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Journal of the Optical Society of America B

Journal of the Optical Society of America B

| OPTICAL PHYSICS

  • Editor: Grover Swartzlander
  • Vol. 30, Iss. 1 — Jan. 1, 2013
  • pp: 71–78
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Efficient entanglement concentration for arbitrary single-photon multimode W state

Lan Zhou, Yu-Bo Sheng, Wei-Wen Cheng, Long-Yan Gong, and Sheng-Mei Zhao  »View Author Affiliations


JOSA B, Vol. 30, Issue 1, pp. 71-78 (2013)
http://dx.doi.org/10.1364/JOSAB.30.000071


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Abstract

We put forward an efficient entanglement concentration protocol (ECP) for recovering the single-photon less-entangled W state into the maximally entangled W state with only two conventional auxiliary photons. The ECP includes two local steps, both of which are based on the weak cross-Kerr nonlinearities and the variable beam splitter (VBS). Benefiting from the cross-Kerr nonlinearities and the VBS, the ECP can be used repeatedly to further concentrate the less-entangled W state. All the advantages indicate that our protocol may be feasible and convenient in current quantum communications areas.

© 2012 Optical Society of America

1. INTRODUCTION

Entanglement is considered to be one of the most important resources in various quantum information processing (QIP) schemes. It not only can hold the power for the quantum nonlocality [1

1. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935). [CrossRef]

], but also can provide wide applications in the QIP [2

2. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

,3

3. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002). [CrossRef]

], such as the quantum teleportation [4

4. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993). [CrossRef]

6

6. F. G. Deng, C. Y. Li, Y. S. Li, H. Y. Zhou, and Y. Wang, “Symmetric multiparty-controlled teleportation of an arbitrary two-particle entanglement,” Phys. Rev. A 72, 022338 (2005). [CrossRef]

], quantum dense coding [7

7. C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992). [CrossRef]

], quantum communication [8

8. G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65, 032302 (2002). [CrossRef]

10

10. C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005). [CrossRef]

], and entanglement-based quantum key distribution [11

11. A. K. Ekert, “Quantum cryptography based on Bells theorem,” Phys. Rev. Lett. 67, 661–663 (1991). [CrossRef]

]. Among numerous entanglement forms, the multimode entanglement is an important entanglement form. In 1991, Tan et al. first articulated the notion of single-particle nonlocality [12

12. S. Tan, D. Walls, and M. Collett, “Nonlocality of a single photon,” Phys. Rev. Lett. 66, 252–255 (1991). [CrossRef]

14

14. A. Peres, “Nonlocal effects in Fock space,” Phys. Rev. Lett. 74, 4571–4571 (1995). [CrossRef]

]. They pointed out that the nonlocality could be determined by a Bell inequality test with a single photon in a superposition of two distinct spatial modes. Since then, many researchers have focused on the single-photon two-mode entanglement with the form
|ϕAB=12(|0A|1B+|1A|0B),
(1)
where A and B represent two different spatial modes, and |0 and |1 mean zero photons and one photon, respectively. The single-photon entanglement between two modes has been proved to be a valuable resource for cryptography [15

15. C. Silberhorn, T. C. Ralph, N. Lütkenhaus, and G. Leuchs, “Continuous variable quantum cryptography: beating the 3 dB loss limit,” Phys. Rev. Lett. 89, 167901–167904 (2002). [CrossRef]

,16

16. Ch. Silberhorn, N. Korolkova, and G. Leuchs, “Quantum key distribution with bright entangled beams,” Phys. Rev. Lett. 88, 167902–167905 (2002). [CrossRef]

], state engineering [17

17. M. G. A. Paris, M. Cola, and R. Bonifacio, “Quantum state engeneering assisted by entanglement,” Phys. Rev. A 67, 042104 (2003). [CrossRef]

], and tomography of states and operations [18

18. G. M. D’Ariano and P. Lo Presti, “Quantum tomography for measuring experimentally the matrix elements of an arbitrary quantum operation,” Phys. Rev. Lett. 86, 4195–4198 (2001). [CrossRef]

,19

19. G. M. D’Ariano, P. Lo Presti, and M. G. A. Paris, “Using entanglement improves the precision of quantum measurements,” Phys. Rev. Lett. 87, 270404–270407 (2001). [CrossRef]

].

