Physica A 390 (2011) 760–768

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Physica A

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Faithful teleportation of multi-particle states involving multi spatiallyremote agents via probabilistic channelsJiang Min a,∗, Li Hui b, Zhang Zeng-ke b, Zeng Jia c

a School of Electronics & Information Engineering, Soochow University, Suzhou 215006, Chinab Department of Automation, Tsinghua University, Beijing 100084, Chinac The 208th Institute of the Second Research Academy, CASIC, Beijing 100854, China

a r t i c l e i n f o

Article history:Received 9 April 2009Received in revised form 26 August 2010Available online 20 October 2010

Keywords:Probabilistic teleportationMulti-particle stateQuantum state transferMulti agentsBell-state measurement

a b s t r a c t

We present an approach to faithfully teleport an unknown quantum state of entangledparticles in amulti-particle system involvingmulti spatially remote agents via probabilisticchannels. In our scheme, the integrity of an entangled multi-particle state can bemaintained even when the construction of a faithful channel fails. Furthermore, in aquantum teleportation network, there are generally multi spatially remote agents whichplay the role of relay nodes between a sender and a distant receiver. Hence, we propose twoschemes for directly and indirectly constructing a faithful channel between the sender andthe distant receiver with the assistance of relay agents, respectively. Our results show thatthe required auxiliary particle resources, local operations and classical communications areconsiderably reduced for the present purpose.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Quantum information is a rapidly evolving research field [1–4]. Since Bennett et al. [5] proposed the quantumteleportation of an unknown one-particle state via a two-particle maximally entangled state, quantum teleportationhas played an important role in the rapidly evolving quantum information and can be used in the context of quantumcommunication [6,7]. Due to thepotential applications to the quantumkeydistribution [1] or future quantumnetworks [8,9],quantum teleportation has acquired lots of interest both theoretically and experimentally.

There are some papers related to the probabilistic teleportation of two-particle and multi-particle states [10–16]with a non-maximally entangled state as the teleportation channel. In practical applications, due to the inevitableenvironmental effects, the initially maximally entangled state may evolve into a non-maximally entangled state or mixedstate. Entanglement distillation [17–19] can be used to prepare a non-maximally entangled state or maximally entangledstate as a quantum channel via the introduction of auxiliary particles.

Differently to the standard deterministic teleportation [5], in the usual probabilistic teleportation protocol [10,11,20–24],the teleportation of unknown states can be successfully realized with a certain probability if the receiver adopts anappropriate unitary-reduction strategy [10,11,20–24]. The receiver will need an auxiliary qubit resource and will performhigh dimensional unitary gates onto particles while high dimensional unitary gates are usually difficult to implementby experiment. When the measurement outcome of the auxiliary particle obtained by the receiver does not meet therequirements, the unknown state of one or two particles will be ruined thoroughly. Furthermore, there is no uniformapproach to achieve the teleportation or probabilistic teleportation of two-particle andmulti-particle states. These potential

∗ Corresponding author.E-mail address: jiangmin03@gmail.com (M. Jiang).

0378-4371/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2010.10.020

M. Jiang et al. / Physica A 390 (2011) 760–768 761

barriers bring inconvenience into the process of the construction of a quantum teleportation network. In the present paper,we will give a generalized scheme to realize the teleportation of any multi-particle state where only two-qubit gates andone-qubit gates are needed to perform by the receiver or the sender.

So far, the current schemes for the probabilistic teleportation via a non-maximally entangled state have been presentedwith only two or three parties involved (when the controllerwas considered) [10,11,20–24]. However, in a realistic situation,the long distance communication network consists ofmany spatially distributed nodes. In a quantum teleportation network,multi spatially remote agents playing the role of relay nodes are necessary between a sender and a distant receiver [25,26].So far, there are no results that use probabilistic channels to realize the deterministic teleportation of an entangled multi-particle system involving multi relay agents. Hence we propose a faithful teleportation method to successfully resolve theabove problems.

