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Three-qubit quantum error-correction scheme based on quantum cloning
Physics Letters A 329 (2004) 294–297 www.elsevier.com/locate/pla
Three-qubit quantum error-correction scheme based on quantum cloning Daxiu Wei ∗ , Jun Luo, Xianping Sun, Xizhi Zeng, Maili Liu State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, PR China Received 16 June 2004; received in revised form 13 July 2004; accepted 14 July 2004 Available online 29 July 2004 Communicated by R. Wu
Abstract This Letter presents a three-qubit quantum error correction scheme that adopts the optimal universal 1 → 2 quantum cloning transformation U to encode an unknown state |φ|0|0 into a three-qubit state U |φ|0|0. The encoded state could protect against collective decoherences that any two qubits in a three-qubit system are affected by either phase error or bit flip error, or simultaneously affected by both of them. To recover the original unknown state, we just need to perform a Bell measurement on the first two qubits and rotate the third qubit along a suitable axis. Specially, our scheme for the first time shows that quantum cloning is useful for quantum error correction. 2004 Elsevier B.V. All rights reserved. PACS: 03.67.Lx; 03.65.Ta Keywords: Quantum cloning; Error correction
Decoherence remains the most severe obstacle in the path towards the dramatic speedup offered by future quantum information processing (QIP) devices [1]. To circumvent this difficulty, various schemes have been developed, including quantum error correction codes (QECC) [2], decoherence-free subspaces (DFS) [1,3], noiseless subsystems (NS) [4], quantum Zeno effect (QZE) [5–7], and others [8,9]. All of these existing approaches are successful in protecting from some special quantum decoherences. For instance, DFS has shown its particular power in avoiding collective decoherence [10,11] and quantum Zeno effect may be effective for fairly stable quantum states by virtue of the consecutive measurements. On the other hand, it seems to be more efficient when one technique combined with an other approach. Such examples include the proposal to concatenate DFSs and QECCs considered in * Corresponding author.
E-mail address: [email protected] (D. Wei). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.07.025
D. Wei et al. / Physics Letters A 329 (2004) 294–297
295
Ref. [12], the scheme to combine DFSs and dynamical decoupling in Ref. [13], the method proposed by WonYoung Hwang et al. in Ref. [14] where the authors concatenated the DFS with the QZE or with QECC to preserve one qubit, and so on [15]. Their results indicated that these combinations could not only protect against more errors, but also reduce the number of the required qubits. Now, the question that we are considering in this Letter is whether some procedures of quantum information processing (such as quantum cloning, quantum teleportation, . . . ) can be helpful for error-correction. The answer is “yes”. Here we for the first time use the optimal universal 1 → 2 quantum cloning transformation to encode an unknown state into a three-qubit state. This simple error-correction scheme is very effective for avoiding collective decoherence where a pair of qubits are affected by bit flip or phase errors. Quantum cloning has been extensively studied. Though the no-cloning theorem [16] forbids perfect cloning, there are several ways of performing imperfect cloning. One concrete instance is the optimal 1 → 2 quantum cloning machine [17] which takes one unknown qubit as input and produces a three-qubit output such that two of them are the clones and the third one is an auxiliary qubit. In this case, a cloning transformation U acts on the two basis states to generate two states as follows, 2 1 |01 02 |13 − |01 12 + |11 02 |03 , U |01 |02 |03 = 3 6 2 1 |11 12 |03 + |11 02 + |01 12 |13 , U |11 |02 |03 = − (1) 3 6 respectively, where the subscripts denote the qubits 1, 2 and 3. Based on the optimal 1 → 2 quantum cloning, the authors in Ref. [18] pointed out that it is possible to restore the original state in one of the output qubits after performing certain measurements on the other outputs and communicating classically. Inspired by their interesting discovery, we find that the optimal universal 1 → 2 quantum cloning can be applied to protect an unknown state |φ = α|0 + β|1 against collective decoherences, including collective phase and bit flip errors. Similar to quantum error correction code, our scheme introduces two auxiliary qubits which originally are in the state |0|0. Firstly, using the optimal 1 → 2 cloning transformation U to encode the initial state |φ|0|0 into U |φ|0|0 which can be rewritten in the following form: U α|0 + β|1 |0|0 1 1 1 = √ φ + −β|0 + α|1 + √ φ − β|0 + α|1 + √ ψ + −α|0 + β|1 , (2) 3 3 3 where |φ ± and |ψ ± are two-qubit Bell states. Then, the encoded state will be affected by decoherences. In this Letter, we only pay attention to one class of decoherence, namely, collective errors which any two of these three qubits are affected by phase or bit flip errors. Concretely, these errors can be expressed by Ei , Ej , and Ek of the forms Ei = ai,0 I1 I2 I3 + ai,1 Z1 Z2 I3 + ai,2 X1 X2 I3 + ai,3 Y1 Y2 I3 , Ej = aj,0 I1 I2 I3 + aj,1 Z1 I2 Z3 + aj,2 X1 I2 X3 + aj,3 Y1 I2 Y3 , Ek = ak,0 I1 I2 I3 + ak,1 I1 Z2 Z3 + ak,2 I1 X2 X3 + ak,3I1 Y2 Y3 .
