A new optical scheme for quantum teleportation of superposed coherent states

Physics Letters A 358 (2006) 115–120 www.elsevier.com/locate/pla

A new optical scheme for quantum teleportation of superposed coherent states Jie-Qiao Liao, Le-Man Kuang ∗ Department of Physics, Hunan Normal University, Changsha 410081, People’s Republic of China Received 21 November 2005; received in revised form 2 May 2006; accepted 5 May 2006 Available online 16 May 2006 Communicated by P.R. Holland

Abstract We propose a new optical scheme for the quantum teleportation of superposed coherent states (SCSs) in terms of optical elements. Our scheme can realize a near-complete quantum teleportation of SCSs with arbitrary coefficients. Different from previous schemes, our protocol need only yes or no measurements of the photon numbers of the related modes instead of the exact measurements of the photon numbers in the previous schemes. © 2006 Elsevier B.V. All rights reserved. PACS: 03.67.Hk; 03.65.Ud; 03.67.Lx

1. Introduction Quantum teleportation, first proposed by Bennett et al. [1], has attracted much attention of both theorists and experimenters in the last decade. The original proposals for quantum teleportation [1,2] focused on teleporting quantum states of a system with a finite-dimensional (discrete variable) state space, such as the two polarizations of a photon or the discrete levels of an atom. Discrete-variable teleportation has been demonstrated experimentally in optical systems [3] and liquid-state nuclear magnetic resonance systems [4]. In recent years, quantum teleportation has been extended to continuous-variable cases corresponding to quantum states of infinite-dimensional (continuous-variable) systems [5,6] such as optical fields or the motion of massive particle. In particular, following the theoretical proposal of Ref. [6], continuousvariable teleportation has been realized for coherent states of a light field [7] by using entangled two-mode squeezed optical beams produced by parametric down-conversion in a sub-threshold optical parametric oscillator. Although coherent * Corresponding author.

E-mail address: [email protected] (L.-M. Kuang). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.05.009

states are continuous and nonorthogonal states, they are very close to classical states. A real challenge for quantum teleportation is to teleport truly nonclassical states like quantum superposition states, entangled states, and squeezed states. Quantum teleportation of a superposed coherent state (SCS), a superposition of two nonorthogonal coherent states with opposite phases, has been widely studied in the past few years [8–12,14–17]. It should be pointed out that the teleported states are generally arbitrary superposition states of two equalamplitude and anti-phase coherent states, they are unknown superposition states of two known coherent states. The state information is encoded in the coefficients of the teleported coherent states. The basis of the teleported state, i.e., |α and |−α, are quasi-orthogonal as the amplitude |α| increases. This is quite similar to the teleportation of the usual qubit states in which one a priori knows that the teleported state is a superposition state of states |0 and |1. Hence, this kind of teleportation protocols can also be regarded as teleportation protocols of quasi-qubit states. Enk and Hirota [8] first proposed a scheme to faithfully teleport a SCS through a quantum channel described by a maximally entangled coherent state with the success probability of 1/2. Then, Jeong et al. [9] presented a maximally-entangled quantum channel scheme with the success probability of approaching 1. Quantum teleportation methods of a SCS via a

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nonmaximally entangled coherent channel were investigated in Refs. [10,11]. An [12,13] developed two protocols for teleporting a SCS within a network by using GHZ-type and W-type entangled coherent states as quantum channels, respectively. Recently, Cheong et al. [14] has proposed a simple scheme to perform a near-complete teleportation of a SCS with the success probability and fidelity of nearly 100%. Wang [16] proposed a simple scheme to teleport bipartite and multipartite entangled coherent states with the probability of success 1/2. It should be noted that all of these schemes need to clearly discriminate odd number of photons and even number of photons by using photon detectors. So, a highly efficient detector which can discriminate odd–even photon numbers is necessary. This becomes the major obstacle to demonstrate them experimentally, since the loss of one photon can change odd and even characteristic of the number of photons, and leads to measurement error. Therefore, it is interesting to find a SCS teleportation protocol which is independent of the parity (even–odd) characteristic of the number of photons. In this Letter, we propose a new optical scheme to faithfully teleport a SCS with the probability of success approaching 1. In our scheme, only measurements of zero photon and nonzero photons are needed. Hence, the major difficulty of clearly measuring odd–even photon numbers in previous schemes are overcome. 2. Teleportation of superposed coherent states We now consider quantum teleportation of the following SCS   |ϕ1 = N x|α + y|−α , −1/2  2 , N = |x|2 + |y|2 + 2e−2|α| Re(xy ∗ ) (1) where |α is the usual Glauber coherent state defined by |α = e

