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Three methods to distill multipartite entanglement over bipartite noisy channels
Physics Letters A 372 (2008) 3157–3161 www.elsevier.com/locate/pla
Three methods to distill multipartite entanglement over bipartite noisy channels Soojoon Lee a,∗ , Jungjoon Park b a Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 130-701, South Korea b Division of Liberal Art in Mathematics, University of Seoul, Seoul 130-743, South Korea
Received 12 April 2007; received in revised form 17 January 2008; accepted 21 January 2008 Available online 6 February 2008 Communicated by P.R. Holland
Abstract We first assume that there are only bipartite noisy qubit channels in a given multipartite system, and present three methods to distill the general Greenberger–Horne–Zeilinger state. By investigating the methods, we show that multipartite entanglement distillation by bipartite entanglement distillation has higher yield than ones in the previous multipartite entanglement distillations. © 2008 Elsevier B.V. All rights reserved. PACS: 03.65.Ud; 03.67.Mn; 03.67.Hk Keywords: Multipartite entanglement distillation; Teleportation; Entanglement swapping; CNOT operation
1. Introduction Quantum entanglement has been considered as one of the most crucial resources in quantum information sciences. In particular, it has been known that entanglement guarantees perfectly secure quantum communications if it is perfect. Unfortunately, it is also known that it is hard for a perfect entanglement to naturally exist in the real world. Thus, in order to perform a faithful quantum communication, one needs a process to make a nearly perfect entanglement out of noisy entanglement, which is called the entanglement distillation. In the case of bipartite systems, it has been shown that the unconditional security of quantum key distribution is closely related with the distillation of entanglement [1,2]. Thus, for faithful multiparty quantum communications based on multipartite entanglement, such as quantum secret sharing protocols [3,4], the distillation of multipartite entanglement should be required. On this account, several methods [5–7] for multipartite entanglement distillation have developed by generaliz-
* Corresponding author.
E-mail addresses: [email protected] (S. Lee), [email protected] (J. Park). 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.01.074
ing the 2-qubit distillation protocol with the hashing protocol in [8]. Remark that multipartite entanglement can be simply generated from bipartite entanglements by various ways, in which there are three typical methods to exploit quantum teleportation [9], entanglement swapping [10,11], and the CNOT operation [11,12], respectively. We now assume that there are bipartite noisy qubit channels in a given multipartite system, such as the identical depolarizing qubit channels. Then, by employing the three typical methods and generalizing them, we present three methods to distill multipartite entanglement in this situation, and by comparing the fidelities between the perfect multipartite entangled state (for example, the Greenberger–Horne– Zeilinger (GHZ) state [13]) and the multipartite states distilled by the three methods, we show that the methods provide us with the same fidelities, and that the fidelities are the same as the fidelity between perfect GHZ state and the state sent through the identical depolarizing qubit channels. We finally show that the multipartite entanglement distillation by bipartite entanglement distillation could be more efficient than the previous multipartite entanglement distillations in [5–7]. This Letter is organized as follows. In Section 2 we introduce our three methods, and show that the noisy GHZ states
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made by all the three methods have the same fidelities. In Section 3 we show that multipartite entanglement distillation by bipartite entanglement distillation has higher yield than that of the previous multipartite entanglement distillations. Finally, in Section 4 we summarize our result. 2. Three methods to distill multipartite entanglement For convenience, we first consider the distillation of the 3qubit GHZ state, 1 |Φ3 = √ |000 + |111 , 2
(1)
and assume that for three systems A, B, and C, there are the identical depolarizing qubit channels DF with fidelity F between A and B and between A and C, respectively. Here, the action of each channel is described as 1−F σx ρσx† + σy ρσy† + σz ρσz† . DF : ρ → Fρ + 3
+ + − − − ΦF = λ + 0 Ψ 0 Ψ0 + λ 0 Ψ 0 Ψ0 +
(2)
|φ +
We remark that if one prepares a Bell state, = (|00 + √ |11)/ 2, and send one qubit of |φ + through the depolarizing channel, then the state becomes 1 − F I ⊗ I − φ + φ + , ρF = F φ + φ + + 3
Fig. 1. The method to use teleportation: The dashed lines, the dotted ellipses and the dotted lines represent nearly perfect entangled states, the 2-qubit orthogonal measurements, and transmissions of classical information, respectively.