These achievements of the single-photon two-mode entanglement have ignited a great interest in the generation and application of the single-photon N-mode entanglement, which is called the single-photon W state with the form
|WN=1N(|1,0,0,,0+|0,1,0,,0++|0,0,1),
(2)
where the single photon is in the superposition of N spatial modes in different locations (N>2). Experimental schemes to generate these W states have already been proposed [20

20. A. Furusawa, J. L. Søensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998). [CrossRef]

23

23. O. Göckl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova, and G. Leuchs, “Experiment towards continuous-variable entanglement swapping: highly correlated four-partite quantum state,” Phys. Rev. A 68, 012319 (2003). [CrossRef]

]. For example, in 2003, Aoki et al. successfully created a fully inseparable three-mode W state and verified it by mixing three independent squeezed vacuum states in a network of beam splitters [22

22. T. Aoki, N. Takey, H. Yonezawa, K. Wakui, T. Hiraoka, A. Furusawa, and P. van Loock, “Experimental creation of a fully inseparable tripartite continuous-variable state,” Phys. Rev. Lett. 91, 080404–080407 (2003). [CrossRef]

]. In addition, in the same year, Göckl et al. created a single-photon four-mode entangled W state to realize entanglement swapping with pulsed beams [23

23. O. Göckl, S. Lorenz, C. Marquardt, J. Heersink, M. Brownnutt, C. Silberhorn, Q. Pan, P. van Loock, N. Korolkova, and G. Leuchs, “Experiment towards continuous-variable entanglement swapping: highly correlated four-partite quantum state,” Phys. Rev. A 68, 012319 (2003). [CrossRef]

]. It has been proven that the W state is robust to decoherence in the noisy environment [24

24. A. SenDe, U. Sen, M. Wieśniak, D. Kaszlikowski, and M. Żukowski, “Multiqubit W states lead to stronger nonclassicality than Greenberger-Horne-Zeilinger states,” Phys. Rev. A 68, 062306 (2003). [CrossRef]

26

26. R. Chaves and L. Davidovich, “Robustness of entanglement as a resource,” Phys. Rev. A 82, 052308 (2010). [CrossRef]

], and it displays an effective all-versus-nothing nonlocality as the number of N delocalizations of the single particle goes up [27

27. L. Heaney, A. Cabello, M. F. Santos, and V. Vedral, “Extreme nonlocality with one photon,” New J. Phys. 13, 053054–053065(2011). [CrossRef]

]. Unfortunately, in the practical application, the maximally entangled W state may inevitably suffer decoherence under realistic conditions, which can make it degrade to the pure less-entangled state with the form
|WN=a1|1,0,0,,0+a2|0,1,0,,0++aN|0,0,1,
(3)
where |a1|2+|a2|2++|aN|2=1, and all the ai(i=1,2,,N) are not equal. In practical application, such a less-entangled state may decrease after the entanglement swapping and cannot ultimately set up the high-quality quantum-entanglement channel [28

28. L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001). [CrossRef]

]. Therefore, in practical applications, it is necessary to recover the less-entangled W state into the maximally entangled W state.

Entanglement concentration is a powerful tool for distilling the maximally entangled state from the pure less-entangled state [29

29. C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052 (1996). [CrossRef]

47

47. B. Gu, “Single-photon-assisted entanglement concentration of partially entangled multiphoton W states with linear optics,” J. Opt. Soc. Am. B 29, 1685–1689 (2012). [CrossRef]

]. In 1996, Bennett et al. proposed the first entanglement concentration protocol (ECP), which is called the Schmidit projection method [29

29. C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052 (1996). [CrossRef]

]. Although some collective measurements in the protocol are hard to manipulate under present experimental conditions, this protocol is regarded as a great start for the ECP. After that, various ECPs have been put forward, successively. For example, in 1999, Bose et al. proposed an ECP based on the quantum swapping [30

30. S. Bose, V. Vedral, and P. L. Knight, “Purification via entanglement swapping and conserved entanglement,” Phys. Rev. A 60, 194–197 (1999). [CrossRef]

], and later it was developed by Shi et al. [31

31. B. S. Shi, Y. K. Jiang, and G. C. Guo, “Optimal entanglement purification via entanglement swapping,” Phys. Rev. A 62, 054301 (2000). [CrossRef]

]. In 2001, Yamamoto et al. and Zhao et al. proposed two similar ECPs based on the linear optics independently [32

32. Z. Zhao, J. W. Pan, and M. S. Zhan, “Practical scheme for entanglement concentration,” Phys. Rev. A 64, 014301 (2001). [CrossRef]

,33

33. T. Yamamoto, M. Koashi, and N. Imoto, “Concentration and purification scheme for two partially entangled photon pairs,” Phys. Rev. A 64, 012304 (2001). [CrossRef]

]. In 2009, Sheng et al. proposed an ECP for single-photon entanglement with the help of cross-Kerr nonlinearity [34

34. Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Single-photon entanglement concentration for long-distance quantum communication,” Quantum Inf. Comput. 10, 272–281 (2010).