−|α|2 /2

∞  αn √ |n. n! n=0

(2)

In previous protocols, Alice and Bob must prepare an entangling quantum channel which belongs both of them, then Alice performs so-called joint Bell measurements with respect to one subsystem involved in the quantum channel. Alice sends the result of the joint measurements to Bob through a classical communication. After Bob makes a proper unitary operation, the unknown quantum state in the Alice’s side is teleported to Bob. Our scheme is little similar with the former schemes. We present a schematic diagram for quantum teleportation of the SCS in Fig. 1. We need five modes of lights in order to teleport an unknown SCS given by Eq. (1) from mode 1 in Alice’s side to the mode 3 in Bob’s side. Three of them are ancillary modes. Two Kerr media, two beam splitters, and four photodetectors are needed to realize the teleportation scheme. As is well known, a cross Kerr interaction is described by the Hamiltonian [18] ˆ Hˆ = −χ aˆ † aˆ bˆ † b,

(3)

Fig. 1. Schematic diagram for quantum teleportation of the superposed coherent state.

which leads to the following unitary evolution operator † ˆ† ˆ Kˆ ab (τ ) = eiτ aˆ aˆ b b ,

(4)

ˆ are and aˆ (b) where τ = χt with t is the evolution time, the creation and annihilation operators for mode 1 (2), respectively. It is easy to see that when χt = π the action of the cross-Kerr unitary evolution operator on two modes with the coherent-state input |αa |βb is given by aˆ †

(bˆ † )

Kˆ ij (π)|αi |βj     1 = |αi |βj + |−βb + |−αi |βj − |−βj , (5) 2 which acts as the quantum channel when i = 3 and j = 2. We assume that mode 1 and 3 are those of the source and target states of the teleportation, mode 2 is an ancillary mode. Mode 1, 2, and 3 are initially in states |ϕ1 given by Eq. (1), |α2 , and |α3 , respectively. Let mode 2 and 3 pass through the cross-Kerr medium K23 with a τ = π interaction, this process can prepare a entangled state which has the form of Eq. (5) with replacement of β by α between modes 2 and 3 and acts the role of the quantum channel for teleportation, this state has special form, the states of mode 2 are not orthogonal each other, while ones of mode 3 do. For teleportation, Bob mix mode 1 and mode 2 by the cross-Kerr medium K12 with a τ = π interaction. After passing two cross Kerr media the output state of modes 1, 2, and 3 becomes |Ψ 123 = Kˆ 12 (π)Kˆ 23 (π)|ϕ1 |α2 |α3   1  = N |α, α12 x|α3 + y|−α3 2   + |α, −α12 x|α3 − y|−α3   + |−α, α12 x|−α3 + y|α3   + |−α, −α12 −x|−α3 + y|α3 .

(6)

From Eq. (6) we can see that quantum teleportation of the SCS given by Eq. (1) from mode 1 to mode 3 may be realized if one can discriminate quantum states |α, α12 , |α, −α12 , |−α, α12 , and |−α, −α12 . Unfortunately, no one can discriminate decisively the four states since they are nonorthogonal with each other. As the increase of the coherent amplitude, these states becoming quasi-orthogonal, so we can discriminate them approximately. In order to overcome this discrimination problem of the four states, we further introduce two

