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Quantum state sharing of an arbitrary m-qudit state with two-qudit entanglements and generalized Bell-state measurements
Physica A 387 (2008) 4716–4722
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Quantum state sharing of an arbitrary m-qudit state with two-qudit entanglements and generalized Bell-state measurements Tie-Jun Wang a,b,c,d , Hong-Yu Zhou a,b,c,d , Fu-Guo Deng a,b,c,d,e,∗ a The Key Laboratory of Beam Technology and Material Modification of Ministry of Education, Beijing Normal University, Beijing 100875, People’s Republic of China b Institute of Low Energy Nuclear Physics, Beijing Normal University, Beijing 100875, People’s Republic of China c Department of Material Science and Engineering, Beijing Normal University, Beijing 100875, People’s Republic of China d Beijing Radiation Center, Beijing 100875, People’s Republic of China e Department of Physics, Beijing Normal University, Beijing 100875, People’s Republic of China
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info
Article history: Received 1 September 2007 Available online 30 March 2008 PACS: 03.67.Hk 03.65.Ud Keywords: Quantum state sharing m-qudit Two-particle entanglement Generalized two-particle Bell-state measurement
a b s t r a c t A scheme for quantum state sharing of an arbitrary m-qudit state is proposed with twoqudit entanglements and generalized Bell-state (GBS) measurements. In this scheme, the sender Alice should perform m two-particle GBS measurements on her 2m qudits, and the controllers also take GBS measurements on their qudits and transfer their quantum information to the receiver with entanglement swapping if the agents cooperate. We discuss two topological structures for this quantum state sharing scheme, a dispersive one and a circular one. The former is better at the aspect of security than the latter as it requires the number of the agents who should cooperate for recovering the quantum secret larger than the other one. © 2008 Elsevier B.V. All rights reserved.
1. Introduction The combination of the principles in quantum mechanics with information supplies some interesting applications for the transmission and processing of information [1]. For example, quantum key distribution [2–9] provides a secure way for creating a private key between two remote legitimate users, and has progressed quickly since Bennett and Brassard proposed an original protocol with nonorthogonal single-photon states in 1984 [3]. Quantum secure direct communication [10–12], a novel branch of quantum communication, is used to communicate messages directly [13–15]. Quantum teleportation [16], a unique quantum phenomenon, has been demonstrated by experiment [17,18]. Quantum secret sharing (QSS), an important branch of quantum communication, attracts a lot of attention. In fact, QSS is the generalization of classical secret sharing [19] into a quantum scenario [20,21]. The basic idea of secret sharing in a simple case is that a secret (MA ) is divided by the sender Alice into two pieces (MB and MC ) which will be distributed to her two remote agents, Bob and Charlie, respectively, and the two agents can recover the secret only when they cooperate; otherwise, none can get useful information about the message MA = MB ⊕ MC . As the classical signal can be copied freely and fully without disturbing it, there is no way for people to accomplish this task unconditionally securely with classical physics in principle. However, when quantum mechanics enters the field of information, the story is changed. ∗ 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). 0378-4371/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2008.03.030
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In 1999, Hillery, Bužek and Berthiaume proposed an original QSS scheme [20] for sharing a private key with threeparticle and four-particle entangled Greenberger-Horne-Zeilinger (GHZ) states. They also discussed the way for sharing an unknown single-particle quantum state. Subsequently, Karlsson, Koashi and Imoto [21] presented a scheme for sharing a private key with a two-particle entangled state. Now, QSS is used to not only create a private key among some parties [22–34], but also share quantum information [35–49]. The branch for sharing an unknown quantum state is called quantum state sharing (QSTS) by Lance et al. [47] in 2004. In 2005, we presented a scheme [38] for sharing an arbitrary two-particle state with Einstein–Podolsky–Rosen (EPR) pairs and a scheme [39] for controlled teleportation of an arbitrary two-particle entanglement with GHZ states. In essence, controlled teleportation is equivalent to quantum state sharing [38–40]. More recently, we have proposed a QSTS scheme for sharing an arbitrary two-qubit state with only two-photon entanglements and Bell-state measurements [41]. Wang et al. [42] proposed a QSTS scheme for sharing an unknown single-particle qutrit state (the state of a three-level quantum system). The way for controlled teleportation of an arbitrary multi-qudit state (the state of a d-level quantum system) with d-dimensional GHZ states is introduced in Ref. [48] and that with a pure entangled multi-particle quantum channel is also discussed in Ref. [49]. In this paper, we want to introduce a way for sharing an arbitrary multi-qudit state with only two-particle entanglements and generalized two-particle Bell-state measurements, following some ideas in Ref. [41]. It requires all the parties only to perform two-particle generalized Bell-state measurements, not d-dimensional GHZ-state measurements. The receiver can recover the unknown state with a suitable unitary operation on each particle held in his hand if he cooperates with the controllers. We discuss two topological structures for this quantum state sharing scheme, a dispersive one and a circular one. The former is better at the aspect of security than the latter as it requires the number of the agents who should cooperate for recovering the quantum secret larger than the other one. No matter what the topological structure of this QSTS scheme is, its intrinsic efficiency for qudits in this scheme in principle approaches 100%. 2. Three-party QSTS of an arbitrary two-qudit state with two-particle entanglements The generalized d-dimensional Bell states are [12,16] d−1 1 X
|ψuv iAB = √
d l=0
e
2πi d
lu
|liA ⊗ |l + viB ,
(1)
where |ui and |vi (u, v = 0, 1, . . . , d − 1) are the eigenvectors of the basis Zd , and the subscripts A and B represent the particles A and B in the generalized d-dimensional Bell state (GBS) |ψuv iAB l + v in |l + vi means mod d. The d-dimensional unitary operation Uuv =
d−1 X
e
2πi
lu
d
(2)
|l + vi hl|
l=0
on the particle B can transform the GBS d−1 1 X
|ψ00 i = √
d l=0
(3)
|li ⊗ |li
into the GBS |ψuv i Uuv |ψ00 i = |ψuv i.
