Simple scheme for expanding a polarization-entangled W state by adding one photon
aa r X i v : . [ qu a n t - ph ] O c t Simple scheme for expanding a polarization-entangled W state by adding one photon Yan-Xiao Gong, ∗ Xu-Bo Zou, Yun-Feng Huang, † and Guang-Can Guo Key Laboratory of Quantum Information, University of Science and Technology of China,CAS, Hefei, 230026, People’s Republic of China (Dated: November 5, 2018)We propose a simple scheme for expanding a polarization-entangled W state. By mixing a singlephoton and one of the photons in an n -photon W state at a polarization-dependent beam splitter(PDBS), we can obtain an ( n + 1)-photon W state after post-selection. Our scheme also opens thedoor for generating n -photon W states using single photons and linear optics. PACS numbers: 03.67.Mn, 03.65.Ud, 42.50.Dv
Entanglement not only plays a central role in funda-mental quantum physics [1, 2], but also has wide applica-tions in quantum information processing, such as quan-tum teleportation [3], dense coding [4], quantum cryptog-raphy [5], and quantum computation [6]. While bipartiteentanglement has been well understood, multipartite en-tanglement offers a very complicated structure. For ex-ample, it was shown that genuine three-particle entangle-ment can be classified into two classes by the equivalenceunder stochastic local operations and classical commu-nication (SLOCC) [7]. One is the Greenberger-Horne-Zeilinger (GHZ) state [8] | GHZ i = 1 √ | i + | i ) . (1)The other is the W state | W i = 1 √ | i + | i + | i ) . (2)The GHZ state is usually taken as “maximally entangled”state in some senses, for instance, it violates Bell inequali-ties maximally. However, it is also maximally fragile, i.e.,if one or more particles are lost or discarded, then all theentanglement is destroyed. The W state is less entan-gled in the sense that its violation is weaker than thatof GHZ state. While, the W state is very robust againstthe loss of one of the particles, namely, two-particle en-tanglement can be observed after one particle is lost ormeasured. Thereby, in this sense, the W state is moreentangled.The entanglement persistency property can be easilyobtained from the representation of the n -particle W state | W n i = 1 √ n ( | · · · i + | · · · i + · · · + | · · · i + | · · · i )= 1 √ n | n − , i , (3) ∗ Electronic address: [email protected] † Electronic address: [email protected] where | n − , i denotes the (unnormalized) totally sym-metric state including n − | i and oneparticle in state | i , e.g., | , i = | i + | i + | i + | i . We can see that any particle is entangled with theother particles and that all the particles are equivalent.In fact, it was shown that the W state has the maximumdegree of entanglement between any pair of particles [9].These interesting features lead the W -class states to ap-plications in a variety quantum information processingtasks, such as quantum teleportation [10, 11, 12], densecoding [13], quantum secret communication [14].Linear optical systems have supplied a broad field forexperimental implementation of multipartite entangledstates. There have been many proposals [15, 16, 17, 18,19, 20, 21] and experimental implementations [22, 23, 24,25] for producing W states. Quite recently, Tashima etal. introduced an interesting optical gate for expandingpolarization-entangled W states [26]. In their scheme,after the operation of the gate on one of the photons inan n -photon W state, an ( n + 2)-photon W state can beobtained after post-selection.In this paper, using a similar expanding principle withthat in Ref. [26], we propose a simple scheme for expand-ing a polarization-entangled W state by adding a singlephoton to the existing state, rather than adding two pho-tons in Ref. [26]. Our scheme needs only a polarization-dependent beam splitter (PDBS), where one of the pho-tons in an n -photon W state interferences with a sin-gle photon and after post-selection an ( n + 1)-photon W state can be obtained.Before introducing our scheme we would like to notethat the qubits here are all encoded in polarization statesof single photons, so that | i ≡ | H i and | i ≡ | V i , where | H i ( | V i ) denotes the horizontal (vertical) polarizationstate. Our scheme for adding a single photon to an n -photon W state is depicted in Fig. 1. The key of ourscheme is an element of PDBS, with reflectivities of η H = 5 − √