(3)
Here, I is unit operator; Z, X, Y denote Pauli operators σz , σx , σy , respectively; The subscript n ∈ {1, 2, 3} stands for the physical qubit; |am,n |2 (m ∈ {i, j, k} and n ∈ {0, 1, 2}) indicates the probability of the errors. Now the question is how to recover the original information, i.e., the state |φ. Our final step is to carry out a Bell measurement on the first two qubits, and then obtain the primal state |φ on the third qubit after operating a suitable single-qubit unitary transformation, as shown in Table 1. In this way, the unknown state |φ is certainly recovered. Moreover, it is interesting to note that if the measurement result of the first two qubits is |ψ − , then we
296
D. Wei et al. / Physics Letters A 329 (2004) 294–297
Table 1 A Bell measurement on the first two qubits and the third qubit collapses to a conditional state which can be transformed to the state |φ by suitable unitary operation Qubits 1,2
Qubit 3
Unitary operation
|φ + |φ − |ψ + |ψ −
β|0 − α|1 −β|0 − α|1 α|0 − β|1 α|0 + β|1
iσy σx σz I
could conclude that the errors certainly affected the third qubit. In other words, parts of information for the errors can be known. To illuminate the above process more clearly, let us consider the concrete process. The errors Ei , Ej , Ek act on the encoded state U (α|0 + β|1)|0|0, and then the states of the system are as follows: |Ψi = Ei U α|0 + β|1 |0|0 1 1 = √ (ai,0 + ai,1 + ai,2 − ai,3 )φ + −β|0 + α|1 + √ (ai,0 + ai,1 + ai,2 − ai,3 )φ − β|0 + α|1 3 3 + 1 + √ (ai,0 − ai,1 + ai,2 + ai,3 )ψ −α|0 + β|1 , (4) 3 |Ψj = Ei U α|0 + β|1 |0|0 1 1 = √ (aj,0 − aj,1 − aj,2 )φ + −β|0 + α|1 + √ (aj,0 − aj,1 − aj,3 )φ − β|0 + α|1 3 3 + 1 1 + √ (aj,0 − aj,2 − aj,3 )ψ −α|0 + β|1 + √ (−aj,1 − aj,2 − aj,3 )ψ − α|0 + β|1 , (5) 3 3 |Ψk = Ei U α|0 + β|1 |0|0 1 1 = √ (ak,0 − ak,1 − ak,2 )φ + −β|0 + α|1 + √ (ak,0 − ak,1 − ak,3 )φ − β|0 + α|1 3 3 + 1 1 + √ (ak,0 − ak,2 − ak,3 )ψ −α|0 + β|1 + √ (ak,1 + ak,2 + ak,3 )ψ − α|0 + β|1 . (6) 3 3 So if we measure the first two particles on the Bell basis, then the third particle will collapse into the corresponding state with certainty. For instance, if the state of the first two particles is |φ + , then we can conclude that the third particle is in the state −β|0 + α|1 and we can recover the original state |φ by applying a unitary operator iσy to the third particle. This example explicitly proved that the state encoded by the cloning transformation is resilient to the collective errors model. Comparing our error model with the one commonly referred to “collective decoherence”, such as the one defined in Ref. [1], the difference is that the collective decoherence considered in Ref. [1] affects the states as an identity in a DFS and one need not do any auxiliary operations. But the collective decoherence described in this Letter has not this property. If we do not measure the prior two particles, the original state will not be recovered. Here, we call the error as “collective decoherence” just because the error discussed in this Letter simultaneously affects any two of the three qubits. In this sense, our encoded state is quite different from the states found in the theory of DFS [1] or NS [4]. In a DFS or a NS, the encoded states are the eigenstates of the errors, whereas the ones encoded by the cloning transformation U are not the eigenstates of our error model. Our method probably could be generalized to the encoding of more than one qubit in terms of the cloning transformation transforming N identical qubits into M identical copies. But the encoding process would be quite complicated. In addition, this encoded state may be less useful for quantum computation than the one described in
D. Wei et al. / Physics Letters A 329 (2004) 294–297
297
DFS or NS for the reason that, in order to obtain the original state, we have to measure parts of the system. The measuring process is like the correction operation in QECC and its complexity will increase when one encodes more than one qubit. These problems are waiting for further study. To conclude, we have provided a three-qubit error correction scheme which uses the optimal 1 → 2 quantum cloning transformation to encode the original state. Any one of the errors Ei , Ej , and Ek can have an effect on the encoded state. But we always can recover the state |φ via measuring the first two qubits in their Bell basis, followed by a corresponding rotating operation on the third qubit. In particular, our scheme explicitly shows that quantum cloning could be of use in quantum error correction.
