to passthrough | H i polarization photons and reflect | V i pho-tons. From the Eq. (11), one can see that if the twodetectors D and D or the two detectors D and D fire, the photon pair a b is left in the state | φ + i a b = 1 √ | H i a | H i b + | V i a | V i b ) . (12)If D and D or D and D fire, the photon pair a b areleft in the state | φ − i a b = 1 √ | H i a | H i b − | V i a | V i b ) . (13)Both of these two states are the maximally entangledones. In order to get the same state of | Φ i + a b , one ofthe two parties Alice and Bob should perform a simplelocal operation of phase rotation on her or his photon.The maximally entangled states are generated with aboveoperations.In our scheme, only one QND is used to detect the par-ity of the two polarization photons. If the two photonsare in the same polarization | HH i or | V V i , the phaseshift of the coherent beam is θ , which is easy to detect bythe homodyne measurement. Furthermore, our schemeis not required to have sophisticated single-photon de-tectors, but only conventional photon detectors. Thisis a good feature of our scheme, compared with otherschemes. B. Reusing resource-based entanglementconcentration of partially entangled photon pairs
With only one QND, our entanglement concentrationhas the same efficiency as that based on linear optics [26,27]. The yield of maximally entangled states Y is | αβ | .Here the yield is defined as the ratio of the number ofmaximally entangled photon pairs, N m , and the numberof originally less entangled photon pairs, N l . That is, theyield of our scheme discussed above is Y = N m N l = | αβ | .In fact, Y is not the maximal value of the yield of theentanglement concentration scheme with the QND.In our entanglement concentration scheme above, thetwo parties Alice and Bob only pick up instances in whichBob gets the phase shift θ on his coherent beam andremoves the other instances. In this way, the photonpairs kept are in the state | Ψ i ′′ . However, if Bob choosesa suitable cross-Kerr medium and controls accurately theinteraction time t , he can make the phase shift θ = χt = π . In this way, 2 θ and 0 represent the same phase shift0. The two photon pairs removed by Alice and Bob inthe scheme above are just in the state | Φ i ′′ = α | H i a | V i a | H i b | V i b + β | V i a | H i a | V i b | H i b . (14)This four-photon system is not in a maximally entangledstate, but it can be used to get some maximally entangledstate with entanglement concentration. In detail, Aliceand Bob take a rotation by 90 ◦ on each photon of thesecond four-photon system and cause the state of thissystem to become | Φ i ′′ = β | H i a ′ | V i a ′ | H i b ′ | V i b ′ + α | V i a ′ | H i a ′ | V i b ′ | H i b ′ . (15)The state of the composite system composed of eight pho-tons becomes | Φ s i ′′ ≡ | Φ i ′′ ⊗ | Φ i ′′ = α β ( | H i a | V i a | H i b | V i b | H i a ′ | V i a ′ | H i b ′ | V i b ′ + | V i a | H i a | V i b | H i b | V i a ′ | H i a ′ | V i b ′ | H i b ′ )+ α | H i a | V i a | H i b | V i b | V i a ′ | H i a ′ | V i b ′ | H i b ′ + β | V i a | H i a | V i b | H i b | H i a ′ | V i a ′ | H i b ′ | V i b ′ . (16)For picking up the first two terms, Bob need only detectthe parities of the two photons b and b ′ with the QND.As the two polarization photons b and b ′ in the first twoterms have the same parity, they will cause the coherentbeam | α i p to have a phase shift θ = π . Those in theother two terms cause the coherent beam | α i p to have aphase shift 0.When Bob gets the phase shift θ = π , the eight photonscollapse to the state | Φ s i ′′′ = 1 √ | H i a | V i a | H i b | V i b | H i a ′ | V i a ′ | H i b ′ | V i b ′ + | V i a | H i a | V i b | H i b | V i a ′ | H i a ′ | V i b ′ | H i b ′ ) . (17)The probability that Alice and Bob get this state is P s = 2 | αβ | ( | α | + | β | ) . (18)They have the probability P ′ f = 1 − P s to obtain theless entangled state | Φ i ′′′ = α | H i a | V i a | H i b | V i b | V i a ′ | H i a ′ | V i b ′ | H i b ′ + β | V i a | H i a | V i b | H i b | H i a ′ | V i a ′ | H i b ′ | V i b ′ (19)which can be used to concentrate entanglement by itera-tion of the process discussed above. In this way, one canobtain easily the probability P s n = 2 | αβ | n ( | α | n + | β | n ) , (20)where n is the iteration number of the entanglement con-centration processes.For the four photons in the state described by Eq.