(4)
Another unbiased measuring basis (MB) Xd can be described as [48,50] 1
|0ix = √ (|0i + |1i + · · · + |d − 1i) , d
1
|1ix = √
d
1
|2ix = √
|0i + e
d
|0i + e
2πi d
4πi d
|1i + · · · + e |1i + · · · + e
(d−1)2πi
|d − 1i ,
(d−1)4πi
|d − 1i ,
d
d
··· 1
|d − 1ix = √ (|0i + e d
2(d−1)πi d
|1i + e
2×2(d−1)πi d
|2i + · · · + e
(d−1)×2(d−1)πi d
|d − 1i).
(5)
The two vectors |ui and |lix coming from two MBs satisfy the relation |hu|lix |2 = 1d . An unknown arbitrary two-qudit state can be described as
|φix1 ,x2 = d−1 X y1 ,y2 =0
d−1 X y1 ,y2 =0
ay1 ,y2 |y1 , y2 ix1 x2 ,
|ay1 ,y2 |2 = 1.
(6)
(7)
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Fig. 1. Quantum state sharing of an arbitrary two-qudit state with two-particle entanglements and generalized d-dimensional Bell-state measurements. The dashed lines connect qudits in generalized Bell states |ψ00 i or the unknown two-qudit state |φix1 ,x2 . The rectangles represent the generalized Bell-state measurements.
For sharing the unknown state |φix1 ,x2 , the three parties, Alice, Bob and Charlie, should first create a maximally entangled quantum channel between each other, shown in Fig. 1(i). That is, Alice prepares a sequence of generalized Bell states (GBSs) |ψ00 iA1 B1 , and sends the particles B1 in each GBS to Bob. They can obtain privately a subset of GBSs with security analysis (similar to that in the two-step protocol [10]) and entanglement purification [51]. Alice also shares a sequence of GBSs |ψ00 iA2 C2 with Bob in the same way. So do Bob and Charlie. Without loss of generalization, let us suppose Bob will recover the unknown two-qudit state with the help of the controller Charlie. In this time, Alice performs a GBS measurement on the particles x1 and A1 , and then another GBS measurement on the particles x2 and A2 . The controller Charlie also performs a GBS measurement on the particles C2 and C 0 , shown in Fig. 1(ii). Bob can reconstruct the unknown state |φix1 ,x2 with some unitary operations when he obtains the outcomes of the GBS measurements done by Alice and Charlie. In detail, the state of the composite quantum system composed of the eight particles x1 , x2 , A1 , B1 , A2 , C2 , B0 , and C 0 can be written as
|Φ ix1 x2 A1 B1 A2 C2 B0 C0 ≡ |φix1 ,x2 ⊗ |ψ00 iA1 B1 ⊗ |ψ00 iA2 C2 ⊗ |ψ00 iC0 B0 =
=
1
d 3/2
d−1 X
y1 ,y2 =0
d−1 X
1
d3 u,v,u0 ,v0 ,u00 ,v00 =0 d−1 X
⊗
y1 ,y2 =0
d−1 X
ay1 ,y2 |y1 , y2 i ⊗
!
|li|li
l=0
⊗ A 1 B1
d−1 X
!
|l0 i|l0 i
l0 =0
⊗ A2 C2
d−1 X l00 =0
!
|l00 i|l00 i C 0 B0
|ψuv ix1 A1 |ψu0 v0 ix2 A2 |ψu00 v00 iC2 C0
ay1 ,y2 e −
2πi d
(y1 u+y2 u0 +y2 u00 +v0 u00 )
|y1 + viB1 |y2 + v0 + v00 iB0 .
(8)
That is to say, if the outcomes of the GBS measurements obtained by Alice are |ψuv ix1 A1 and |ψu0 v0 ix2 A2 and that obtained by Charlie is |ψu00 v00 iC2 C0 , the quantum system composed of the two particles B1 and B0 collapses to the state
|ζiB1 B0 =
d−1 X y1 ,y2 =0
ay1 ,y2 e −
2πi d
(y1 u+y2 u0 +y2 u00 +v0 u00 )
⊗ |y1 + viB1 ⊗ |y2 + v0 + v00 iB0 .
(9)
P Bob can reconstruct the unknown two-qudit state |φiB1 ,B0 = dy1−,1y2 =0 ay1 ,y2 |y1 , y2 iB1 B0 by performing a unitary operation Uu,−v on the particle B1 and another one Uu0 +u00 ,−v0 −v00 on the particle B0 , i.e., 0
|φiB1 B0 = (Uu,1−v ⊗ UuB0 +u00 ,−v0 −v00 )|ζiB1 B0 = F B
d−1 X y1 ,y2 =0
ay1 ,y2 |y1 , y2 iB1 B0 ,
(10)
where F=e
2πi d
(uv+u0 v0 +u00 v00 +u0 v00 )
(11) 0
is a whole phase of the two-qudit quantum system B1 B and it does not affect the feature of the state in physics. In this QSTS scheme, Bob can get nothing if he does not cooperate with Charlie. That is, Bob can recover the unknown state |φix1 ,x2 only after he gets the information about the GBS measurement on the particles C2 and C 0 done by Charlie. After Alice performs the GBS measurements on the particle pairs x1 A1 and x2 A2 , the state of the unknown quantum system x1 x2 is transferred to the quantum system composed of the particles B1 and C2 . In other words, Bob only controls one particle of the entangled two-particle quantum system B1 C2 . It is well known that one cannot obtain the whole information from the measurement on a part of entangled quantum system as the density matrix of the particle B1 is
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Fig. 2. Quantum state sharing of an arbitrary m-qudit state with two-particle entanglements and generalized d-dimensional Bell-state measurements. Alice splits her unknown quantum system into two pieces: one is composed of mB particles and sends it to Bob, and the other is made up of mC = m − mB particles and sends it to Charlie. The dashed lines connect qudits in generalized Bell states |ψ00 i or the unknown m-qudit state |φix1 ,x2 ,...,xm . The rectangles with real lines represent the generalized Bell-state measurements. The rectangles with dashed-dot lines represent the spaces controlled by the parties securely.