(3)
which is called the isotropic state [14]. Thus, if two |φ + ’s are prepared in the system A and one qubit of each |φ + is sent to the systems B and C through the two depolarizing channels, then one can obtain two isotropic states as nearly perfect entangled states in the compound systems AB and AC, respectively (if F is close to 1). Hence, the present three methods begin on the assumption that the subsystems AB and AC have already shared two nearly perfect isotropic states, respectively. In all three methods, it is also assumed that the three systems can have sufficiently many resources and can deal with all kinds of local quantum operations and classical communication.
3
λj Ψj+ Ψj+ + Ψj− Ψj− ,
(5)
j =1
√ where |Ψj± = (|j ± |7 − j )/ 2 are the GHZ-basis states (note that |Ψ0+ = |Φ3 ), and
1 1 + 2F 2 4F − 1 2 ± λ0 = ± , 2 3 3
1 − F 1 + 2F = λ3 , λ1 = 3 3 2(1 − F )2 . (6) 9 Thus, by the first method, a nearly perfect GHZ state can be distilled with fidelity
1 1 + 2F 2 4F − 1 2 Φ3 |ΦF |Φ3 = + 2 3 3 1 = 10F 2 − 2F + 1 . (7) 9 λ2 =
2.2. The method to use entanglement swapping 2.1. The method to use teleportation We firstly take account of the method to use teleportation, which is illustrated in Fig. 1 and is described as follows: (i) Prepare the GHZ state in the party A. (ii) Teleport two qubits of the GHZ state over nearly perfect bipartite states ρF ; Perform the 2-qubit orthogonal measurements (with respect to the Bell basis) on each qubit of the GHZ state and one qubit of each ρF , and then apply proper local unitary operations on the systems BC depending on the measurement outcomes. In the step (i), the total state can be written as |Φ3 Φ3 | ⊗ ρF ⊗ ρF ,
(4)
and it is straightforward to obtain that after the step (ii) the resulting 3-qubit state becomes a state in the class of 4-parameter states presented in [15] as follows:
We secondly consider the method to use entanglement swapping, which is seen in Fig. 2 and is described as follows: (i) Prepare one of the Bell states |φ + in the system A. (ii) Perform the 3-qubit orthogonal measurement (with respect to the GHZ basis) on one qubit of |φ + and one qubits of each ρF in the system A, and then apply proper local unitary operations on the systems BC depending on the measurement outcome, respectively. In the step (i) of the second method, the total state can be written as + + φ φ ⊗ ρF ⊗ ρF . (8) Then it can be shown that the 3-qubit state shared by the systems ABC after the step (ii) becomes the same state as one obtained from the first method in Eq. (5). Hence, we can see that from the second method a nearly perfect GHZ state can be
S. Lee, J. Park / Physics Letters A 372 (2008) 3157–3161
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Fig. 2. The method to use entanglement swapping: The dashed lines, the dotted circle and the dotted lines represent nearly perfect entangled states, the 3-qubit orthogonal measurement, and transmissions of classical information, respectively.
Fig. 3. The method to use the CNOT operation: The dashed lines, the dotted circle and the dotted lines represent nearly perfect entangled states, the 1-qubit orthogonal measurement, and a transmission of classical information, respectively.
also distilled with fidelity
2.4. Multipartite cases
1 2
1 + 2F 3
2
4F − 1 + 3
2 =
1 10F 2 − 2F + 1 , 9
(9)
which is the same as the first one in Eq. (7). Therefore, the second method is essentially equivalent to the first one.
