Schemes for entanglement concentration of two unknown partially entangled states with cross-kerr nonlinearity

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1 Schemes for entanglement concentration of two unknown partially entangled states with cross-kerr nonlinearity Wei Xiong, and Liu Ye School of Physics and Material Science, Anhui University, Hefei 30039, People s Repulic of China Astract. We propose practical schemes for concentrating entanglement of a pair of unknown partially entangled Bell states and three-photon W states with cross-kerr nonlinearity. In the schemes, utilizing local operations and classical communication, two separated parties can otain one maximally entangled photon pair from two previously shared partially entangled photon pairs, and three separated parties can otain one maximally entangled three-photon W state and a maximally entangled cluster state from two identical partially entangled three-photon W state with a certain success proaility. Finally, we discuss the influences of sources of errors and de-coherence on the schemes. The proposed setup is very simple, just employing some linear optical elements and cross-kerr medium, which greatly simplifies the experimental realization of the schemes. The schemes are feasile within current experimental technology,. PACS numer. Keywords. concentration, linear optics, weak cross-kerr nonlinearity. Ⅰ. Introduction Quantum entanglement etween distriuted quantum systems is of fundamental importance to the future implementation of various quantum information processing, such as quantum teleportation [1-3], quantum secret key distriution [4-7], quantum computation [8], and quantum dense coding [9,10]. All the aove mentioned applications which can e realized ideally with unit success proaility and unit fidelity require that two or many distant parties share maximally entangled states. Actually, however, the two or many distant parties can not share maximally entangled

2 states faithfully ut some forms of non-maximally entangled pure states can e otained due to the influences of de-coherence and the imperfection at the sources. Under this condition, the success proaility of the implementation will e less than one. For this reason, entanglement concentration [11] schemes are of practical significance ecause they can extract maximally entangled states from some partially entangled states via applying local operations and classical communication (LOCC). Many theoretical and experimental schemes for otaining maximally entangled particles y LOCC have een proposed [1-9]. In 1996, Bennett et al. [1] proposed an original entanglement purification scheme for purifying Bell states y use of local operations on copies of noisy Bell pairs and classical communication etween two parties. After that, Zhao et al [17] proposed a proailistic scheme for entanglement concentration ased on the principle of quantum erasure and the Schmidt projection method. In their scheme, one can concentrate entanglement from aritrary identical non-maximally entangled pairs at distant locations. Yamamoto et al. [1] proposed an experimentally feasile concentration and purification scheme with linear optical elements such as polarizing eam splitters (PBSs) and quarter wave plates (QWPs). Yang et al [] and Cao et al [3] proposed schemes for entanglement concentration of unknown atomic entangled states via entanglement swapping and cavity decay with a certain success proaility in cavity QED, respectively. Although the entanglement concentration in a ipartite system has een studied intensively, there are few schemes for concentrating the non-maximally entangled states of the ipartite system and tripartite system exploiting cross-kerr nonlinearity. To the est of our knowledge, cross-kerr nonlinearity provides a good tool to construct nondestructive quantum non-demolition (QND) detectors, which have the potential availale of eing ale to condition the evolution of our system ut without necessarily destroying the single photons [30, 31]. Such QND detector can determine whether there are photons after the PBS or not, which cannot e accomplished only with PBS. For these reasons, exploiting cross-kerr nonlinearity is full of significances to realize entanglement concentration schemes for a pair of unknown pure non-maximally entangled Bell states and three-photon W states with cross-kerr

