Optics Communications 316 (2014) 26–30
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Generation of three-photon polarization-entangled GHZ state via linear optics and weak cross-Kerr nonlinearity Chun-Hong Zheng a,b, Jie Zhao a, Peng Shi a, Wen-Dong Li a, Yong-Jian Gu a,n a b
Department of Physics, Ocean University of China, Qingdao 266100, PR China School of Science, Qingdao Technological University, Qingdao 266033, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 3 August 2013 Received in revised form 21 November 2013 Accepted 23 November 2013 Available online 6 December 2013
We propose a simple scheme for generating three-photon polarization-entangled GHZ state via crossKerr medium, linear optical elements and P homodyne measurement. In our protocol, the two-photon interference is used rather than the single-photon interference, which decreases the difﬁculty in experimental realization. Besides, the generation scheme proposed doubles the phase shift θ, which will result in lower error probability and decoherence effect. These advantages make our scheme more feasible in the regime of weak cross-Kerr nonlinearity with current experimental technologies. In addition, we give two efﬁcient schemes for the generation of arbitrary multi-photon GHZ state, which is signiﬁcant to large-scale quantum information processing (QIP). & 2013 Elsevier B.V. All rights reserved.
Keywords: GHZ state Weak cross-Kerr nonlinearity P homodyne measurement
1. Introduction In recent years, quantum information processing (QIP) has been investigated quite extensively and intensively. Entanglement, as one of the most striking features of quantum mechanics, not only provides the possibilities to test quantum mechanics against local hidden theory, but also offers an indispensable resource in QIP. In order to complete some QIP tasks, such as quantum teleportation [1,2], quantum dense coding [3–5], quantum cryptography [6,7], quantum secret sharing , and quantum cloning machine , successfully, maximally entangled states are usually required. For a tripartite system, it has been known that there exist at least two different types of entanglement: namely, the Greenberger–Horne–Zeilinger-type (GHZ-type) entanglement and the W-type entanglement. The GHZ state is usually referred to as “maximally entangled” in several senses because it violates Belltype inequalities maximally, and it is the most stable one against white noise. As an important quantum resource, great effort has been taken to study the generation of GHZ state in different physical systems [10–15]. Among all these protocols, the one based on linear optical system has an inherent advantage due to the possibility of long-distance quantum information transmission with relatively low decoherence and quite simple single-qubit manipulations of photonic states. However, these schemes are usually based on the detection of one or two photons [14,15], the success probability will decrease as the detection efﬁciency of the photon