J.-Q. Liao, L.-M. Kuang / Physics Letters A 358 (2006) 115–120

ancillary modes 4 and 5. Let mode 1 and 4, mode 2 and 5 mix at lossless symmetric 50/50 beam splitters BS1 and BS2 , respectively. The beam splitter transformation on mode i and j is described by Bij = exp[i(π/4)(aˆ i† aˆ j + aˆ j† aˆ i )]. Each beam splitter is equipped together with a pair of π/2 phase shifters described by the unitary operator Pi = exp(−iπ aˆ i† aˆ i /2) with i being mode label. Then we have the total unitary operator Bij = Pj Bij Pj , which transforms the state |α, βij as √  √   Bi,j |αi |βj = (α + β)/ 2 i (α − β)/ 2 j . (7) After passing the cross-Kerr media, beam splitters and phase shifts, the output state of the five modes is given by |Φ12345 = B14 B25 |Ψ 123 |α4 |α5   1 √  √  = N  2α 1  2α 2 |04 |05 x|α3 + y|−α3 2 √   √    + |01  2α 2 − 2α 4 |05 x|−α3 + y|α3  √   √   +  2α 1 |02 |04 − 2α 5 x|α3 − y|−α3  √   √    + |01 |02 − 2α 4 − 2α 5 −x|−α3 + y|α3 . (8) From Eq. (8) it can be seen that near-complete quantum teleportation of the SCS given by Eq. (1) can be realized through photon-number measurements of mode 1, 2, 4, and 5, classical communication between Alice and Bob, and Bob’s proper operations. In what follows, we investigate the probability of success of the teleportation scheme for various different measurement cases of photon numbers. We denote the photon numbers counted by detectors D1 , D4 , D2 , and D5 by n1 , n4 , n2 , and n5 , respectively. All of these detectors are in Alice’s side. Looking carefully at Eq. (8), we recognize that the perfect or approximate teleportation of the SCS can succeed with respect to the outcomes of the following four cases of photon-number measurements made by detectors D1 , D4 , D2 , and D5 , n1 > 0,

n2 > 0,

n4 = n5 = 0,

(9)

n2 > 0,

n4 > 0,

n1 = n5 = 0,

(10)

n1 > 0,

n5 > 0,

n2 = n4 = 0,

(11)

n4 > 0,

n5 > 0,

n1 = n2 = 0.

(12)

The analysis below indicate that corresponding to the outcomes given by Eqs. (9) and (10), a perfect teleportation of the SCS can be realized. Indeed, for the outcomes of the photonnumber measurements given by Eq. (9), from Eq. (8) we can see that Alice sends Bob outcomes (n1 > 0, n2 > 0, n4 = n5 = 0) of D1 , D2 , D4 and D5 through a classical communication, a perfect teleportation of the SCS is realized. In this case, Bob need not do any other operation. The probability of success is P1 =

∞    1 n|2 m|4 0|5 0|Φ12345 2 .

(13)

n,m=1

The outcome given by Eq. (10) corresponds to the second term on the right-hand side of Eq. (8). In this case, in order to get a complete teleported state, Bob has to make a π phase shift ˆ which transforms the transformation given by Pˆ = exp(iπ aˆ † a),

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coherent state |α as Pˆ |α = |−α. In this case, the probability of success is P2 =

∞    1 0|2 n|4 m|5 0|Φ12345 2 .

(14)

n,m=1

Making use of Eq. (8) we find the probabilities of success to be 1 2 2 (15) 1 − e−2|α| . 4 However, corresponding to the outcomes given by Eqs. (11) and (12), the situation is different. Only an approximate teleportation of the SCS is possible in this case. In fact, when the outcomes given by Eqs. (11) and (12) appear, the mode 3 collapses to the following states, respectively,   |ϕ3 = M x|α3 − y|−α3 , (16)   |ϕ4 = M −x|−α3 + y|α3 , (17) P1 = P2 =

where the normalization constant is given by −1/2  2 . M = |x|2 + |y|2 − 2e−2|α| Re(xy ∗ )

(18)

The corresponding probabilities of success are given by P3 = P4 =

∞    1 n|4 0|2 0|5 m|Φ12345 2 , n,m=1 ∞ 

  1 0|2 0|4 n|5 m|Φ12345 2 .

(19)

(20)

n,m=1

Making use of Eq. (8) we find the corresponding probabilities of success to be 1 2 2 P3 = P4 = 1 − e−2|α| . (21) 4 From Eqs. (16) and (17) we can see that in order to realize a perfect teleportation of the SCS one has to require the following state transformation [8]     M x|α3 − y|−α3 ⇒ N x|α3 + y|−α3 , (22) which implies that the following transformation between coherent states |−α → −|−α,

|α → |α,

(23)

which is generally not a unitary transformation except for the limit case of |α| → ∞. Finally, we calculate the average fidelity of entire teleportation process. In order to do this, we use the displacement transformation with the parameter value iπ/(4α) to approximately realize the transformation (22). Then, when the input state is |ϕ3 given by Eq. (16), the normalized output state becomes

  iπ |β = D3 M x|α3 − y|−α3 4α    

 iπ  iπ   = M x (24) + α + y −α , 4α 4α 3 3

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Fig. 2. The fidelity given by Eq. (25) as a function of the parameters α and θ .