Acknowledgements This work has been supported by the National Natural Science Foundation of China under Grant No. 10374103, the National Basic Research Programme of China under Grant No. 2001CB309309, and the National Knowledge Innovation Program of the Chinese Academy of Sciences under Grant No. KJCX2-W1-3.
References [1] D.A. Lidar, I.L. Chuang, K.B. Whaley, Phys. Rev. Lett. 81 (1998) 2594. [2] E. Knill, E. Laflamme, Phys. Rev. A 55 (1997) 900; D. Gottesman, Phys. Rev. A 54 (1996) 1862; A.R. Calderbank, et al., Phys. Rev. Lett. 79 (1997) 405. [3] P. Zanardi, M. Rasetti, Mod. Phys. Lett. B 25 (1997) 1085; L.M. Duan, G.C. Guo, Phys. Rev. Lett. 79 (1997) 1953; D. Bacon, J. Kempe, D.A. Lidar, K.B. Whaley, Phys. Rev. Lett. 85 (2000) 1758. [4] E. Knill, R. Laflamme, L. Viola, Phys. Rev. Lett. 84 (2000) 2525. [5] B. Misra, E.C.G. Sudersan, J. Math. Phys. 18 (1977) 756. [6] W.H. Zurek, Phys. Rev. Lett. 53 (1984) 391. [7] P. Facchi, D.A. Lidar, S. Pascazio, Phys. Rev. A 69 (2004) 032314. [8] L. Viola, S. Lioyd, Phys. Rev. A 58 (1998) 2733; L. Viola, E. Knill, S. Lioyd, Phys. Rev. Lett. 82 (1999) 2417; D. Vitali, P. Tombesi, Phys. Rev. A 59 (1999) 4178; P. Zanardi, Phys. Lett. A 258 (1999) 77. [9] A. Kitaev, quant-ph/9707021; P. Zanardi, S. Lloyd, Phys. Rev. Lett. 90 (1999) 067902. [10] J.E. Ollerenshaw, D.A. Lidar, L.E. Kay, Phys. Rev. Lett. 91 (2003) 217904. [11] J.B. Altepeter, P.G. Hadley, S.M. Wendelken, A.J. Berglund, P.G. Kwiat, Phys. Rev. Lett. 92 (2004) 147901; L.-A. Wu, D.A. Lidar, Phys. Rev. Lett. 88 (2002) 207902. [12] D.A. Lidar, D. Bacon, K.B. Whaley, Phys. Rev. Lett. 82 (1999) 4556. [13] M.S. Byrd, D.A. Lidar, Phys. Rev. Lett. 89 (2002) 047901. [14] W.Y. Hwang, H. Lee, D. Ahn, S.W. Hwang, Phys. Rev. A 62 (2000) 062305. [15] M.S. Byrd, L.-A. Wu, D.A. Lidar, quant-ph/0402098. [16] W.K. Wootters, W.H. Zurek, Nature 299 (1982) 802. [17] D. Bruß, D.P. DiVincenzo, A. Ekert, C.A. Fuchs, C. Macchiavello, J.A. Smolin, Phys. Rev. A 57 (1998) 2368. [18] D. Bruß, J. Calsamiglia, N. Lükenhaus, Phys. Rev. A 63 (2001) 042308.