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detectors decreases. On the other hand, these protocols also contain a signiﬁcant problem concerning scalability in practice due to the destruction of photons. Once the photons are detected, they are destroyed, which makes it impossible to utilize the output entangled states for further usage. In an attempt to overcome this problem, a quantum non-demolition detection (QND) based on cross-Kerr nonlinear interaction has been proposed, and a host of applications have been studied, such as the implementation of quantum logic gate [16,17], quantum teleportation , entanglement puriﬁcation and concentration [19–22], quantum entanglement state engineering [23–29]. In this paper, we propose an improved scheme for generating three-photon polarization-entangled GHZ state with the aid of linear optical elements, cross-Kerr nonlinearity, and P homodyne measurement, and generalize it to the generation of n-photon GHZ state. The remainder of this paper is organized as follows. In Section 2, we brieﬂy introduce the model of cross-Kerr nonlinear interaction. In Section 3, a simpliﬁed scheme for generating three-photon GHZ state is proposed, and the advantages of this scheme are discussed. In Section 4, two extended schemes for generating arbitrary multi-photon GHZ states are given. The discussion and conclusion will be shown in Section 5. 2. Weak cross-Kerr nonlinear interaction Let us ﬁrst review the model of cross-Kerr nonlinear interaction, which was introduced by Munro et al.  in 2003. This innovation moved optical QIP beyond linear optics and brought the hope of deterministic optical QIP. As the key element of our protocol, the cross-Kerr nonlinearity involves two optical ﬁeld
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modes, one is called signal mode and the other probe mode. Under the single-mode approximation, the interaction Hamiltonian has the form bs n bp ; H ¼ ℏκ n
þjVH〉12 ðjαei2θ 〉 þ j αei2θ 〉Þ ðjH〉 þ jV〉Þ3 :
Next let photons 2 and 3 pass PBS3 and then interact with the other two weak cross-Kerr nonlinearities with the phase shift 3θ and 3θ. After PBS4, the joint state of the whole system becomes
here κ is the coupling strength of the nonlinearity which is determined by the property of the nonlinear weak cross-Kerr b s and n b p are the photon number operators for the signal medium, n mode and the probe mode respectively. If the signal mode is in state jn〉 and the probe mode is in an initial coherent state jα〉p with amplitude α. After photons pass through the weak cross-Kerr medium, the joint state of the combined system will be
c jϕ〉out ¼ pﬃﬃﬃ½jHHH〉 þjVVV 〉ðjα〉 þ j α〉Þ 2 2 þjHVV〉ðjαe i2θ 〉þ j αe i2θ 〉Þ
n sb np jφ〉out ¼ eiκ tb jn〉s jα〉p ¼ jn〉s jαeinθ 〉p ;
Then, we perform P homodyne measurement on the probe mode, which projects the probe mode to momentum space and the signal mode to the entangled state. With real α, the state shown in Eq. (5) will collapse into
where θ ¼ κ t represents the nonlinear phase shift and t refers to the interaction time for the signal and probe modes with the nonlinear medium. It is clear that the Fock state is unaffected and the coherent state picks up a phase shift which is directly proportional to the number of photons in the signal mode. One can exactly obtain the information of photons in the Fock state without destroying them by detecting phase shift of the probe mode . The key advantages of QND measurement, at least in the optical regime, are that it is highly efﬁcient  and is a standard tool of continuous variable experimentalists.
Now, let us move our attention to the generation of the threephoton polarization-entangled GHZ state by using the cross-Kerr nonlinearity, PBSs, coherent superposition state, P homodyne measurement and appropriate single photon local operations. The schematic setup of our protocol is illustrated in Fig. 1. First, the photons in the signal mode are initially prepared in state j þ 〉j ðj ¼ 1; 2; 3Þ and the probe beam is in a coherent superposition state (CSS) cðjα〉 þ j α〉Þ with a normalization factor pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c ¼ 1= 2 þ 2e 2α2 . Thus, the initial state of the whole composite system can be expressed as ð3Þ
