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Reply to: “Comment on: ‘Efficient high-capacity quantum secret sharing with two-photon entanglement’ [Phys. Lett. A 372 (2008) 1957]” [Phys. Lett. A 373 (2009) 396]
Physics Letters A 373 (2009) 399–400
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Reply
Reply to: “Comment on: ‘Efficient high-capacity quantum secret sharing with two-photon entanglement’ [Phys. Lett. A 372 (2008) 1957]” [Phys. Lett. A 373 (2009) 396] Fu-Guo Deng a,b,c,d,∗ , Xi-Han Li a,b,c , Hong-Yu Zhou a,b,c a
The Key Laboratory of Beam Technology and Material Modification of Ministry of Education, Beijing Normal University, Beijing 100875, People’s Republic of China Institute of Low Energy Nuclear Physics, and Department of Material Science and Engineering, Beijing Normal University, Beijing 100875, People’s Republic of China c Beijing Radiation Center, Beijing 100875, People’s Republic of China d Department of Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, People’s Republic of China b
a r t i c l e
i n f o
Article history: Received 3 September 2008 Received in revised form 22 October 2008 Accepted 27 October 2008 Available online 30 October 2008 Communicated by P.R. Holland
a b s t r a c t A corresponding Comment, raised by Yang and Wen, claimed that (i) our scheme cannot complete the task of secret sharing well even if the agents obtain the information about their measuring bases; (ii) they give a feasible improvement of our protocol. In this Reply, we show that their first judgment is wrong and their improvement is insecure. © 2007 Elsevier B.V. All rights reserved.
PACS: 03.67.Dd 03.67.Hk 03.65.Ud 89.70.+c
Our Letter [1] proposed a scheme for quantum secret sharing (QSS) with two-photon entanglements. It has the advantage of a high capacity and a high intrinsic efficiency as it has the feature of quantum dense code and works in a deterministic way. Moreover, it is more convenient than others [2,3]. In the preceding comment [4], Yang and Wen think that our QSS scheme cannot complete the task of secret sharing very well when the message sender uses the nonorthogonal entangled states as the quantum information carriers. There are two points in their comment. First, they think that our QSS scheme is invalid if the sender uses the nonorthogonal entangled states shown in Eq. (2) and Eq. (5) in our Letter [1] as the quantum information carriers because they think that the sender Alice cannot deduce deterministically the combination of the keys obtained by the two agents Bob and Charlie even if they obtain the information about the measuring bases (MBs) of the nonorthogonal states. Second, they give an improvement of
our scheme with a little modification, i.e., using another set of basis obtained by rotating the first photon in the states shown in Eq. (2) by 45◦ , instead of the basis shown in Eq. (5) in our Letter. Hereinafter, we refute the Comment from point to point. First, our scheme is valid if the sender announces the measuring basis (MB) of each Einstein–Podolsky–Rosen (EPR) pair after the transmission, not the case claimed by Yang and Wen [4]. There is only a small neglect in the description for the steps of our original scheme. That is, in the first step, the two agents Bob and Charlie should encode their local unitary operations U 0 , U 1 , U 2 and U 3 according to the information of the MB of the EPR pair transmitted. Suppose that both Bob and Charlie encode the four operations U 0 , U 1 , U 2 and U 3 as 00, 11, 01 and 10, respectively, when the sender Alice uses the first set of entangled states MB1 = {|φ ± = √1 (|+ z B |+ zC ± |− z B |− zC ), |ψ ± = 2
√1
2
DOI of original article: 10.1016/j.physleta.2007.10.066. DOI of comment: 10.1016/j.physleta.2008.10.055. Corresponding author at: The Key Laboratory of Beam Technology and Material Modification of Ministry of Education, Beijing Normal University, Beijing 100875, People’s Republic of China. E-mail address: [email protected] (F.-G. Deng).
