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Loss of entanglement after propagation in a quantum noisy channel modeled by a canonical unitary operation in two qubits
Physics Letters A 360 (2006) 251–255 www.elsevier.com/locate/pla
Loss of entanglement after propagation in a quantum noisy channel modeled by a canonical unitary operation in two qubits Wellington Alves de Brito, Rubens Viana Ramos ∗ Department of Teleinformatic Engineering, Federal University of Ceara, Campus do Pici, 710, C.P. 6007, 60755-640 Fortaleza-Ceará, Brazil Received 19 June 2006; received in revised form 9 August 2006; accepted 11 August 2006 Available online 22 August 2006 Communicated by P.R. Holland
Abstract In this work, we analyze the loss of entanglement of bipartite states after propagation in a noisy channel modeled by the interaction between the bipartite state and the environment through a canonical unitary form of a two-qubit gate. An analytic expression for the entanglement loss is found. © 2006 Elsevier B.V. All rights reserved. PACS: 03.65.Ta; 03.67.-a; 03.65.Bz Keywords: Entanglement; Canonical unitaries; Quantum noisy channel
Entanglement is a very important property of some composite quantum systems that plays a key role in many of the most important application of quantum computation and quantum information processing, as quantum key distribution [1,2] and quantum teleportation [3,4]. Despite of its importance for quantum information protocols, entanglement is a fragile property that is easily lost when the quantum system interacts with the environment, a process called decoherence [4,5]. Long distance quantum teleportation, for example, is limited by this effect. In fact, the quantum system–environment interaction represents a quantum noisy channel. The larger the noisy the faster the loss of entanglement and the increasing of entropy during channel propagation. Further, the lower the entanglement the larger (toward the value 0.5) the error rate in the quantum communication or computation systems. In this work, our aim is to analyze the entanglement variation of a bipartite state during propagation in a quantum noisy channel. A similar task was realized in [6]. However, differently from their approach, we will use for the quantum noisy channel model the system–environment interaction through a unitary evolution. The unitary evolution considered here is the canonical unitary evolution discussed in [7]. Having this, we have been able to find an analytical expression for the loss of entanglement during channel propagation. Basically, the quantum noise appears due to the fact that one cannot access all existing variables, being required to trace out part of the whole system. Hence, we can see a quantum noisy channel for a bipartite state as a unitary interaction between the bipartite state and a single qubit representing the environment and, after the interaction, tracing out the environment. Hence, we are going to analyze the bipartite state obtained from a pure tripartite C2 ⊗ C2 ⊗ C2 state. Let us consider the quantum noisy channel modeled as shown in Fig. 1.
* Corresponding author.
E-mail addresses: [email protected] (W. Alves de Brito), [email protected] (R. Viana Ramos). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.08.039
252
W. Alves de Brito, R. Viana Ramos / Physics Letters A 360 (2006) 251–255
Fig. 1. Quantum noisy channel modeled by the canonical unitary operation.
As can be seen, the bipartite state interacts with channel’s qubit, assumed being the state |0, through a canonical unitary operation, UD , which is given by [8] 3 θ k σ k ⊗ σk UD = exp −i (1) k=1
where σk are the Pauli matrices: 0 1 0 σ1 = , σ2 = 1 0 i
−i , 0
σ3 =
1 0
0 . −1
(2)
The θk s obey the relations π/4 θ1 θ2 θ3 0. Since any two-qubit gate can be written as U = (UA ⊗ UB )UD (VA ⊗ VB ) where UA , UB , VA and VB are single-qubit gates, there are no restrictions on the results that will be obtained with the choice of UD to represent the unitary interaction. In other words, those local unitaries can be thought of as a local change of basis before and after the interaction, and√therefore can have no effect on the entanglement of the state. The state at channel’s input is the Bell state + = (|00 + |11)/ 2 while the state at the channel’s output is the mixed state ΓAB given by: |ΦAB
+ + ΓAB = TrC (I2 ⊗ UD ) ΦAB (3) ΦAB ⊗ ρC (I2 ⊗ UD )+ , ⎤ ⎡ cos2 (θ1 −θ2 ) (1,4) 0 0 ΓAB 2 ⎥ ⎢ (2,3) sin2 (θ1 −θ2 ) ⎥ ⎢ 0 ΓAB 0 ⎥ ⎢ 2 ΓAB = ⎢ (4) ⎥, 2 (2,3) ∗ sin (θ1 +θ2 ) ⎥ ⎢ 0 (ΓAB ) 0 ⎦ ⎣ 2 (ΓAB )∗ (1,4)
0
0
cos2 (θ1 +θ2 ) 2
[cos(2θ1 ) + cos(2θ2 )] −i2θ3 , e 4 [cos(2θ2 ) − cos(2θ1 )] −i2θ3 (2,3) , ΓAB = e 4 where I2 is the 2 × 2 identity matrix. The concurrence [9,10] of state (3) is simply given by (1,4)
ΓAB =
C(ΓAB ) = cos(2θ1 ).