Now let the signal photons 1 and 2 pass through PBS1, and then interact with two separate weak cross-Kerr nonlinearities with the phase shift to be θ and θ; as shown in Fig. 1. The subsequent PBS2 is used to separate two photons into different spatial modes, after which, the whole system evolves into the following state: c jϕ〉 ¼ pﬃﬃﬃ½jHH〉12 þ jVV〉12 ðjα〉þ j α〉Þ þ jHV 〉12 ðjαe i2θ 〉 þ j αe i2θ 〉Þ 2 2
c( α + − α )
þjHVH〉ðjαei4θ 〉þ j αei4θ 〉Þ þ jHHV〉ðjαe i6θ 〉 þ j αe i6θ 〉Þ þjVVH〉ðjαei6θ 〉 þ j αei6θ 〉Þ:
jϕ〉P 〈Pjϕ〉out ¼
jmj2 ¼ 4s20 cos 2 τ0 þs22 þ þ s22 þs24 þ þ s24 þ s26 þ þ s26 þ 2s2 þ s2 cos ðτ2 þ þ τ2 Þ þ2s4 þ s4 cos ðτ4 þ þ τ4 Þ þ 2s6 þ s6 cos ðτ6 þ þ τ6 Þ; pﬃﬃ 2 where sn 7 ¼ π 1=4 e ð1=2Þðp 8 2α sin nθÞ and
jG〉4 7 ¼ p1ﬃﬃ2 ðe 7 iτ4 7 jVHV〉þ e 8 iτ4 7 jHVH〉Þ; jG〉6 7 ¼ p1ﬃﬃ2 ðe 7 iτ6 7 jHHV〉 þ e 8 iτ6 7 jVVH〉Þ; h i pﬃﬃ pﬃﬃﬃ τn 7 ¼ 2α cos nθ p 8 22 α sin nθ ; n ¼ 0; 2; 4; 6; here sn 7 are the Gaussian probability amplitudes of the states j G〉n 7 ðn ¼ 0; 2; 4; 6Þ whose variation concerning the variant p is depicted in Fig. 2. It is noteworthy that the difference between state jϕ〉P and the standard GHZ state jG〉0 varies depending on the homodyne measurement result p. From Fig. 2, we can see that when different values of p are taken, the composite system after the measurement collapses into certain state jG〉n 7 ðn ¼ 2; 4; 6Þ accordingly. Without loss of generality, we suppose that the measurement pﬃﬃﬃ ′ result p is in the regime pﬃﬃﬃ of p4 o p o′ p4 (where p4 ¼ 200 2 sin 0:04 δ4 and p′4 ¼ 200 2 sin 0:04 þ δ4 , which are near the peak of s4 þ ) as illustrated in Fig. 2. In this case, after the homodyne measurement, the corresponding polarizational entangled photonic state jϕ〉P will collapse into j G〉4 þ . In the next step, we need to further eliminate the phase factor e 7 iτ4 þ with the help of singlequbit phase shifter and the result of homodyne measurement via a feed-forward process (the dashed line in Fig. 1). Under this condition, we can set Φ1 ¼ 0 and Φ2 ¼ τ4 þ as illustrated in the P homodyne measurement
jG〉2 7 ¼ p1ﬃﬃ2 ðe 7 iτ2 7 jHVV〉þ e 8 iτ2 7 jVHH〉Þ;
1 ½2s0 cos τ 0 jG〉0 þ s2 þ jG〉2 þ þs2 jG〉2 m þ s4 þ jG〉4 þ þ s4 jG〉4 þ s6 þ jG〉6 þ þ s6 jG〉6 ;
where m is a normalization factor with the form
KM KM 3θ −3θ
jG〉0 ¼ p1ﬃﬃ2 ðjHHH〉 þ jVVV 〉Þ;
3. Scheme for generating the three-photon GHZ state
3 1 jϕ〉in ¼ pﬃﬃﬃðjH〉 þjV〉Þj cðjα〉þ j α〉Þ: j¼1 2
þjVHH〉ðjαei2θ 〉þ j αei2θ 〉Þ þ jVHV〉ðjαe i4θ 〉 þ j αe i4θ 〉Þ
PS 2 3
Fig. 1. Schematic setup for generation of the three-photon polarization-entangled GHZ state. KM refers to cross-Kerr nonlinear medium; six polarization beam splitter (PBS) are included to transmit the horizontal polarization and reﬂect the vertical one; CF refers to classical feed-forward and PSs are single-qubit phase shifters.
C.-H. Zheng et al. / Optics Communications 316 (2014) 26–30
note of Fig. 1. It is obvious that the jH〉 component of photon 2 and the jV〉 component of photon 3 would pick up the phase τ4 þ when the photon pulses pass through PS1 and PS2. The state jG〉4 þ then becomes jG〉′4 þ ¼ p1ﬃﬃ2 ½eiτ4 þ ðe i2τ4 þ jVHV〉Þ þ e iτ4 þ jHVH〉;