We now generalize these three methods into general multipartite cases. Assume that there are n − 1 nearly perfect Bell states ρF in the subsystems AB1 , AB2 , . . . , and ABn−1 of the total system AB1 B2 · · · Bn−1 . Then, through the three methods, one tries to distill the n-qubit GHZ state,
2.3. The method to use CNOT operation
1 |Φn = √ 0n + 1n , 2
We finally take the method to use the CNOT operation into account, which is illustrated in Fig. 3 and is described as follows: (i) Apply the CNOT operation on two 1-qubit states of each ρF in the system A. (ii) Perform the 1-qubit orthogonal measurement (with respect to the standard basis) on one qubit of the target qubit in the system A, and then apply a proper local unitary operation on the system C depending on the measurement outcome. Then, it follows from simple calculations that the resulting 3-qubit state after the step (ii) of the third method becomes
in the system AB1 B2 · · · Bn−1 . In this situation, the first method is as follows: (i) Prepare |Φn in the system A. (ii) Teleport n − 1 qubits of |Φn through n − 1 nearly perfect Bell states ρF , and the second one is executed as follows: (i) Prepare one of the Bell states |φ + in the system A. (ii) Perform the n-qubit orthogonal measurement (with respect to the n-qubit GHZ basis) on one qubit of |φ + and each one qubit of n − 1 states of ρF , and then apply proper local unitary operations on the systems B1 B2 · · · Bn−1 depending on the measurement outcome. Here, the n-qubit GHZ-basis states are
+ + (3) − − − ΨF = λ+ 0 Ψ 0 Ψ 0 + λ 0 Ψ 0 Ψ0 +
2
|j ± |2n − 1 − j √ 2
λ1 Ψj+ Ψj+ + Ψj− Ψj−
+ λ2 Ψ3+ Ψ3+ + Ψ3− Ψ3− ,
(10)
which is slightly different from the first and second one in Eq. (5). Nevertheless, we can readily see that after executing the third method, a nearly perfect GHZ state can be distilled with fidelity
2
4F − 1 + 3 1 = 10F 2 − 2F + 1 , 9 1 2
1 + 2F 3
(13)
for j = 0, 1, . . . , 2n−1 − 1. Then the total states in the step (i) of the first and the second methods can be described as
j =1
(3) Φ3 |ΨF |Φ3 =
(12)
2
(11)
which is exactly the same as ones in the first and second methods.
|Φn Φn | ⊗ ρF ⊗ ρF ⊗ · · · ⊗ ρF , + + φ φ ⊗ ρF ⊗ ρF ⊗ · · · ⊗ ρF ,
(14)
respectively. Although the total states in the two methods are slightly different as seen in Eq. (14), one can obtain that the two methods are essentially equivalent as in the 3-qubit case, since the resulting n-qubit states after the protocols are the same. The third method is inductively described as follows: Let (k) k 3. (i) Assume that a nearly perfect k-qubit GHZ state ΨF (3) is shared in the systems AB1 B2 · · · Bk−1 , since one can get ΨF in the systems AB1 B2 from the third method in the 3-qubit case. (ii) In the system A, apply the CNOT operation on one qubit of (k) ΨF and one qubit of ρF of the systems ABk . (iii) Perform
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the 1-qubit orthogonal measurement (with respect to the standard basis) on one qubit of the target qubit in the system A, and then apply a proper local unitary operation on the system Bk depending on the measurement outcome. Then after the iterations of the steps (ii) and (iii), a nearly perfect n-qubit GHZ state can be clearly obtained. Even though the three methods seem to be different, by straightforward but tedious calculations, it can be obtained that the three methods have all the same fidelities with the n-qubit GHZ state,
1 − 4F n−1 1 1 + 2F n−1 + . (15) 2 3 3 3. Comparison with other multipartite distillation protocols For convenience, we consider the case of n = 4, which could be generalized into any multipartite cases. We have seen in the previous section that if F is close to 1 then by our three methods one can distill a nearly perfect 4-qubit GHZ state with the fidelity in Eq. (15) over the depolarizing qubit channels with fidelity F . We note that this fidelity in Eq. (15) can be also obtained simply by sending each of the three qubits in the 4-qubit GHZ state |Φ4 from the system A to each system Bk through the depolarizing channels, respectively. In fact, it can be readily shown that Φ4 |I ⊗ DF ⊗ DF ⊗ DF |Φ4 Φ4 | |Φ4
1 1 + 2F 3 1 − 4F 3 = (16) + , 2 3 3
Fig. 4. Comparison of the yields of the protocols when n = 4: The dotted line, the dashed line, and the solid line represent the yields of the protocols of Refs. [5–7], respectively, and the bold line represents that of the 2-qubit protocol in Ref. [8].