3 nonlinearity in our letter. But interestingly, we not only acquire the maximally entangled Bell states and W states, ut also a genuine cluster state, which is essential to the one-way computer [3], can e otained y LOCC. On the other hand, three-particle W state is easier to prepare than four-particle cluster state. Thus our schemes provide a new way to generate multiple-particle entanglement, which is profound to quantum computer in the further. In the scheme, we use the polarization of photons as quit and define horizontally (vertically) linear polarization H (V ) as the quit 0 (1 ). Ⅱ. Entanglement concentration with weak cross-kerr nonlinearity Before we outline our schemes of entanglement concentration, we riefly review the principle of QND measurement using weak cross-kerr nonlinearity first presented y Nemoto and Munro [30]. The Hamiltonian of a cross-kerr nonlinear medium can e descried y the form as follows (setting = 1): H QND χ n a c = n (1) where ( ) denotes the numer operator for mode a (c) and n a n c χ is the coupling strength of the nonlinearity, which is decided y the property of material. If we consider a signal state to have the form ψ = a with the proe eam a a initially in a coherent state α c, the cross-kerr interaction causes the comined system composed of a single photon and a coherent state to evolve as [30] ih t QND iθ U ψ α = e ( a ) α = a 0 α + 1 αe () ck c a a c a c a c whereθ = χt is introduced y the nonlinearity and t is the interaction time. We oserve immediately that the Fock state n a is unaffected y the interaction ut the coherent state α picks up a phase shift directly proportional to the numer of c photons na in the n state. For photons in the signal mode, the proe eam a n a in evolves to α e θ a. Through a general homodyne-heterodyne measurement (X c

4 homodyne measurement) of the phase of the coherent state, the signal state ψ will e projected into a definite numer state or superposition of numer states. Because the measurement can e performed with high fidelity, the projection is nearly deterministic. This technique was first used to realize a CNOT gate [30], a parity projector [33], and the Bell state [33]. It provides a new route to new quantum computation [34]. The requirement for this technique is αθ 1[34], whereα is the amplitude of the coherent state. Even with the weak nonlinearity (θ is small), this requirement can e satisfied with large amplitude of the coherent state. Then this requirement may e feasile with current experimental technology. Our schemes of entanglement concentration also work with the weak cross-kerr nonlinearity. A. Concentration scheme for two unknown partially entangled photon pairs In this section, we show how the two separated parties Alice and Bo can concentrate a maximally entangled photon pair from two identical partially entangled photon pairs y LOCC. We assume that Alice and Bo are given two pairs of photons in the following polarization entangled states ψ ψ = ( α H H + β V V )) ( α H H + β V V ) (3) where α and β are aritrary complex numers satisfying α + β = 1. Alice holds photons 1 and 3, Bo holds photons and 4, respectively. Alice and Bo can transform these photons into a maximally entangled photon pair in modes ' and3' in the following approach. To egin with, we expand Eq. (3) as φ = α H H H H β V V V V αβ( H H V V + V V H H ) (4) We note that the third term and the fourth term in Eq. (4) have the same coefficientsαβ. So we desire to distinguish the third term and the fourth term from the first term and the second term. We let photons in modes 1 and 3 pass through PBS1, PBS, PBS 3 and PBS 4. The state of photons is split individually on the PBSs into two spatial modes, which transmit H and reflect V. The proe eam then

5 interacts with the photons in the horizontal mode and vertical mode through the α θ θ X X R 45 ' 1 a 1 a 5 1 D 1 H a D 1 V R 45 R 90 D 3 H a 3 a 6 3' D 3 V a 4 Figure 1. (Color online) The schematic diagram of the proposed entanglement concentration for two unknown partially entangled photon pairs. Polarization eam splitters(pbs) transmit H photons and reflect V photons. HD denotes homodyne measurement, R denotes detector. λ wave plate, P represents π -phase shifter, and D denotes cross-kerr nonlinear medium, the photons on the modes and a gets the same phase a1 3 shift θ with their Homodyne measurements on their coherent states, as shown in figure 1. So Eq. (4) can e written as φ = α H H H H α e θ i a5 a6 4 + β V V V V a5 a6 4 ( ) i + αβ H V H V + V H V H αe θ (5) a5 a6 4 a5 a6 4 Through a general X homodyne measurement, if we find that proe mode is in the coherent state α e iθ, the four-photon state will e projected into the following state: φ = αβ( H V H V + V H V H ) (6) 4 3 a5 a6 4 a5 a6 In the following, Alice and Bo rotate the polarizations of their photons in modes 4, and a y 45 and90 using λ wave plates ( R45 and R 90 ), the action of R 45 is a5 6 given y

