Generalized tripartite scheme for sharing arbitrary 2n-qudit state

Optics Communications 283 (2010) 4108–4112

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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m

Generalized tripartite scheme for sharing arbitrary 2n-qudit state Xue-qin Zuo a, Yi-min Liu b, Chun-jie Xu a, Zi-yun Zhang a,⁎, Zhan-jun Zhang a,⁎ a b

School of Physics & Material Science, Anhui University, Hefei 230039, China Department of Physics, Shaoguan University, Shaoguan 512005, China

a r t i c l e

i n f o

Article history: Received 17 March 2010 Received in revised form 29 May 2010 Accepted 29 May 2010 Keywords: Generalized quantum state sharing Non-maximally entangled state Generalized Bell-state measurement

a b s t r a c t Utilizing three non-maximally entangled qutrit pairs as quantum channels, we first propose a generalized tripartite scheme for sharing an arbitrary two-qutrit state with generalized Bell-state measurements. In the scheme if and only if the two recipients collaborate together, they can recover the split qutrit state with the probability determined uniquely by the smallest coefficients of the non-maximally entangled pairs. Afterwards, we further extend the scheme for sharing an arbitrary 2n-qudit state by taking 3n nonmaximally entangled qudit pairs as quantum channels. Moreover, the scheme success probability relative to the inherent entanglement in quantum channels and its structure is simply discussed. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Secret sharing was first proposed by Shamir [1] in 1979. The basic idea of it can be regarded as that a secret (a classical message) is divided into two pieces and each piece is distributed to a receiver by the sender. The secret can be reconstructed if and only if both receivers cooperate together and neither of them can get anything about the original message solely. Twenty years later, this concept was generalized to quantum scenario by Hillery, Bŭzek and Berthiaume (HBB) [2]. They endowed it a novel concept of quantum secret sharing (QSS). Different from classical secret sharing, QSS is used to deal with not only classical messages (i.e., bits) but also quantum information (i.e., a quantum state). The latter is conventionally termed as quantum state sharing (QSTS) or quantum information splitting. So far, QSTS has already attracted much attention in both theoretical and experimental aspects [2–41]. It is worth pointing out that, almost all of the above protocols [2–33] utilized maximally entangled states as quantum channels (QCs). However, in a realistic situation, it is quite possible that the QC states are non-maximally entangled due to the decoherence effects from environments. In this case, one feasible solution is to use the so-called quantum purification technique [42], which allows to extract a maximally entangled one from a large ensemble of non-maximally entangled states. Nevertheless, in majority conditions the shared

⁎ Corresponding authors. E-mail addresses: [email protected] (Z.Y Zhang), [email protected] (Z.J. Zhang). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.05.079

resource is limited and insufficient, hence a maximally entangled state cannot be deterministically obtained. In this case, one may prefer to utilize the non-maximally entangled states to achieve the teleportation probabilistically instead of doing nothing, especially in some urgent requirements. Therefore, it is necessary and significant to investigate QSTS with non-maximally entangled state as QCs directly. As a matter of fact, people have already paid more attention on non-maximally entangled states recently [34–41,43,44]. For instances, in 2006, inspired by the probabilistic quantum teleportation protocol of Agrawal and Pati [45], Gordon and Rigolin [36] first presented two QSTS protocols by using non-maximally entangled GHZ states and non-maximally entangled Bell states as QCs respectively. Soon after, Wang et al. [37] proposed a scheme for probabilistically implementing QSTS of an arbitrary two-qubit state, where two non-maximally entangled threequbit states are employed as QCs. In 2008 Liu et al. [38] and Yuan et al. [39] further considered the non-maximally entangled qutrit states and accordingly presented tripartite schemes for splitting arbitrary unknown single-qutrit and two-qutrit states, respectively. At the same year, Tao et al. [40] utilized a non-maximally entangled GHZ state as QC to put forward a protocol for splitting an arbitrary unknown single-qudit state. Besides, we made use of three two-qubit partially entangled states as QCs to present a QSTS scheme of an arbitrary unknown two-qubit state [41]. Note that all these QSTS protocols except for that in Ref. [40] treat only qubit or qutrit states. In this work we further extend the previous works. Specifically, we will present a generalized scheme for sharing an arbitrary 2n-qudit state with 3n two-qudit non-maximally entangled pairs as QCs. The rest of this paper is organized as follows. In Section 2, a generalized tripartite QSTS scheme is presented for sharing an arbitrary two-qutrit state with three non-maximally two-qutrit