*
0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.10.056
(|+ z B |−zC ±|− z B |+zC )}. However, Bob encodes the four op-
erations U 0 , U 1 , U 2 and U 3 as 00, 01, 10 and 11, respectively, when Alice uses the second set of entangled states {|Φ ± , |Ψ ± }. That is, Bob and Charlie do the codes U 0B = U 0C = 00, U 1B = U 3C = 10, U 2B = U 1C = 11 and U 3B = U 2C = 01 when Alice exploits the states {|Φ ± , |Ψ ± } shown in Eq. (5) in our Letter [1]. Here U B and U C represent the local operations done by Bob and Charlie, respectively. Of course, Alice should tell Bob and Charlie which way they
400
F.-G. Deng et al. / Physics Letters A 373 (2009) 399–400
choose for their codes at the end. With just these modifications, Alice can get a key K A = K B ⊕ K C in a deterministic way as she can deduce the correlation of the unitary operations performed by Bob and Charlie, which means that the first point in the comment [4] is wrong. Certainly, we should confess that there is a small neglect in the description for the steps of our original scheme [1] and thank Yan and Wen for their comment as it attracts our attention on this neglect. Second, in the improvement by Yang and Wen, they replace our second set of entangled states MB2 = {|Φ ± , |Ψ ± } with another one MB2 = {|Φ ± , |Ψ ± }. Here
+ Φ = √1 |+x B |+zC + i |−x B |−zC , 2
− Φ = √1 |+x B |+zC − i |−x B |−zC , 2
+ Ψ = √1 |+x B |−zC + i |−x B |+zC , 2
− Ψ = √1 |+x B |−zC − i |−x B |+zC , 2
+ 1 Φ = √ |+x B |+zC + |−x B |−zC , 2
− 1 Φ = √ |+x B |+zC − |−x B |−zC , 2
+ 1 Ψ = √ |+x B |−zC + |−x B |+zC , 2
− 1 Ψ = √ |+x B |−zC − |−x B |+zC . 2
It is not difficult to prove that the projection of the states in MB2 on MB1 is not symmetric, which will decrease the security of this scheme with a practical channel, similar to the Bennett 1992 quantum key distribution protocol [5], when the parties exploit the nonorthogonal character of the two sets of the entangled states to ensure the security of their transmission (before and after the coding). For example, |Φ + = 12 (|φ − + |ψ − ) and it dose not contain the items |φ + and |ψ + . This is the reason that we replace MB2 with MB2 in the revision of our Letter [1] according to the referee’s report. It is important to point out that the feasible improvement claimed by Yang and Wen in the comment is insecure in a lossy channel as the three parties in the QSS scheme cannot exploit the nonorthogonal character of the two MBs MB1 and MB2 (or MB1 and MB2 ) to prevent a dishonest agent from eavesdropping freely as a dishonest agent can attack the key with a fake signal and cheating [6]. In detail, the dishonest, say Bob first intercepts the photon C sent from Alice to Charlie, and replaces it with a fake photon C which is one particle in the Bell state |φ + B C = √1 (|+ z B |+ zC + |− z B |− zC ) prepared by Bob 2
himself. If the photon pair BC is chosen by Alice as a sample for eavesdropping check, Bob performs a Bell-state measurement on the photons C and B . If Bob gets the outcome |φ + B C = √1 (|+ z B |+ zC + |− z B |− zC ), Bob measures the photon B with 2
the basis Z or X randomly, and he needs no additional operations for concealing his eavesdropping. If he gets the outcome
|ψ − B C =
√1
2
(|+ z B |−zC − |−z B |+zC ), he needs a unitary op-
eration i σ y = |+ z− z| − |− z+ z| on the photon B and then measure it as the case he does not eavesdrop the transmission. If he gets the other two outcomes (|φ − B C and |ψ + B C ), he declares that he does not receive the photon B as the loss arisen from the channel or the efficiency of his detector. It is not difficult to prove that just this disguise makes Bob free from the first eavesdropping check in the improvement proposed by Yang and Wen. When the photon C is used to carry message and is return Alice from Charlie after a local unitary operation, Bob intercepts it and measures