]. The ECPs described above focus on the two-mode entanglement. There are also some ECPs for the less-entangled W state [43

43. A. Yildiz, “Optimal distillation of three-qubit W states,” Phys. Rev. A 82, 012317 (2010). [CrossRef]

47

47. B. Gu, “Single-photon-assisted entanglement concentration of partially entangled multiphoton W states with linear optics,” J. Opt. Soc. Am. B 29, 1685–1689 (2012). [CrossRef]

], such as the optimal distillation of three-qubit symmetry W states [43

43. A. Yildiz, “Optimal distillation of three-qubit W states,” Phys. Rev. A 82, 012317 (2010). [CrossRef]

], the concentration of special unknown partially entangled three-photon W states with linear optics [44

44. H. F. Wang, S. Zhang, and K. H. Yeon, “Linear optical scheme for entanglement concentration of two partially entangled threephoton W states,” Eur. Phys. J. D 56, 271–275 (2010). [CrossRef]

], the ECP for polarized less-entangled W states [45

45. Y. B. Sheng, L. Zhou, and S. M. Zhao, “Efficient two-step entanglement concentration for arbitrary W states,” Phys. Rev. A 85, 044302 (2012). [CrossRef]

], and the ECP for unknown special partially entangled N-photon W states with both linear optics and cross-Kerr nonlinearity [46

46. F. F. Du, T. Li, B. C. Ren, H. R. Wei, and F. G. Deng, “Single-photon-assisted entanglement concentration of a multi-photon system in a partially entangled W state with weak cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 29, 1399–1405 (2012). [CrossRef]

,47

47. B. Gu, “Single-photon-assisted entanglement concentration of partially entangled multiphoton W states with linear optics,” J. Opt. Soc. Am. B 29, 1685–1689 (2012). [CrossRef]

]. However, most ECPs described above focus on the W state in polarization degree of freedom, and they cannot deal with the less-entangled single-photon W state. Though the ECPs for less-entangled single-photon two-mode state α|0|1+β|1|0 have been proposed, these ECPs cannot be extended to concentrate the less–entangled single-photon multimode W state [34

34. Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Single-photon entanglement concentration for long-distance quantum communication,” Quantum Inf. Comput. 10, 272–281 (2010).

].

In this paper, we put forward an efficient ECP for recovering the less-entangled single-photon three-mode W state into the maximally entangled W state, inspired by the ECPs for less-entangled W state in polarization degree of freedom [43

43. A. Yildiz, “Optimal distillation of three-qubit W states,” Phys. Rev. A 82, 012317 (2010). [CrossRef]

]. The weak cross-Kerr nonlinearity is the key element of our ECP, which is adopted to carry out the nondestructive photon-number measurement. The ECP includes two steps. With the help of the variable beam splitter (VBS) and the weak cross-Kerr nonlinearity, only a conventional auxiliary photon is needed to complete each concentration step. The ECP has several advantages. First, different from the ECP in [44

44. H. F. Wang, S. Zhang, and K. H. Yeon, “Linear optical scheme for entanglement concentration of two partially entangled threephoton W states,” Eur. Phys. J. D 56, 271–275 (2010). [CrossRef]

], we do not require two copies of less-entangled W state. Second, different from the ECP in [34

34. Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Single-photon entanglement concentration for long-distance quantum communication,” Quantum Inf. Comput. 10, 272–281 (2010).

], we do not need to send back the second pair of less-entangled states, which can largely simplify the practical operation. Third, we do not require the large cross-Kerr nonlinearities, and the weak cross-Kerr nonlinearities can complete the task. Forth, our ECP can be used repeatedly to get a high success probability. All the advantages make our ECP useful and convenient in the current QIP. The paper is organized as follows: in Section 2, we explain the details of the ECP, and in Section 3, we calculate the approximate success probability and make a discussion and summary.

2. ECP FOR ARBITRARY LESS-ENTANGLED SINGLE-PHOTON W STATE

Before explaining our ECP detachedly, let us briefly introduce the key element of our protocol, the cross-Kerr nonlinearity. In the recent years, the cross-Kerr nonlinearity has been widely used to construct the Controlled-not (CNOT) gate [48

48. K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-nOT gate,” Phys. Rev. Lett. 93, 250502(2004). [CrossRef]

,49

49. Q. Lin and J. Li, “Quantum control gates with weak cross-Kerr nonlinearity,” Phys. Rev. A 79, 022301 (2009). [CrossRef]

] and Bell-state analyzer [50

50. S. D. Barrett, P. Kok, K. Nemoto, R. G. Beausoleil, W. J. Munro, and T. P. Spiller, “Symmetry analyzer for nondestructive Bell-state detection using weak nonlinearities,” Phys. Rev. A 71, 060302(R) (2005). [CrossRef]

,51

51. Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement,” Phys. Rev. A 81, 032307 (2010). [CrossRef]

], and so on [52

52. Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-Kerr nonlinearity,” Phys. Rev. A 77, 042308 (2008). [CrossRef]

57

57. B. He, Q. Lin, and C. Simon, “Cross-Kerr nonlinearity between continuous-mode coherent states and single photons,” Phys. Rev. A 83, 053826 (2011). [CrossRef]

]. In particular, it has become a powerful tool in the construction of the quantum nondemolition detector (QND), which is widely used in the quantum purification and concentration [37

37. F. G. Deng, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A 85, 022311 (2012). [CrossRef]

,42

42. W. Xiong and L. Ye, “Schemes for entanglement concentration of two unknown partially entangled states with cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 28, 2030–2037 (2011). [CrossRef]