1/d
ρB1
0 = ···
0 1/d
··· ··· ··· ···
···
0
0
0 0
···
=
1
I . d d×d
The GBS measurement on the particles C2 and C 0 done by Charlie is used to transfer the
1/d
unknown state |φix1 ,x2 to the particle pair B1 B0 with entanglement swapping [52]. Bob can reconstruct the unknown twoqudit state only when he gets the outcome of the GBS measurement done by Charlie as the density matrix of the particle pair 2 1/d
B1 B0 is ρB1 B0
0 = ··· 0 0
0
1/d2
··· 0 0
··· ··· ··· ··· ···
0
0
··· 1/d2 0
0
0
=
··· 0
1
I 2 2 for d2 d ×d
Bob if he has no information about the GBS measurement done
1 /d 2
by Charlie. In a word, this QSTS scheme is secure if the process for setting up the quantum channel with GBSs is secure. In practice, the three parties can set up the quantum channels between each other securely if they use the methods in Ref. [10] to share a sequence of GBSs and do quantum privacy amplification with entanglement purification [51]. So this QSTS scheme can be made to be secure with present technology. 3. Three-party QSTS of an arbitrary m-qudit state with two-particle entanglements In fact, in this QSTS scheme Alice splits her unknown two-qudit state into two pieces and distributes them to her two agents respectively. When Bob wants to reconstruct the unknown state, he should require Charlie to teleport her piece to him with entanglement swapping. In the case with an unknown arbitrary m-qudit state, Alice can also split it into two pieces and then send them to her two agents. Bob can reconstruct the unknown state when he cooperates with Charlie, see Fig. 2. Similar to the case with an arbitrary two-qudit state, Alice first shares a sequence of GBSs |ψ00 i with Bob privately, say SB = [A1 B1 , A2 B2 , . . . , AmB BmB ], and then shares another sequence of GBSs with Charlie privately, say SC = [AmB +1 C1 , AmB +2 C2 , . . . , AmB +mC CmC ]. Also, Charlie shares a sequence of GBSs with Bob. All these processes equal to setting up three secure quantum channels between the parties. As the quantum channel between two parties can be created securely with security analysis and entanglement purification, all these processes can be in principle made to be secure [10]. An unknown arbitrary m-qudit state can be written as
|φix1 ,x2 ,...,xm = d−1 X y1 ,y2 ,...,ym =0
d−1 X y1 ,y2 ,...,ym =0
ay1 ,y2 ,...,ym |y1 , y2 , . . . , ym ix1 x2 ···xm ,
|ay1 ,y2 ,...,ym |2 = 1.
(12)
(13)
For sharing the unknown state |φix1 ,x2 ,...,xm , Alice can first perform GBS measurements on the pairs xi Ai (i = 1, 2, . . . , mB + mC ), and then announce the outcomes in public. Also, Charlie performs GBS measurements on the pairs Cj Cj0 (j = 1, 2, . . . , mC ). When Bob cooperates with Charlie, he can reconstruct the unknown state |φix1 ,x2 ,...,xm with some unitary operations,
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according to the information about the GBS measurements published by Charlie. In detail, the state of the composite system composed of all the particles can be described as
|Φ is ≡ |φix1 ,x2 ,...,xm ⊗
mB Y
|ψ00 iAi Bi ⊗
mC Y
i=1
1
= d
y1 ,y2 ,...,ym =0
2
dm+mC
l0j =0
×e
mC Y
⊗
u,v,u0 ,v0 =0
j=1
m P l=0
m PC
yl ul +
l0 =0
d−1 X
|ψuj vj ixj Aj
!
mC Y k=1
(ym
0 0 +vm +l0 )ul0 B +l B
!
|li i|li i
l=0
i=1
Ai Bi
|l00k i|l00k i
=0 l00 k
AmB +j Cj m Y
⊗
mB d−1 Y X
k=1
d−1 X
− 2dπi
|ψ00 iCk0 B0k
x1 x2 ···xm
1
=
k=1
ay1 ,y2 ,...,ym |y1 , y2 , . . . , ym i
mC d−1 Y X 0 0 |lj i|lj i j=1
mC Y
d−1 X
m+mC
⊗
j=1
|ψ00 iAmB +j Cj ⊗
Ck0 B0k
!
|ψu0k v0k iCk Ck0 ⊗
d−1 X y1 ,y2 ,...,ym =0
ay1 ,y2 ,...,ym
! mB Y
!
!
mC Y
|yn + vn iBn
|ymB +n0 + vmB +n0 + vn0 iB0n . 0
(14)
n0 =1
n=1
That is to say, if the outcomes of GBS measurements on the pairs xj Aj (j = 1, 2, . . . , m) obtained by Alice are |ψuj vj ixj Aj and those obtained by Charlie on the pairs Ck Ck0 are |ψu0k v0k iCk Ck0 (k = 1, 2, . . . , mC ), the quantum system composed of the particles Bl and Bl0 (l = 1, 2, . . . , mB and l0 = 1, 2, . . . , mC ) collapses to the state d−1 X
|ζ iB1 B2 ···Bm B01 B02 ···B0mC = 0
y1 ,y2 ,...,ym =0 mB Y
×
ay1 ,y2 ,...,ym e
!
|yn + vn iBn
− 2dπi
mC Y
m P l=0
yl ul +
m PC
l0 =0
(ym
0 0 +vm +l0 )ul0 B
!
B +l
!
|ymB +n0 + vmB +n0 + v0n0 iB0n .
(15)
n0 =1
n=1
When Bob wants to recover the unknown state, he can perform the unitary operation Uul ,−vl (l = 1, 2, . . . , mB ) on his particle Bl and the unitary operation Uum +l0 +u00 ,−vm +l0 −v00 on the particle B0l0 (l0 = 1, 2, . . . , mC ) if he obtains the help of the controller l
B
B
l
Charlie, i.e., mB Y
|φiB1 B2 ···BmB B01 B02 ···B0mC =
! B Uull ,−vl
⊗
l0 =1
l=1
= F0
mC Y
d−1 X y1 ,y2 ,...,ym =0
!
