Vol 14 o 5, May 005 cfl 005 Chin. Phys. Soc. 1009-1963/005/14(05)/0974-06 Chinese Physics and IOP Publishing Ltd Proposal of many-party controlled teleportation for multi-qubit entangled W state * Huang Zhi-Ping( ΠΞ) and Li Hong-Cai(Λ ) y Department of Physics, Fujian ormal University, Fuzhou 350007, China (Received 1 ovember 004; revised manuscript received 7 December 004) A scheme of M-party controlled teleportation for one -qubit entangled W state via (-1) Einstein Podolsky Rosen (EPR) pairs and one (M+)-qubit Greenberger Horne Zeilinger (GHZ) state is proposed. We achieve the teleportation in such a way that M agents can execute the Hadamard transformation, perform the measurement on their qubits and inform the receiver of their measurements. Then we discuss that the receiver cannot fully recover the state from the sender if one agent does not co-operate with him. Keywords: many-party controlled teleportation, multi-qubit entangled W state, Hadamard transformation, EPR pair, GHZ state PACC: 450 1. Introduction Quantum teleportation has received much attention because it plays an important role in quantum information and communication. Since Bennett et al [1] teleported an unknown quantum state of a two-state particle from one place to another with the aid of long-range Einstein Podolsky Rosen (EPR) correlations, scientists have made dramatic progress in the field of quantum teleportation. Quantum teleportation has been extended from discrete-variable systems to continuous-variable systems [ 6] and also from single particle to multi-particles. [7 10] Experimental demonstration of quantum teleportation has been realized with photon polarized states, [11] optical coherent states, [1] and nuclear magnetic resonance [13] recently. In 000, Zhou et al presented a theoretical scheme for controlled quantum teleportation. [14] According to the scheme, a third side is included beside asenderand a receiver, and the quantum channel is supervised by this additional side. The signal state cannot be transmitted unless all three sides agree to co-operate with each other. Recently, ang et al have presented a way to teleport multi-qubit quantum information via the control of many agents in a network. [15] de Sen Aditi et Λ Project supported by the Fujian atural Science Foundation (Grant o A010014). y Corresponding author: E-mail: hcli45@fjnu.edu.cn http://www.iop.org/journals/cp al have found that the -qubit states of W class, [16] for > 10, lead to more robust violations of local realism than the Greenberger Horne Zeilinger (GHZ) states. In this paper, we present a simple scheme to teleport an unknown -qubit entangled W state from a sender to a remote receiver via the control of M agents in a network by using (-1) EPR pairs and one (M+)-qubit GHZ state [17] as the quantum channels. We show that the receiver can successfully get access to the original W state if all the agents collaborate with each other through their local operation and classical communication with the receiver; however, if even one of agents does not co-operate withthe others, the original W state cannot be fully recovered by the receiver. Finally, we give an example of the controlled teleportation for three-qubit entangled W states..many-party controlled teleportation for multi-qubit entangled W state We suppose that the sender Alice wants to teleport an unknown multi-qubit entangled W state to the receiver Bob. The W state of message qubits labelled