Three-qubit quantum error-correction scheme based on quantum cloning Daxiu Wei ∗ , Jun Luo, Xianping Sun, Xizhi Zeng, Maili Liu State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, PR China Received 16 June 2004; received in revised form 13 July 2004; accepted 14 July 2004 Available online 29 July 2004 Communicated by R. Wu
Abstract This Letter presents a three-qubit quantum error correction scheme that adopts the optimal universal 1 → 2 quantum cloning transformation U to encode an unknown state |φ|0|0 into a three-qubit state U |φ|0|0. The encoded state could protect against collective decoherences that any two qubits in a three-qubit system are affected by either phase error or bit flip error, or simultaneously affected by both of them. To recover the original unknown state, we just need to perform a Bell measurement on the first two qubits and rotate the third qubit along a suitable axis. Specially, our scheme for the first time shows that quantum cloning is useful for quantum error correction. 2004 Elsevier B.V. All rights reserved. PACS: 03.67.Lx; 03.65.Ta Keywords: Quantum cloning; Error correction
Decoherence remains the most severe obstacle in the path towards the dramatic speedup offered by future quantum information processing (QIP) devices [1]. To circumvent this difficulty, various schemes have been developed, including quantum error correction codes (QECC) [2], decoherence-free subspaces (DFS) [1,3], noiseless subsystems (NS) [4], quantum Zeno effect (QZE) [5–7], and others [8,9]. All of these existing approaches are successful in protecting from some special quantum decoherences. For instance, DFS has shown its particular power in avoiding collective decoherence [10,11] and quantum Zeno effect may be effective for fairly stable quantum states by virtue of the consecutive measurements. On the other hand, it seems to be more efficient when one technique combined with an other approach. Such examples include the proposal to concatenate DFSs and QECCs considered in * Corresponding author.
E-mail address: [email protected] (D. Wei). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.07.025
D. Wei et al. / Physics Letters A 329 (2004) 294–297
295
Ref. [12], the scheme to combine DFSs and dynamical decoupling in Ref. [13], the method proposed by WonYoung Hwang et al. in Ref. [14] where the authors concatenated the DFS with the QZE or with QECC to preserve one qubit, and so on [15]. Their results indicated that these combinations could not only protect against more errors, but also reduce the number of the required qubits. Now, the question that we are considering in this Letter is whether some procedures of quantum information processing (such as quantum cloning, quantum teleportation, . . . ) can be helpful for error-correction. The answer is “yes”. Here we for the first time use the optimal universal 1 → 2 quantum cloning transformation to encode an unknown state into a three-qubit state. This simple error-correction scheme is very effective for avoiding collective decoherence where a pair of qubits are affected by bit flip or phase errors. Quantum cloning has been extensively studied. Though the no-cloning theorem [16] forbids perfect cloning, there are several ways of performing imperfect cloning. One concrete instance is the optimal 1 → 2 quantum cloning machine [17] which takes one unknown qubit as input and produces a three-qubit output such that two of them are the clones and the third one is an auxiliary qubit. In this case, a cloning transformation U acts on the two basis states to generate two states as follows, 2 1 |01 02 |13 − |01 12 + |11 02 |03 , U |01 |02 |03 = 3 6 2 1 |11 12 |03 + |11 02 + |01 12 |13 , U |11 |02 |03 = − (1) 3 6 respectively, where the subscripts denote the qubits 1, 2 and 3. Based on the optimal 1 → 2 quantum cloning, the authors in Ref. [18] pointed out that it is possible to restore the original state in one of the output qubits after performing certain measurements on the other outputs and communicating classically. Inspired by their interesting discovery, we find that the optimal universal 1 → 2 quantum cloning can be applied to protect an unknown state |φ = α|0 + β|1 against collective decoherences, including collective phase and bit flip errors. Similar to quantum error correction code, our scheme introduces two auxiliary qubits which originally are in the state |0|0. Firstly, using the optimal 1 → 2 cloning transformation U to encode the initial state |φ|0|0 into U |φ|0|0 which can be rewritten in the following form: U α|0 + β|1 |0|0 1 1 1 = √ φ + −β|0 + α|1 + √ φ − β|0 + α|1 + √ ψ + −α|0 + β|1 , (2) 3 3 3 where |φ ± and |ψ ± are two-qubit Bell states. Then, the encoded state will be affected by decoherences. In this Letter, we only pay attention to one class of decoherence, namely, collective errors which any two of these three qubits are affected by phase or bit flip errors. Concretely, these errors can be expressed by Ei , Ej , and Ek of the forms Ei = ai,0 I1 I2 I3 + ai,1 Z1 Z2 I3 + ai,2 X1 X2 I3 + ai,3 Y1 Y2 I3 , Ej = aj,0 I1 I2 I3 + aj,1 Z1 I2 Z3 + aj,2 X1 I2 X3 + aj,3 Y1 I2 Y3 , Ek = ak,0 I1 I2 I3 + ak,1 I1 Z2 Z3 + ak,2 I1 X2 X3 + ak,3I1 Y2 Y3 .