6 1 H ( H + V ), 1 V ( H V ) (7) and the action of R90 makes H V. Therefore, Eq.(6) is changed into the state φ = 1 4 ( H H H H H V H H + V H H H V V H H + H H V V + H V V V V H V V V V V V (8) ) 1 4 3' Finally, let the photons in modes and pass through PBS 5 and PBS 6, 1 4 respectively. Apparently, if Alice and Bo detect the photons in the polarization state H H ( V V ), then the remaining two photons in modes 3' and ' are left 1 4 in the state 1 4 φ + 1 = ( 3'' H H + V V ) (9) Similarly, if Alice and Bo detect the photons in the polarization state H V ( V H ), then the remaining two photons in modes 3' and ' are left 1 4 in the state 1 4 φ 1 = ( 3'' H H V V ) (10) which can e transformed into equation (9) y applying a π -phase shifter P to change the sign of the polarization state V. Therefore, the total proaility of sharing a maximally entangled photon pair in the state φ + is α (1 α ), which is plotted in figure 4(a). B. Concentration scheme for two partially entangled three-photon W state In the following, we assume that three parties Alice, Bo and Charlie share an unknown three-photon polarization entangled states: ϕ = γ H H V + δ( H V H + V H H ) (11)

7 where γ andδ are aritrary complex numers satisfying γ + δ = 1. Alice holds photons 6, Bo and Charlie hold photons 4 and 5, respectively. Furthermore, Alice also holds an identical state (11) on her hand ϕ. Therefore, the state of the whole 13 system is given y ϕ ϕ = [ γ H H V + δ( H V H + V H H )] [ γ H H V + δ( H V H + V H H ) ] (1) Alice, Bo and Charlie can transform these photons into a maximally entangled three-photon W state in modes ', 4' and 5' in the following way. First, we expand Eq. (1) as Φ = γ H H V H H V γδ ( H H V H V H + H H V V H H H V H H H V + V H H H H V ) δ ( H V H H V H + H V H V H H V H H H V H + V H H V H H ) (13) We oserve that there are four terms which have the same coefficientγδ. Therefore, we want to distinguish them from the state (13). We send photons 1,, 3 and 6 to pass through PBS j(j=1, 8), then the proe eam interacts with the photons in the horizontal mode and vertical mode through the cross-kerr nonlinear medium, the photons in modes and a get the same phase shift θ, the photon in mode gets a a a 5 phase shift θ and the photon in mode a6 picks up a phase shift θ with their x Homodyne measurements on their coherent states, as shown in figure. Consequently, equation (13) can e written as Φ = γ H H V V H H αe i 1 a1 a3 a6 a8 + γδ + ( H H V H H V H H V H V H a1 a3 a6 a7 a1 a3 a6 a7 + H V H V H H + V H H V H H α ) a1 a4 a5 a8 a a3 a5 a8 θ