510 and η V = 5 + √ , (4)for horizontally ( H ) and vertically ( V ) polarized photons,respectively. Such class of elements has been used in sev-eral experiments [27, 28, 29, 30]. One of the photons instate | W n i and a single photon in state | H i meet at thePDBS, which are input in modes a and b , respectively. FIG. 1: Scheme for adding a single photon to an n -photon W state. The polarization-dependent beam splitter (PDBS) hasreflectivities of η H = (5 − √ /
10 and η V = (5 + √ /
10 forhorizontally ( H ) and vertically ( V ) polarized photons, respec-tively. One of the photons in the W state is input in mode a , and a single photon in state | H i is added in mode c . Ahalf-wave plate (HWP) oriented at 0 ◦ can introduce a phaseshift of π between H and V polarized photons. This schemesucceeds in the case of twofold coincidence detection in theoutput modes c and d . If they are indistinguishable except the degrees of pathand polarization (fourth-order interference will happenfor the same polarization photons), the state transfor-mations at the PDBS can be expressed as | H i a →√ η H | H i c + p − η H | H i d , (5) | V i a →√ η V | V i c + p − η V | V i d , (6) | H i b → p − η H | H i c − √ η H | H i d . (7)After the PDBS, we use a half-wave plate (HWP) setto 0 ◦ to introduce a phase shift of π between H and V polarized photons, with the transformations | H i c → | H i c , | V i c → −| V i c . (8)Therefore, if we post-select the successful events, i.e.,twofold coincidence detection at the output modes c and d , we can obtain the state transformations as follows, | H i a | H i b → √ | H i c | H i d , (9) | V i a | H i b → √ | H i c | V i d + | V i c | H i d ) . (10)Next we explain how a single photon can be added toa state | W n i through our scheme. Since all the photonsin the W state are equivalent, we can choose any photonto inject in mode a , for instance, mode n , so that we canrewrite the W state given by Eq. (3) as follows, | W n i = 1 √ n | n − , i = 1 √ n h | n − , i| H i n + | H i ⊗ ( n − | V i n i −→ √ n h | n − , i| H i a + | H i ⊗ ( n − | V i a i . (11) FIG. 2: Schematic of preparing an N -photon W state usingsingle photons and the scheme shown in Fig. 1. Then we can write the state evolution of the photon-added process as | W n i| H i b → √ n h | n − , i| H i c | H i d + | H i ⊗ ( n − ⊗ (cid:0) | H i c | V i d + | V i c | H i d (cid:1)i + | Φ i = r n + 15 n | W n +1 i + | Φ i , (12)where | Φ i is an unnormalized state including the ampli-tudes that would not lead to the successful events.From Eq. (12) we can see that the success probabilityfor adding a single photon to a state | W n i is ( n + 1) / (5 n ),which approaches a constant 1 / n becomes large.It is not difficult to find that if we generate an N -photon W state using single photons and our scheme (seeFig. 2 for the schematic), the total success probability is N/ N − ( N > η H = η V = 1 /
2, theprobability of success can be improved to N/ (4 × N − ).Alternatively, if we do not restrict our sources to singlephotons and EPR states are available, we can get higherprobability. Explicitly, with our scheme, we can first adda single photon to an EPR state ( | HV i + | V H i ) / √ | W i and then add single photons one by oneas the way in Fig. 2. In this case we can obtain the state | W N i with the success probability N/ (2 × N − ). In par-ticular, the success probability for preparing state | W i is 3 /
10, which is highest compared with other linear op-tical schemes, as the most efficient one at present is 3 / | W i is 2 /
25, which is lower than 1 / /