This paper is organized as follows. In Section 2, we introduce the teleportation of an unknown state carried by oneentangled particle in a multi-particle system via an EPR channel and demonstrate that an EPR channel can be used tofaithfully teleport unknown states of a multi-particle system. Section 3 focuses on the construction of an EPR channel vianon-maximally two-particle entangled states and Section 4 considers the construction of the EPR channel involving threeagents. In Sections 5 and 6, the indirect construction and direct construction of an EPR channel involving multi agents arepresented, respectively. The conclusion is given in Section 7.

2. Teleportation of an unknown quantum state of an entangled multi-particle system via an EPR channel

Firstly, we assume that the unknown quantum state of a multi-particle which will be teleported in Alice’s place can beexpressed as the following form:

|ϕ⟩1,2,...,n =

−a{i}|{i}⟩1,2,...,n−1

|0⟩n +

−b{i}|{i}⟩1,2,...,n−1

|1⟩n (1)

with∑

|a{i}|2+∑

|b{i}|2

= 1, where |{i}⟩1,2,...,n−1(i ∈ {0, 1}) is the computational basis of particles 1, 2, . . . , n − 1.We further assume that Alice and Bob share one EPR pair as the quantum channel

|φ⟩AnBn =1

√2(|00⟩AnBn + |11⟩AnBn) (2)

where particles An and Bn belong to Alice and Bob, respectively. The initial quantum state of the joint system reads

|ϕ⟩1,2,...,nAnBn =1

√2

−ai|xi⟩1,2,...,n−1

|0⟩n +

−bi|xi⟩1,2,...,n−1

|1⟩n

(|00⟩AnBn + |11⟩AnBn). (3)

Now let Alice make a Bell state measurement on particles n and An, which will project particles 1, 2, . . . , n − 1 and Bninto one of the following forms

φ±|n Anϕ

1,2,...,nAnBn

=12

−a{i}|{i}⟩1,2,...,n−1|0⟩Bn ±

−b{i}|{i}⟩1,2,...,n−1|1⟩Bn

(4)

ψ±

|n Anϕ1,2,...,nAnBn

=12

−a{i}|{i}⟩1,2,...,n−1|1⟩Bn ±

−b{i}|{i}⟩1,2,...,n−1|0⟩Bn

. (5)

Here |φ±⟩ =

1√2(|00⟩ ± |11⟩), |ψ±

⟩ =1

√2(|01⟩ ± |10⟩).

After that, Alice informs Bob of the outcome of the measurement through the classical channel. Bob then determineshis action based on the received result. If the result is |φ+

⟩nAn , |φ−⟩nAn , |ψ

+⟩nAn or |ψ−

⟩nAn , Bob respectively applies thecorresponding unitary operators IBn , XBn , ZBn or ZBnXBn onto the particle Bn. As a result, the state of particles 1, 2, . . . , n − 1and Bn become

|ϕ⟩1,2,...,n−1Bn =

−a{i}|{i}⟩1,2,...,n−1

|0⟩Bn +

−b{i}|{i}⟩1,2,...,n−1

|1⟩Bn . (6)

Comparing the state (6) with the state (1), it is clear that particle n held by Alice in the initial entangled system is nowrestored by particle Bn held by Bob.

Thus one by one, during the whole process, Bob only performs one qubit unitary particles on the particles composed ofthe corresponding EPR channel instead of applying multi qubit unitary operations onto n particles owned by him composedof EPR channels as in papers [10,20]. As a consequence, the unknown quantum state of a multi-particle can be faithfullyteleported by only one qubit unitary operation on particles composed of corresponding EPR channels respectively, whichmakes it much easier for the physical realization.

In the following sections, to ensure the integrity of the unknown state of an entangled particle in amulti-particle system,we will focus our interests on constructing the EPR channels. Once the construction fails, the original channel will bediscarded and another will be tried.