X. Zuo et al. / Optics Communications 283 (2010) 4108–4112

entangled pairs. In Section 3, the tripartite scheme is extended to 2nqudit state. In Section 4, the relation between the entanglement structure of QC and the success probability is briefly discussed. In addition, the security check as the precondition of splitting is also concisely depicted. Finally, a brief summary is given in Section 5.

4109

To achieve her goal, Alice first measures her two qutrit pairs ðx1 ; a1 Þ and ðx2 ; a2 Þ with the generalized three-dimensional Bell-state (TGBS) bases. The TGBS bases are defined as 2πi lm 3 jl〉x ⊗jl

1 2 jψmn 〉xa = pffiffiffi ∑ e 3 l=0

+ n〉a ;

ð4Þ

2. Generalized tripartite QSTS of an arbitrary two-qutrit state Let us first introduce our generalized tripartite scheme for sharing an arbitrary two-qutrit state in this section. Suppose there are three legitimate parties, say, Alice, Bob and Charlie. Alice is the initial owner of the secret quantum information (QI). The QI is in Alice's two qutrits x1 and x2, i.e., jχ〉x1 x2 =

where m; n∈f0; 1; 2g and l + n≡ðl + nÞ mod 3. They form a complete orthogonal basis set of a two-qutrit Hilbert space. Naturally, Alice's measurements collapse the joint state of the two qutrit pairs ðb1 ; c1 Þ   and b′1 ; c′1 to

jτ〉b1 c1 b′ c′ = 1 1

2

∑

y1 ;y2 = 0

ξy1 y2 jy1 y2 〉x1 x2 ;

ð1Þ



2 where ξ's are complex and satisfy the relation ∑2y1 ;y2 = 0
ξy1 y2
= 1. Bob and Charlie are Alice's two agents. Alice wants to let them share the QI and she can assign either of them to finally reconstruct it. The precondition of such reconstruction is that the two agents should cooperate to conclusively get it provided that the QCs are secure. Otherwise, neither can get it. Additionally, Alice shares two nonmaximally two-qutrit entangled pairs with Bob and Charlie, and Bob also shares a non-maximally two-qutrit entangled pair with Charlie. The schematic demonstration is illustrated in Fig. 1. The QCs consist of three non-maximally entangled pairs

jϕ〉a1 a2 b1 c1 b′ c′ = jϕ1 〉a1 b1 ⊗ jϕ2 〉a2 c1 ⊗jϕ3 〉b′ c′ ; 1 1

ð2Þ

1 1

where jϕi 〉 = ∑2j = 0 βij jjj〉ði = 1; 2; 3Þ, β's satisfy the usual normalization condition, and |βi0| b |βi1| b |βi2| is assumed without loss of     generality. The three qutrit pairs ða1 ; a2 Þ, b1 ; b′1 and c1 ; c′1 belong to Alice, Bob and Charlie, respectively. The joint state of the 8-qutrit system is

jω〉x1 x2 ;a1 a2 b1 c1 b′ c′ = jχ〉x1 x2 ⊗jϕ〉a1 a2 b1 c1 b′ c′ 1 1

ð3Þ

1 1

2πiðy1 m1 + y2 m2 Þ 3

1 2 − ∑ e 3 y1 ;y2 = 0 ξy1 y2 β1y1

+ n1 β 2y2 + n2

jy1 + n1 ; y2 + n2 〉b1 c1 ⊗jϕ3 〉b′ c′ ; 1 1

ð5Þ where m1 ; n1 ; m2 ; n2 ∈f0; 1; 2g. From Eq. (5), one can see that Alice may get any one of 34 possible results. For instance, suppose Alice   obtains |ψ00〉x1a1|ψ00〉x2a2, then the state of the subsystem b1 ; c1 ; b′1 ; c′1 becomes jτ〉b1 c1 b′ c′ = 1 1