(5) (6)
(7)
Although cos(2θ1 ) is always non-negative for 0 θ1 π/4, at this moment it is more interesting to use the tangle (square of concurrence) to measure the entanglement τAB = C 2 (ΓAB ) = cos2 (2θ1 ). In this way, the loss of entanglement during channel propagation is given by E = 1 − τAB = 1 − cos2 (2θ1 ) = sin2 (2θ1 ).
(8)
The loss of entanglement obviously belongs to the interval [0, 1], depending on the value of θ1 . When θ1 = 0 we have also θ2 = θ3 = 0 and UD is the 4 × 4 identity matrix, I4 , hence, the entanglement does not vary. On the other hand, when θ1 = π/4, the canonical unitary takes the form ⎡ ⎤ A1 e−iθ3 0 0 iA2 e−iθ3 √ ⎢ ⎥ 0 A2 eiθ3 −iA1 eiθ3 0 2⎢ ⎥ UD = (9) ⎢ ⎥, ⎦ 2 ⎣ A2 eiθ3 0 0 −iA1 eiθ3 −iA2 e−iθ3 A1 = cos(θ2 ) + sin(θ2 ) ,
0 0 A1 e−iθ3 A2 = cos(θ2 ) − sin(θ2 ) .
(10)
W. Alves de Brito, R. Viana Ramos / Physics Letters A 360 (2006) 251–255
In this case, the state at channel’s output is ⎡ [c(θ2 )+s(θ2 )]2 0 4 ⎢ 2 [c(θ )−s(θ 2 2 )] ⎢ 0 ⎢ 4 ΓAB = ⎢ [c(θ3 )+is(θ3 )]2 x ⎢ 0 ⎣ 4 x=
[c(θ3 )+is(θ3 )]2 x 4 −1 + 2c2 (θ2 ) ,
[−c(θ3 )+is(θ3 )]2 x 4
0 [−c(θ3 )+is(θ3 )]2 x 4 [c(θ2 )+s(θ2 )]2 4
0 c(θ2(3) ) = cos(θ2(3) ),
0
0 0
253
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦
(11)
[c(θ2 )−s(θ2 )]2 4
s(θ2(3) ) = sin(θ2(3) ).
(12)
TA are [1/2, 1/2, 0, 0], hence, as expected, the Using Peres–Horodecki separability criterion [11,12] we find the eigenvalues of ΓAB output state is disentangled for any values of θ2 and θ3 . + Although (7) has been found for the particular case of having the Bell state |ΦAB at the channel’s input, it can be numerically checked that it is valid for any maximally entangled state and, further, (7) is a particular case of C0 = Ci cos(2θ1 ) (13)
that gives the amount of entanglement, measured by the concurrence, of the state at the noisy channel’s output (modeled by bipartite state–environment interaction via canonical unitary operation) for any pure bipartite state at channel’s input. Ci is the concurrence of the input state. Let us now analyze the same problem from the point of view of C2 ⊗C2 ⊗C2 pure tripartite states. The three-way entanglement of tripartite pure states can be calculated using the 3-tangle, τ3 , an entanglement monotone that can be calculated using the expression [13–16] τ3 = τa(b,c) − τab(ab,bc) − τac(bc,ac) ,
(14)
τa(b,c) = 4 det |ρa(b,c) |,
(15)
ρa(b,c) = Trbc(ac,ab) (Ψabc ),
(16)
where, for the pure tripartite state Ψabc , τab , τbc and τac are tangles of the bipartite states Φab = Trc (Ψabc ), Φbc = Tra (Ψabc ) and Φac = Trb (Ψabc ), respectively, and they can be calculated using Wootters’ equation [9]. At last, the tangles τa , τb and τc measure the bipartite entanglement between a bipartite and a single qubit. Now, let us check the tripartite state produced by the interaction, through canonical unitary operation, of the input bipartite state (σ00 |00 + σ01 |01 + σ10 |10 + σ11 |11) and the single-qubit (|0) environment |Ψin = σ00 |00 + σ01 |01 + σ10 |10 + σ11 |11 ⊗ |0, (17) |Ψout = (I2 ⊗ UD )|Ψin .