where the displacement operator D(iπ/4α) can be effectively performed using a beam splitter with the transmission coefficient T close to unity and a high-intensity coherent field [19]. The fidelity of the state (24) with respect to the target state (1) is given by  2 F3 = β|ϕ1 

π2 2 2 = |N | |M| exp − 16α 2   2  2  ×  i |x|2 + |y|2 + 2e−2iα Im(xy ∗ )  , (25) where we have assumed α to be real. It is straightforward to show that under the above beam splitter when the input state is |ϕ4 given by (17), the same fidelity can be obtained, i.e., F4 = F3 . Let x = cos θ , y = sin θ , in Fig. 2 we plot the fidelity as a function of θ and α. Fig. 2 indicates that the fidelity is independent of the parameter θ , and it approaches 1 for a large coherent amplitude α. The average fidelity Fav of the whole teleportation process can be defined as Fav =

4 

Pi (x, y)Fi (x, y),

(26)

i=1

where P1 , P2 , P3 and P4 correspond to the probabilities of the outcome from case (9) to case (12), and F1 = F2 = 1, and F3 = F4 . We plot the average fidelity Fav as a function of α and θ for x = cos θ , y = sin θ in Fig. 3. From Fig. 3 we can see that the average fidelity of the whole teleportation process is independent of the parameter θ . This means that the present teleportation scheme has the same quality for an arbitrary teleported state with the form of the state given by Eq. (1). We can also see that the average fidelity of the scheme approaches 1 for a large coherent amplitude α, as the increase of the amplitude of coherent states |±α, they becoming quasi-orthogonal, states of modes 1 and 2 in Eq. (3) can be discriminated approximately, and the transformation described by Eq. (23) can be implemented approximately, this scheme corresponds to teleportation of quasi-qubit states with the logic bases encoded by |α and |−α. Therefore, we conclude that the present scheme can realize a near-complete quantum teleportation of superposed coherent states with a large coherent amplitude.

Fig. 3. The average fidelity of the teleportation as a function of the parameters α and θ .

In order to observe advantages with respect to the two-mode squeezed-vacuum-state quantum channel teleportation scheme in Ref. [6] which demonstrated by Furusawa and coworkers [7], we make a comparison upon teleportation quality of the our present protocol and the scheme proposed in Ref. [6] through calculating the fidelity of corresponding protocols for the same teleported state with a given fixed total energy (i.e., a given fixed α). For simplicity and without loss of generality, we con√ sider the case of x = y = 1/ 2 and α being real in the teleported state (1). In this case, from Eq. (26) we can find the average fidelity of our present protocol to be

π2  2 Fav = exp − 1 − e−4α . 2 16α

(27)

On the other hand, for the two-mode squeezed-vacuum-state quantum channel teleportation scheme in Ref. [6], when we consider ideal detectors with efficiency η = 1, from Eq. (12) in Ref. [6] we can get the fidelity of the same teleported state as F (r) =

 −4α 2   2 e−2r  −4α 2 − 1 + exp 1+e exp −4α −2r − e 1+e−2r 2(1 + e−2r )(1 + e−2α ) 2

+

1 , 1 + e−2r

(28)

which indicates that the fidelity of the teleported states depends on not only the squeezing parameter of the quantum channel r but also the parameter of the teleported state α. In order to evaluate the quality of the two protocols in terms of Eqs. (27) and (28), we plot the fidelity given in Eqs. (27) and (28) as a function of the teleported-state parameter α in Fig. 4. For the two-mode squeezed vacuum state protocol, we plot five (solid) curves which correspond to five different values of the squeezed parameter r = 0.1, 1, 2, 3, and 6, respectively. The dashed curve corresponds to our protocol. From Fig. 4 it is easy to see that the fidelity of our protocol increases with the increase of coherent amplitude α and reaches the upper bound in the limit of α → ∞. At the same time, for a small r, the fidelity of protocol proposed in Ref. [6] decreases with the increase of coherent amplitude α. And the larger of the squeezed parameter r, the larger of the fidelity. For a large squeezed parameter r,

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entangled coherent states |α, −α, . . . , −α+|−α, α, . . . , −α+ |−α, −α, . . . , α, respectively. In the scheme in Ref. [10] a non-maximally entangled coherent state cos θ |α|α − sin θ × |−α|−α is used to act as the quantum channel to teleport the SCS. 3. Concluding remarks