presented three methods to distill the general GHZ state, by employing three typical methods to generate the GHZ state from bipartite entanglements and generalizing them. We have shown that the three methods have the same fidelities, and have shown that the fidelities are the same as the fidelity between the perfect GHZ state and the state sent through the identical depolarizing qubit channels. We have finally concluded that multipartite entanglement distillation by bipartite entanglement distillation has higher yield than ones in the previous protocols. Hence, the distillable entanglement for the multipartite entanglement could be explained by means of the distillable entanglement of each bipartite subsystem.
which is the same as one in Eq. (15). On the assumption that there are given the identical depolarizing channels with fidelity F (here, F is not necessarily close to 1) between each of two qubits in the total system, the noisy GHZ state I ⊗ DF ⊗ DF ⊗ DF (|Φ4 Φ4 |) can be also distilled into a nearly perfect GHZ state by the previous multipartite distillation protocols with hashing protocols in [5–7], and the yields of the protocols can be expressed as functions of the fidelity F of the depolarizing channels, as plotted in Fig. 3 of Ref. [7]. However, the yields are less than the yield of the 2-qubit protocol in [8], which can be seen in Fig. 4. Therefore, if we regard a faithful sharing of a perfect multipartite entangled state over a noisy environment (in our case, the identical depolarizing qubit channels) as a sort of the multipartite entanglement distillation, then we can conclude that the previous multipartite entanglement distillation protocols in [5–7] is less efficient than our protocols following the 2-qubit entanglement distillation protocol.
Acknowledgement
4. Conclusions
[10]
In conclusion, on the assumption that there are only bipartite noisy qubit channels in a given multipartite system, we have
This work was supported by the Kyung Hee University Research Fund in 2007 (KHU-20070659). References [1] [2] [3] [4] [5] [6] [7] [8]
[9]
[11] [12]
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[13] D.M. Greenberger, M.A. Horne, A. Zeilinger, in: M. Kafatos (Ed.), Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, Kluwer, Dordrecht, 1989, p. 69. [14] M. Horodecki, P. Horodecki, Phys. Rev. A 59 (1999) 4206; B.M. Terhal, K.G.H. Vollbrecht, Phys. Rev. Lett. 85 (2000) 2625;
S. Lee, D.P. Chi, S.D. Oh, J. Kim, Phys. Rev. A 68 (2003) 062304. [15] W. Dür, J.I. Cirac, R. Tarrach, Phys. Rev. Lett. 83 (1999) 3562; S. Lee, J. Joo, J. Kim, Phys. Rev. A 76 (2007) 012311.
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Three methods to distill multipartite entanglement over bipartite noisy channels Soojoon Lee a,∗ , Jungjoon Park b a Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 130-701, South Korea b Division of Liberal Art in Mathematics, University of Seoul, Seoul 130-743, South Korea
Received 12 April 2007; received in revised form 17 January 2008; accepted 21 January 2008 Available online 6 February 2008 Communicated by P.R. Holland
Abstract We first assume that there are only bipartite noisy qubit channels in a given multipartite system, and present three methods to distill the general Greenberger–Horne–Zeilinger state. By investigating the methods, we show that multipartite entanglement distillation by bipartite entanglement distillation has higher yield than ones in the previous multipartite entanglement distillations. © 2008 Elsevier B.V. All rights reserved. PACS: 03.65.Ud; 03.67.Mn; 03.67.Hk Keywords: Multipartite entanglement distillation; Teleportation; Entanglement swapping; CNOT operation
1. Introduction Quantum entanglement has been considered as one of the most crucial resources in quantum information sciences. In particular, it has been known that entanglement guarantees perfectly secure quantum communications if it is perfect. Unfortunately, it is also known that it is hard for a perfect entanglement to naturally exist in the real world. Thus, in order to perform a faithful quantum communication, one needs a process to make a nearly perfect entanglement out of noisy entanglement, which is called the entanglement distillation. In the case of bipartite systems, it has been shown that the unconditional security of quantum key distribution is closely related with the distillation of entanglement [1,2]. Thus, for faithful multiparty quantum communications based on multipartite entanglement, such as quantum secret sharing protocols [3,4], the distillation of multipartite entanglement should be required. On this account, several methods [5–7] for multipartite entanglement distillation have developed by generaliz-
* Corresponding author.