762 M. Jiang et al. / Physica A 390 (2011) 760–768

3. Construction of the EPR channel via a non-maximally entangled two-particle state

Suppose that a quantum channel between Alice and Bob is set up as

|φ′⟩AnBn = (α|00⟩AnBn + β|11⟩AnBn) (7)

where

|α|2+ |β|

2= 1, |α| < |β|. (8)

Here both parameters α and β are restricted to take real values.In order to realize the faithful teleportation, Alice introduces an ancillary particle with the initial state |0⟩r . The joint

system of particles An, Bn and r may be expressed as

|ϕ⟩AnBnr =α|00⟩AnBn + β|11⟩AnBn

|0⟩r . (9)

Alice performs the following unitary operation onto her particles An and r

UAnr =

1 0 0 00 1 0 0

0 0α

β

1 −

α2

β2

0 0

1 −

α2

β2−α

β

. (10)

Then the state of particles An, Bn and r will be transformed into the following form

|ϕ⟩AnBnr = α(|00⟩AnBn + |11⟩AnBn)|0⟩r +

β2 − α2|111⟩AnBnr . (11)

Now Alice makes a von Neumann measurement on the ancillary particle r . If the result is |0⟩r , the state of particles Anand Bn becomes

|ϕ⟩AnBn = α|00⟩AnBn + |11⟩AnBn

(12)

which means that a faithful channel can be established between Alice and Bob. Here we do not normalize the state in (12)for convenience. If the result is |1⟩r , the establishment process fails and we should start another establishment.

From (11), we can find that the success probability to obtain |0⟩r is 2|α|2. If |α| = |β|, the success probability becomes 1

and the auxiliary particle is not needed.Similarly, Bob can also adjust the above two-particle state in (7) to establish an EPR channel. Thus, comparing with

probabilistic teleportation where auxiliary particle resources are added and the unitary operation is performed only by thereceiver [20], our scheme is much easier to achieve in experiments since we can choose either the receiver or the sender toadjust the probabilistic channel.

4. Construction of an EPR channel involving three agents

Now consider this following situation: Alice wants to teleport an unknown quantum state of one entangled particle in amulti-particle system to Charlie while there is no direct quantum channel between them. However, a quantum channel

|φ′⟩A1B1 = α|00⟩A1B1 + β|11⟩A1B1 (13)

shared by Alice and Bob in two different places has been set up.Bob and Charlie share the following quantum channel

|φ′⟩B2C1 = γ |00⟩B2C1 + δ|11⟩B2C1 (14)

where

|γ |2+ |δ|2 = 1, |γ | < |δ|. (15)

Here both parameters γ and δ are restricted to take real values.Particles A1, B1 and B2, C1 belong to Alice, Bob and Charlie, respectively. The initial state of the joint system reads

|ϕ⟩A1B1B2C1 = (α|00⟩A1B1 + β|11⟩A1B1)(γ |00⟩B2C1 + δ|11⟩B2C1)

= αγ |0000⟩A1B1B2C1 + αδ|0011⟩A1B1B2C1 + βγ |1100⟩A1B1B2C1 + βδ|1111⟩A1B1B2C1 . (16)

Firstly, an ancillary particle |0⟩r is added by Bob. The initial state of whole system then can be expressed as

|ϕ⟩A1B1B2C1r =αγ |0000⟩A1B1B2C1 + αδ|0011⟩A1B1B2C1 + βγ |1100⟩A1B1B2C1 + βδ|1111⟩A1B1B2C1

|0⟩r . (17)

M. Jiang et al. / Physica A 390 (2011) 760–768 763

After that, Bob performs a unitary transformation

UB1B2r =

1 0 0 0 0 0 0 00 1 0 0 0 0 0 0

0 0γ

δ

1 −

γ 2

δ20 0 0 0

0 0

1 −

γ 2

δ2−γ

δ0 0 0 0

0 0 0 0α

β

1 −

α2

β20 0

0 0 0 0

1 −

α2

β2−α

β0 0

0 0 0 0 0 0αγ

βδ

1 −

α2γ 2

βδ

0 0 0 0 0 0

1 −

α2γ 2

β2δ2−αγ

βδ

. (18)

It is easy to obtain that

|ϕ⟩A1C1B1B2r = αγ (|0000⟩A1C1B1B2 + |0101⟩A1C1B1B2 + |1010⟩A1C1B1B2 + |1111⟩A1C1B1B2)|0⟩r

+αδ2 − γ 2|01011⟩A1C1B1B2r + γ

β2 − α2|10101⟩A1C1B1B2r +

β2δ2 − α4γ 2|11111⟩A1C1B1B2r . (19)