1 2 ∑ ξ β β jy ; y 〉 ⊗ jϕ3 〉b′ c′ : 1 1 3 y1 ;y2 = 0 y1 y2 1y1 2y2 1 2 b1 c1

ð6Þ

After Alice's measurements the original QI has already been transferred to the qutrits b1 and c1. That is to say, the QI has already been split into the subsystem shared by Bob and Charlie. In essence, this splitting process swaps quantum entanglements in a nondeterministic manner [46]. Consequently, Bob and Charlie can subsequently retrieve Alice's QI in a probabilistic way via their mutual assistance. As mentioned before, either Bob or Charlie can reconstruct the original QI with the other's help. Without loss of generality, suppose Charlie is the assignee. In this case, Bob performs a TGBS measure  ment on his qutrit pair b1 ; b′1 . See Fig. 1 for illustration. His   measurement will collapse the state of the qutrit pair c1 ; c′1 to 2 1 ′ jτ 〉c1 c′ = pffiffiffi3 ∑ f ðm1 ; n1 ; m2 ; n2 Þjy2 + n2 ; y1 + n1 + n3 〉c1 c′ ; 1 1 ;y y 1 2 =0 3

ð7Þ

= jχ〉x1 x2 ⊗jϕ1 〉a1 b1 ⊗ jϕ2 〉a2 c1 ⊗jϕ3 〉b′ c′ : 1 1

2πiðy1 m1 + y2 m2 + y1 + n1 m3 Þ

−

3 where the function f ðm1 ; n1 ; m2 ; n2 Þ≡e β1y1 + n1 β2y2 + n2 β3y1 + n1 + n3 ξy1 y2 and m3 ; n3 ∈f0; 1; 2g. It is easily

found that the coefficients ξs initially in the QI |χ〉 now appear in the joint state |τ ′ 〉c1c1′. This indicates that the joint state |τ ′ 〉c1c1′ has already contained the information about the original QI. According to their prior agreement, Alice announces her measurement results |ψm1n1〉x1a1 and |ψm2n2〉x2a2 by using a 4-trit message “m1n1m2n2” via a public channel. If Bob collaborates with Charlie, then he tells Charlie his measurement result |ψm3n3〉b1b1′ via a 2-trit message “m3n3”. Once receiving Alice's and Bob's messages, Charlie knows exactly which   state her qutrit pair c1 ; c′1 has already collapsed to. To recover the QI in his site, Charlie does as follows. Firstly, he performs the unitary operation 2

Vc1 c′ = 1

Fig. 1. Schematic demonstration of sharing an arbitrary two-qutrit state with three twoqutrit non-maximally entangled pairs. The solid rectangles denote Alice's and Bob's TGBS measurements on their qutrit pairs, respectively. The dotted ellipse represents Charlie's collective unitary operations Ω on the two qutrits c1, c 1′ and the auxiliary qubit d. The dashed square characterizes Charlie's single-qubit operation on the qubit d.