(18)
Having (18), one can easily find the partial bipartite (Φab , Φbc , Φac ) and single-qubit (ρa , ρb , ρc ) states and determine τa,(b,c) , using (15)–(16), τbc(ac,ab) using Wootters’ expression and τ3 using (14). The important results are: 2 C ≡ C |Ψin = 4 (σ00 σ11 − σ01 σ10 ) , (19) τ3 = C cos(4θ1 ) − cos(4θ2 ) /2, (20) τa = C,
(21)
τab = C cos (2θ1 ). 2
(22)
Eq. (20) shows us that the canonical unitary operation can produce an entangled tripartite state only if θ1 = θ2 . On the other hand, the entanglement between the first qubit of the input bipartite state and the bipartite state composed by the second qubit of the input bipartite state and the environment, τa , is equal to the entanglement of the input bipartite state. At last, the entanglement of the bipartite state Φab (the state at channel output), τab , is exactly the same given by (13). At last, we would like to consider the recovery of the entangled state based on the error correction procedure proposed in [17]. The optical setup for error correction is shown in Fig. 2. In the setup shown in Fig. 2, PC are Pockels cells, PBS are polarizing beam splitters and L and S mean long and short path. At the sender, Alice, the input polarization qubit is transformed in a time-bin qubit by the unbalanced polarization interferometer. The input horizontal part takes the short path while the vertical part takes the long path. The delayed part has it polarization rotated of π/2 by the Pockels cell. Hence, at channel input there exist only horizontal states. After channel propagation, at the receiver, Bob, at the first beam splitter, the horizontal part takes the lower path while the vertical part takes the upper path. The Pockels cell PCB1 is activated only in the delayed time while PCB2 is activated in the earlier time. At last, at the two unbalanced polarization interferometers at Bob, the vertical part takes the short path while the horizontal part takes the long path. The input state is |Ψi = σ00 |H H + σ01 |H V + σ10 |V H + σ11 |V V |0. (23)
254
W. Alves de Brito, R. Viana Ramos / Physics Letters A 360 (2006) 251–255
Fig. 2. Optical setup for error correction.
At the channel input (or after the encoder) the state is: |Ψ = σ00 |H HS 0 + σ01 |H HL 0 + σ10 |V HS 0 + σ11 |V HL 0.
(24)
After channel propagation the state is: |Ψo = |H σ00 α|HS 0 + σ00 β|VS 1 + σ01 α|HL 0 + σ01 β|VL 1 + |V σ10 α|HS 0 + σ10 β|VS 1 + σ11 α|HL 0 + σ11 β|VL 1 ,
(25)
where the following relations have been used |H H 0 |H H 0 |H V 1 (I ⊗ UD ) =α +β , |V H 0 |V H 0 |V V 1 1 cos(θ1 − θ2 + θ3 ) + cos(θ1 − θ2 − θ3 ) + i sin(θ1 − θ2 − θ3 ) − sin(θ1 − θ2 + θ3 ) , α= 2 1 cos(θ1 − θ2 − θ3 ) − cos(θ1 − θ2 + θ3 ) + i sin(θ1 − θ2 + θ3 ) + sin(θ1 − θ2 − θ3 ) . β =− 2 At the decoder output the state is: |Ψf = |H α σ00 |HSL + σ01 |VLS 1 |0 + |H β σ00 |HSL + σ01 |VLS 2 |1 + |V α σ10 |HSL + σ11 |VLS 1 |0 + |V β σ10 |HSL + σ11 |VLS 2 |1
(26) (27) (28)
(29)
or, after a little algebra and dropping out the SL/LS lower-index (SL (LS) mean short (long) path at Alice and long (short) path at Bob), the final state is |Ψf = α σ00 |H H + σ01 |H V + σ10 |V H + σ11 |V V 1 |0 + β σ00 |H H + σ01 |H V + σ10 |V H + σ11 |V V 2 |1. (30) Hence, from (30) one sees that the correct input state is obtained, however, there is an uncertainty if the second qubit is at Bob’s output 1 or 2. With probability |α|2 (|β|2 ) the photon leaves Bob at output 1 (2). However, using a multiplexer composed by an optical delay and an electro-optic switch, one can place Bob’s photon always at the same output. In the real world, the unitary operation modeling the channel varies randomly. The optical setup shown in Fig. 2 works properly, recovering the input state, if the separation of the time-bin qubit is lower than the characteristic time of fluctuation of channel’s parameters. In summary, we considered the loss of entanglement of a bipartite state in a quantum noisy channel modeled by the interaction between the bipartite state and a qubit representing the channel state, through a general canonical unitary operation. We have found an analytical expression for the entanglement of the state at the channel’s output, when the state at the channel’s input is any pure state. The final entanglement depends only of the entanglement of the input state and the component θ1 of the canonical unitary operation modeling the channel. That result was obtained looking directly the output bipartite state and observing the entanglements of a pure tripartite of qubit states. At last, we showed an optical setup for recovery the input state and its entanglement after propagation at the noisy channel of the type here discussed. Acknowledgements This work was supported by the Brazilian agency CNPq. The authors thank helpful discussion with José Cláudio and Daniel Barbosa. References [1] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Rev. Mod. Phys. 74 (2002) 145. [2] S.J.D. Phoenix, P.D. Townsend, Contemp. Phys. 36 (1995) 165.