Then, let Bob make a von Neumann measurement on the ancillary particle. If the result is |1⟩r , it fails. Otherwise, thestate of particles A1, B1, B2 and C1 is

|ϕ⟩A1C1B1B2 = αγ|0000⟩A1C1B1B2 + |0101⟩A1C1B1B2 + |1010⟩A1C1B1B2 + |1111⟩A1C1B1B2

= αγ

|00⟩A1B1 + |11⟩A1B1

|00⟩B2C1 + |11⟩B2C1

(20)

where we do not normalize the state for convenience. From (20), we can see that two EPR channels can be establishedindirectly between Alice and Bob, and between Bob and Charlie, respectively. The probability to obtain the state (20), thatis, the probability of success can be calculated as 4|αγ |

2.The above method needs to operate an eight-dimensional unitary operation which may be very difficult to implement

in real experiments. In the following, we will show another scheme which is much easier to implement.Instead of adding an ancillary particle, at the first step Bob makes a Bell state measurement on his particles B1 and B2

which will project the joint system of particles A1, B1, B2 and C1 into one of the following statesφ±

B1,B2

ϕ

A1B1B2C1

=1

√2(αγ |00⟩A1C1 ± βδ|11⟩A1C1) (21)

ψ±

B1B2

ϕ

A1B1B2C1

=1

√2(αδ|01⟩A1C1 ± βγ |10⟩A1C1) (22)

where the probabilities to obtain the Bell state |φ±⟩B1B2 and |ψ±

⟩B1B2 are |α|2|γ |

2+ |β|

2|δ|2 and |α|

2|δ|2 + |β|

2|γ |

2,respectively. Bob will then inform Alice or Charlie of the outcome of the measurement.

According to the measurement results |φ+⟩B1B2 , |φ

−⟩B1B2 , |ψ

+⟩B1B2 and |ψ−

⟩B1B2 , Charlie will perform the I, Z, X or XZgate on the particle C1. Then the channel between Alice and Charlie will be adjusted as one of the following forms:

|ϕ⟩A1C1 =1

√2(αγ |00⟩A1C1 + βδ|11⟩A1C1) (23)

|ϕ⟩A1C1 =1

√2(αδ|00⟩A1C1 + βγ |11⟩A1C1). (24)

It is clear that the above states of Eqs. (23) and (24) are consistent with Eq. (13). Thus, as soon as Alice or Charlie receivesBob’s result, they can use the method proposed by Section 2 to adjust the channel.

Therefore, if the outcome of Bob’s measurement is |φ±⟩B1B2 , the probability of success can be calculated as P1 =

(|α|2|γ |

2+ |β|

2|δ|2) ·

2|α|2|γ |

2

|α|2|γ |2+|β|2|δ|2= 2|α|

2|γ |

2.

764 M. Jiang et al. / Physica A 390 (2011) 760–768

Fig. 1. Indirect construction of the faithful EPR channel involving multi agents.

If |αδ| ≤ |βγ |(|αδ| > |βγ |) and the outcome of the measurement is |ψ±⟩B1B2 , the probability of success based on the

state (22) can be calculated as

P2 = (|α|2|δ|2 + |β|

2|γ |

2) ·2|α|

2|δ|2

|α|2|δ|2 + |β|2|γ |2= 2|α|

2|δ|2

P ′

2 = (|α|2|δ|2 + |β|

2|γ |

2) ·2|β|

2|γ |

2

|α|2|δ|2 + |β|2|γ |2= 2|β|

2|γ |

2.

(25)

The total probability of success is

P = P1 + P2 = 2|α|2|γ |

2+ 2|α|

2|δ|2 = 2|α|

2 |γ |

2+ |δ|2

> 4|α|

2|γ |

2P ′

= P1 + P ′

2 = 2|α|2|γ |

2+ 2|β|

2|γ |

2= 2(|α|

2+ |β|

2)|γ |2 > 4|α|

2|γ |

2 . (26)

If we make use of the probabilistic teleportation scheme in Ref. [20], an ancillary particle will be added and a reductionunitary operation will be implemented by Bob and Charlie, respectively. The successful probability is 4|α|

2|γ |

2. In the abovescheme, Alice or Bob only needs to introduce an ancillary particle and perform a two-particle unitary operation. Thus, thisscheme can efficiently reduce ancillary particle resources. In the aspect of efficiency, the probability can be enhanced asshown as in (26). Hence, it will be more easily implemented by experimental physicists.