o. 5 Proposal of many-party controlled teleportation for... 975 by (C 1 ;C ; ;C ) can be expressed as [16] j' w i (c 0 j000 :::0i + c 1 j100 :::0i + c j010 :::0i where + + c j000 :::1i) C 1C:::C jffi ki i Ci ; (1) 1, 6 0, ffi ki < : 0; if k 6 i; 1; if k i: Firstly, Alice prepares one (-1) EPR pairs jffii AB and one (M+)-qubit GHZ state jffii A B D : jffii AB 1 jffi + i AiBi 1 1 p (j00i + j11i) AiBi ; () jffii A B D 1 p (j000 :::0i + j111 :::1i) A B D 1 DM : (3) And then she sends the first (-1) EPR qubits (B 1 ;B ;:::;B 1 ) and one GHZ qubit B to Bob and M GHZ qubits to the M agents while keeps the GHZ qubit A and the other (-1) EPR qubits (A 1 ;A ;:::;A 1 ) by herself. The state of the whole system is given by jψ T i j' w ijffii AB jffii A B D 1 jffi ki i Ci p (j00i + j11i) AiBi 1 p (j000 :::0i + j111 :::1i) A B D 1:::DM : (4) Secondly, Alice performs a series of joint Bellstate measurements on one EPR qubit and one message qubit (A i ;C i ); (i 1; ; ; 1) and informs Bob of her measurements through a classical channel. According to Alice's measurements, Bob performs a series of single-qubit unitary transformations From Eq.(), we have j i Bi >< >: U i (i 1; ; ; 1) on his corresponding EPR qubits B i (i 1; ; ; 1). Thirdly, Alice performs a Bell-state measurement on one message qubit and one GHZ qubit (A ;C ) and broadcasts her measurements. Then according to Alice's measurements, Bob performs a single-qubit unitary transformation U on his qubit B and all of M agents perform a series of single-qubit unitary transformations (U D ;U 1 D ; ;U DM ) on their qubits (D 1 ;D ; ;D M ). After the preceding manipulations, the state becomes jψ T i 1 U 1 U U U U D 1 U DM A 1C1hψj A Chψj A C hψjψ T i 1 U iaici hψjffi ki i Ci jffii AiBi U U D 1 U DM A C hψjffi k i C jffii A B D ; (5) where jψi AiCi are four Bell states involved in Alice's Bell-state measurements on her qubits (A i ;C i ), which are expressed as jffi ± i AiCi 1 p (j00i±j11i) AiCi ; (6) j' ± i AiCi 1 p (j01i±j10i) AiCi : (7) The U i and U Dj (j 1; ; ;M) performed by Bob and one of M agents are Pauli matrices, which are shown in the following equation. For simplicity, we define j i Bi U iaici hψjffi ki i Ci jffii AiBi ; ( > i 1); () j i B D U U D 1 U DM A C hψjffi k i C jffii A B D : I Bi ( Ci h0jffi ki i Ci j0i Bi + Ci h1jffi ki i Ci j1i Bi ); for jψi AiCi jffi + i AiCi ;U i I Bi ; (ff z ) Bi ( Ci h0jffi ki i Ci j0i Bi Ci h1jffi ki i Ci j1i Bi ; for jψi AiCi jffi i AiCi ;U i (ff z ) Bi ; (ff x ) Bi ( Ci h1jffi ki i Ci j0i Bi + Ci h0jffi ki i Ci j1i Bi ); for jψi AiCi j' + i AiCi ;U i (ff x ) Bi ; (iff y ) Bi ( Ci h1jffi ki i Ci j0i Bi Ci h0jffi ki i Ci j1i Bi ); for jψi AiCi j' i AiCi ;U i (iff y ) Bi : By calculation, we find that Eq.(10) can be reduced to the following expression: j i Bi Ci h0jffi ki i Ci j0i Bi + Ci h1jffi ki i Ci j1i Bi < : j0i Bi ; k 6 i; ffi ki 0; j1i Bi ; k i; ffi ki 1; (9) (10) ( > i 1): (11)
976 Huang Zhi-Ping et al Vol. 14 From Eq.(9), we have I B M j1 I Dj ( C h0jffi k i C j00 0i B D 1 DM C + C h1jffi k i C j11 1i B D 1 DM ); for jψi A C (jffi + i A C ; U I B ;U D 1 I D 1 ; ;U DM I DM ; j i B D >< (ff z ) B M j1 I Dj ( C h0jffi k i C j00 0i B D 1 DM C h1jffi k i C j11 1i B D 1 DM ); for jψi A C jffi i A C ; U (ff z ) B ;U D 1 I D 1 ; ;U DM I DM ; (ff x ) B M j1 (ff x ) Dj ( C h1jffi k i C j00 0i B D 1 DM C + h0jffi k i C j11 1i B D 1 DM ); >: for jψi A C j' + i A C ; U (ff x ) B ;U D 1 (ff x ) D 1 ; ;U DM (ff x ) DM ; (iff y ) B M j1 (ff x ) Dj ( C h1jffi k i C j00 0i B D 1 DM C h0jffi k i C j11 1i B D 1 DM ); for jψi A C j' i A C ; U (iff y ) B ;U D 1 (ff x ) D ; ;U 1 DM (ff x ) DM : (1) By calculation, we find that Eq.