W. Alves de Brito, R. Viana Ramos / Physics Letters A 360 (2006) 251–255
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
M.B. Plenio, V. Vedral, Contemp. Phys. 39 (1998) 431. C.H. Bennett, Phys. Scr. T 76 (1998) 210. B.J. Dalton, J. Mod. Opt. 52 (2005) 2563. K. Zyczkowski, P. Horodecki, M. Horodecki, R. Horodecki, quant-ph/0008115. C. Wang, C. Li, G. Guo, Phys. Lett. A 337 (2005) 29. B. Kraus, J.I. Cirac, Phys. Rev. A 63 (2001) 062309. W.K. Wootters, Phys. Rev. Lett. 80 (1998) 2245. J.A. Smolin, Quantum gates, mixed-states entanglement and error-correcting codes, Doctorate thesis, University of California, 1996. A. Peres, Phys. Rev. Lett. 77 (1996) 1413. M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 223 (1996) 8. V. Coffman, J. Kundu, W.K. Wootters, Phys. Rev. A 61 (2000) 052306. C.S. Yu, H. Song, Phys. Lett. A 330 (2004) 377. A. Acín, D. Bruß, M. Lewenstein, A. Sanpera, quant-ph/0103025. V.M. Kendon, K. Nemoto, W.J. Munro, quant-ph/0106023. D. Kalamidas, Phys. Lett. A 343 (2005) 331.
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Loss of entanglement after propagation in a quantum noisy channel modeled by a canonical unitary operation in two qubits Wellington Alves de Brito, Rubens Viana Ramos ∗ Department of Teleinformatic Engineering, Federal University of Ceara, Campus do Pici, 710, C.P. 6007, 60755-640 Fortaleza-Ceará, Brazil Received 19 June 2006; received in revised form 9 August 2006; accepted 11 August 2006 Available online 22 August 2006 Communicated by P.R. Holland
Abstract In this work, we analyze the loss of entanglement of bipartite states after propagation in a noisy channel modeled by the interaction between the bipartite state and the environment through a canonical unitary form of a two-qubit gate. An analytic expression for the entanglement loss is found. © 2006 Elsevier B.V. All rights reserved. PACS: 03.65.Ta; 03.67.-a; 03.65.Bz Keywords: Entanglement; Canonical unitaries; Quantum noisy channel
Entanglement is a very important property of some composite quantum systems that plays a key role in many of the most important application of quantum computation and quantum information processing, as quantum key distribution [1,2] and quantum teleportation [3,4]. Despite of its importance for quantum information protocols, entanglement is a fragile property that is easily lost when the quantum system interacts with the environment, a process called decoherence [4,5]. Long distance quantum teleportation, for example, is limited by this effect. In fact, the quantum system–environment interaction represents a quantum noisy channel. The larger the noisy the faster the loss of entanglement and the increasing of entropy during channel propagation. Further, the lower the entanglement the larger (toward the value 0.5) the error rate in the quantum communication or computation systems. In this work, our aim is to analyze the entanglement variation of a bipartite state during propagation in a quantum noisy channel. A similar task was realized in [6]. However, differently from their approach, we will use for the quantum noisy channel model the system–environment interaction through a unitary evolution. The unitary evolution considered here is the canonical unitary evolution discussed in [7]. Having this, we have been able to find an analytical expression for the loss of entanglement during channel propagation. Basically, the quantum noise appears due to the fact that one cannot access all existing variables, being required to trace out part of the whole system. Hence, we can see a quantum noisy channel for a bipartite state as a unitary interaction between the bipartite state and a single qubit representing the environment and, after the interaction, tracing out the environment. Hence, we are going to analyze the bipartite state obtained from a pure tripartite C2 ⊗ C2 ⊗ C2 state. Let us consider the quantum noisy channel modeled as shown in Fig. 1.