(17),Alice and Bob can obtain a maximally entangled pho-ton pair with some single-photon measurements on theother six photons by choosing the basis X = {| ± x i = √ ( | H i ± | V i ) } . That is, Alice and Bob first rotatetheir polarization photons a , b , a ′ , b ′ , a ′ and b ′ by 45 ◦ , similar to the case discussed above (shown inFig.2), and then measure these six photons. If thenumber of the antiparallel outcomes obtained by Aliceand Bob is even, the photon pair a b collapses to thestate | φ + i a b = √ ( | H i a | H i b + | V i a | V i b ); otherwisethe photon pair a b collapses to the state | φ − i a b = √ ( | H i a | H i b − | V i a | V i b ).With the iteration of the entanglement concentrationprocess, the yield of our scheme is improved to be Y —i.e., Y = n X i =1 Y i , (21)where Y = | αβ | ,Y = 12 (1 − | αβ | ) | αβ | ( | α | + | β | ) ,Y = 12 (1 − | αβ | )[1 − | αβ | ( | α | + | β | ) ] | αβ | ( | α | + | β | ) ,. . .Y n = 12 n − (1 − | αβ | ) n − Y j =3 [1 − | αβ | j − ( | α | j − + | β | j − ) ] | αβ | n ( | α | n + | β | n ) . (22)The yield is shown in Fig.3 with the change of the iter-ation number of entanglement concentration processes n and the coefficient α ∈ [0 , √ ]. FIG. 3: (Color online) The yield ( Y ) is altered with the iter-ation number of entanglement concentration processes n andthe coefficient α ∈ [0 , √ ]. Certainly, Alice and Bob can also accomplish the itera-tion of the entanglement concentration by first measuringthe two photons a and b in the state | Φ i ′′ described byEq. (14) with the basis X and then concentrating somemaximally entangled states from the partially entangledquantum systems composed of the pairs a b . In fact, af-ter the measurements of the two photons with the basis X , Alice and Bob can transfer the state of photon pair a b to α | H i a | H i b + β | V i a | V i b with or without aunitary operation. Alice and Bob can accomplish the en-tanglement concentration with the same way discussedin Sec. II A.The same as the entanglement concentration schemeswith linear optical elements [26, 27], the present schemehas the advantage that the two parties of quantum com-munication are not required to know the coefficients of the less entangled states in advance in order to recon-struct some maximally entangled states. Moreover, thisscheme does not require sophisticated single-photon de-tectors and has a higher yield of maximally entangledstates than those based on linear optical elements [26, 27]as the efficiency in the latter is just | αβ | [the probabil-ity that Alice and Bob get an Einstein- Podolsky-Rosen(EPR) pair from two partially entangled photon pairsis 2 | αβ | in Refs. [26, 27]]. These good features makethe present entanglement concentration scheme more ef-ficient and more convenient than others in practical ap-plications. III. ENTANGLEMENT CONCENTRATION OFLESS ENTANGLED MULTIPARTITEGHZ-CLASS STATES
It is straightforward to generalize our entanglementconcentration scheme to reconstruct maximally entan-gled multipartite GHZ states from partially entangledGHZ-class states.Suppose the partially entangled N -particle GHZ-classstates are described as follows: | Φ ′ + i = α | HH · · · H i + β | V V · · · V i , (23)where | α | + | β | = 1. For two GHZ-class states, thecomposite state can be written as | Ψ ′ i = | Φ ′ + i ⊗ | Φ ′ + i = ( α | H i | H i · · · | H i N + β | V i | V i · · · | V i N ) ⊗ ( α | H i N +1 | H i N +2 · · · | H i N + β | V i N +1 | V i N +2 · · · | V i N ) . (24) G H Z-classG H Z-class R R PB S D QND R R R R D D D D D A lice B ob C harlie
FIG. 4: Schematic diagram of the multipartite entanglementconcentration scheme. 2 N particles in two partially entangled N -particle GHZ-class states are sent to N parties of quan-tum communication—say Alice, Bob, Charlie, etc. Photons 2and N + 2 are sent to Bob and enter into QND to completea parity-check measurement. After the QND measurement,Bob asks the others to retain their photons if his two photonshave the same parity ( | HH i or | V V i ) and remove them fornext iteration if Bob gets an odd parity ( | HV i or | V H i ). The principle of our entanglement concentrationscheme for multipartite GHZ-class states is shown inFig.4. 