∑

y1 ;y2 = 0

2πiðy1 m1 + y2 m2 + y1 + n1 m3 Þ 3 jy1 〉c

e

 1

y2 + n2 j y2

c′1

〈y1 + n1 + n3 j

ð8Þ   on his qutrit pair c1 ; c′1 . Note that corresponding to different measurement results Vc1c1′ is different. For example, if Alice and Bob

4110

X. Zuo et al. / Optics Communications 283 (2010) 4108–4112

measure |ψ00〉x1a1 |ψ11〉x2a2 and |ψ22〉b1b1′, the unitary operation Vc1c1′ which Charlie should perform takes the form as 2πi = 3

Vc1 c ′ = j00〉c1 c1′ 〈12 j + e 1

4πi = 3

+e

2πi = 3

+e

4πi = 3

j01〉c1 c1′ 〈22j + e

2πi = 3

4πi = 3

j20〉c1 c1′ 〈11j + e

1 Ωc1 c′ d Vc1 c′ jτ′ 〉c1 c′ j0〉d = pffiffiffi 3 ½β10 β20 β30 jχ〉c1 c′ j0〉d 1 1 1 1 ð 3Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + ðξ01 β221 −β220 β10 β30 j01〉

j02〉c1 c1′ 〈02 j

j10〉c1 c1′ 〈10j + j11〉c1 c1′ 〈20 j + e

j12〉c1 c1′ 〈00j

j21〉c1 c1′ 〈21 j + j22〉c1 c1′ 〈01 j:

The operations Vc1c1′ and Ωc1c1′d transform the joint state |τ ′ 〉c1c1′|0〉d into

+ ξ02

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β222 −β220 β10 β30 j02〉

+ ξ10

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β211 β231 −β210 β230 β20 j10〉

+ ξ11

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β211 β221 β231 −β210 β220 β230 j11〉

+ ξ12

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β211 β212 β231 −β210 β220 β230 j12〉

+ ξ20

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β212 β232 −β210 β230 β20 j20〉

+ ξ21

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β212 β221 β232 −β210 β220 β230 j21〉

+ ξ22

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β212 β222 β232 −β210 β220 β230 j22〉Þc1 c′ j1〉d :

ð9Þ

The operation Vc1c1′ converts the collapsed state |τ ′ 〉c1c1′ into 1 ″ ′ jτ 〉c1 c′ = Vc1 c′ ⊗ jτ 〉c1 c′ = pffiffiffi3 1 1 1 3 2

× ∑

y1 ;y2 = 0

β1y1

+ n1 β 2y2 + n2 β 3y1 + n1 + n3 ξy1 y2

ð10Þ

jy1 y2 〉c1 c′ : 1

Secondly, he introduces an auxiliary qubit d initially in the state |0〉 and carries out a collective unitary transformation Ωc1c1′d on his three particles c1, c 1′ and d. Under the ordering eighteen basis vectors {|000〉, |010〉, |020〉, |100〉, |110〉, |120〉, |200〉, |210〉, |220〉, |001〉, |011〉, |021〉, |101〉, |111〉, |121〉, |201〉, |211〉, |221〉}, the collective unitary operation Ωc1c1′d takes the matrix form of

Ωc1 c′ d = 1

 ρ2 ; −ρ1

ρ1 ρ2

where ρ2 =

λ21, λ22) is a diagonal matrix. The diagonal elements λ′s are uniformly   expressed as λy1 y2 = β10 β20 β30 = β1y1 + n1 β2y2 + n2 β3y1 + n1 + n3 . Apparently, they are functions of n1, n2 and n3. As indicates that they are dependent of Alice's and Bob's TGBS measurement outcomes. As an example, if Alice's and Bob's measurement results are |ψ  00〉x1a1|ψ00〉x2a2 and |ψ00〉b1b′1 respectively, then Charlie's qutrit pair c1 ; c′1 is in the state 2 1 ′ j τ 〉c1 c1′ = pffiffiffi3 ∑ ξy1 y2 β1y1 β2y2 β3y1 jy2 ; y1 〉c1 c′ 1 3 y1 ;y2 = 0

1 = pffiffiffi3 ðξ00 β10 β20 β30 j00〉c1 c′ + ξ01 β10 β21 β30 j01〉c1 c′ 1 1 3 + ξ02 β10 β22 β30 j02〉c1 c′ + ξ10 β11 β20 β31 j10〉c1 c′ 1