(3)
Here, I is unit operator; Z, X, Y denote Pauli operators σz , σx , σy , respectively; The subscript n ∈ {1, 2, 3} stands for the physical qubit; |am,n |2 (m ∈ {i, j, k} and n ∈ {0, 1, 2}) indicates the probability of the errors. Now the question is how to recover the original information, i.e., the state |φ. Our final step is to carry out a Bell measurement on the first two qubits, and then obtain the primal state |φ on the third qubit after operating a suitable single-qubit unitary transformation, as shown in Table 1. In this way, the unknown state |φ is certainly recovered. Moreover, it is interesting to note that if the measurement result of the first two qubits is |ψ − , then we
296
D. Wei et al. / Physics Letters A 329 (2004) 294–297
Table 1 A Bell measurement on the first two qubits and the third qubit collapses to a conditional state which can be transformed to the state |φ by suitable unitary operation Qubits 1,2
Qubit 3
Unitary operation
|φ + |φ − |ψ + |ψ −
β|0 − α|1 −β|0 − α|1 α|0 − β|1 α|0 + β|1
iσy σx σz I
could conclude that the errors certainly affected the third qubit. In other words, parts of information for the errors can be known. To illuminate the above process more clearly, let us consider the concrete process. The errors Ei , Ej , Ek act on the encoded state U (α|0 + β|1)|0|0, and then the states of the system are as follows: |Ψi = Ei U α|0 + β|1 |0|0 1 1 = √ (ai,0 + ai,1 + ai,2 − ai,3 )φ + −β|0 + α|1 + √ (ai,0 + ai,1 + ai,2 − ai,3 )φ − β|0 + α|1 3 3 + 1 + √ (ai,0 − ai,1 + ai,2 + ai,3 )ψ −α|0 + β|1 , (4) 3 |Ψj = Ei U α|0 + β|1 |0|0 1 1 = √ (aj,0 − aj,1 − aj,2 )φ + −β|0 + α|1 + √ (aj,0 − aj,1 − aj,3 )φ − β|0 + α|1 3 3 + 1 1 + √ (aj,0 − aj,2 − aj,3 )ψ −α|0 + β|1 + √ (−aj,1 − aj,2 − aj,3 )ψ − α|0 + β|1 , (5) 3 3 |Ψk = Ei U α|0 + β|1 |0|0 1 1 = √ (ak,0 − ak,1 − ak,2 )φ + −β|0 + α|1 + √ (ak,0 − ak,1 − ak,3 )φ − β|0 + α|1 3 3 + 1 1 + √ (ak,0 − ak,2 − ak,3 )ψ −α|0 + β|1 + √ (ak,1 + ak,2 + ak,3 )ψ − α|0 + β|1 . (6) 3 3 So if we measure the first two particles on the Bell basis, then the third particle will collapse into the corresponding state with certainty. For instance, if the state of the first two particles is |φ + , then we can conclude that the third particle is in the state −β|0 + α|1 and we can recover the original state |φ by applying a unitary operator iσy to the third particle. This example explicitly proved that the state encoded by the cloning transformation is resilient to the collective errors model. Comparing our error model with the one commonly referred to “collective decoherence”, such as the one defined in Ref. [1], the difference is that the collective decoherence considered in Ref. [1] affects the states as an identity in a DFS and one need not do any auxiliary operations. But the collective decoherence described in this Letter has not this property. If we do not measure the prior two particles, the original state will not be recovered. Here, we call the error as “collective decoherence” just because the error discussed in this Letter simultaneously affects any two of the three qubits. In this sense, our encoded state is quite different from the states found in the theory of DFS [1] or NS [4]. In a DFS or a NS, the encoded states are the eigenstates of the errors, whereas the ones encoded by the cloning transformation U are not the eigenstates of our error model. Our method probably could be generalized to the encoding of more than one qubit in terms of the cloning transformation transforming N identical qubits into M identical copies. But the encoding process would be quite complicated. In addition, this encoded state may be less useful for quantum computation than the one described in
D. Wei et al. / Physics Letters A 329 (2004) 294–297
297
DFS or NS for the reason that, in order to obtain the original state, we have to measure parts of the system. The measuring process is like the correction operation in QECC and its complexity will increase when one encodes more than one qubit. These problems are waiting for further study. To conclude, we have provided a three-qubit error correction scheme which uses the optimal 1 → 2 quantum cloning transformation to encode the original state. Any one of the errors Ei , Ej , and Ek can have an effect on the encoded state. But we always can recover the state |φ via measuring the first two qubits in their Bell basis, followed by a corresponding rotating operation on the third qubit. In particular, our scheme explicitly shows that quantum cloning could be of use in quantum error correction.