2 N photons in two pairs of N -particle non-maximally entangled GHZ-class states are sent to Alice,Bob, Charlie, ect. (i.e., the N parties of quantum com-munication). Each party gets two photons. One comesfrom the state | Φ + i and the other comes from | Φ + i ,shown in Fig.4. Suppose Alice gets photon 1 and the pho-ton N + 1 and Bob gets photon 2 and photon N + 2. Be-fore entanglement concentration, each party rotates hissecond polarization photon by 90 ◦ , similar to the case forconcentrating two-photon pairs. After the 90 ◦ rotations,the state of the 2 N photons becomes | Ψ ′ i ′ = α | H i | H i · · · | H i N | V i N +1 | V i N +2 · · · | V i N + αβ | H i | H i · · · | H i N | H i N +1 | H i N +2 · · · | H i N + αβ | V i | V i · · · | V i N | V i N +1 | V i N +2 · · · | V i N + β | V i | V i · · · | V i N | H i N +1 | H i N +2 · · · | H i N . (25)Bob lets photons 2 and N + 2 pass through his QND de-tector whose principle is shown in Fig.2. For | HH i and | V V i , Bob gets the result with an X homodyne measure-ment θ ; for | HV i , the result is 2 θ and | V H i will make nophase shift. By choosing the phase shift θ , Bob asks theothers to retain their photons; otherwise, all the partiesremove the photons. In this way, the whole state of theretained photons can be described as | Ψ ′ i ′′ = 1 √ | H i | H i · · · | H i N | H i N +1 | H i N +2 · · · | H i N + | V i | V i · · · | V i N | V i N +1 | V i N +2 · · · | V i N ) . (26)The success probability is 2 | αβ | , the same as that fortwo-photon pairs P s . The above state is a maximallyentangled 2 N -particle state. By measuring each of thephotons coming from the second GHZ-class state withbasis X , the parties will obtain a maximally entangled N -particle state, as after the photons N + 1, N + 2, . . . ,and 2 N pass through the R plates, which rotate the po-larizations of photons by 45 ◦ , the state of the compositesystem becomes | Ψ ′ i ′′′ = 1 √ | H i | H i · · · | H i N ( 1 √ ⊗ N ( | H i + | V i ) ⊗ N + | V i | V i · · · | V i N ( 1 √ ⊗ N ( | H i − | V i ) ⊗ N ] . (27)By measuring the N photon with the conventional pho-ton detectors, the N parties will obtain a maximally en-tangled state | GHZ + i ··· N if the number of parties whoobtain a single-photon measurement outcome | V i is even;otherwise, they will obtain the maximally entangled state | GHZ − i ··· N . Here | GHZ + i = 1 √ | H i | H i · · · | H i N + | V i | V i · · · | V i N ) , (28) and | GHZ − i = 1 √ | H i | H i · · · | H i N − | V i | V i · · · | V i N ) . (29)For the photons removed by the parties, the methoddiscussed in Sec. II B also works for improving the effi-ciency of a successful concentration of GHZ-class statesand the yield. In this time, one need only replace | HH i and | V V i in Sec. II B with | HH · · · H i and | V V · · · V i ,respectively. IV. DISCUSSION AND SUMMARY
Compared with the entanglement concentrationschemes [23, 24, 25] by evolving the composite systemand an auxiliary particle, the present scheme does not re-quire the parties of quantum communication to know ac-curately information about the less entanglement states.This good feature makes the present scheme more effi-cient than those in Refs. [23, 24, 25] as the decoherenceof entangled quantum systems depends on the noise ofquantum channels or the interaction with the environ-ment, which causes the two parties to be blind to theinformation about the state. With sophisticated single-photon detectors, entanglement concentration schemes[26, 27] with linear optical elements are efficient for con-centrating some partially entangled states. With the de-velopment of technology, sophisticated single-photon de-tectors may be obtained in the future even though theyare far beyond what is experimentally feasible at present.Cross-Kerr nonlinearity provides a good QND with whicha parity-check measurement can be accomplished per-fectly in principle [33]. With the QND, our entangle-ment concentration scheme has a higher efficiency andyield than those with linear optical elements [26, 27].In summary, we propose a different scheme for nonlocalentanglement concentration of partially entangled multi-partite states. We exploit cross-Kerr nonlinearities todistinguish the parity of two polarization photons. Com-pared with other entanglement concentration schemes,this scheme does not require a collective measurementand does not require the parties of quantum communica-tion to know the coefficients α and β of the less entan-gled states. This advantage makes our scheme have thecapability of distilling arbitrary multipartite GHZ-classstates. Moreover, it does not require the parties to adoptsophisticated single-photon detectors, which makes thisscheme feasible with present techniques. By iteration ofentanglement concentration processes, this scheme has ahigher efficiency than those with linear optical elements.All these advantages make this scheme more convenientin practical applications than others. ACKNOWLEDGEMENTS
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