B00
Uu
l
mB +l0 +ul0 ,−vmB +l0 −vl0
0
0
|ζ0 iB1 B2 ···Bm B01 B02 ···B0mC
ay1 ,y2 ,...,ym |y1 , y2 , . . . , ym iB1 B2 ···Bm
B0 B0 ···B0mC
B 1 2
,
(16)
where 2πi
0
F =e
d
m P l=0
ul vl +
m PC
(um
l0 =0
0 0 0 +ul0 )vl0
!
B +l
(17) 0
0
0
is a whole phase of the m-qudit quantum system B1 B2 · · · BmB B1 B2 · · · BmC . Similar to the case for sharing an arbitrary two-qudit state, Bob cannot obtain the unknown m-qudit state if he does not cooperate with the controller Charlie. Even though Bob knows the fact that Charlie has performed the GBS measurements on the pairs Cj Cj0 (j = 1, 2, . . . , mC ), he cannot reconstruct the unknown state. As there are d2mC outcomes obtained by Charlie with GBS measurements, Bob has only the probability d21mC to get the correct result. The larger the number of the particles controlled by Charlie mC is, the less the probability that Bob can reconstruct the unknown state without the help of Charlie is. This result is also kept for the case that the receiver is Charlie and the controller is Bob. For symmetry, the integer number mC 1 m+1 is optimal if it does not beyond the range [ m− , 2 ]. Certainly, Bob will get nothing if he does not know whether Charlie has 2 performed the GBS measurements on her particle pairs Cj Cj0 (j = 1, 2, . . . , mC ). In fact, Bob only gets a part of the unknown quantum system if Charlie does not perform the GBS measurements on her pairs and transfer the entanglement of the unknown quantum system to the particles controlled by Bob, and he cannot reconstruct the unknown m-qudit state in this time. In a word, Bob can reconstruct the unknown m-qudit state after the three parties have set up their secure quantum channel if and only if he cooperates with Charlie. That is, this QSTS scheme for an arbitrary unknown m-qudit state is secure if the quantum channels are secure. 4. Discussion and summary It is straightforward to generalize this QSTS scheme to the case with N + 1 agents, say the receiver Bob and the controllers Charliei (i = 1, 2, . . . , N). In this time, Alice shares a sequence of GBSs |ψ00 i with each agent, and Bob also shares a sequence
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Fig. 3. Circular topological structure for QSTS. Each agent shares a sequence of GBSs with his two neighboring agents, or one agent and the sender Alice.
of GBSs |ψ00 i with each of the controllers Charliei . Suppose that Alice splits mB particles to Bob and mCi particles to the controller Charliei . In this way, all the controllers Charliei act as the role of the controller Charlie in the case with two agents discussed in the Section 3. This is a dispersive topological structure for QSTS. Same as [41], another interesting topological structure of the QSTS is a circular one. That is, each agent shares privately a sequence of GBSs |ψ00 i with only his/her two neighboring agents (or one agent and the sender Alice), not each of the agents, shown in Fig. 3. Alice first splits the unknown state into two pieces and then transfers them into the particles controlled by the receiver Bob and one of the controllers Charlie1 . The other agents (Charliej (j = 2, 3, . . . , N)) only provide their quantum channels for Charlie1 to transfer her piece of unknown quantum information into the particles controlled by Bob with entanglement swapping. Surely, Bob can reconstruct the unknown state only when he gets all the outcomes of GBS measurements done by all the controllers if the controllers are honest. As the goal of QSTS is that Alice can prevent the dishonest man (no more than one) in the agents from eavesdropping the quantum secret, a QSTS scheme is secure if Alice can find out the eavesdropping done by the dishonest agent. In this QSTS scheme for sharing an arbitrary m-qudit state, Alice can in principle forbid any agents from eavesdropping her quantum information freely as she can share privately a sequence of GBSs with each of her agents. After setting up their quantum channels securely, none of the agents has the chance to eavesdrop the quantum secret freely. Obviously, in the dispersive topological structure, Bob can reconstruct the quantum secret only when he cooperates with all the other agents if mB 6= 0 and mCi 6= 0 (for i = 1, 2, . . . , N). In the circular topological structure, the N controllers do not play an equivalent role in QSTS. In detail, Charlie1 has the superpower for controlling the reconstruction of the quantum secret as she holds a piece of the original quantum secret split by Alice, different from the other controllers Charliej (j = 2, 3, . . . , N). That is to say, Bob can recover the originally unknown state if he obtains the help of Charlie1 in this time, not resorting to the help of the other controllers. Although this QSTS scheme with a circular topological structure is secure if there is no more than one dishonest agent, it cannot prevent two potentially dishonest agents from eavesdropping the quantum secret freely, especially the two agents who obtain the pieces of the originally unknown m-qudit split by the sender Alice. In the case with a dispersive topological structure, Alice can split symmetrically (or nearly symmetrically) her quantum secret and then distribute it to her agents, which will require the receiver Bob to cooperate with all the controllers (N + 1 ≤ m) or m agents (N + 1 > m) for recovering the quantum secret. No matter what the topological structure of this QSTS scheme is, its intrinsic efficiency for qudits ηq ≡ qqut approaches 100% in principle as almost all the qudits are useful. Here qu is the number of the useful qudits and qt is the number of the qudits transmitted. In summary, we have presented a QSTS scheme for sharing an arbitrary m-qudit state based on two-qudit entanglements and generalized Bell-state measurements, following some ideas in Ref. [41]. We give a general form for N + 1 agents to share the unknown m-qudit state securely with entanglement swapping. In this scheme, the sender Alice performs m generalized Bell-state measurements and the controllers also take some generalized Bell-state measurements on their particles. The receiver can recover the unknown state if he cooperates with the controllers. Also, we discuss the difference between a dispersive topological structure and a circular one. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant Nos. 10604008 and 10435020, A Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant No. 200723, and Beijing Education Committee under Grant No. XK100270454. References [1] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2000. [2] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Rev. Modern Phys. 74 (2002) 145. [3] C.H. Bennett, G. Brassard, Proceeding of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, IEEE, New York, 1984, p. 175. [4] A.K. Ekert, Phys. Rev. Lett. 67 (1991) 661. [5] C.H. Bennett, G. Brassard, N.D. Mermin, Phys. Rev. Lett. 68 (1992) 557.