1

ð12Þ

+ ξ11 β11 β21 β31 j11〉c1 c′ + ξ12 β11 β22 β31 j12〉c1 c′ 1

1

+ ξ20 β12 β20 β32 j20〉c1 c′ + ξ21 β12 β21 β32 j21〉c1 c′ 1

1

+ ξ22 β12 β22 β32 j22〉c1 c′ Þ: 1

Charlie performs the unitary operation 2

V c1 c′ = 1

∑

y1 ;y2 = 0

ð13Þ

ð11Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffi 1−ρ21 and ρ 1 = diag(λ 00 , λ 01 , λ 02 , λ 10 , λ 11 , λ 12 , λ 20 ,

jy1 y2 〉c1 c′ 〈y2 ; y1 j

1

Apparently, the operation Ωc1c1′d has already entangled the qutrit   pair c1 ; c′1 with the ancilla d. In particular, one can see that, the entangled three-particle state can be decomposed into the superposition of the two-qutrit state and the auxiliary qubit, and the first-term component comprises the conceivable state |χ〉c1c1′. Consequently, in the last step Charlie measures the ancilla d in the bases f j0〉; j1〉g. If he measures |0〉, then Charlie knows the original QI  |χ〉 has already been successfully reconstructed on his qutrit pair c1 ; c′1 . Otherwise, the reconstruction fails. Hence the entangling and measuring operations play a crucial role in the whole reconstruction process. Their essence is to project the collapsed state onto the conceivable state by measuring the ancilla. Therefore it is crucial to finally determine an appropriate entangling operation, which completely decides the success probability of the scheme with a given QC. Incidentally, if it is optimal, then the success probability will be uniquely decided by the entanglement structure of the QC. We will briefly analyze this problem in the last section. Through comparatively complicated calculations one can further work out the success probability, it is 2 1 27 ðβ 10 β 20 β 30 Þ in this situation. Obviously, the success probability is determined by the three smallest coefficients β10, β20 and β30. Provided that other measurement cases are taken into account also, 1 the total success probability is 27 ðβ10 β20 β30 Þ2 × 36 = 27ðβ10 β20 β30 Þ2 . pffiffiffi Incidentally, if the coefficients satisfy jβ10 j = jβ 20 j = j β30 j = 1 = 3,

then the total successful probability is 1. In this case, the present generalized tripartite QSTS scheme is reduced to the usual standard QSTS scheme. 3. Extension for sharing an arbitrary 2n-qudit state

1

  on his qutrit pair c1 ; c′1 . Afterwards, Charlie incorporates an ancilla d and executes a collective unitary operation Ωc1c1′d on his three particles c1, c′1 and d. The 9 × 9 matrix ρ1 of Ωc1c1′d can be written as

 β β β β β β ρ1 = diag 1; 10 20 30 ; ⋯; 10 20 30 : β10 β21 β30 β12 β22 β32

In this section, we will simultaneously generalize the above tripartite 2-qutrit sharing scheme from both aspects, i.e., the particle degree and the particle number. The schematic demonstration is shown in Fig. 2. Suppose that the 2n-qudit state Alice wants to send is

′

jχ 〉x1 x2 ⋯x2n =

d−1

∑

y1 ;y2 ;⋯;y2n = 0

Cy1 y2 ⋯y2n jy1 y2 ⋯y2n 〉x1 x2 ⋯x2n ;