,46

46. F. F. Du, T. Li, B. C. Ren, H. R. Wei, and F. G. Deng, “Single-photon-assisted entanglement concentration of a multi-photon system in a partially entangled W state with weak cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 29, 1399–1405 (2012). [CrossRef]

,51

51. Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement,” Phys. Rev. A 81, 032307 (2010). [CrossRef]

,52

52. Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-Kerr nonlinearity,” Phys. Rev. A 77, 042308 (2008). [CrossRef]

]. The schematic drawing of the QND constructed by cross-Kerr nonlinearity can be described in Fig. 1. During the cross-Kerr interaction process, a coherent beam in the state |αp and a single photon with the form |φ=γ|0+δ|1 interact with the cross-Kerr material. The cross-Kerr nonlinearity has a Hamiltonian of the form
Hck=χna^nb^,
(4)
where χ is the coupling strength of the nonlinearity, which depends on the cross-Kerr material. na^ and nb^ are the photon-number operators for mode a and mode b, respectively. After the interaction, the system can evolve to
Uck|ψ|α=(γ|0+δ|1)|αγ|0|α+δ|1|αeiθ.
(5)
Here, |0 and |1 mean zero photons and one photon, respectively. θ=χt, where t means the interaction time for the signal with the nonlinear material. According to Eq. (5), we can easily find that the phase shift of the coherent state is directly proportional to the photon number in the spatial mode. Therefore, by measuring the phase shift, we can check the photon number without destroying the photons.

Fig. 1. Schematic drawing of the photon-number QND based on the weak cross-Kerr nonlinearity. Here, the photons in the spatial modes a1 and a2 pass through the cross-Kerr material. A photon in mode a1 can cause the phase shift of θ on the coherent state |αp, while a photon in mode a2 can cause the phase shift of θ. The phase shift can be determined with a momentum quadrature measurement, so that the photon number in the modes a1 and a2 can be checked without destroying the photons.

Now we start to explain our ECP for the arbitrary less-entangled single-photon three-mode W state. The schematic drawing of the ECP is shown in Fig. 2. We suppose a single photon source, here named S1, emits a photon and sends it to the three parties, say Alice, Bob, and Charlie in the spatial mode a1, b1, and c1, respectively. Therefore, a less-entangled single-photon three-mode W state shared by the three parties is created, which can be written as
|ψa1b1c1=α|1,0,0a1b1c1+β|0,1,0a1b1c1+γ|0,0,1a1b1c1.
(6)
α, β, and γ are the coefficients of the initial W state, where |α|2+|β|2+|γ|2=1 and αβγ.

Fig. 2. Schematic drawing of our ECP for distilling the maximally entangled single-photon W state from arbitrary single-photon less-entangled W state. The protocol includes two similar steps. In each concentration step, only a conventional auxiliary photon is needed to complete the task. In both steps, the QND is adopted to carry out the nondestructive photon-number measurement. The VBS is used to adjust the coefficients of the entangled state and ultimately obtain the maximally entangled state [58]. Moreover, with the help of the QND and VBS, the protocol can be used repeatedly to get a high success probability.

Our ECP includes two steps. In the first step, the single-photon source, here named S2, emits an auxiliary single photon and sends it to Bob. Bob makes this single photon pass through a VBS, here named as VBS1, with the transmission of t1. The VBS is a common optical element in current technology. Recently, Osorio et al. reported their results about heralded photon amplification for quantum communication with the help of the VBS [58

58. C. I. Osorio, N. Bruno, N. Sangouard, H. Zbinden, N. Gisin, and R. T. Thew, “Heralded photon amplification for quantum communication,” Phys. Rev. A 86, 023815 (2012). [CrossRef]

]. In their protocol, they can adjust the splitting ratio of VBS from 5050 to 9010 to increase the visibility from 46.7±3.1% to 96.3±3.8%. Using VBS1, an entangled single-photon state between the spatial modes d1 and d2 can be created with the form
|ψd1d2=1t|1,0d1d2+t|0,1d1d2.
(7)

Therefore, the whole two-photon system can be described as
|Ψa1b1c1d1d2=|ψa1b1c1|ψd1d2=(α1t|1,0,0,1,0+αt|1,0,0,0,1+β1t|0,1,0,1,0+βt|0,1,0,0,1+γ1t|0,0,1,1,0+γt|0,0,1,0,1)a1b1c1d1d2.
(8)

Then, Bob makes the photons in the spatial modes b1 and d1 pass through the QND, whose structure is shown in Fig. 1. After the QND, the whole system combined with the coherent state can evolve to
|Ψa1b1c1d1d2|αα1t|1,0,0,1,0a1b1c1d1d2|αeiθ+αt|1,0,0,0,1a1b1c1d1d2|α+β1t|0,1,0,1,0a1b1c1d1d2|α+βt|0,1,0,0,1a1b1c1d1d2|αeiθ+γ1t|0,0,1,1,0a1b1c1d1d2|αeiθ+γt|0,0,1,0,1a1b1c1d1d2|α.
(9)