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Quantum state sharing of an arbitrary m-qudit state with two-qudit entanglements and generalized Bell-state measurements Tie-Jun Wang a,b,c,d , Hong-Yu Zhou a,b,c,d , Fu-Guo Deng a,b,c,d,e,∗ a The Key Laboratory of Beam Technology and Material Modification of Ministry of Education, Beijing Normal University, Beijing 100875, People’s Republic of China b Institute of Low Energy Nuclear Physics, Beijing Normal University, Beijing 100875, People’s Republic of China c Department of Material Science and Engineering, Beijing Normal University, Beijing 100875, People’s Republic of China d Beijing Radiation Center, Beijing 100875, People’s Republic of China e Department of Physics, Beijing Normal University, Beijing 100875, People’s Republic of China
article
info
Article history: Received 1 September 2007 Available online 30 March 2008 PACS: 03.67.Hk 03.65.Ud Keywords: Quantum state sharing m-qudit Two-particle entanglement Generalized two-particle Bell-state measurement
a b s t r a c t A scheme for quantum state sharing of an arbitrary m-qudit state is proposed with twoqudit entanglements and generalized Bell-state (GBS) measurements. In this scheme, the sender Alice should perform m two-particle GBS measurements on her 2m qudits, and the controllers also take GBS measurements on their qudits and transfer their quantum information to the receiver with entanglement swapping if the agents cooperate. We discuss two topological structures for this quantum state sharing scheme, a dispersive one and a circular one. The former is better at the aspect of security than the latter as it requires the number of the agents who should cooperate for recovering the quantum secret larger than the other one. © 2008 Elsevier B.V. All rights reserved.
1. Introduction The combination of the principles in quantum mechanics with information supplies some interesting applications for the transmission and processing of information [1]. For example, quantum key distribution [2–9] provides a secure way for creating a private key between two remote legitimate users, and has progressed quickly since Bennett and Brassard proposed an original protocol with nonorthogonal single-photon states in 1984 [3]. Quantum secure direct communication [10–12], a novel branch of quantum communication, is used to communicate messages directly [13–15]. Quantum teleportation [16], a unique quantum phenomenon, has been demonstrated by experiment [17,18]. Quantum secret sharing (QSS), an important branch of quantum communication, attracts a lot of attention. In fact, QSS is the generalization of classical secret sharing [19] into a quantum scenario [20,21]. The basic idea of secret sharing in a simple case is that a secret (MA ) is divided by the sender Alice into two pieces (MB and MC ) which will be distributed to her two remote agents, Bob and Charlie, respectively, and the two agents can recover the secret only when they cooperate; otherwise, none can get useful information about the message MA = MB ⊕ MC . As the classical signal can be copied freely and fully without disturbing it, there is no way for people to accomplish this task unconditionally securely with classical physics in principle. However, when quantum mechanics enters the field of information, the story is changed. ∗ 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). 0378-4371/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2008.03.030
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In 1999, Hillery, Bužek and Berthiaume proposed an original QSS scheme [20] for sharing a private key with threeparticle and four-particle entangled Greenberger-Horne-Zeilinger (GHZ) states. They also discussed the way for sharing an unknown single-particle quantum state. Subsequently, Karlsson, Koashi and Imoto [21] presented a scheme for sharing a private key with a two-particle entangled state. Now, QSS is used to not only create a private key among some parties [22–34], but also share quantum information [35–49]. The branch for sharing an unknown quantum state is called quantum state sharing (QSTS) by Lance et al. [47] in 2004. In 2005, we presented a scheme [38] for sharing an arbitrary two-particle state with Einstein–Podolsky–Rosen (EPR) pairs and a scheme [39] for controlled teleportation of an arbitrary two-particle entanglement with GHZ states. In essence, controlled teleportation is equivalent to quantum state sharing [38–40]. More recently, we have proposed a QSTS scheme for sharing an arbitrary two-qubit state with only two-photon entanglements and Bell-state measurements [41]. Wang et al. [42] proposed a QSTS scheme for sharing an unknown single-particle qutrit state (the state of a three-level quantum system). The way for controlled teleportation of an arbitrary multi-qudit state (the state of a d-level quantum system) with d-dimensional GHZ states is introduced in Ref. [48] and that with a pure entangled multi-particle quantum channel is also discussed in Ref. [49]. In this paper, we want to introduce a way for sharing an arbitrary multi-qudit state with only two-particle entanglements and generalized two-particle Bell-state measurements, following some ideas in Ref. [41]. It requires all the parties only to perform two-particle generalized Bell-state measurements, not d-dimensional GHZ-state measurements. The receiver can recover the unknown state with a suitable unitary operation on each particle held in his hand if he cooperates with the controllers. We discuss two topological structures for this quantum state sharing scheme, a dispersive one and a circular one. The former is better at the aspect of security than the latter as it requires the number of the agents who should cooperate for recovering the quantum secret larger than the other one. No matter what the topological structure of this QSTS scheme is, its intrinsic efficiency for qudits in this scheme in principle approaches 100%. 2. Three-party QSTS of an arbitrary two-qudit state with two-particle entanglements The generalized d-dimensional Bell states are [12,16] d−1 1 X
|ψuv iAB = √
d l=0
e
2πi d
lu
|liA ⊗ |l + viB ,
(1)
where |ui and |vi (u, v = 0, 1, . . . , d − 1) are the eigenvectors of the basis Zd , and the subscripts A and B represent the particles A and B in the generalized d-dimensional Bell state (GBS) |ψuv iAB l + v in |l + vi means mod d. The d-dimensional unitary operation Uuv =
d−1 X
e
2πi
lu
d
(2)
|l + vi hl|
l=0
on the particle B can transform the GBS d−1 1 X
|ψ00 i = √
d l=0
(3)
|li ⊗ |li
into the GBS |ψuv i Uuv |ψ00 i = |ψuv i.