ð14Þ

X. Zuo et al. / Optics Communications 283 (2010) 4108–4112



2

= 1. The where the coefficients c′s satisfy ∑d−1 y1 ;y2 ;⋯;y2n = 0 Cy1 y2 ⋯y2n QCs contain 3n two-qudit non-maximally entangled pairs, i.e., n

j Φ〉a1 a2 ⋯a2n b1 b′ c1 c′ b2 b′ c2 c′ ⋯bn b′n cn c′n = ∏ j Φr 〉ar br ⊗ j Φn 1

1

2

2

r=1

d−1

n

= ∏

∑

r = 1 j1 ;j2 ;j3 = 0

× an

+ r 〉an

+ r cr

⊗ j Φ2n + r 〉b′r c′r

Prj1 jj1 j1 〉ar br ⊗Pðn +

+ r cr ⊗Pð2n + rÞj3

rÞj2

j j2 j2 〉

j j3 j3 〉b′ c′ ;

d−1 1 jΓ′ 〉c1 c′ c2 c′ ⋯cn c′n = pffiffiffi3n ∑ E1 E2 Cy1 y2 ⋯y2n 1 2 ;y y 1 2 ;⋯;y2n = 0 d

r r

n

ð15Þ

∏ PrΛðrÞ Pðn

r=1

where P′s satisfy the normalization condition and jPi′ 0 jbjPi′ 1 j  b⋯bjPi′ d−1 j i′ = r; n + r; 2n + r is assumed without loss of generality. The joint state of the 8n-qudit system is jW〉x1 a1 x2 a2 ⋯x2n a2n b1 b′ c1 c′ b2 b′ c2 c′ ⋯bn b′n cn c′n 1

1

2

2

′

= j χ 〉x1 x2 ⋯x2n ⊗ jΦ〉a1 a2 ⋯a2n b1 b′ c1 c′ b2 b′ c2 c′ ⋯bn b′n cn c′n 1

d−1

n

′

= j χ 〉x1 x2 ⋯x2n ⊗ ∏

∑

r = 1 j1 ;j2 ;j3 = 0

1

2

+ rÞj2

jj2 j2 〉an + r cr ⊗Pð2n

+ rÞj3

jj3 j3 〉b′r c′r :

ð16Þ Similar to the case of sharing an arbitrary two-qutrit state, Alice first measures her 2n qudit pairs (x1, a1), (x2, a2), ⋯, (x2n, a2n) with the d-dimensional GBS (DGBS) bases. The DGBS bases are defined as 1 d−1 2πiLM jψMN 〉xa = pffiffiffi ∑ e d jL〉x ⊗jL + N〉a ; d L=0

2πi∑nr= 1 ðΛðrÞ + M2n

−

2

2

n

∏ PrΛðrÞ jΛðr Þ〉br Pðn +

r=1

n

+ rÞ

n

1 n ∏ jyr yn + r 〉cr c′r 〈Λðn + r ÞΘðr Þj ð20Þ E1 E2 r = 1 r=1       ′ ′ ′ on his qudit pairs c1 ; c1 , c2 ; c2 ,⋯, and cn ; cn . The operations ∏ nr = 1Ucrcr′ convert the collapsed state |Γ ′ 〉c1c1′c2c 2′ ⋯ cnc n′ into n

″

n

ð18Þ

rÞΛðn + rÞ jΛðn + r Þ〉cr ∏ jΦ2n + r 〉b′r c′r ; r=1

+ r Mn + r Þ

, the function ΛðrÞ =

′

jΓ 〉c1 c′ c2 c′ ⋯cn c′n = ∏ Ucr c′r jΓ 〉c1 c′ c2 c′ ⋯cn c′n 2

r=1

1

2

d−1 1 = pffiffiffi3n ∑ Cy1 y2 ⋯y2n y ;y 1 2 ;⋯;y2n = 0 d

n × ∏ PrΛðrÞ Pðn + rÞΛðn + rÞ Pð2n + rÞΘðrÞ jyr yn

ð17Þ

d−1 1 ∑ E C dn y1 ;y2 ;⋯;y2n = 0 1 y1 y2 ⋯y2n

2πi∑r = 1 ðyr Mr + yn d

−

where E1 = e

+ rÞΘðrÞ jΘðrÞ〉c′r ;

∏ Ucr c′r =

where L + N≡ðL + NÞ mod d. After Alice's measurements, the joint state of those qudits in Bob's and Charlie's sites collapses to