it with the photon B together. The outcome of the Bell-state measurement on the photons B and C will leak Charlie’s message fully. That is to say, the improvement in the comment is insecure in a practical application. As pointed out in the second step in our Letter [1], Alice should exploit the decoy-photon technique [7] to ensure the security of the transmission between her and each of her agents. In this way, it is unnecessary for Alice to use two sets of entangled states as her quantum information carries, just one. That is, the simplest way for improving our original QSS scheme with two-photon entanglements is that Alice uses the four Bell states {|φ ± , |ψ ± } to carry the information and uses some decoy photons to ensure the security of the transmission. It means that we only replace the eight nonorthogonal entangled states {|φ ± , |ψ ± , |Φ ± , |Ψ ± } with the four orthogonal states {|φ ± , |ψ ± } in the step (2) in our original scheme [1]. To summarize, the two most important points in the comment raised by Yang and Wen are wrong. That is, our QSS scheme is valid if the two agents obtain the information about the measuring bases of the nonorthogonal states, and the feasible improvement claimed in the comment is insecure. Moreover, we give a simplest way for dealing with our neglect, only replacing the eight nonorthogonal entangled states {|φ ± , |ψ ± , |Φ ± , |Ψ ± } with the four orthogonal states {|φ ± , |ψ ± } in the step (2) in our original scheme [1]. With just this little modification, our scheme is more efficient than the original one [1] as it holds all the advantages in the latter (such as a high capacity, a high intrinsic efficiency and a less classical communication) and it does not require the sender Alice to prepare two sets of nonorthogonal entangled states, which will decrease the difficulty of the preparation of the entangled states. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No. 10604008 and A Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant No. 200723. References [1] [2] [3] [4] [5] [6] [7]
F.G. Deng, X.H. Li, H.Y. Zhou, Phys. Lett. A 372 (2008) 1957. M. Hillery, V. Bužek, A. Berthiaume, Phys. Rev. A 59 (1999) 1829. A. Karlsson, M. Koashi, N. Imoto, Phys. Rev. A 59 (1999) 162. Y.G. Yang, Q.Y. Wen, Phys. Lett. A 373 (2009) 396. C.H. Bennett, Phys. Rev. Lett. 68 (1992) 3121. F.G. Deng, X.H. Li, H.Y. Zhou, arXiv: 0705.0279. C.Y. Li, et al., Chin. Phys. Lett. 22 (2005) 1049; C.Y. Li, et al., Chin. Phys. Lett. 23 (2006) 2896.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Reply
Reply to: “Comment on: ‘Efficient high-capacity quantum secret sharing with two-photon entanglement’ [Phys. Lett. A 372 (2008) 1957]” [Phys. Lett. A 373 (2009) 396] Fu-Guo Deng a,b,c,d,∗ , Xi-Han Li a,b,c , Hong-Yu Zhou a,b,c a
The Key Laboratory of Beam Technology and Material Modification of Ministry of Education, Beijing Normal University, Beijing 100875, People’s Republic of China Institute of Low Energy Nuclear Physics, and Department of Material Science and Engineering, Beijing Normal University, Beijing 100875, People’s Republic of China c Beijing Radiation Center, Beijing 100875, People’s Republic of China d Department of Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, People’s Republic of China b
a r t i c l e
i n f o
Article history: Received 3 September 2008 Received in revised form 22 October 2008 Accepted 27 October 2008 Available online 30 October 2008 Communicated by P.R. Holland
a b s t r a c t A corresponding Comment, raised by Yang and Wen, claimed that (i) our scheme cannot complete the task of secret sharing well even if the agents obtain the information about their measuring bases; (ii) they give a feasible improvement of our protocol. In this Reply, we show that their first judgment is wrong and their improvement is insecure. © 2007 Elsevier B.V. All rights reserved.