(4)
Another unbiased measuring basis (MB) Xd can be described as [48,50] 1
|0ix = √ (|0i + |1i + · · · + |d − 1i) , d
1
|1ix = √
d
1
|2ix = √
|0i + e
d
|0i + e
2πi d
4πi d
|1i + · · · + e |1i + · · · + e
(d−1)2πi
|d − 1i ,
(d−1)4πi
|d − 1i ,
d
d
··· 1
|d − 1ix = √ (|0i + e d
2(d−1)πi d
|1i + e
2×2(d−1)πi d
|2i + · · · + e
(d−1)×2(d−1)πi d
|d − 1i).
(5)
The two vectors |ui and |lix coming from two MBs satisfy the relation |hu|lix |2 = 1d . An unknown arbitrary two-qudit state can be described as
|φix1 ,x2 = d−1 X y1 ,y2 =0
d−1 X y1 ,y2 =0
ay1 ,y2 |y1 , y2 ix1 x2 ,
|ay1 ,y2 |2 = 1.
(6)
(7)
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Fig. 1. Quantum state sharing of an arbitrary two-qudit state with two-particle entanglements and generalized d-dimensional Bell-state measurements. The dashed lines connect qudits in generalized Bell states |ψ00 i or the unknown two-qudit state |φix1 ,x2 . The rectangles represent the generalized Bell-state measurements.
For sharing the unknown state |φix1 ,x2 , the three parties, Alice, Bob and Charlie, should first create a maximally entangled quantum channel between each other, shown in Fig. 1(i). That is, Alice prepares a sequence of generalized Bell states (GBSs) |ψ00 iA1 B1 , and sends the particles B1 in each GBS to Bob. They can obtain privately a subset of GBSs with security analysis (similar to that in the two-step protocol [10]) and entanglement purification [51]. Alice also shares a sequence of GBSs |ψ00 iA2 C2 with Bob in the same way. So do Bob and Charlie. Without loss of generalization, let us suppose Bob will recover the unknown two-qudit state with the help of the controller Charlie. In this time, Alice performs a GBS measurement on the particles x1 and A1 , and then another GBS measurement on the particles x2 and A2 . The controller Charlie also performs a GBS measurement on the particles C2 and C 0 , shown in Fig. 1(ii). Bob can reconstruct the unknown state |φix1 ,x2 with some unitary operations when he obtains the outcomes of the GBS measurements done by Alice and Charlie. In detail, the state of the composite quantum system composed of the eight particles x1 , x2 , A1 , B1 , A2 , C2 , B0 , and C 0 can be written as
|Φ ix1 x2 A1 B1 A2 C2 B0 C0 ≡ |φix1 ,x2 ⊗ |ψ00 iA1 B1 ⊗ |ψ00 iA2 C2 ⊗ |ψ00 iC0 B0 =
=
1
d 3/2
d−1 X
y1 ,y2 =0
d−1 X
1
d3 u,v,u0 ,v0 ,u00 ,v00 =0 d−1 X
⊗
y1 ,y2 =0
d−1 X
ay1 ,y2 |y1 , y2 i ⊗
!
|li|li
l=0
⊗ A 1 B1
d−1 X
!
|l0 i|l0 i
l0 =0
⊗ A2 C2
d−1 X l00 =0
!
|l00 i|l00 i C 0 B0
|ψuv ix1 A1 |ψu0 v0 ix2 A2 |ψu00 v00 iC2 C0
ay1 ,y2 e −
2πi d
(y1 u+y2 u0 +y2 u00 +v0 u00 )
|y1 + viB1 |y2 + v0 + v00 iB0 .
(8)
That is to say, if the outcomes of the GBS measurements obtained by Alice are |ψuv ix1 A1 and |ψu0 v0 ix2 A2 and that obtained by Charlie is |ψu00 v00 iC2 C0 , the quantum system composed of the two particles B1 and B0 collapses to the state
|ζiB1 B0 =
d−1 X y1 ,y2 =0
ay1 ,y2 e −
2πi d
(y1 u+y2 u0 +y2 u00 +v0 u00 )
⊗ |y1 + viB1 ⊗ |y2 + v0 + v00 iB0 .
(9)
P Bob can reconstruct the unknown two-qudit state |φiB1 ,B0 = dy1−,1y2 =0 ay1 ,y2 |y1 , y2 iB1 B0 by performing a unitary operation Uu,−v on the particle B1 and another one Uu0 +u00 ,−v0 −v00 on the particle B0 , i.e., 0
|φiB1 B0 = (Uu,1−v ⊗ UuB0 +u00 ,−v0 −v00 )|ζiB1 B0 = F B
d−1 X y1 ,y2 =0
ay1 ,y2 |y1 , y2 iB1 B0 ,
(10)
where F=e
2πi d
(uv+u0 v0 +u00 v00 +u0 v00 )
(11) 0
is a whole phase of the two-qudit quantum system B1 B and it does not affect the feature of the state in physics. In this QSTS scheme, Bob can get nothing if he does not cooperate with Charlie. That is, Bob can recover the unknown state |φix1 ,x2 only after he gets the information about the GBS measurement on the particles C2 and C 0 done by Charlie. After Alice performs the GBS measurements on the particle pairs x1 A1 and x2 A2 , the state of the unknown quantum system x1 x2 is transferred to the quantum system composed of the particles B1 and C2 . In other words, Bob only controls one particle of the entangled two-particle quantum system B1 C2 . It is well known that one cannot obtain the whole information from the measurement on a part of entangled quantum system as the density matrix of the particle B1 is
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Fig. 2. Quantum state sharing of an arbitrary m-qudit state with two-particle entanglements and generalized d-dimensional Bell-state measurements. Alice splits her unknown quantum system into two pieces: one is composed of mB particles and sends it to Bob, and the other is made up of mC = m − mB particles and sends it to Charlie. The dashed lines connect qudits in generalized Bell states |ψ00 i or the unknown m-qudit state |φix1 ,x2 ,...,xm . The rectangles with real lines represent the generalized Bell-state measurements. The rectangles with dashed-dot lines represent the spaces controlled by the parties securely.