1

+ r Þ〉cr ⊗Pð2n

d and ΘðrÞ = ΛðrÞ + N2n + r . By where E2 = e the way, Eq. (19) is reduced to Eq. (7) in the case of n = 1. Charlie can reconstruct the original QI by the following method similar to that described in Section 2. First, he implements n two-qudit unitary operations

1

1

+ rÞΛðn + rÞ jΛðn

ð19Þ

2

Prj1 jj1 j1 〉ar br ⊗Pðn

j Γ〉b1 b′ c1 c′ b2 b′ c2 c′ ⋯bn b′n cn c′n =

4111

    DGBS measurements on his qudit pairs b1 ; b′1 , b2 ; b′2 ; ⋯, and   bn ; b′n . His measurements collapse the joint state of the n qudit pairs       ′ ′ ′ c1 ; c1 , c2 ; c2 ,⋯, and cn ; cn to

r=1

+ r 〉cr c′r

 :

ð21Þ Second, he incorporates n auxiliary qubits d1, d2, ⋯, and dn each initially in the state |0〉, then carries out n collective unitary operations ∏ nr = 1Ξcrcr′dr on the 3n particles c1, c′1, d1, c2, c′2, d2, ⋯, cn, c′n and dn. See Fig. 2 for illustration. Under the ordering 2d2 basis vectors {|000〉, |010〉, ⋯, j0ðd−1Þ0〉, |100〉, |110〉,⋯, j1ðd−1Þ0〉,⋯, j ðd−1Þ00〉, j ðd−1Þ10〉,⋯, j ðd−1Þðd−1Þ0〉, |001〉, |011〉,⋯, j0ðd−1Þ1〉, |101〉, |111〉,⋯, j1ðd−1Þ1〉,⋯, j ðd−1Þ01〉, j ðd−1Þ11〉,⋯, j ðd−1Þðd−1Þ1〉g, the collective unitary operation Ξcrc r′dr takes the following matrix form

yr + Nr , and Mr ; Nr ∈f0; 1; ⋯d−1g. Subsequently, Bob performs n

Ξcr cr dr =

ϱ1 ϱ2

 ϱ2 ; −ϱ1

where ϱ1 = ðζ1 ; ζ2 ; ⋯; ζd2 Þ is a d2 ×d2 diagonal matrix and

ð22Þ ϱ2 =

qffiffiffiffiffiffiffiffiffiffiffiffiffi 1−ϱ21 .

The diagonal elements ζ's can be uniformly expressed as Pr0 Pðn + rÞ0   Pð2n + rÞ0 = PrΛðrÞ Pðn + rÞΛðn + rÞ Pð2n + rÞΘðrÞ . Apparently, the elements depend on the joint state of the three particles cr, c′r and dr. At last, he measures the qubits d1, d2, …, dn in the bases f j00⋯0〉; j11⋯1〉g. If the measurement result is |00 ⋯ 0〉, he knows he has already successfully reconstructed the original state |χ ′ 〉 on his qudits c1, c 1′, c2, c 2′, ⋯, cn,  2 and c′n with the probability of d13n ∏3n s = 1 Ps0 . Otherwise, the outcome is |11 ⋯ 1〉 and in this case Charlie cannot reconstruct the original state. 4. Discussions Fig. 2. Schematic demonstration of sharing an arbitrary 2n-qudit state with 3n twoqudit non-maximally entangled pairs as QCs. The solid rectangles denote Alice's and Bob's DGBS measurements on their qudit pairs. The dotted ellipses represent Charlie's collective unitary operations on the qudit pairs (c1, c 1′), (c2, c 2′),⋯, (cn, c n′ ) and auxiliary qubits d1, d2, ⋯,dn. The dashed squares characterize Charlie's single-qubit operations on the qubits d1, d2, ⋯,dn.