PACS: 03.67.Dd 03.67.Hk 03.65.Ud 89.70.+c
Our Letter [1] proposed a scheme for quantum secret sharing (QSS) with two-photon entanglements. It has the advantage of a high capacity and a high intrinsic efficiency as it has the feature of quantum dense code and works in a deterministic way. Moreover, it is more convenient than others [2,3]. In the preceding comment [4], Yang and Wen think that our QSS scheme cannot complete the task of secret sharing very well when the message sender uses the nonorthogonal entangled states as the quantum information carriers. There are two points in their comment. First, they think that our QSS scheme is invalid if the sender uses the nonorthogonal entangled states shown in Eq. (2) and Eq. (5) in our Letter [1] as the quantum information carriers because they think that the sender Alice cannot deduce deterministically the combination of the keys obtained by the two agents Bob and Charlie even if they obtain the information about the measuring bases (MBs) of the nonorthogonal states. Second, they give an improvement of
our scheme with a little modification, i.e., using another set of basis obtained by rotating the first photon in the states shown in Eq. (2) by 45◦ , instead of the basis shown in Eq. (5) in our Letter. Hereinafter, we refute the Comment from point to point. First, our scheme is valid if the sender announces the measuring basis (MB) of each Einstein–Podolsky–Rosen (EPR) pair after the transmission, not the case claimed by Yang and Wen [4]. There is only a small neglect in the description for the steps of our original scheme. That is, in the first step, the two agents Bob and Charlie should encode their local unitary operations U 0 , U 1 , U 2 and U 3 according to the information of the MB of the EPR pair transmitted. Suppose that both Bob and Charlie encode the four operations U 0 , U 1 , U 2 and U 3 as 00, 11, 01 and 10, respectively, when the sender Alice uses the first set of entangled states MB1 = {|φ ± = √1 (|+ z B |+ zC ± |− z B |− zC ), |ψ ± = 2
√1
2
DOI of original article: 10.1016/j.physleta.2007.10.066. DOI of comment: 10.1016/j.physleta.2008.10.055. Corresponding author at: The Key Laboratory of Beam Technology and Material Modification of Ministry of Education, Beijing Normal University, Beijing 100875, People’s Republic of China. E-mail address: [email protected] (F.-G. Deng).
*
0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.10.056
(|+ z B |−zC ±|− z B |+zC )}. However, Bob encodes the four op-
erations U 0 , U 1 , U 2 and U 3 as 00, 01, 10 and 11, respectively, when Alice uses the second set of entangled states {|Φ ± , |Ψ ± }. That is, Bob and Charlie do the codes U 0B = U 0C = 00, U 1B = U 3C = 10, U 2B = U 1C = 11 and U 3B = U 2C = 01 when Alice exploits the states {|Φ ± , |Ψ ± } shown in Eq. (5) in our Letter [1]. Here U B and U C represent the local operations done by Bob and Charlie, respectively. Of course, Alice should tell Bob and Charlie which way they
400
F.-G. Deng et al. / Physics Letters A 373 (2009) 399–400
choose for their codes at the end. With just these modifications, Alice can get a key K A = K B ⊕ K C in a deterministic way as she can deduce the correlation of the unitary operations performed by Bob and Charlie, which means that the first point in the comment [4] is wrong. Certainly, we should confess that there is a small neglect in the description for the steps of our original scheme [1] and thank Yan and Wen for their comment as it attracts our attention on this neglect. Second, in the improvement by Yang and Wen, they replace our second set of entangled states MB2 = {|Φ ± , |Ψ ± } with another one MB2 = {|Φ ± , |Ψ ± }. Here
+ Φ = √1 |+x B |+zC + i |−x B |−zC , 2
− Φ = √1 |+x B |+zC − i |−x B |−zC , 2
+ Ψ = √1 |+x B |−zC + i |−x B |+zC , 2
− Ψ = √1 |+x B |−zC − i |−x B |+zC , 2
+ 1 Φ = √ |+x B |+zC + |−x B |−zC , 2
− 1 Φ = √ |+x B |+zC − |−x B |−zC , 2
+ 1 Ψ = √ |+x B |−zC + |−x B |+zC , 2
− 1 Ψ = √ |+x B |−zC − |−x B |+zC . 2