* Corresponding author.
E-mail addresses: [email protected] (W. Alves de Brito), [email protected] (R. Viana Ramos). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.08.039
252
W. Alves de Brito, R. Viana Ramos / Physics Letters A 360 (2006) 251–255
Fig. 1. Quantum noisy channel modeled by the canonical unitary operation.
As can be seen, the bipartite state interacts with channel’s qubit, assumed being the state |0, through a canonical unitary operation, UD , which is given by [8] 3 θ k σ k ⊗ σk UD = exp −i (1) k=1
where σk are the Pauli matrices: 0 1 0 σ1 = , σ2 = 1 0 i
−i , 0
σ3 =
1 0
0 . −1
(2)
The θk s obey the relations π/4 θ1 θ2 θ3 0. Since any two-qubit gate can be written as U = (UA ⊗ UB )UD (VA ⊗ VB ) where UA , UB , VA and VB are single-qubit gates, there are no restrictions on the results that will be obtained with the choice of UD to represent the unitary interaction. In other words, those local unitaries can be thought of as a local change of basis before and after the interaction, and√therefore can have no effect on the entanglement of the state. The state at channel’s input is the Bell state + = (|00 + |11)/ 2 while the state at the channel’s output is the mixed state ΓAB given by: |ΦAB
+ + ΓAB = TrC (I2 ⊗ UD ) ΦAB (3) ΦAB ⊗ ρC (I2 ⊗ UD )+ , ⎤ ⎡ cos2 (θ1 −θ2 ) (1,4) 0 0 ΓAB 2 ⎥ ⎢ (2,3) sin2 (θ1 −θ2 ) ⎥ ⎢ 0 ΓAB 0 ⎥ ⎢ 2 ΓAB = ⎢ (4) ⎥, 2 (2,3) ∗ sin (θ1 +θ2 ) ⎥ ⎢ 0 (ΓAB ) 0 ⎦ ⎣ 2 (ΓAB )∗ (1,4)
0
0
cos2 (θ1 +θ2 ) 2
[cos(2θ1 ) + cos(2θ2 )] −i2θ3 , e 4 [cos(2θ2 ) − cos(2θ1 )] −i2θ3 (2,3) , ΓAB = e 4 where I2 is the 2 × 2 identity matrix. The concurrence [9,10] of state (3) is simply given by (1,4)
ΓAB =
C(ΓAB ) = cos(2θ1 ).
(5) (6)
(7)
Although cos(2θ1 ) is always non-negative for 0 θ1 π/4, at this moment it is more interesting to use the tangle (square of concurrence) to measure the entanglement τAB = C 2 (ΓAB ) = cos2 (2θ1 ). In this way, the loss of entanglement during channel propagation is given by E = 1 − τAB = 1 − cos2 (2θ1 ) = sin2 (2θ1 ).
(8)
The loss of entanglement obviously belongs to the interval [0, 1], depending on the value of θ1 . When θ1 = 0 we have also θ2 = θ3 = 0 and UD is the 4 × 4 identity matrix, I4 , hence, the entanglement does not vary. On the other hand, when θ1 = π/4, the canonical unitary takes the form ⎡ ⎤ A1 e−iθ3 0 0 iA2 e−iθ3 √ ⎢ ⎥ 0 A2 eiθ3 −iA1 eiθ3 0 2⎢ ⎥ UD = (9) ⎢ ⎥, ⎦ 2 ⎣ A2 eiθ3 0 0 −iA1 eiθ3 −iA2 e−iθ3 A1 = cos(θ2 ) + sin(θ2 ) ,
0 0 A1 e−iθ3 A2 = cos(θ2 ) − sin(θ2 ) .