and the total phase factor e iτ4 þ could be extracted and ignored. That is, the form of jG〉′4 þ could be transformed into jG〉″4 þ ¼ p1ﬃﬃ2 ðjVHV〉 þ jHVH〉Þ:
After that, we choose s1 ¼ s3 ¼ I and s2 ¼ sx . Thus, the state jG〉″4 þ will evolve into the standard GHZ state jG〉0 . In the following, we shall discuss the probability and ﬁdelity of the scheme. It can be calculated that the probability of getting jG〉4 þ state is pﬃﬃ ′ Z P4 ¼
200 2 sin 0:04 þ δ4
pﬃﬃ 200 2 sin 0:04 δ4
〈 pjtrsignal ðjϕ〉outout 〈ϕjÞjp〉 dp
1 ′ ½erf ðδ4 Þ þ erf ðδ4 Þ; 16
pﬃﬃﬃ where erf(x) is the error function. As the distance 2dn ¼ 2α ½ sin nθ sin ðn 2Þθ (shown in Fig. 2) between adjacent peaks that can be regarded as the distinguishability of the measurement ′ increases, this probability will approach 18 . The values of δ4 and δ4 depend on the ﬁdelity we require, which should meet the following formulas: F jG〉4 þ ðp4 Þ ¼ j4 þ 〈 Gjϕ〉P j2
1 ; 1 þ e 2d4 ðd4 δ4 Þ
and F jG〉4 þ ðp′4 Þ
1 þ e 2d6 ðd6 δ4 Þ
For example, if the smallest ﬁdelity required is 0.9999, from Eqs. ′ (11) and (12), we can get δ4 0:857d4 , δ4 0:856d6 . In short, the standard GHZ state jG〉0 could be generated with the probability 1/ 8 as well as the ﬁdelity more than 0.9999 when the measurement result p is in the regime of p4 o p o p′4 . Similar analysis can be made when the measurement result p is near the peaks of the other ﬁve side functions in Fig. 2, and we can also gain the state jG〉0 with the same probability by implementing simple local operations according to certain outcome of the homodyne measurement through a feed-forward. The correspondence between
Table 1 Correspondence between the measurement result p and local operations on the signal photons (where scn 7 represent some values near the peaks of sn 7 ðn ¼ 2; 4; 6Þ respectively). p
Local operations on the signal photons
sc4 þ sc4 sc6 þ sc6 sc2 þ sc2
Phase shift of PSj ðj ¼ 1; 2Þ
Operation sj (j¼ 1,2,3)
Φ1 ¼ 0; Φ2 ¼ τ4 þ Φ1 ¼ 0; Φ2 ¼ τ4 Φ1 ¼ 0; Φ2 ¼ τ6 þ Φ1 ¼ 0; Φ2 ¼ τ6 Φ1 ¼ 2τ2 þ ; Φ2 ¼ 0 Φ1 ¼ 2τ2 ; Φ2 ¼ 0
s1 ¼ s3 ¼ I, s2 ¼ sx s1 ¼ s3 ¼ I, s2 ¼ sx s1 ¼ s2 ¼ I, s3 ¼ sx s1 ¼ s2 ¼ I, s3 ¼ sx s1 ¼ sx , s2 ¼ s3 ¼ I s1 ¼ sx , s2 ¼ s3 ¼ I
the measurement result p and the local operations on the signal photons is shown in Table 1. Due to the rapid oscillation near p¼0, we simply discard the measurement results near the center of s0 . Therefore, the total success probability of obtaining jG〉0 will approach 3/4 when good distinguishabilities are achieved. Compared with the scheme proposed by Jin et al. , we double the phase shift θ, which decreases the error probability as discussed below. pﬃﬃﬃ The error probability is deﬁned to be P error ¼ ð1=2Þerfcð2dn = 2Þ, where 2dn is the aforementioned distinguishability. For the regime of weak cross-Kerr nonlinearity θ 5 1, the 1 error pﬃﬃﬃ probability in our scheme turns out to be P′error ¼ 2 erfc ð 2αθÞ, pwhich is less than that of the latter P error ¼ ð1=2Þ ﬃﬃﬃ erfcðαθ= 2Þ. If we choose the same probe beam and weak crossKerr nonlinearities as the latter α ¼ 200 and θ ¼ 0:01, P error ¼ 3:17 10 5 , while P ′error ¼ 2:31 10 2 . On the other hand, for the same error probability, the nonlinear phase shift or the strength of coherent probe beam required in our protocol will reduce to half, this advantage will improve the feasibility and reduce the decoherence effect when its realization is considered. In conclusion, the three-photon standard GHZ state can be generated in our scheme with high ﬁdelity and success probability in a more feasible and simpler way.