According to Eq. (9), we can easily find that both the items α1t|1,0,0,1,0a1b1c1d1d2 and γ1t|0,0,1,1,0a1b1c1d1d2 can make the coherent state pick up the phase shift of θ, while the item βt|0,1,0,0,1a1b1c1d1d2 can make it pick up θ. The other items make the coherent state pick up no phase shift. In the homodyne measurement process, ±θ is undistinguishable. After the measurements, Bob only selects the items corresponding to the phase shift ±θ and discards other items. Therefore, the Eq. (9) can collapse to
|Ψ1a1b1c1d1d2=α1t|1,0,0,1,0a1b1c1d1d2+βt|0,1,0,0,1a1b1c1d1d2+γ1t|0,0,1,1,0a1b1c1d1d2.
(10)

Then Bob needs to make the photons in the spatial modes d1 and d2 pass through the 5050 beam splitter (BS1), which makes
d^1|0=12(e^1|0+e^2|0),d^2|0=12(e^1|0e^2|0).
(11)

Here, d^j and e^j (j=1, 2) are the creation operators of the spatial mode dj and ej, respectively. From Fig. 2, the e1 and e2 are the output spatial modes. The creation operators obey the rules that i^j|0=|1ij and (i^j)2|0=2|2ij. Therefore, after passing through the BS, Eq. (10) can evolve to
|Ψ1a1b1c1e1e2=(α1t2|1,0,0a1b1c1+βt2|0,1,0a1b1c1+γ1t2|0,0,1a1b1c1)|1e1+(α1t2|1,0,0a1b1c1βt2|0,1,0a1b1c1+γ1t2|0,0,1a1b1c1)|1e2.
(12)

It can be found that if the detector D1 fires, Eq. (12) will collapse to
|ψ1a1b1c1=α1t|1,0,0a1b1c1+βt|0,1,0a1b1c1+γ1t|0,0,1a1b1c1,
(13)
while if the detector D2 fires, Eq. (12) will collapse to
|ψ1a1b1c1=α1t|1,0,0a1b1c1βt|0,1,0a1b1c1+γ1t|0,0,1a1b1c1.
(14)

There is only a phase difference between Eq. (13) and Eq. (14). If Bob gets Eq. (14), he only needs to carry out a phase-flip operation with the help of a half-wave plate, and Eq. (14) can be converted to Eq. (13). Meanwhile, it can be found that if we can adopt a suitable VBS with the transmission t1=(|α|2)/(|α|2+|β|2), where the subscript “1” means in the first concentration step, Eq. (13) can evolve to
|ψ1a1b1c1=αβα2+β2|1,0,0a1b1c1+αβα2+β2|0,1,0a1b1c1+γβα2+β2|0,0,1a1b1c1,
(15)
which can be rewritten as
|ψ1a1b1c1=α|1,0,0a1b1c1+α|0,1,0a1b1c1+γ|0,0,1a1b1c1.
(16)

So far, we have completed the first concentration step. In this step, with the help of the cross-Kerr nonlinearities and the VBS, we successfully make the entanglement coefficients of the items |1,0,0a1b1c1 and |0,1,0a1b1c1 to be the same, with the success probability of
P11=|β|2(2|α|2+|γ|2)|α|2+|β|2.
(17)

Interestingly, we will prove that the first concentration step can be reused to further concentrate the less-entangled W state. Under the case that t1=(|α|2)/(|α|2+|β|2), the discarded items in Eq. (9) that make the coherent state pick up no phase shift can evolve to
|Ψa1b1c1d1d2=α2|1,0,0,0,1a1b1c1d1d2+β2|0,1,0,1,0a1b1c1d1d2+αγ|0,0,1,0,1a1b1c1d1d2.
(18)

Then, Bob still makes the photons in spatial modes d1 and d2 pass through the BS1. After BS1, it can be found if the detector D1 fires, Eq. (18) will collapse to
|ψ2a1b1c1=α2|1,0,0a1b1c1+β2|0,1,0a1b1c1+αγ|0,0,1a1b1c1,
(19)
while if the detector D2 fires, Eq. (18) will collapse to
|ψ2a1b1c1=α2|1,0,0a1b1c1β2|0,1,0a1b1c1+αγ|0,0,1a1b1c1.
(20)

Similarly, Eq. (20) can be easily converted to Eq. (19) by a phase-flip operation from Bob. Equation (19) can be rewritten as
|ψ2a1b1c1=α2α4+β4+α2γ2|1,0,0a1b1c1+β2α4+β4+α2γ2|0,1,0a1b1c1+αγα4+β4+α2γ2|0,0,1a1b1c1.
(21)