1/d
ρB1
0 = ···
0 1/d
··· ··· ··· ···
···
0
0
0 0
···
=
1
I . d d×d
The GBS measurement on the particles C2 and C 0 done by Charlie is used to transfer the
1/d
unknown state |φix1 ,x2 to the particle pair B1 B0 with entanglement swapping [52]. Bob can reconstruct the unknown twoqudit state only when he gets the outcome of the GBS measurement done by Charlie as the density matrix of the particle pair 2 1/d
B1 B0 is ρB1 B0
0 = ··· 0 0
0
1/d2
··· 0 0
··· ··· ··· ··· ···
0
0
··· 1/d2 0
0
0
=
··· 0
1
I 2 2 for d2 d ×d
Bob if he has no information about the GBS measurement done
1 /d 2
by Charlie. In a word, this QSTS scheme is secure if the process for setting up the quantum channel with GBSs is secure. In practice, the three parties can set up the quantum channels between each other securely if they use the methods in Ref. [10] to share a sequence of GBSs and do quantum privacy amplification with entanglement purification [51]. So this QSTS scheme can be made to be secure with present technology. 3. Three-party QSTS of an arbitrary m-qudit state with two-particle entanglements In fact, in this QSTS scheme Alice splits her unknown two-qudit state into two pieces and distributes them to her two agents respectively. When Bob wants to reconstruct the unknown state, he should require Charlie to teleport her piece to him with entanglement swapping. In the case with an unknown arbitrary m-qudit state, Alice can also split it into two pieces and then send them to her two agents. Bob can reconstruct the unknown state when he cooperates with Charlie, see Fig. 2. Similar to the case with an arbitrary two-qudit state, Alice first shares a sequence of GBSs |ψ00 i with Bob privately, say SB = [A1 B1 , A2 B2 , . . . , AmB BmB ], and then shares another sequence of GBSs with Charlie privately, say SC = [AmB +1 C1 , AmB +2 C2 , . . . , AmB +mC CmC ]. Also, Charlie shares a sequence of GBSs with Bob. All these processes equal to setting up three secure quantum channels between the parties. As the quantum channel between two parties can be created securely with security analysis and entanglement purification, all these processes can be in principle made to be secure [10]. An unknown arbitrary m-qudit state can be written as
|φix1 ,x2 ,...,xm = d−1 X y1 ,y2 ,...,ym =0
d−1 X y1 ,y2 ,...,ym =0
ay1 ,y2 ,...,ym |y1 , y2 , . . . , ym ix1 x2 ···xm ,
|ay1 ,y2 ,...,ym |2 = 1.
(12)
(13)
For sharing the unknown state |φix1 ,x2 ,...,xm , Alice can first perform GBS measurements on the pairs xi Ai (i = 1, 2, . . . , mB + mC ), and then announce the outcomes in public. Also, Charlie performs GBS measurements on the pairs Cj Cj0 (j = 1, 2, . . . , mC ). When Bob cooperates with Charlie, he can reconstruct the unknown state |φix1 ,x2 ,...,xm with some unitary operations,
T.-J. Wang et al. / Physica A 387 (2008) 4716–4722
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according to the information about the GBS measurements published by Charlie. In detail, the state of the composite system composed of all the particles can be described as
|Φ is ≡ |φix1 ,x2 ,...,xm ⊗
mB Y
|ψ00 iAi Bi ⊗
mC Y
i=1
1
= d
y1 ,y2 ,...,ym =0
2
dm+mC
l0j =0
×e
mC Y
⊗
u,v,u0 ,v0 =0
j=1
m P l=0
m PC
yl ul +
l0 =0
d−1 X
|ψuj vj ixj Aj
!
mC Y k=1
(ym
0 0 +vm +l0 )ul0 B +l B
!
|li i|li i
l=0
i=1
Ai Bi
|l00k i|l00k i
=0 l00 k
AmB +j Cj m Y
⊗
mB d−1 Y X
k=1
d−1 X
− 2dπi
|ψ00 iCk0 B0k
x1 x2 ···xm
1
=
k=1
ay1 ,y2 ,...,ym |y1 , y2 , . . . , ym i
mC d−1 Y X 0 0 |lj i|lj i j=1
mC Y
d−1 X
m+mC
⊗
j=1
|ψ00 iAmB +j Cj ⊗
Ck0 B0k
!
|ψu0k v0k iCk Ck0 ⊗
d−1 X y1 ,y2 ,...,ym =0
ay1 ,y2 ,...,ym
! mB Y
!
!
mC Y
|yn + vn iBn
|ymB +n0 + vmB +n0 + vn0 iB0n . 0
(14)
n0 =1
n=1
That is to say, if the outcomes of GBS measurements on the pairs xj Aj (j = 1, 2, . . . , m) obtained by Alice are |ψuj vj ixj Aj and those obtained by Charlie on the pairs Ck Ck0 are |ψu0k v0k iCk Ck0 (k = 1, 2, . . . , mC ), the quantum system composed of the particles Bl and Bl0 (l = 1, 2, . . . , mB and l0 = 1, 2, . . . , mC ) collapses to the state d−1 X
|ζ iB1 B2 ···Bm B01 B02 ···B0mC = 0
y1 ,y2 ,...,ym =0 mB Y
×
ay1 ,y2 ,...,ym e
!
|yn + vn iBn
− 2dπi
mC Y
m P l=0
yl ul +
m PC
l0 =0
(ym
0 0 +vm +l0 )ul0 B
!
B +l
!
|ymB +n0 + vmB +n0 + v0n0 iB0n .
(15)
n0 =1
n=1
When Bob wants to recover the unknown state, he can perform the unitary operation Uul ,−vl (l = 1, 2, . . . , mB ) on his particle Bl and the unitary operation Uum +l0 +u00 ,−vm +l0 −v00 on the particle B0l0 (l0 = 1, 2, . . . , mC ) if he obtains the help of the controller l
B
B
l
Charlie, i.e., mB Y
|φiB1 B2 ···BmB B01 B02 ···B0mC =
! B Uull ,−vl
⊗
l0 =1
l=1
= F0
mC Y
d−1 X y1 ,y2 ,...,ym =0
!