4. Extension of the scheme for generating arbitrary n-photon GHZ state In Section 2, we have obtained the three-photon GHZ state. Straightforwardly, it can be generalized to the generation of arbitrary n-particle GHZ states. The schematic setup is illustrated in Fig. 3. Suppose the initial state of the whole system is n 1 jϕ〉in ¼ pﬃﬃﬃðjH〉 þ jV〉Þj cðjα〉 þ j α〉Þ: j¼1 2
After passing the device in Fig. 3, Eq. (13) turns to be c jϕ〉out ¼ pﬃﬃﬃﬃﬃn ½jHH⋯H〉 þ jVV⋯V〉ðjα〉þ j α〉Þ 2 n n þ jHH⋯V〉ðjαe ið2 2Þθ 〉 þ j αe ið2 2Þθ 〉Þ þ jVV ⋯H〉ðjαeið2 þ ⋯⋯
〉þ j αeið2
þ jHV ⋯V〉ðjαe i2θ 〉 þ j αe i2θ 〉Þ þ jVH⋯H〉ðjαei2θ 〉 þ j αei2θ 〉Þ:
Thus, we can obtain the standard n-photon GHZ states jG〉n ¼ p1ﬃﬃ2 ðjHH⋯H〉 þ jVV⋯V 〉Þn ; Fig. 2. 2s0 j cos τ0 j and sn 7 ðn ¼ 2; 4; 6Þ as functions of the homodyne measurement result p with α ¼ 200 and θ ¼ 0:01.
by performing P homodyne measurement and simple local singlephoton operations via a feed-forward. The success probability is 1 1=2n 1 when good distinguishability is achieved, which will
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KM KM +θ −θ
c( α + − α )
P homodyne measurement P P
Fig. 3. The setup for generating an n-photon GHZ state.
c( α + − α )
P homodyne measurement
KM KM +θ
P P CF
ϕ(Ρ) PBS 1
Fig. 4. (a) The parity check gate for photonic qubit encoded in the polarization degree of freedom. (b) The simpliﬁed presentation of the setup in Fig. 4(a).
equal to 1 as n increases. However, a drawback of this generalization is that the phase shift θ increases exponentially with the photon number n, which will result in an increase of the decoherence effect , and a more serious problem is that the phase shift will increase beyond the regime of the weak cross-Kerr nonlinearity quickly. As demonstrated by Gea-Banacloche in Ref. , strong cross-Kerr nonlinearity is impossible, while the weak cross-Kerr nonlinearity is theoretically feasible . Therefore, this extension scheme is only available for few photons GHZ state generation. In fact, in our scheme, the two-photon interference associated with cross-Kerr nonlinear interaction acts as the so-called twoqubits parity-check operation (PC), and for GHZ state generation, the parity-check operation is enough. The original idea for n-photon GHZ state generation with parity-check operation can be found in Ref. , where a PBS acts as a parity-check operation in linear optics. In 2004, a nearly deterministic parity-check operation with the aid of weak cross-Kerr nonlinearity was proposed by Nemoto and Munro in Ref. . Over the past few years, several schemes for the preparation of multi-photon entangled states, such as GHZ states, cluster states, and graph states, based on the paritycheck operation were proposed [37–39]. Now, let us ﬁrst review how our PC works. Assume that two photons initialized in the polarizational superposition state p1ﬃﬃ ðjH〉 þ jV〉Þ ði ¼ 1; 2Þ are set to pass through the PC system i 2 (as shown in Fig. 4(a)), the state of the signal photons will then collapse into either p1ﬃﬃ2 jHH〉 þ jVV〉 or p1ﬃﬃ2 ðe 7 iτ2 7 jHV 〉 þ e 8 iτ2 7 jVH〉Þ according to the measurement result p of P homodyne measurement. When we obtain the even parity state p1ﬃﬃ2 jHH〉 þ jVV〉, let PC ¼0; otherwise, by performing simple local rotation via a feedforward process to remove the relative phase τ2 7 , we will get the state p1ﬃﬃ2 jHV〉 þ jVH〉, and then let PC ¼ 1. Taking the advantage of P homodyne measurement the PC system can implement the parity check operation on the signal photons with high efﬁciency. When PC ¼1, we can also obtain the same state p1ﬃﬃ2 jHH〉 þ jVV〉 via a sx operation on the second photon. An arbitrary multi-photon GHZ state can be generated easily using PC systems in series (as shown in Fig. 5) as described below. Suppose the initial input state is as
Fig. 5. The setup for generating an n-photon GHZ state, i.e. n PC systems are cascaded in series.