If we make α=(α2)/(α4+β4+α2γ2), β=(β2)/(α4+β4+α2γ2), and γ=(αγ)/(α4+β4+α2γ2), Eq. (21) will have the same form as Eq. (6). That is to say, following the first concentration step described above, Eq. (21) can be reconcentrated for the second round. In the second concentration round, the photon source S2 emits a single photon and Bob makes it pass though another VBS with the transmission of t12, where the subscript “1” means in the first concentration step and the superscript “2” means in the second concentration round. Bob still makes the photons in the spatial modes b1 and d1 pass through the QND, and selects the items that make the coherent state pick up the phase shift of ±θ. Finally, by making the photons in the spatial modes d1 and d2 pass through the BS1 and detecting the output state in the spatial modes e1 and e2, the new single-photon system can ultimately evolve to
|ψ3a1b1c1=α21t12|1,0,0a1b1c1+β2t12|0,1,0a1b1c1+αγ1t12|0,0,1a1b1c1.
(22)

If the transmission t12=(|α|2)/(|α|2+|β|2)=(|α|4)/(|α|4+|β|4), Eq. (22) can be converted into Eq. (16), with the success probability of
P12=|β|4(|αγ|2+2|α|4)(|α|2+|β|2)(|α|4+|β|4).
(23)

Similarly, by making the discarded items in the second round pass through the BS1, we can get a new less-entangled W state as
|ψ4a1b1c1=α4|1,0,0a1b1c1+β4|0,1,0a1b1c1+α3γ|0,0,1a1b1c1,
(24)
which can be reconcentrated for the third round.

Therefore, we have proved that if we can provide suitable VBS with the transmission t1K=(|α|2K)/(|α|2K+|β|2K), where the superscript “K” means the iteration time, the first concentration step can be used repeatedly to further concentrate the less-entangled W state to Eq. (16). The success probability in each concentration round can be written as
P1K=|β|2K(|α|2K2|γ|2+2|α|2K)(|α|2+|β|2)(|α|4+|β|4)(|α|2K+|β|2K),
(25)
and the total success probability of the first concentration step equals the sum of the probability in each concentration round, which can be described as
Ptotal1=P11+P12++P1K=K=1P1K.
(26)

Then, we start to explain the second concentration step of our ECP. Here, another single-photon source S2 emits a single photon and sends it to Charlie. Charlie also makes the photon pass through a VBS, here named VBS2, with a transmission of t2. Therefore, he can create another entangled single-photon state in the spatial modes g1 and g2 with the form
|ψg1g2=1t2|1,0g1g2+t2|0,1g1g2.
(27)

Then Charlie needs to make the photons in the spatial modes g1 and c1 pass through the QND. The new whole two-photon system combined with the coherent state can be described as
|ψ1a1b1c1|ψg1g2|αα1t2|1,0,0,1,0a1b1c1g1g2|αeiθ+αt2|1,0,0,0,1a1b1c1g1g2|α+α1t2|0,1,0,1,0a1b1c1g1g2|αeiθ+αt2|0,1,0,0,1a1b1c1g1g2|α+γ1t2|0,0,1,1,0a1b1c1g1g2|α+γt2|0,0,1,0,1a1b1c1g1g2|αeiθ.
(28)

In this way, as ±θ are undistinguishable during the measurement process, Charlie still selects the items that make the coherent state pick up the phase shift of ±θ and discards other items, so that Eq. (28) can collapse to
|Ψ2a1b1c1g1g2=α1t2|1,0,0,1,0a1b1c1g1g2+α1t2|0,1,0,1,0a1b1c1g1g2+γt2|0,0,1,0,1a1b1c1g1g2.
(29)

Then Charlie makes the photons in the spatial modes g1 and g2 pass through another 5050 BS, here named BS2, which makes
g^1|0=12(f^1|0+f^2|0),g^2|0=12(f^1|0f^2|0).
(30)

After the BS2, Eq. (29) can evolve to
|Ψ2a1b1c1f1f2=(α1t2|1,0,0+α1t2|0,1,0+γt2|0,0,1)a1b1c1|1f1+(α1t2|1,0,0+α1t2|0,1,0γt2|0,0,1)a1b1c1|1f2.
(31)

It is obvious that if the detector D3 fires, Eq. (31) can collapse to
|ψ5a1b1c1=α1t2|1,0,0a1b1c1+α1t2|0,1,0a1b1c1+γt2|0,0,1a1b1c1,
(32)
while if the detector D4 fires, Eq. (31) can collapse to
|ψ5a1b1c1=α1t2|1,0,0a1b1c1+α1t2|0,1,0a1b1c1γt2|0,0,1a1b1c1.
(33)

Equation (33) can be easily converted to Eq. (32) by a phase-flip operation from Charlie. According to Eq. (32), we can find if the transmission of the VBS2 is t2=(|α|2)/(|α|2+|γ|2), Eq. (32) can evolve to
|ψ5a1b1c1=13(|1,0,0a1b1c1+|0,1,0a1b1c1+|0,0,1a1b1c1),
(34)
which is the maximally entangled W state. So far, we have completed the second concentration step and finally obtain the maximally entangled W state with the success probability of
P21=3|α|2|γ|2(|α|2+|γ|2)(2|α|2+|γ|2),
(35)
where the subscript “2” means the second concentration step and the superscript “1” means in the first concentration round.