Acknowledgements This work has been supported by the National Natural Science Foundation of China under Grant No. 10374103, the National Basic Research Programme of China under Grant No. 2001CB309309, and the National Knowledge Innovation Program of the Chinese Academy of Sciences under Grant No. KJCX2-W1-3.
References [1] D.A. Lidar, I.L. Chuang, K.B. Whaley, Phys. Rev. Lett. 81 (1998) 2594. [2] E. Knill, E. Laflamme, Phys. Rev. A 55 (1997) 900; D. Gottesman, Phys. Rev. A 54 (1996) 1862; A.R. Calderbank, et al., Phys. Rev. Lett. 79 (1997) 405. [3] P. Zanardi, M. Rasetti, Mod. Phys. Lett. B 25 (1997) 1085; L.M. Duan, G.C. Guo, Phys. Rev. Lett. 79 (1997) 1953; D. Bacon, J. Kempe, D.A. Lidar, K.B. Whaley, Phys. Rev. Lett. 85 (2000) 1758. [4] E. Knill, R. Laflamme, L. Viola, Phys. Rev. Lett. 84 (2000) 2525. [5] B. Misra, E.C.G. Sudersan, J. Math. Phys. 18 (1977) 756. [6] W.H. Zurek, Phys. Rev. Lett. 53 (1984) 391. [7] P. Facchi, D.A. Lidar, S. Pascazio, Phys. Rev. A 69 (2004) 032314. [8] L. Viola, S. Lioyd, Phys. Rev. A 58 (1998) 2733; L. Viola, E. Knill, S. Lioyd, Phys. Rev. Lett. 82 (1999) 2417; D. Vitali, P. Tombesi, Phys. Rev. A 59 (1999) 4178; P. Zanardi, Phys. Lett. A 258 (1999) 77. [9] A. Kitaev, quant-ph/9707021; P. Zanardi, S. Lloyd, Phys. Rev. Lett. 90 (1999) 067902. [10] J.E. Ollerenshaw, D.A. Lidar, L.E. Kay, Phys. Rev. Lett. 91 (2003) 217904. [11] J.B. Altepeter, P.G. Hadley, S.M. Wendelken, A.J. Berglund, P.G. Kwiat, Phys. Rev. Lett. 92 (2004) 147901; L.-A. Wu, D.A. Lidar, Phys. Rev. Lett. 88 (2002) 207902. [12] D.A. Lidar, D. Bacon, K.B. Whaley, Phys. Rev. Lett. 82 (1999) 4556. [13] M.S. Byrd, D.A. Lidar, Phys. Rev. Lett. 89 (2002) 047901. [14] W.Y. Hwang, H. Lee, D. Ahn, S.W. Hwang, Phys. Rev. A 62 (2000) 062305. [15] M.S. Byrd, L.-A. Wu, D.A. Lidar, quant-ph/0402098. [16] W.K. Wootters, W.H. Zurek, Nature 299 (1982) 802. [17] D. Bruß, D.P. DiVincenzo, A. Ekert, C.A. Fuchs, C. Macchiavello, J.A. Smolin, Phys. Rev. A 57 (1998) 2368. [18] D. Bruß, J. Calsamiglia, N. Lükenhaus, Phys. Rev. A 63 (2001) 042308.