5. Indirect construction of an EPR channel involving multi agents

Similarly to a classical communicationnetwork [15,16],multi spatially remote agents play the role of relay nodes betweena sender and a receiver in a quantum teleportation network. Thus, it is necessary to generalize the construction of an EPRchannel involving three relay agents to multi agents.

Assume that there is a quantum teleportation network withm agents where the sender ‘‘Alice’’, receiver ‘‘Bob’’ andm−2intermediate spatially separated agents are involved. As a sender and a receiver, Alice and Bob play the role of agent 1 andm respectively. Now, the purpose of the scheme is to teleport the unknown quantum state of an entangled particle in Alice’splace to the particle in Bob’s place assisted bym − 2 intermediate agents, whose whole process is illustrated by Fig. 1.

Here agent i (i = 1, 2, . . . ,m − 1) shares the following channel with the next adjacent agent i + 1

|φ′⟩AiBi = αi|00⟩AiBi + βi|11⟩AiBi . (27)

M. Jiang et al. / Physica A 390 (2011) 760–768 765

Agent i (i = 2, 3, . . . ,m − 2) holds the particles Bi−1 and Ai. Particles A1 and Bm−1 belong to Alice and Bob, respectively.At first, the initial state of the joint system composed of particle Ai and Bi (i = 1, 2, . . . ,m − 1) can be expressed as

|ϕ⟩A1B1A2B2···Am−1Bm−1 =

m−1∏i=1

(αi|00⟩AiBi + βi|11⟩AiBi). (28)

As shown in Fig. 1, agent 2 makes a Bell state measurement on his particles B1 and A2. The whole system then will beprojected into one of the following forms

φ±

|B1A2ϕA1B1A2B2···Am−1Bm−1

=1

√2(α1α2|00⟩A1B2 ± β1β2|11⟩A1B2)

m−1∏i=3

(αi|00⟩AiBi + βi|11⟩AiBi) (29)

ψ±

|B1A2ϕA1B1A2,B2···Am−1Bm−1

=1

√2(α1β2|01⟩A1B2 ± α2β1|10⟩A1B2)

m−1∏i=3

(αi|00⟩AiBi + βi|11⟩AiBi). (30)

Agent 2 informs his next adjacent node (agent 3) of his measurement outcome. According to the received results|φ+

⟩B1A2 , |φ−⟩B1A2 , |ψ

+⟩B1A2 or |ψ−

⟩B1A2 , agent 3 determines his action and correspondingly applies IB2 , ZB2 , XB2 or ZB2XB2onto his particle B2.

As a matter of fact, when the measurement outcome is |φ±⟩B1A2 , the whole system will become

|ϕ⟩A1A3A4···Am−1B2B3···Bm−1 =1

√2(α1α2|00⟩A1B2 + β1β2|11⟩A1B2)

m−1∏i=3

(αi|00⟩AiBi + βi|11⟩AiBi). (31)

When the measurement outcome is |ψ±⟩B1,A2 , the whole system will become

|ϕ⟩A1A3A4···Am−1B2B3···Bm−1 =1

√2(α1β2|01⟩A1B2 + α2β1|10⟩A1B2)

m−1∏i=3

(αi|00⟩AiBi + βi|11⟩AiBi). (32)

Next, agent i (i = 3, 4, . . . ,m − 1) performs a similar operation which includes the following three steps: (1) accordingto the previous Bell state measurement outcome, perform an appropriate operation onto his particle Bi−1; (2) make a Bellstate measurement onto his particles Bi−1 and Ai; (3) inform his next adjacent agent of the outcome of the measurement bya classical channel.