B00
Uu
l
mB +l0 +ul0 ,−vmB +l0 −vl0
0
0
|ζ0 iB1 B2 ···Bm B01 B02 ···B0mC
ay1 ,y2 ,...,ym |y1 , y2 , . . . , ym iB1 B2 ···Bm
B0 B0 ···B0mC
B 1 2
,
(16)
where 2πi
0
F =e
d
m P l=0
ul vl +
m PC
(um
l0 =0
0 0 0 +ul0 )vl0
!
B +l
(17) 0
0
0
is a whole phase of the m-qudit quantum system B1 B2 · · · BmB B1 B2 · · · BmC . Similar to the case for sharing an arbitrary two-qudit state, Bob cannot obtain the unknown m-qudit state if he does not cooperate with the controller Charlie. Even though Bob knows the fact that Charlie has performed the GBS measurements on the pairs Cj Cj0 (j = 1, 2, . . . , mC ), he cannot reconstruct the unknown state. As there are d2mC outcomes obtained by Charlie with GBS measurements, Bob has only the probability d21mC to get the correct result. The larger the number of the particles controlled by Charlie mC is, the less the probability that Bob can reconstruct the unknown state without the help of Charlie is. This result is also kept for the case that the receiver is Charlie and the controller is Bob. For symmetry, the integer number mC 1 m+1 is optimal if it does not beyond the range [ m− , 2 ]. Certainly, Bob will get nothing if he does not know whether Charlie has 2 performed the GBS measurements on her particle pairs Cj Cj0 (j = 1, 2, . . . , mC ). In fact, Bob only gets a part of the unknown quantum system if Charlie does not perform the GBS measurements on her pairs and transfer the entanglement of the unknown quantum system to the particles controlled by Bob, and he cannot reconstruct the unknown m-qudit state in this time. In a word, Bob can reconstruct the unknown m-qudit state after the three parties have set up their secure quantum channel if and only if he cooperates with Charlie. That is, this QSTS scheme for an arbitrary unknown m-qudit state is secure if the quantum channels are secure. 4. Discussion and summary It is straightforward to generalize this QSTS scheme to the case with N + 1 agents, say the receiver Bob and the controllers Charliei (i = 1, 2, . . . , N). In this time, Alice shares a sequence of GBSs |ψ00 i with each agent, and Bob also shares a sequence
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Fig. 3. Circular topological structure for QSTS. Each agent shares a sequence of GBSs with his two neighboring agents, or one agent and the sender Alice.
of GBSs |ψ00 i with each of the controllers Charliei . Suppose that Alice splits mB particles to Bob and mCi particles to the controller Charliei . In this way, all the controllers Charliei act as the role of the controller Charlie in the case with two agents discussed in the Section 3. This is a dispersive topological structure for QSTS. Same as [41], another interesting topological structure of the QSTS is a circular one. That is, each agent shares privately a sequence of GBSs |ψ00 i with only his/her two neighboring agents (or one agent and the sender Alice), not each of the agents, shown in Fig. 3. Alice first splits the unknown state into two pieces and then transfers them into the particles controlled by the receiver Bob and one of the controllers Charlie1 . The other agents (Charliej (j = 2, 3, . . . , N)) only provide their quantum channels for Charlie1 to transfer her piece of unknown quantum information into the particles controlled by Bob with entanglement swapping. Surely, Bob can reconstruct the unknown state only when he gets all the outcomes of GBS measurements done by all the controllers if the controllers are honest. As the goal of QSTS is that Alice can prevent the dishonest man (no more than one) in the agents from eavesdropping the quantum secret, a QSTS scheme is secure if Alice can find out the eavesdropping done by the dishonest agent. In this QSTS scheme for sharing an arbitrary m-qudit state, Alice can in principle forbid any agents from eavesdropping her quantum information freely as she can share privately a sequence of GBSs with each of her agents. After setting up their quantum channels securely, none of the agents has the chance to eavesdrop the quantum secret freely. Obviously, in the dispersive topological structure, Bob can reconstruct the quantum secret only when he cooperates with all the other agents if mB 6= 0 and mCi 6= 0 (for i = 1, 2, . . . , N). In the circular topological structure, the N controllers do not play an equivalent role in QSTS. In detail, Charlie1 has the superpower for controlling the reconstruction of the quantum secret as she holds a piece of the original quantum secret split by Alice, different from the other controllers Charliej (j = 2, 3, . . . , N). That is to say, Bob can recover the originally unknown state if he obtains the help of Charlie1 in this time, not resorting to the help of the other controllers. Although this QSTS scheme with a circular topological structure is secure if there is no more than one dishonest agent, it cannot prevent two potentially dishonest agents from eavesdropping the quantum secret freely, especially the two agents who obtain the pieces of the originally unknown m-qudit split by the sender Alice. In the case with a dispersive topological structure, Alice can split symmetrically (or nearly symmetrically) her quantum secret and then distribute it to her agents, which will require the receiver Bob to cooperate with all the controllers (N + 1 ≤ m) or m agents (N + 1 > m) for recovering the quantum secret. No matter what the topological structure of this QSTS scheme is, its intrinsic efficiency for qudits ηq ≡ qqut approaches 100% in principle as almost all the qudits are useful. Here qu is the number of the useful qudits and qt is the number of the qudits transmitted. In summary, we have presented a QSTS scheme for sharing an arbitrary m-qudit state based on two-qudit entanglements and generalized Bell-state measurements, following some ideas in Ref. [41]. We give a general form for N + 1 agents to share the unknown m-qudit state securely with entanglement swapping. In this scheme, the sender Alice performs m generalized Bell-state measurements and the controllers also take some generalized Bell-state measurements on their particles. The receiver can recover the unknown state if he cooperates with the controllers. Also, we discuss the difference between a dispersive topological structure and a circular one. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant Nos. 10604008 and 10435020, A Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant No. 200723, and Beijing Education Committee under Grant No. XK100270454. References [1] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2000. [2] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Rev. Modern Phys. 74 (2002) 145. [3] C.H. Bennett, G. Brassard, Proceeding of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India, IEEE, New York, 1984, p. 175. [4] A.K. Ekert, Phys. Rev. Lett. 67 (1991) 661. [5] C.H. Bennett, G. Brassard, N.D. Mermin, Phys. Rev. Lett. 68 (1992) 557.
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