8 + δ + ( H V H H H V H V H H V H a1 a4 a5 a7 a1 a4 a5 a7 + V H H H H V + V H H H V H e θ (14) ) i α a a3 a5 a7 a a3 a5 a7 If we find that proe mode is in the coherent state α, the aove state will e projected to the state: Φ = + γδ ( H H V H H V H H V H V H a9 a10 a11 a1 a9 a10 a11 a1 + H V H V H H + V H H V H H (15) ) a9 a10 a11 a1 a9 a10 a11 a1 In the following, to egin with, Alice rotates the polarizations of her photons in mode a11 y 90 using λ wave plates ( R 90 ), then she rotates the polarizations of her photons in mode a11 y 45 using λ wave plates ( R 45 ), respectively, as shown in figure. So equation (15) will e given y γδ Φ = [( H + V ) 3 a H H H H V + ( H + V ) 9' a10 a11' a1 a H H H V H 9' a10 a11' a1 + ( H + V ) V V V H H + ( H V ) H V V H H )] a9' a10 a11' a1 a9' a10 a11' a1 Then we send photons in modes,, and a to pass through PBS i ( i a9' a10 a11' 1 =9,10, 16), after that, the proe eam interacts with the photons in the horizontal mode and vertical mode through the cross-kerr nonlinear medium, and we find the (16) photons in modes and a get the same phase shift a14 15 θ, the photon in modes a 17 and a0 pick up the same phase shift θ. Therefore, Eq.(16) evolves to the state γδ Φ = [( H H H H H V + H H H H V H 4 1' ' 3' 6' 1' ' 3' 6' V V V V H H ) α + ( V H H H V H 1' ' 3' 6' 1' ' 3' 6' + V H H H H V e + H V V V H H iθ ) α ( 1' ' 3' 6' 1' ' 3' 6' V H V V H H e + H H V V H H e (17) iθ iθ ) α α ] 1' ' 3' 6' 1' ' 3' 6'

9 α θ θ θ θ θ θ θ θ X X R 45 1 a 1 a 9 a 9' a 13 1' 1' D F ' 4 4 a a 3 a 10 a 14 a 15 ' '' ' D S a 4 a 16 R 90 ' a 5 a11 a11' a 17 3' 3' D F a 6 a 18 3' D S 6 a 7 a 1 a 19 6' 6' D F a 8 a 0 6' D S Figure. (Color online) The schematic diagram of the proposed entanglement concentration for two unknown partially entangled three-photon W state. Rotated Polarization eam splitters(fs-pbs) transmit F polarization photons and reflect S polarization photons. HD denotes homodyne measurement, R denotes λ wave plate, P represents π -phase shifter, and D denotes detector. Through a general X homodyne measurement, we find if the proe eam is in the coherent state α, the state (17) will e projected to the state γδ Φ = ( 5 1' ' 3' 6' 1' ' 3' 6' 4 H H H H H V + H H H H V H 5 + V V V V H H ) (18) 1' ' 3' 6' Finally, let the photons in modes1', 3' and 6' pass through a series of rotated polarizing eam splitters (FS-PBS) k (k=1,, 3), which change V and H into a new frame as 1 V F S = ( + ), H ( F S ) = 1 (19) and always reflect S-polarizing photons and transmit F-polarizing photons. In the new frame, the state given in Eq. (18) can ecome into γδ Φ = [ F F F ( H H V + H V H + V H H ) 6 1' 3' 6' 4 '4'6'

10 + F S S ( H H V + H V H + V H H ) 1' 3' 6' '4'6' + S S F ( H H V + H V H + V H H ) 1' 3' 6' '4'6' + S F S ( H H V + H V H + V H H ) 1' 3' 6' '4'6' F F S ( H H V + H V H V H H ) 1' 3' 6' '4'6' F S F ( H H V + H V H V H H ) 1' 3' 6' '4'6' S F F ( H H V + H V H V H H ) 1' 3' 6' '4'6' S S S ( H H V + H V H V H H ) ] (0) 1' 3' 6' '4'6' Oviously, when Alice detects the photons in the polarization state F F F ( F S S, S S F or S F S ), equation (0) will e 1' 3' 6' 1' 3' 6' 1' 3' 6' 1' 3' 6' projected into the state + 1 W = ( H H V + H V H + V H H )'4'6' (1) 3 Similarly, if Alice detects the photons in the polarization state S S S ( F F S, F S F, S F F ), equation (0) will 1' 3' 6' 1' 3' 6' 1' 3' 6' 1' 3' 6' e projected into the state 1 W = ( H H V + H V H V H H )'4'6' () 3 which can e transformed into equation (1) y utilizing a π -phase shifter P to change the sign of the polarization state V into V. Therefore, the total success ' '' 3 proaility of otaining three-photon W state is γδ, which is plotted in figure 4(a). Compared to [35], the proaility of our scheme for concentrating a maximally entangled three-photon W state is aout 9times than theirs, and our scheme is easier to operate in practical realization, which is feasile within current technology. C. Extracting a maximally entangled four-photon cluster state from two unknown partially entangled three-photon W state