In the last two sections the conclusion that the success probability of our schemes is completely determined by the smallest coefficients of the QC wave-functions, has been drawn. In fact, this conclusion is under the given precondition that the entangling operation should take the form we choose and expressed as Eqs. (11) or (22). As

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mentioned before, if the choice of the entangling operation is optimal, then the success probability can be maximal. In this sense, it is intriguing to ask whether the present choice is optimal? As a matter of fact, in QSTS the capability of a QC is its inherent property. In Ref. [47], the authors have already revealed that the success probability of a non-maximally entangled state can be converted into a maximally entangled one is fully decided by the smallest coefficient of the nonmaximally entangled state. This means the maximal capability is only the function of the smallest coefficient. Consequently, the success probabilities in our present schemes are indeed maximal. However, it is worthy pointing out that the entanglement in the QC is not determined only by the smallest coefficient for any qudit entangled state. Hence, for a QC with a given entanglement, one can not directly infer its capability in QSTS. This indicates that the inherent entanglement of a qudit QC is insufficient to characterize its capability. To characterize it, one needs further to consider the specific coefficients of the QC state corresponding to a given entanglement. Now let us briefly discuss the issue of entanglement structure. In fact, there is another different form of QSTS schemes named generalized asymmetric QSTS schemes usually. In a generalized asymmetric QSTS scheme, the QC particle number in each participant's site may be different. Moreover, maybe only a part of agents can be the potential assignee while others should act as the assistants. The asymmetry is essentially determined by the entanglement structure of the QC employed [33]. In Ref. [33], the authors have extensively studied such issue and revealed the relation between the shared QI and the QC entanglement structure. However, it is worth to emphasize that our present schemes are conventionally the symmetric ones. Alternatively, the QC particle number is same for any participant, and any agent can be assigned to be the final assignee who can conclusively reconstruct the original secret QI with other agents' assistances. This fact has been stressed before. The symmetric property of our schemes is essentially determined by the QC qutrit (qidit) distribution among all participants (i.e., the entanglement structure). To assure such symmetry, in our schemes more particles should be involved in when comparing to the so-called asymmetric QSTS. In other words, more quantum resources should be consumed to assure the symmetry. As mentioned before, the precondition of our QSTS schemes is the QCs are assumed secure. Hence, it is necessary to consider the issue of the QC security before the sender's splitting. In our schemes there are three independent and parallel quantum channels linking Alice and Bob, Alice and Charlie, and Bob and Charlie as well. The present QCs are very similar to those in Refs. [15,16,23,27,48,49] to some extent. Consequently, the mature sampling method [15,16,23,27,48,49] combining with the block transmission technique [50] can be taken for the QC security check. Here we do not repeat it anymore. 5. Summary To summarize, in this paper we have proposed a tripartite symmetric QSTS scheme for probabilistically splitting an unknown two-qutrit state with three two-qutrit non-maximally entangled pairs as QCs. In order to share the qutrit state, Alice first performs two TGBS measurements on her particle pairs and assigns either agent to reconstruct the unknown state. Before the assignee resumes the original state, the other agent should also perform TGBS measurement on his particle pair. Finally, the quantum state can be resumed by a collective unitary operation and a single-qutrit measurement.

Moreover, the generalization of the tripartite scheme to the case of sharing 2n-qudit state is concisely introduced. In addition, the success probability of our present scheme relative to the QC inherent entanglement and its structure is also sketched. Acknowledgements Authors are grateful to the anonymous referees for their constructive suggestions and recommendation. This work is partly supported by the program for New Century Excellent Talents at the University of China under Grant No. NCET-06-0554, the National Natural Science Foundation of China under Grant Nos. 10975001, 60677001, 10747146 and 10874122, the Science-Technology Fund of Anhui province for Outstanding Youth under Grant No. 06042087, the key fund of the Ministry of Education of China under Grant No. 206063, the Talent Foundation of High Education of Anhui Province for Outstanding Youth under Grant No. 2009SQRZ018, and the Natural Science Foundation of Guangdong Province under Grant Nos. 06300345 and 7007806. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]

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