Similar with the first concentration step, we will prove that the second concentration step can also be used repeatedly to get a higher success probability. Under the case that t2=(|α|2)/(|α|2+|γ|2), the discarded items in Eq. (28) that make the coherent state pick up no phase shift can be written as
|Ψ2a1b1c1g1g2=α2|1,0,0,0,1a1b1c1g1g2+α2|0,1,0,0,1a1b1c1g1g2+γ2|0,0,1,1,0a1b1c1g1g2.
(36)

Charlie still makes the photons in the spatial modes g1 and g2 pass through the BS2. After the BS2, Eq. (36) can ultimately evolve to
|ψ6a1b1c1=α2|1,0,0a1b1c1+α2|0,1,0a1b1c1+γ2|0,0,1a1b1c1.
(37)

If Charlie can provide the suitable VBS with t22=(|α|4)/(|α|4+|γ|4), Eq. (38) can be converted to Eq. (34) with the success probability of
P22=3|α|4|γ|4(|α|2+|γ|2)(|α|4+|γ|4)(2|α|2+|γ|2).
(39)

Similarly, by making the discarded items in the second concentration round pass through the BS2, the discarded items can evolve to
|ψ8a1b1c1=α4|1,0,0a1b1c1+α4|0,1,0a1b1c1+γ4|0,0,1a1b1c1,
(40)
which can be reconcentrated in the third round. Therefore, we have proved that by providing the suitable VBS2 with t2K=(|α|2K)/(|α|2K+|γ|2K), where the superscript K′ means the iteration time, the second concentration step can be reused to get a higher success probability. The success probability of the second concentration step in each round can be written as
P2K=3|α|2K|γ|2K(|α|2+|γ|2)(|α|4+|γ|4)(|α|2K+|γ|2K)(2|α|2+|γ|2).
(41)

The total success probability of the second concentration step equals the sum of the probability in each concentration round, which can be described as
Ptotal2=P21+P22++P2K=K=1P2K.
(42)

So far, we have completed our whole concentration protocol and ultimately recover the less-entangled W state into the maximally entangled W state. The total success probability of the whole protocol equals the product of the probability in each concentration step, which can be written as
Ptotal=Ptotal1Ptotal2=K=1P1KK=1P2K.
(43)

3. DISCUSSION AND SUMMARY

It is interesting to calculate the success probability of our protocol. According to Eqs. (25), (41), and (43), we can make a numerical simulation of the Ptotal. During the simulation, we make γ=(1)/(3) and α[0,(2)/(3)], and we suppose both of the two concentration steps are operated for N times. In Fig. 3, we display three curves, which represent the relationship between the Ptotal and α under the conditions that N=1, 3, 5, respectively. It is obvious that Ptotal largely depends on the origin coefficient α. The higher initial entanglement leads to the greater Ptotal. Moreover, it is obvious that repeating the protocol can increase Ptotal largely, which indicates by repeating the protocol, we can recover the less-entangled W state into the maximally entangled W state with a high success probability.

Fig. 3. Success probability (Ptotal) of the concentration protocol for obtaining a maximally entangled single-photon three-mode W state. We make γ=(1)/(3) and α[0,(2)/(3)]. For numerical simulation, we suppose that both the two concentration steps are operated N times. Here, the three curves represent the relationship between the Ptotal and α under the conditions that N=1, 3, 5, respectively. It can be found that Ptotal largely depends on the initial coefficient α. Moreover, it is obvious that by repeating the protocol, the success probability can be largely increased.

In this paper, we propose an efficient ECP for recovering the less-entangled single-photon three-mode W state into the maximally entangled W state. In this ECP, the single photon is distributed into the different locations, say Alice, Bob, and Charlie. In a sense, it is essentially the entanglement transformation for single-photon entanglement. Interestingly, based on the early work of Reck et al. [59

59. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994). [CrossRef]

], He et al. discussed a nonunitary transformation that makes the coefficients of the state equal that can be implemented by a linear optical circuit with a certain success probability [60

60. B. He, J. A. Bergou, and Z. Wang, “Implementation of quantum operations on single-photon qudits,” Phys. Rev. A 76, 042326 (2007). [CrossRef]

]. In their protocol, the spatial modes of the photon are local, while in this protocol, they are nonlocal. So we resort to the local operation and classical communication to obtain the maximally entangled state. This protocol is not only the entanglement transformation, but also the entanglement concentration. On the other hand, the cross-Kerr nonlinearity is the key element of our protocol. In conventional linear optical protocols, the photon-number measurement always depends on the sophisticated single-photon detectors. However, the sophisticated photon-number detection is hard to realize under present experimental conditions. Moreover, the photon will be destroyed if it is detected by the detector, which is called the postselection principle. In our protocol, with the help of the cross-Kerr nonlinearity, we can check the photon number without destroying the photons. Interestingly, we only require the weak cross-Kerr nonlinearity to complete the task. As pointed out by [48