(1) can be expressed as j i B D C h0jffi k i C j00 0i B D 1 DM + C h1jffi k i C j11 1i B D 1 DM < : j00 0i B D1 DM ; k 1; ; 1; ffi k 0; j11 1i B D 1 DM ; k ; ffi k 1: (13) Then, Eq.(5) can be rewritten as 1 jψ T i 1 c j11 1i B D 1 DM + j00 0i B D 1 DM : (14) Lastly, all of the M agents execute the Hadamard transformation on their qubits in the following forms: j0i Dj! p 1 (j0i + j1i) Dj and j1i Dj! p 1 (j0i j1i) Dj (j 1; ; ;M). Then the state of Eq.(14) becomes M 1 jψ T i (c j1i B (j0i j1i) Dj + j1 c j1i B 1 + j0i B M j1 jde i D1 DM jd o i D1 DM j0i B jde i D 1 DM + jd o i D1 DM jd e i D 1 DM c j1i B + j0i B 1 (j0i + j1i) Dj 1 1 + jd o i D c 1 DM j1i B + j0i B ; (15) where jd e i D 1 DM is a sum over all possible binary-computational bases of M GHZ qubits (D 1, D ; ;D M ), for which each basis jd e i contains even number 1 (i.e. the number of j1i contained in each jd e i is even), while jd o i D represents a sum 1 DM over all possible binary-computational bases of qubits (D 1 ;D ; ;D M ), for which each basis jd o i contains odd number 1 (i.e. the number of j1i contained in each jd o i is odd). And then each agent makes a measurement on his GHZ qubit in the single-qubit computational basis fj0i; j1ig and sends their measurements to receiver
o. 5 Proposal of many-party controlled teleportation for... 977 Bob. From Eq.(15) one can see that if Bob knows the M agents' measurements for their qubits contain even number 1 (i.e. jd e i), he can predicate that his qubits must be in the state jψ T i 3 c j1i B + j0i B : (16) On the other hand, if Bob knows that the M agents' measurements include odd number 1 (i.e. jd o i), he knows that his qubits must be in the state jψ T i 4 1 c j1i B + j0i B : (17) Then Bob executes the unitary transformation (ff z ) B on the qubit B (ff z ) B jψ T i 4 jψ T i 3 : (1) From Eq(16) and Eq(1), it is evident that Bob fully recovers the original state of Alice. So the manyparty controlled teleportation of multi-qubit entangled W state is successfully achieved. 3. The many-party controlled teleportation is a failure if even one of agents does not cooperate with the others In this section, our purpose is to show that in the case when even one of agents does not collaborate with the others, it is impossible for Bob to gain the full quantum message. Assume that one agent, say agent D M, does not co-operate with other agents. In this case, after other agents (D 1 ;D ; ;D M 1 ) have performed a Hadamard transformation on their GHZ qubits in the following forms: j0i Dj! p 1 (j0i + j1i) Dj and j1i Dj! 1 p (j0i j1i) Dj (j 1; ;:::;M 1), one can see that the state of Eq.(14) will be transformed into M 1 jψti 0 c j11i B DM (j0i j1i) Dj + M 1 j00i B DM j1 j1 c j11i B DM jde i D 1 DM 1 jd o i D1 DM 1 1 + j00i B DM jde i D 1 DM 1 + jd o i D1 DM 1 jd 0 ei D 1 DM 1 1 c j11i B DM + 1 j00i B DM (j0i + j1i) Dj + jdoi 0 D c 1 DM 1 j11i B DM + j00i B DM ; (19) where jdei 0 D 1 DM 1 represents a sum over all possible binary-computational bases of the (M 1) GHZ qubits belonging to the agents (D 1 ;D ;:::;D M 1 ), for which each binary-computational P bases jdei 0 contains even number 1; while jdoi 0 D 1 DM 1 represents a sum over all possible binary-computational basis of the (M 1) GHZ qubits (D 1 ;D ; ;D M 1 ), for which each binary-computational basis jdoi 0 contains odd number 1. From Eq.(19) one can see that for every measurement on the (M 1) GHZ qubits (D 1 ;D ;:::;D M 1 ), the density operator ρ B of the qubits (B 1 ;B ; ;B ) belonging to Bob, after having traced over the qubit D M, can be written as where jψ 0 Ti 3 ρ B Tr DM (jψ 0 Ti 33 hψ 0 Tj); (0) c j11i B DM + 1 1 j00i B DM : (1)