(10)
W. Alves de Brito, R. Viana Ramos / Physics Letters A 360 (2006) 251–255
In this case, the state at channel’s output is ⎡ [c(θ2 )+s(θ2 )]2 0 4 ⎢ 2 [c(θ )−s(θ 2 2 )] ⎢ 0 ⎢ 4 ΓAB = ⎢ [c(θ3 )+is(θ3 )]2 x ⎢ 0 ⎣ 4 x=
[c(θ3 )+is(θ3 )]2 x 4 −1 + 2c2 (θ2 ) ,
[−c(θ3 )+is(θ3 )]2 x 4
0 [−c(θ3 )+is(θ3 )]2 x 4 [c(θ2 )+s(θ2 )]2 4
0 c(θ2(3) ) = cos(θ2(3) ),
0
0 0
253
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦
(11)
[c(θ2 )−s(θ2 )]2 4
s(θ2(3) ) = sin(θ2(3) ).
(12)
TA are [1/2, 1/2, 0, 0], hence, as expected, the Using Peres–Horodecki separability criterion [11,12] we find the eigenvalues of ΓAB output state is disentangled for any values of θ2 and θ3 . + Although (7) has been found for the particular case of having the Bell state |ΦAB at the channel’s input, it can be numerically checked that it is valid for any maximally entangled state and, further, (7) is a particular case of C0 = Ci cos(2θ1 ) (13)
that gives the amount of entanglement, measured by the concurrence, of the state at the noisy channel’s output (modeled by bipartite state–environment interaction via canonical unitary operation) for any pure bipartite state at channel’s input. Ci is the concurrence of the input state. Let us now analyze the same problem from the point of view of C2 ⊗C2 ⊗C2 pure tripartite states. The three-way entanglement of tripartite pure states can be calculated using the 3-tangle, τ3 , an entanglement monotone that can be calculated using the expression [13–16] τ3 = τa(b,c) − τab(ab,bc) − τac(bc,ac) ,
(14)
τa(b,c) = 4 det |ρa(b,c) |,
(15)
ρa(b,c) = Trbc(ac,ab) (Ψabc ),
(16)
where, for the pure tripartite state Ψabc , τab , τbc and τac are tangles of the bipartite states Φab = Trc (Ψabc ), Φbc = Tra (Ψabc ) and Φac = Trb (Ψabc ), respectively, and they can be calculated using Wootters’ equation [9]. At last, the tangles τa , τb and τc measure the bipartite entanglement between a bipartite and a single qubit. Now, let us check the tripartite state produced by the interaction, through canonical unitary operation, of the input bipartite state (σ00 |00 + σ01 |01 + σ10 |10 + σ11 |11) and the single-qubit (|0) environment |Ψin = σ00 |00 + σ01 |01 + σ10 |10 + σ11 |11 ⊗ |0, (17) |Ψout = (I2 ⊗ UD )|Ψin .