48. K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-nOT gate,” Phys. Rev. Lett. 93, 250502(2004). [CrossRef]

,49

49. Q. Lin and J. Li, “Quantum control gates with weak cross-Kerr nonlinearity,” Phys. Rev. A 79, 022301 (2009). [CrossRef]

], in a practical operation, we should require αpθ21. So in the condition of weak cross-Kerr nonlinearity (θ is small), it can be satisfied with large amplitude of the coherent state. In a practical experiment, optical nonlinearity provided by the giant Kerr effect is achievable with ac Stark-shifted electromagnetically induced transparency (EIT) [61

61. H. Schmidt and A. Imamoğlu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936–1938 (1996). [CrossRef]

]. As discussed in [62

62. C. Wang, Y. Zhang, and G. S. Jin, “Polarization-entanglement purification and concentration using cross-Kerr nonlinearity,” Quantum Inf. Comput. 11, 988–1002 (2011).

,63

63. W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum- nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005). [CrossRef]

], current nonlinearities in a single-photon level could be θ102 with EIT. Compared with the ECP in [44

44. H. F. Wang, S. Zhang, and K. H. Yeon, “Linear optical scheme for entanglement concentration of two partially entangled threephoton W states,” Eur. Phys. J. D 56, 271–275 (2010). [CrossRef]

], we only require one pair of less-entangled states, while they need two pairs of less-entangled states, which indicates our ECP is more economic. Compared with the [46

46. F. F. Du, T. Li, B. C. Ren, H. R. Wei, and F. G. Deng, “Single-photon-assisted entanglement concentration of a multi-photon system in a partially entangled W state with weak cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 29, 1399–1405 (2012). [CrossRef]

,47

47. B. Gu, “Single-photon-assisted entanglement concentration of partially entangled multiphoton W states with linear optics,” J. Opt. Soc. Am. B 29, 1685–1689 (2012). [CrossRef]

], our ECP can concentrate an arbitrary less-entangled state with the initial coefficients of α, β, and γ, while they only concentrate some special W state with the initial coefficients of α and β. Certainly, before successfully performing this ECP, we should know the initial coefficients α, β, and γ to prepare the suitable single-photon state with the help of VBS. In a practical operation, we can measure enough samples of the less-entangled states to obtain the values of α, β, and γ [37

37. F. G. Deng, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A 85, 022311 (2012). [CrossRef]

,46

46. F. F. Du, T. Li, B. C. Ren, H. R. Wei, and F. G. Deng, “Single-photon-assisted entanglement concentration of a multi-photon system in a partially entangled W state with weak cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 29, 1399–1405 (2012). [CrossRef]

,47

47. B. Gu, “Single-photon-assisted entanglement concentration of partially entangled multiphoton W states with linear optics,” J. Opt. Soc. Am. B 29, 1685–1689 (2012). [CrossRef]

].

In conclusion, we put forward an efficient ECP for recovering the less-entangled single-photon three-mode W state into the maximally entangled W state with a high success probability. Our protocol includes two steps, and in each step, we only need one conventional auxiliary photon to complete the task. The operation in both steps is based on the local operation and classical communication. In the protocol, we do not require the large cross-Kerr nonlinearity; only the weak cross-Kerr nonlinearity can complete the task. Benefiting from the cross-Kerr nonlinearities and the VBS, our protocol can be used repeatedly to further concentrate the less-entangled W state, which makes the protocol able to obtain a high success probability. All the advantages above indicate that our protocol may be feasible and convenient in current quantum communications areas.

ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China under Grant No. 11104159, the Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics Scientific, Tsinghua University, the Open Research Fund Program of the National Laboratory of Solid State Microstructures under Grant Nos. M25020 and M25022, Nanjing University, the University Natural Science Research Foundation of JiangSu Province under Grant No. 11KJA510002, and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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C. Wang, Y. Zhang, and G. S. Jin, “Polarization-entanglement purification and concentration using cross-Kerr nonlinearity,” Quantum Inf. Comput. 11, 988–1002 (2011).

63.

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum- nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71, 033819 (2005). [CrossRef]

OCIS Codes
(270.4180) Quantum optics : Multiphoton processes
(270.5565) Quantum optics : Quantum communications
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Quantum Optics

History
Original Manuscript: August 16, 2012
Revised Manuscript: October 18, 2012
Manuscript Accepted: October 28, 2012
Published: December 6, 2012

Citation
Lan Zhou, Yu-Bo Sheng, Wei-Wen Cheng, Long-Yan Gong, and Sheng-Mei Zhao, "Efficient entanglement concentration for arbitrary single-photon multimode W state," J. Opt. Soc. Am. B 30, 71-78 (2013)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-30-1-71


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