same as Eq. (13), after ðn 2Þ successive successful PC systems and feed-forward, i.e., PC i ¼ 0 and si ¼ I ði ¼ 1; 2; …; n 2Þ, the state of the whole system becomes c jϕ〉′out ¼ pﬃﬃﬃﬃﬃn ½jHH⋯H〉 þ jVV⋯V 〉ðjα〉þ j α〉Þ 2 n n þ jHH⋯V〉ðjαe ið2 2Þθ 〉 þ j αe ið2 2Þθ 〉Þ þ jVV ⋯H〉ðjαeið2
〉þ j αeið2
Then, the P homodyne measurement is performed on the last probe beam. If PC n 1 ¼ 0, we successfully obtain the standard n-photon GHZ state as Eq. (15). Otherwise, by performing simple local rotation via a feed-forward process and a subsequent sx operation on the nth photon, the same state can also be acquired. This extension scheme can efﬁciently avoid the disadvantage existing in the ﬁrst extension scheme, i.e. the phase shift θ becomes so large that the decoherence increases and should be more feasible experimentally.
5. Discussion and conclusion Before giving the conclusion, it is better to brieﬂy discuss the experimental feasibility of our scheme. The feasibility mainly depends on ﬁve factors, that is, cross-Kerr medium, homodyne detection, the generation of coherent superposition state (CSS),
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the stability of the photon interference and decoherence effect. Though nature cross-Kerr nonlinearities are extremely small, with a typical magnitude of θ 10 18 [40,41], it has been suggested recently that the nonlinearity magnitude might be largely improved with the help of EIT (electromagnetically induced transparency), say θ 10 2 [42–44], which could be used effectively in our proposal. The scheme uses P homodyne measurement which has higher efﬁciency and requires considerably smaller strength of the coherent state in the probe mode than the X homodyne measurement in other schemes [26,29] under the same condition. For weak cross-Kerr nonlinearity (θ 5 1), the error pﬃﬃﬃ probability of P homodyne measurement is P error ¼ 12 erfcð 2αθÞ, which can be less than 10 5 as long as αθ 4 2. Thus, on the condition that θ 10 2 , a probe coherent superposition state cðjα〉 þ j α〉Þ with an amplitude of α 102 will satisfy our scheme well. Several schemes to produce a CSS state have been proposed [45–47]. In our protocol, we replace the three single photon interference through M-Z interferometers by two two-photon interference, which will decrease the difﬁculty in experimental realization, since the difference of two paths in the single photon interference is required to be less than the wave length of single photon, while the difference in the two-photon interference is only required to be less than the coherent length of two single photons. Another important point with respect to the practical implementation of our scheme is the decoherence effect during the entanglement generation process in nonlinear medium which should be set as low as possible . Qualitatively, decoherence depends on the interaction time t and the travel path αθ of the coherent state . Compared with the scheme in Ref. , our scheme can further reduce the nonlinear phase shift or the strength of coherent probe beam if we choose the same error probability, which will not only increase the feasibility of the experimental realization, but also enhance the robustness of the generation process against decoherence effect. In summary, we have presented an improved scheme to generate three-photon and arbitrary multi-photon GHZ state with less error probability and high ﬁdelity. Compared with other schemes proposed in Refs. [24,26,29], our schemes have the following advantages: (i) These approaches take advantage of P homodyne measurement, which has lower error probability when we choose the same coherent probe beam and weak crossKerr nonlinearities. (ii) The one-photon interference is replaced by two-photon interference, which reduces the decoherence effect efﬁciently and makes the schemes more feasible experimentally. (iii) These schemes use only basic linear optical elements and weak cross-Kerr nonlinearity. All these advantages are valuable to large-scale quantum information processing. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant nos. 60677044, 11005099 and 11304174), the
Fundamental Research Funds for the Central University under Grant no. 201313012, and the Natural Science Foundation of Shandong Province (Grant no. ZR2013AQ010).
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