E-mail addresses: [email protected] (S. Lee), [email protected] (J. Park). 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.01.074
ing the 2-qubit distillation protocol with the hashing protocol in [8]. Remark that multipartite entanglement can be simply generated from bipartite entanglements by various ways, in which there are three typical methods to exploit quantum teleportation [9], entanglement swapping [10,11], and the CNOT operation [11,12], respectively. We now assume that there are bipartite noisy qubit channels in a given multipartite system, such as the identical depolarizing qubit channels. Then, by employing the three typical methods and generalizing them, we present three methods to distill multipartite entanglement in this situation, and by comparing the fidelities between the perfect multipartite entangled state (for example, the Greenberger–Horne– Zeilinger (GHZ) state [13]) and the multipartite states distilled by the three methods, we show that the methods provide us with the same fidelities, and that the fidelities are the same as the fidelity between perfect GHZ state and the state sent through the identical depolarizing qubit channels. We finally show that the multipartite entanglement distillation by bipartite entanglement distillation could be more efficient than the previous multipartite entanglement distillations in [5–7]. This Letter is organized as follows. In Section 2 we introduce our three methods, and show that the noisy GHZ states
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S. Lee, J. Park / Physics Letters A 372 (2008) 3157–3161
made by all the three methods have the same fidelities. In Section 3 we show that multipartite entanglement distillation by bipartite entanglement distillation has higher yield than that of the previous multipartite entanglement distillations. Finally, in Section 4 we summarize our result. 2. Three methods to distill multipartite entanglement For convenience, we first consider the distillation of the 3qubit GHZ state, 1 |Φ3 = √ |000 + |111 , 2
(1)
and assume that for three systems A, B, and C, there are the identical depolarizing qubit channels DF with fidelity F between A and B and between A and C, respectively. Here, the action of each channel is described as 1−F σx ρσx† + σy ρσy† + σz ρσz† . DF : ρ → Fρ + 3
+ + − − − ΦF = λ + 0 Ψ 0 Ψ0 + λ 0 Ψ 0 Ψ0 +
(2)
|φ +
We remark that if one prepares a Bell state, = (|00 + √ |11)/ 2, and send one qubit of |φ + through the depolarizing channel, then the state becomes 1 − F I ⊗ I − φ + φ + , ρF = F φ + φ + + 3
Fig. 1. The method to use teleportation: The dashed lines, the dotted ellipses and the dotted lines represent nearly perfect entangled states, the 2-qubit orthogonal measurements, and transmissions of classical information, respectively.
(3)
which is called the isotropic state [14]. Thus, if two |φ + ’s are prepared in the system A and one qubit of each |φ + is sent to the systems B and C through the two depolarizing channels, then one can obtain two isotropic states as nearly perfect entangled states in the compound systems AB and AC, respectively (if F is close to 1). Hence, the present three methods begin on the assumption that the subsystems AB and AC have already shared two nearly perfect isotropic states, respectively. In all three methods, it is also assumed that the three systems can have sufficiently many resources and can deal with all kinds of local quantum operations and classical communication.