97 Huang Zhi-Ping et al Vol. 14 The trace of ρ B is Trρ B Tr[Tr DM (jψ 0 Ti 3 hψ 0 Tj)] 1 1 c k c < 1: () From Eqs.(19), (0), (1) and (), one can see that Bob cannot fully recover the original state of Alice if even one of agents does not co-operate with the others. 4. The case of three-qubit entangled W state via three agents For example, Alice wants to teleport three-qubit entangled W state j' W i to Bob via three agents. The state j' W i is j' W i (c 0 j000i + c 1 j100i + c j010i + c 3 j001i) C 1CC3 : (3) Firstly Alice prepares two EPR pairs, jffii AiBi 1 p (j00i + j11i) AiBi (i 1; ) (4) and a five-qubit GHZ state. jffii A 3B3D 1 p (j00000i + j11111i) A 3B3D1DD3 : (5) Then Alice sends the first two EPR qubits (B 1 ;B ) and one GHZ qubit B 3 to Bob and 3 qubits (D 1 ;D ;D 3 ) to the three agents while keeps qubits (A 1 ;A ;A 3 ) by herself. The state of the whole system is given by jψ T i j' W ijffii A 1B1 jffii A B jffii A 3B3D (c 0 j000i + c 1 j100i + c 0 j010i + c 0 j001i) C 1CC3 1 p (j00i + j11i) A 1B1 1 p (j00i + j11i) A B 1 p (j00000i + j11111i) A 3B3D1DD3 : (6) Secondly, Alice performs a joint Bell-state measurement on her qubits (A 1, C 1 ), (A, C ), (A 3, C 3 ), and informs Bob and the three agents of her measurements through a classical channel. According to Alice's measurements, Bob performs three corresponding unitary transformations on his qubits (B 1 ;B ;B 3 ), and the three agents perform a corresponding unitary transformation on their qubits (D 1 ;D ;D 3 ). After the above manipulations, the state becomes jψ T i 1 (c 0 j000i + c 1 j100i + c j010i) B 1BB3 j000i D 1DD3 + c 3 j001i B 1BB3 j111i D 1DD3 : (7) Lastly the three agents execute the Hadamard transformation on their qubits in the following forms j0i Dj! p 1 (j0i+j1i) Dj and j1i Dj! p 1 (j0i j1i) Dj (j1,, 3). The Eq.(5) transfers to jψ T i (c 0 j000i + c 1 j100i + c j010i + c 3 j001i) B 1BB3 (j000i + j011i + j101i + j110i) D 1DD3 +(c 0 j000i + c 1 j100i + c j010i c 3 j001i) B 1BB3 (j001i + j010i + j100i + j111i) D 1DD3 : () Then, the agents perform a measurement on their qubits in the single-qubit computational basis fj0i; j1ig and send their measurements to Bob. If their GHZ qubits contain even number j1i (i.e. j000i, j011i, j101i, j110i), Bob can predicate that his three qubits must be in the state, jψ T i 3 (c 0 j000i + c 1 j100i + c j010i + c 3 j001i) B 1BB3 : (9) If their GHZ qubits contains odd number j1i (i.e. j001i, j010i, j100i, j111i), Bob can predicate that his three qubits must be in the state, jψ T i 4 (c 0 j000i + c 1 j100i + c j010i + c 3 j001i) B 1BB3 : (30) Then Bob executes the unitary transformation (ff z ) B 3 on the qubit B 3, (ff z ) B 3jψ T i 4 jψ T i 3 : (31) From Eq.(7) and Eq.(9), it is evident that Bob can fully recover the original three-qubit entangled W state of Alice under the co-operation with three agents. 5. Conclusion We have proposed a simple scheme for the M- party controlled teleportation of -particle entangled W state. First, the sender (Alice) and the receiver (Bob) need to share (-1)-EPR pairs, and Alice, Bob and M controlled-sides need to share a (M +)-qubit GHZ state. In order to implement such a teleportation, it requires that Alice perform -times two-qubit
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