(18)
Having (18), one can easily find the partial bipartite (Φab , Φbc , Φac ) and single-qubit (ρa , ρb , ρc ) states and determine τa,(b,c) , using (15)–(16), τbc(ac,ab) using Wootters’ expression and τ3 using (14). The important results are: 2 C ≡ C |Ψin = 4 (σ00 σ11 − σ01 σ10 ) , (19) τ3 = C cos(4θ1 ) − cos(4θ2 ) /2, (20) τa = C,
(21)
τab = C cos (2θ1 ). 2
(22)
Eq. (20) shows us that the canonical unitary operation can produce an entangled tripartite state only if θ1 = θ2 . On the other hand, the entanglement between the first qubit of the input bipartite state and the bipartite state composed by the second qubit of the input bipartite state and the environment, τa , is equal to the entanglement of the input bipartite state. At last, the entanglement of the bipartite state Φab (the state at channel output), τab , is exactly the same given by (13). At last, we would like to consider the recovery of the entangled state based on the error correction procedure proposed in [17]. The optical setup for error correction is shown in Fig. 2. In the setup shown in Fig. 2, PC are Pockels cells, PBS are polarizing beam splitters and L and S mean long and short path. At the sender, Alice, the input polarization qubit is transformed in a time-bin qubit by the unbalanced polarization interferometer. The input horizontal part takes the short path while the vertical part takes the long path. The delayed part has it polarization rotated of π/2 by the Pockels cell. Hence, at channel input there exist only horizontal states. After channel propagation, at the receiver, Bob, at the first beam splitter, the horizontal part takes the lower path while the vertical part takes the upper path. The Pockels cell PCB1 is activated only in the delayed time while PCB2 is activated in the earlier time. At last, at the two unbalanced polarization interferometers at Bob, the vertical part takes the short path while the horizontal part takes the long path. The input state is |Ψi = σ00 |H H + σ01 |H V + σ10 |V H + σ11 |V V |0. (23)
254
W. Alves de Brito, R. Viana Ramos / Physics Letters A 360 (2006) 251–255
Fig. 2. Optical setup for error correction.
At the channel input (or after the encoder) the state is: |Ψ = σ00 |H HS 0 + σ01 |H HL 0 + σ10 |V HS 0 + σ11 |V HL 0.
(24)
After channel propagation the state is: |Ψo = |H σ00 α|HS 0 + σ00 β|VS 1 + σ01 α|HL 0 + σ01 β|VL 1 + |V σ10 α|HS 0 + σ10 β|VS 1 + σ11 α|HL 0 + σ11 β|VL 1 ,
(25)
where the following relations have been used |H H 0 |H H 0 |H V 1 (I ⊗ UD ) =α +β , |V H 0 |V H 0 |V V 1 1 cos(θ1 − θ2 + θ3 ) + cos(θ1 − θ2 − θ3 ) + i sin(θ1 − θ2 − θ3 ) − sin(θ1 − θ2 + θ3 ) , α= 2 1 cos(θ1 − θ2 − θ3 ) − cos(θ1 − θ2 + θ3 ) + i sin(θ1 − θ2 + θ3 ) + sin(θ1 − θ2 − θ3 ) . β =− 2 At the decoder output the state is: |Ψf = |H α σ00 |HSL + σ01 |VLS 1 |0 + |H β σ00 |HSL + σ01 |VLS 2 |1 + |V α σ10 |HSL + σ11 |VLS 1 |0 + |V β σ10 |HSL + σ11 |VLS 2 |1
(26) (27) (28)
(29)
or, after a little algebra and dropping out the SL/LS lower-index (SL (LS) mean short (long) path at Alice and long (short) path at Bob), the final state is |Ψf = α σ00 |H H + σ01 |H V + σ10 |V H + σ11 |V V 1 |0 + β σ00 |H H + σ01 |H V + σ10 |V H + σ11 |V V 2 |1. (30) Hence, from (30) one sees that the correct input state is obtained, however, there is an uncertainty if the second qubit is at Bob’s output 1 or 2. With probability |α|2 (|β|2 ) the photon leaves Bob at output 1 (2). However, using a multiplexer composed by an optical delay and an electro-optic switch, one can place Bob’s photon always at the same output. In the real world, the unitary operation modeling the channel varies randomly. The optical setup shown in Fig. 2 works properly, recovering the input state, if the separation of the time-bin qubit is lower than the characteristic time of fluctuation of channel’s parameters. In summary, we considered the loss of entanglement of a bipartite state in a quantum noisy channel modeled by the interaction between the bipartite state and a qubit representing the channel state, through a general canonical unitary operation. We have found an analytical expression for the entanglement of the state at the channel’s output, when the state at the channel’s input is any pure state. The final entanglement depends only of the entanglement of the input state and the component θ1 of the canonical unitary operation modeling the channel. That result was obtained looking directly the output bipartite state and observing the entanglements of a pure tripartite of qubit states. At last, we showed an optical setup for recovery the input state and its entanglement after propagation at the noisy channel of the type here discussed. Acknowledgements This work was supported by the Brazilian agency CNPq. The authors thank helpful discussion with José Cláudio and Daniel Barbosa. References [1] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Rev. Mod. Phys. 74 (2002) 145. [2] S.J.D. Phoenix, P.D. Townsend, Contemp. Phys. 36 (1995) 165.
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