3
λj Ψj+ Ψj+ + Ψj− Ψj− ,
(5)
j =1
√ where |Ψj± = (|j ± |7 − j )/ 2 are the GHZ-basis states (note that |Ψ0+ = |Φ3 ), and
1 1 + 2F 2 4F − 1 2 ± λ0 = ± , 2 3 3
1 − F 1 + 2F = λ3 , λ1 = 3 3 2(1 − F )2 . (6) 9 Thus, by the first method, a nearly perfect GHZ state can be distilled with fidelity
1 1 + 2F 2 4F − 1 2 Φ3 |ΦF |Φ3 = + 2 3 3 1 = 10F 2 − 2F + 1 . (7) 9 λ2 =
2.2. The method to use entanglement swapping 2.1. The method to use teleportation We firstly take account of the method to use teleportation, which is illustrated in Fig. 1 and is described as follows: (i) Prepare the GHZ state in the party A. (ii) Teleport two qubits of the GHZ state over nearly perfect bipartite states ρF ; Perform the 2-qubit orthogonal measurements (with respect to the Bell basis) on each qubit of the GHZ state and one qubit of each ρF , and then apply proper local unitary operations on the systems BC depending on the measurement outcomes. In the step (i), the total state can be written as |Φ3 Φ3 | ⊗ ρF ⊗ ρF ,
(4)
and it is straightforward to obtain that after the step (ii) the resulting 3-qubit state becomes a state in the class of 4-parameter states presented in [15] as follows:
We secondly consider the method to use entanglement swapping, which is seen in Fig. 2 and is described as follows: (i) Prepare one of the Bell states |φ + in the system A. (ii) Perform the 3-qubit orthogonal measurement (with respect to the GHZ basis) on one qubit of |φ + and one qubits of each ρF in the system A, and then apply proper local unitary operations on the systems BC depending on the measurement outcome, respectively. In the step (i) of the second method, the total state can be written as + + φ φ ⊗ ρF ⊗ ρF . (8) Then it can be shown that the 3-qubit state shared by the systems ABC after the step (ii) becomes the same state as one obtained from the first method in Eq. (5). Hence, we can see that from the second method a nearly perfect GHZ state can be
S. Lee, J. Park / Physics Letters A 372 (2008) 3157–3161
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Fig. 2. The method to use entanglement swapping: The dashed lines, the dotted circle and the dotted lines represent nearly perfect entangled states, the 3-qubit orthogonal measurement, and transmissions of classical information, respectively.
Fig. 3. The method to use the CNOT operation: The dashed lines, the dotted circle and the dotted lines represent nearly perfect entangled states, the 1-qubit orthogonal measurement, and a transmission of classical information, respectively.
also distilled with fidelity
2.4. Multipartite cases
1 2
1 + 2F 3
2
4F − 1 + 3
2 =
1 10F 2 − 2F + 1 , 9
(9)
which is the same as the first one in Eq. (7). Therefore, the second method is essentially equivalent to the first one.
We now generalize these three methods into general multipartite cases. Assume that there are n − 1 nearly perfect Bell states ρF in the subsystems AB1 , AB2 , . . . , and ABn−1 of the total system AB1 B2 · · · Bn−1 . Then, through the three methods, one tries to distill the n-qubit GHZ state,
2.3. The method to use CNOT operation
1 |Φn = √ 0n + 1n , 2
We finally take the method to use the CNOT operation into account, which is illustrated in Fig. 3 and is described as follows: (i) Apply the CNOT operation on two 1-qubit states of each ρF in the system A. (ii) Perform the 1-qubit orthogonal measurement (with respect to the standard basis) on one qubit of the target qubit in the system A, and then apply a proper local unitary operation on the system C depending on the measurement outcome. Then, it follows from simple calculations that the resulting 3-qubit state after the step (ii) of the third method becomes
in the system AB1 B2 · · · Bn−1 . In this situation, the first method is as follows: (i) Prepare |Φn in the system A. (ii) Teleport n − 1 qubits of |Φn through n − 1 nearly perfect Bell states ρF , and the second one is executed as follows: (i) Prepare one of the Bell states |φ + in the system A. (ii) Perform the n-qubit orthogonal measurement (with respect to the n-qubit GHZ basis) on one qubit of |φ + and each one qubit of n − 1 states of ρF , and then apply proper local unitary operations on the systems B1 B2 · · · Bn−1 depending on the measurement outcome. Here, the n-qubit GHZ-basis states are
+ + (3) − − − ΨF = λ+ 0 Ψ 0 Ψ 0 + λ 0 Ψ 0 Ψ0 +
2
|j ± |2n − 1 − j √ 2
λ1 Ψj+ Ψj+ + Ψj− Ψj−
+ λ2 Ψ3+ Ψ3+ + Ψ3− Ψ3− ,
(10)
which is slightly different from the first and second one in Eq. (5). Nevertheless, we can readily see that after executing the third method, a nearly perfect GHZ state can be distilled with fidelity
2
4F − 1 + 3 1 = 10F 2 − 2F + 1 , 9 1 2
1 + 2F 3
(13)
for j = 0, 1, . . . , 2n−1 − 1. Then the total states in the step (i) of the first and the second methods can be described as
j =1
(3) Φ3 |ΨF |Φ3 =
(12)
2
(11)
which is exactly the same as ones in the first and second methods.