11 From equation (14), we find that if proe mode is in the coherent state iθ αe through X homodyne measurement, the state in equation (14) will e projected to the state: Ψ = δ H V H H V H + H V H V H H ( V H H H V H + V H H V H H ) (3) The setup is shown in figure 3. Susequently, Alice and Charile send photons in modes and to pass through 5 π cross-kerr medium. The action of π-cross-kerr medium evolves the modes and as follows from Eq. (1), which in the 5 polarization asis produces a π phase shift on the VV term. Conceptually, the simplest such two-quit gate is the CZ gate, i.e., HH HH ; HV HV ; VH VH ; VV VV. (4) After that, Alice and Charlie rotate the polarizations of her photons in mode 1 and 4 y 90 using λ wave plates ( R 90 ), as shown in figure 3. Thus Eq. (3) will evolve to the state Ψ = δ V V H V V H + V V H H H H ( 1' ' 3 4' 5' 6 1' 4' 3 4' 5' 6 + H H H V V H + H H H H H H ) (5) 1' ' 3 4' 5' 6 1' ' 3 4' 5' 6 Finally, Alice detects the photons in modes and, and we find Alice s result is 3 6 H H. 3 6 Following this way, Alice, Bo and Charlie share a maximally entangled cluster state 1 C = ( H H H H + V V H H 1' ' 4' 5' 1' 4' 4' 5' + H H V V V V V V ) (6) 1' ' 4' 5' 1' ' 4' 5' The success proaility is 4 4 δ, which is also plotted in figure 4(a).

12 α θ θ θ θ X X R 90 1 a 1 1' 4' 4 4 a 3 a 4 a 3 a 5 D 3 H π C K M ' 5' 5 a 6 D 3 V 6 a 7 6 D 6 H a 8 5 D 6 V Figure 3. (Color online) The schematic diagram of the proposed entanglement concentration for two unknown partially entangled four-photon cluster state. π CKM: π cross-kerr medium. Ⅲ. Discussion and Analysis In this section, let us riefly analyze and discuss some practical issues in relation to the experimental feasiility of our schemes. Firstly, we simply discuss the sources of errors and their effects. There are two types of errors on the proe mode: (1) an intrinsic measurement error which arises from the fact that phase-shifted coherent i states α e ± θ and coherent state α of the proe mode are not completely orthogonal and a measurement result in one parity suspace could have come form the opposite parity state. This intrinsic error, given y P ( θ) = erfc[ α sin θ / ]/ which can e err suppressed (made small) when αθ 1[30]. For instance withαθ π, P err Choosing the mean photon numer per pulse to e on the order of 1 10 (corresponding to 6 α 10 ) in the realistic pumps, a weak nonlinearity 6 θ should e sufficient to satisfyαθ π. () Errors due to photon loss, decoherence or phase noise on the proe mode. In the real situation, decoherence is inevitale, the photon loss may occur when the coherent state transmitted through an

13 optical fier. When photon loss occurs, the quit states will evolve to the mixed states after the homodyne detection [36-38], after which the fidelity of the proposed schemes will degrade. As descried aove, the amplitude of the coherent state α may e large enough to satisfy the requirement αθ > 1 when the cross-kerr nonlinearity is small. However, as the increasing of the amplitude of the coherent states, the fidelity of these schemes will decrease simultaneously due to the decoherence (photon loss). Fortunately, the decoherence can e made aritrarily small simply y an aritrary strong coherent state associated with a displacement D( α ) performed on the coherent state and the QND photon-numer-resolving detection [37]. Additionally, the photon loss also causes de-phasing, corresponding to phase flip errors, in the original two-quit state [36]. The degree of de-phasing is characterized y the parameters γ = η α θ / where η is the percentage of photons lost from the proe mode. We plot it to analyze more distinctly in figure 4(). γ must e keep small for the de-phasing to have a negligile effect. This in effect requiresη 1/ αθ which can e simply satisfied as long 1 αθ< 10.For instance with αθ π andη The error due to de-phasing is of the order10. Now as we make α θ γ αθ η (a) () Figure 4: (Color on line) (a) The proailities of concentrating Bell state (red curve), W state (lue curve) and cluster state(green curve). () The degree of de-phasing γ as the function of η andαθ.