It is not difficult to prove that the projection of the states in MB2 on MB1 is not symmetric, which will decrease the security of this scheme with a practical channel, similar to the Bennett 1992 quantum key distribution protocol [5], when the parties exploit the nonorthogonal character of the two sets of the entangled states to ensure the security of their transmission (before and after the coding). For example, |Φ + = 12 (|φ − + |ψ − ) and it dose not contain the items |φ + and |ψ + . This is the reason that we replace MB2 with MB2 in the revision of our Letter [1] according to the referee’s report. It is important to point out that the feasible improvement claimed by Yang and Wen in the comment is insecure in a lossy channel as the three parties in the QSS scheme cannot exploit the nonorthogonal character of the two MBs MB1 and MB2 (or MB1 and MB2 ) to prevent a dishonest agent from eavesdropping freely as a dishonest agent can attack the key with a fake signal and cheating [6]. In detail, the dishonest, say Bob first intercepts the photon C sent from Alice to Charlie, and replaces it with a fake photon C which is one particle in the Bell state |φ + B C = √1 (|+ z B |+ zC + |− z B |− zC ) prepared by Bob 2
himself. If the photon pair BC is chosen by Alice as a sample for eavesdropping check, Bob performs a Bell-state measurement on the photons C and B . If Bob gets the outcome |φ + B C = √1 (|+ z B |+ zC + |− z B |− zC ), Bob measures the photon B with 2
the basis Z or X randomly, and he needs no additional operations for concealing his eavesdropping. If he gets the outcome
|ψ − B C =
√1
2
(|+ z B |−zC − |−z B |+zC ), he needs a unitary op-
eration i σ y = |+ z− z| − |− z+ z| on the photon B and then measure it as the case he does not eavesdrop the transmission. If he gets the other two outcomes (|φ − B C and |ψ + B C ), he declares that he does not receive the photon B as the loss arisen from the channel or the efficiency of his detector. It is not difficult to prove that just this disguise makes Bob free from the first eavesdropping check in the improvement proposed by Yang and Wen. When the photon C is used to carry message and is return Alice from Charlie after a local unitary operation, Bob intercepts it and measures it with the photon B together. The outcome of the Bell-state measurement on the photons B and C will leak Charlie’s message fully. That is to say, the improvement in the comment is insecure in a practical application. As pointed out in the second step in our Letter [1], Alice should exploit the decoy-photon technique [7] to ensure the security of the transmission between her and each of her agents. In this way, it is unnecessary for Alice to use two sets of entangled states as her quantum information carries, just one. That is, the simplest way for improving our original QSS scheme with two-photon entanglements is that Alice uses the four Bell states {|φ ± , |ψ ± } to carry the information and uses some decoy photons to ensure the security of the transmission. It means that we only replace the eight nonorthogonal entangled states {|φ ± , |ψ ± , |Φ ± , |Ψ ± } with the four orthogonal states {|φ ± , |ψ ± } in the step (2) in our original scheme [1]. To summarize, the two most important points in the comment raised by Yang and Wen are wrong. That is, our QSS scheme is valid if the two agents obtain the information about the measuring bases of the nonorthogonal states, and the feasible improvement claimed in the comment is insecure. Moreover, we give a simplest way for dealing with our neglect, only replacing the eight nonorthogonal entangled states {|φ ± , |ψ ± , |Φ ± , |Ψ ± } with the four orthogonal states {|φ ± , |ψ ± } in the step (2) in our original scheme [1]. With just this little modification, our scheme is more efficient than the original one [1] as it holds all the advantages in the latter (such as a high capacity, a high intrinsic efficiency and a less classical communication) and it does not require the sender Alice to prepare two sets of nonorthogonal entangled states, which will decrease the difficulty of the preparation of the entangled states. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No. 10604008 and A Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant No. 200723. References [1] [2] [3] [4] [5] [6] [7]
F.G. Deng, X.H. Li, H.Y. Zhou, Phys. Lett. A 372 (2008) 1957. M. Hillery, V. Bužek, A. Berthiaume, Phys. Rev. A 59 (1999) 1829. A. Karlsson, M. Koashi, N. Imoto, Phys. Rev. A 59 (1999) 162. Y.G. Yang, Q.Y. Wen, Phys. Lett. A 373 (2009) 396. C.H. Bennett, Phys. Rev. Lett. 68 (1992) 3121. F.G. Deng, X.H. Li, H.Y. Zhou, arXiv: 0705.0279. C.Y. Li, et al., Chin. Phys. Lett. 22 (2005) 1049; C.Y. Li, et al., Chin. Phys. Lett. 23 (2006) 2896.