|Φn Φn | ⊗ ρF ⊗ ρF ⊗ · · · ⊗ ρF , + + φ φ ⊗ ρF ⊗ ρF ⊗ · · · ⊗ ρF ,
(14)
respectively. Although the total states in the two methods are slightly different as seen in Eq. (14), one can obtain that the two methods are essentially equivalent as in the 3-qubit case, since the resulting n-qubit states after the protocols are the same. The third method is inductively described as follows: Let (k) k 3. (i) Assume that a nearly perfect k-qubit GHZ state ΨF (3) is shared in the systems AB1 B2 · · · Bk−1 , since one can get ΨF in the systems AB1 B2 from the third method in the 3-qubit case. (ii) In the system A, apply the CNOT operation on one qubit of (k) ΨF and one qubit of ρF of the systems ABk . (iii) Perform
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the 1-qubit orthogonal measurement (with respect to the standard basis) on one qubit of the target qubit in the system A, and then apply a proper local unitary operation on the system Bk depending on the measurement outcome. Then after the iterations of the steps (ii) and (iii), a nearly perfect n-qubit GHZ state can be clearly obtained. Even though the three methods seem to be different, by straightforward but tedious calculations, it can be obtained that the three methods have all the same fidelities with the n-qubit GHZ state,
1 − 4F n−1 1 1 + 2F n−1 + . (15) 2 3 3 3. Comparison with other multipartite distillation protocols For convenience, we consider the case of n = 4, which could be generalized into any multipartite cases. We have seen in the previous section that if F is close to 1 then by our three methods one can distill a nearly perfect 4-qubit GHZ state with the fidelity in Eq. (15) over the depolarizing qubit channels with fidelity F . We note that this fidelity in Eq. (15) can be also obtained simply by sending each of the three qubits in the 4-qubit GHZ state |Φ4 from the system A to each system Bk through the depolarizing channels, respectively. In fact, it can be readily shown that Φ4 |I ⊗ DF ⊗ DF ⊗ DF |Φ4 Φ4 | |Φ4
1 1 + 2F 3 1 − 4F 3 = (16) + , 2 3 3
Fig. 4. Comparison of the yields of the protocols when n = 4: The dotted line, the dashed line, and the solid line represent the yields of the protocols of Refs. [5–7], respectively, and the bold line represents that of the 2-qubit protocol in Ref. [8].
presented three methods to distill the general GHZ state, by employing three typical methods to generate the GHZ state from bipartite entanglements and generalizing them. We have shown that the three methods have the same fidelities, and have shown that the fidelities are the same as the fidelity between the perfect GHZ state and the state sent through the identical depolarizing qubit channels. We have finally concluded that multipartite entanglement distillation by bipartite entanglement distillation has higher yield than ones in the previous protocols. Hence, the distillable entanglement for the multipartite entanglement could be explained by means of the distillable entanglement of each bipartite subsystem.
which is the same as one in Eq. (15). On the assumption that there are given the identical depolarizing channels with fidelity F (here, F is not necessarily close to 1) between each of two qubits in the total system, the noisy GHZ state I ⊗ DF ⊗ DF ⊗ DF (|Φ4 Φ4 |) can be also distilled into a nearly perfect GHZ state by the previous multipartite distillation protocols with hashing protocols in [5–7], and the yields of the protocols can be expressed as functions of the fidelity F of the depolarizing channels, as plotted in Fig. 3 of Ref. [7]. However, the yields are less than the yield of the 2-qubit protocol in [8], which can be seen in Fig. 4. Therefore, if we regard a faithful sharing of a perfect multipartite entangled state over a noisy environment (in our case, the identical depolarizing qubit channels) as a sort of the multipartite entanglement distillation, then we can conclude that the previous multipartite entanglement distillation protocols in [5–7] is less efficient than our protocols following the 2-qubit entanglement distillation protocol.
Acknowledgement
4. Conclusions
[10]
In conclusion, on the assumption that there are only bipartite noisy qubit channels in a given multipartite system, we have
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