14 larger, η needs to decrease to keep the de-phasing error small. Other potential errors worth mentioning are associated with differences in θ etween the various quits and uncertainty in the value of θ. Both of these errors can e managed and are small when Δθ / θ 1. Secondly, we analyze some self-kerr nonlinearity introduced y cross-kerr nonlinearity to our schemes. Self-kerr nonlinearity will cause self-phase modulation (SPM). In the continuous-time framework [39], the phase noises, due to SPM and dispersion etc., will e seen to severely degrade the fidelity of the proposed schemes. (However, Matsuda et al. [40] found that free-carrier dispersion as well as the optical kerr effect contriutes to the XPM.) It will also e shown that the phase noise is proportional to the response function s amplitude, implying that stronger nonlinearity is accompanied y increased phase noise. In the aove analysis, we have assumed that there is no SPM in our nonlinear material. It turns out that even if the SPM effect is present in the medium, it can e suppressed y operating in the slow-response regime. In Ref. [41], one scheme for the avoidance of the self-modulation effect is to use a resonant (3) χ medium. The measurement accuracy and the imposed phase noise on the signal wave satisfy the Heisenerg uncertainty principle. This demonstrates that the minimum product of the measurement accuracy and the imposed uncertainty on the conjugate oservale is achievale in the proposed QND measurement. Therefore, we only consider the ideal conditions, and the phase noise is not taken into account in our schemes. Furthermore, schemes need not use collective measurement y photon detectors and need not the states to e known eforehand, so we can use the conventional photon detectors that can only distinguish the vacuum and non-vacuum Fock numer states instead of using the sophisticated single-photon detectors distinguishing one or two photon states. The total success proailities of our schemes are P EPR = μαβ, Pw = γδ μ and Pcluster = 4 δ μ, with μ eing the quantum efficiency. For scalale linear optics quantum computation, the required quantum efficiency μ of the

15 single-photon detectors is extremely high, e.g. for gate success with proaility P 0.99, μ [4]. Although experiments for single-photon detectors have made tremendous progress, such detectors still go eyond the current experimental technologies. This greatly decreases the high-quality requirements of photon detectors in practical realization. For the giant cross-kerr nonlinearities, i.e.π, Ref. [43, 44] has shown that the nonlinear interaction etween weak optical pulses can e dramatically enhanced y a technique for generating stationary pulses, where a π phase shift is achievale. The technique used in Ref. [44] has een experimentally demonstrated [45]. Therefore, our schemes are experimentally feasile and can e implemented. Ⅳ. Conclusion We have proposed experimentally feasile schemes to realize entanglement concentration of two unknown partially entangled states with cross-kerr nonlinearity. In the schemes, separated parties can ontain a maximally entangled photon state from two identical partially entangled photon states y LOCC. The effects of sources of errors and de-coherence have een investigated. We have shown that most effects of de-coherence can e suppressed (or negligile) under current experimental technologies. Additionally, just employing some linear optical elements and cross-kerr medium, the schemes are feasile in current experimental technology. Following the technology developed in experiments on multi-photon entanglement engineering [46-49] and quantum communication [], our schemes may e useful for long-distance quantum information processing and quantum communication in the future. This work was supported y the National Science Foundation of China under Grant No , the Doctoral Foundation of the Ministry of Education of China under Grant No , and also y the Personal Development Foundation of Anhui Province (008Z018). Reference

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