27 in Ref.[15]). Therefore, we believe our scheme is experimentalfeasible for preparing states | W i and | W i . However, asthe number of photons increases the success probabilitydecreases exponentially, so experimental preparing more-photon W states would be still difficult. Actually, this isa common problem in many linear optical schemes.Finally, we would like to give a brief discussion onthe comparison of our scheme with the scheme in Ref.[26]. The two schemes are based on similar expandingprinciples (this can be seen from Eqs. (9) and (10) andEqs. (3) and (4) in their scheme), but a single photon isadded in our scheme rather than two photons are addedin their scheme. This leads to different experimental re-quirements, i.e., we need single photons while they needtwo-photon Fock states. In their scheme, the successprobability is (2 k + 1)2 − k for preparing state | W k +1 i and ( k + 1)2 − k for preparing state | W k +1) i . There-fore, our scheme is more efficient for preparing state | W i but less efficient for preparing more-photon W state.In conclusion, we propose a simple scheme to expanda polarization-entangled W state by adding a single pho-ton. This method should be very helpful in quantum in-formation processing in the future when quantum mem-ory and nondemolition measurements are available, be- cause we can add a single photon to an existing W stateeasily to get a larger one. Furthermore, our method givesa new way to prepare W state using single photons andlinear optical elements.This work was funded by National FundamentalResearch Program (Grant No. 2006CB921907), Na-tional Natural Science Foundation of China (Grants No.60621064, No. 10674128 and No. 10774139), InnovationFunds from Chinese Academy of Sciences,“ Hundreds ofTalents” program of Chinese Academy of Sciences, Pro-gram for New Century Excellent Talents in University,A Foundation for the Author of National Excellent Doc-toral Dissertation of PR China (grant 200729). [1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. ,777 (1935).[2] J. S. Bell, Physics (Long Island City, N. Y.) , 195 (1964).[3] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa,A. Peres, and W. K. Wootters, Phys. Rev. Lett. , 1895(1993).[4] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. ,2881 (1992).[5] A. K. Ekert, Phys. Rev. Lett. , 661 (1991).[6] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information (Cambridge University, Cam-bridge, England, 2000).[7] W. D¨ur, G. Vidal, and J. I. Cirac, Phys. Rev. A ,062314 (2000).[8] D. M. Greenberger, M. A. Horne, and A. Zeilinger, in Bell’s Theorem, Quantum Theory, and Conceptions ofthe Universe , edited by M. Kafatos (Kluwer, Dordrecht,1989), p. 69.[9] M. Koashi, V. Buˇzek, and N. Imoto, Phys. Rev. A ,050302 (2000).[10] B.-S. Shi and A. Tomita, Phys. Lett. A , 161 (2002).[11] J. Joo, Y.-J. Park, S. Oh, and J. Kim, New J. Phys. ,136 (2003).[12] Y. Yeo, arXiv:quant-ph/0302030.[13] V. N. Gorbachev, A. I. Trubilko, A. A. Rodichkina, andA. I. Zhiliba, Phys. Lett. A , 267 (2003).[14] J. Joo, J. Lee, J. Jang, and Y.-J. Park, arXiv:quant-ph/0204003.[15] X. Zou, K. Pahlke, and W. Mathis, Phys. Rev. A ,044302 (2002).[16] T. Yamamoto, K. Tamaki, M. Koashi, and N. Imoto, Phys. Rev. A , 064301 (2002).[17] G.-Y. Xiang, Y.-S. Zhang, J. Li, and G.-C. Guo, J. Opt.B: Quantum Semiclassical Opt. , 208 (2003).[18] Y. Li and T. Kobayashi, Phys. Rev. A , 014301 (2004).[19] H. Mikami, Y. Li, and T. Kobayashi, Phys. Rev. A ,052308 (2004).[20] B.-S. Shi and A. Tomita, J. Mod. Opt. , 765 (2005).[21] B. H. Liu, F. W. Sun, Y. F. Huang, and G. C. Guo, Phys.Lett. A , 389 (2007).[22] N. Kiesel, M. Bourennane, C. Kurtsiefer, H. Weinfurter,D. Kaszlikowski, W. Laskowski, and M. Zukowski, J.Mod. Opt. , 1131 (2003).[23] M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, andH. Weinfurter, Phys. Rev. Lett. , 077901 (2004).[24] H. Mikami, Y. Li, K. Fukuoka, and T. Kobayashi, Phys.Rev. Lett. , 150404 (2005).[25] B. P. Lanyon and N. K. Langford, arXiv:0802.3161.[26] T. Tashima, S¸ahin Kaya ¨Ozdemir, T. Yamamoto,M. Koashi, and N. Imoto, Phys. Rev. A , 030302(2008).[27] N. Kiesel, C. Schmid, U. Weber, G. T´oth, O. G¨uhne,R. Ursin, and H. Weinfurter, Phys. Rev. Lett. , 210502(2005).[28] N. K. Langford, T. J. Weinhold, R. Prevedel, K. J. Resch,A. Gilchrist, J. L. O’Brien, G. J. Pryde, and A. G. White,Phys. Rev. Lett. , 210504 (2005).[29] N. Kiesel, C. Schmid, U. Weber, R. Ursin, and H. Wein-furter, Phys. Rev. Lett. , 210505 (2005).[30] R. Okamoto, H. F. Hofmann, S. Takeuchi, and K. Sasaki,Phys. Rev. Lett.95