Fig. 4. The fidelity of the teleportation of state |α + |−α as a function of the parameter α for two different protocols. The dashed line represents the result of our protocol while the solid lines show the results of the two-mode squeezed-vacuum-state protocol in Ref. [6], which correspond to five different values of the squeezed parameter r = 0.1, 1, 2, 3, and 6, respectively.

say r = 6 in Fig. 4, the fidelity approaches unit which is independent of the teleported-state parameter α. However, it is very difficulty to experimentally obtain so large squeezed parameter r with current experimental technology. In fact, due to the technical limit, the squeezing used for teleportation was limited to 3 dB (r = 0.34) [20,21] although the maximum squeezing of 6 dB (r = 0.69) was already observed in the same experimental setup [22]. It should be mentioned that in the limit of infinite squeezing r → ∞, the two-mode squeezed vacuum state corresponds to the original perfectly correlated and maximally entangled Einstein–Podolski–Rosen state and, in this limit, the teleportation fidelity is perfect with the fidelity of one which is independent on the input state. However, the maximally entangled quantum channel implies an infinitely squeezed state which is a unphysical state since it requires infinite energy [23,24]. Through above analysis of the fidelity, we can see that our scheme has higher fidelity in the larger α and the smaller r regime. We can also see that the fidelity of the teleported state depends on not only the parameter of the quantum channel but also the state parameter of the teleported state for both our present scheme and the protocol in Ref. [6]. Although we need the knowledge of the coherent amplitude of the state for both Alice and Bob. Moreover, our scheme has the same quality for an arbitrary teleported state with the form given by Eq. (1) while the protocol with the two-mode squeezed-vacuum-state quantum channel has not this property. In general, the teleportation quality of a quantum state depends upon parameters of quantum channels. When different quantum channels are used to teleport the same quantum state, the teleportation quality may different. The quantum channel which our present protocol uses is given by Eq. (5) is different from those quantum channels in previous protocols. Our quantum channel is an entangled coherent state which consists of a pair of anti-phase coherent states and a pair of cat states. Moreover, Bell-type entangled coherent states |α|−α ± |−α|α and |α|α ± |−α|−α are used to teleport the SCS in Ref. [8,14]. In An’s many-partite protocols [12,13] quantum channels are GHZ-type entangled coherent states |α|α · · · |α ± |−α|−α · · · |−α and W-type

In conclusion, we have proposed an optical scheme for the quantum teleportation of superposed coherent states by using optical elements such as nonlinear Kerr media, beam splitters, phase shifters, and photon detectors. We have shown that this scheme has the same teleportation quality (i.e., the same average fidelity) for arbitrary superposed coherent states. Although our present protocol is probabilistic, the average fidelity can approach unity for a large amplitude of the superposed coherent states. Hence, this is a near-complete scheme. Comparing with previous optical schemes, our scheme has advantages. As is well known, one has to make the exact photon-number measurements to distinguish between even and odd photons in all of previous optical schemes on teleportation of the superposed coherent states. However, in our protocol what we need is only the yes or no measurements of the photon number of the related modes. Finally, we discuss the feasibility of the present scheme. In our teleportation protocol, except linear optical elements, we need the nonlinear one, cross-Kerr medium. It is a greater challenge to experimentally produce large Kerr nonlinearities. Although sufficiently large Kerr nonlinearities have been difficult to produce, significant progress is being made in this area. In particular, recent progress on atomic quantum coherence [25– 28] indicates that it is possible to prepare Kerr medium with the giant Kerr nonlinearities through using the electromagnetically induced transparency (EIT) technology. Paternostro and coworkers [25] proposed a double EIT scheme which can enhance the cross-Kerr effect in a dense atomic medium in the EIT regime. Especially, it has been proved that the interaction of two travelling fields of light in an atomic medium is able to show giant Kerr nonlinearities by means of the so-called crossphase-modulation. Measured values of the χ 3 parameter are up to six orders of magnitude larger than usual [26]. Therefore, our protocol is at the reach of current experiment. We would emphasize that our proposed protocol can potentially be applied to quantum information processing based on continuous variables. Actually, there have appeared tendencies to encode information in quantum states with continuous variables, since such an encoding allows the information to be manipulated much more efficiently than with traditional discrete variable states (qubits). Acknowledgements This work is supported by the National Fundamental Research Program Grant No. 2001CB309310, the National Natural Science Foundation under Grant Nos. 10325523, 90203018 and 10075018, the Foundation of the Education Ministry of China, and the Scientific Research Fund of Hunan Provincial Education Department under Grant Nos. 200248 and 02A026.

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