Finally, particles A1 and Bm−1 are in the following state

|ϕ⟩A1Bm−1 =1

(√2)m−2

α1

m−1∏i=2

ki|00⟩A1Bm−1 + β1

m−1∏i=2

k̄i|11⟩A1Bm−1

. (33)

When the Bell state measurement is |φ±⟩Bi−1,Ai (i = 2, 3, . . . ,m − 1)(|ψ±

⟩Bi−1,Ai (i = 2, 3, . . . ,m − 1)),

ki = αi, k̄i = βi (ki = βi, k̄i = αi). (34)

The probability to obtain the state (31) is

p1 =

α1

n−1∏i=2

ki

2

+

β1

n−1∏i=2

k̄i

2

. (35)

The total probability then can be calculated as

P =

2m−2−x=1

min

α1

m−1∏i=2

ki

2

+

β1

m−1∏i=2

k̄i

2 ·

2|α1|2m−1∏i=2

|kx(i)|2, 2|β1|2m−1∏i=2

|kx(i)|2α1

m−1∏i=2

ki

2 +

β1

m−1∏i=2

k̄i

2

≥ 2m−1m−1∏i=2

|αi|2. (36)

Here 2m−1∏m−1i=2 |αi|

2 is the total probability of one-by-one probabilistic teleportation method [20].Comparedwith the one-by-one probabilistic teleportationmethod [20], the above scheme clearly shows that the indirect

establishment of an EPR channel between Alice and Bob can enhance the success probability and avoid ancillary particles aswell as unitary operations.

766 M. Jiang et al. / Physica A 390 (2011) 760–768

Fig. 2. Direct construction of an EPR channel involving multi agents.

6. Direct construction of an EPR channel involving multi nodes

In Section 5, it is necessary for each intermediate agent to apply a unitary operation onto his particle and tell hismeasurement outcome to his next adjacent agent. However, more unitary operations will accumulate more errors.Moreover, since each intermediate agent must perform the unitary operation and make his Bell state measurement afterobtaining the Bell state measurement outcome provided by his previous agent, it will consume some idle time. Hence, wedevelop an improved scheme which can directly establish an EPR channel between Alice and Bob with the same probabilityas in Section 5.

As illustrated in Fig. 2, agent i (i = 2, 3, . . . ,m) simultaneously makes a Bell state measurement onto his own particlesAi and Bi−1, then informs the sender or the receiver his measurement outcome.

Considering all the measurement outcomes, if the total number of |ψ−⟩AiBi−1 and |φ−

⟩AiBi−1 as well as the total numberof |ψ−

⟩AiBi−1 and |ψ+⟩AiBi−1 is even, the state of particles A1 and Bm−1 will become

|ϕ⟩Bm−1A1 =1

(√2)m−2

αm−1

m−2∏i=1

ki|00⟩Bn−1A1 + βm−1

m−2∏i=1

k̄i|11⟩Bn−1A1

. (37)

If the total number of |ψ−⟩AiBi−1 and |φ−

⟩AiBi−1 is odd and the total number of |ψ−⟩AiBi−1 and |ψ+

⟩AiBi−1 is even, the stateof particles A1 and Bm−1 will become

|ϕ⟩Bm−1A1 =1

(√2)m−2

αm−1

m−2∏i=1

ki|00⟩Bm−1A1 − βm−1

m−2∏i=1

k̄i|11⟩Bm−1A1

. (38)

If the total number of |ψ−⟩AiBi−1 and |φ−

⟩AiBi−1 is even and the total number of |ψ−⟩AiBi−1 and |ψ+

⟩AiBi−1 is odd, the systemis

|ϕ⟩Bn−1A1 =1

(√2)m−2

αm−1

m−2∏i=1

ki|01⟩Bn−1A1 + βm−1

m−2∏i=1

k̄i|10⟩Bn−1A1

. (39)

M. Jiang et al. / Physica A 390 (2011) 760–768 767

If the total number of |ψ−⟩AiBi−1 and |φ−

⟩AiBi−1 as well as the total number of |ψ−⟩AiBi−1 and |ψ+

⟩AiBi−1 is odd, the systemis

|ϕ⟩Bm−1A1 =1

(√2)m−2

αm−1

m−2∏i=1

ki|01⟩Bm−1A1 − βm−1

m−2∏i=1

k̄i|10⟩Bm−1A1

. (40)

In Eqs. (37)–(40), the coefficients ki and k̄i depend on the measurement results obtained by agents f (f = i, i +

1, . . . ,m − 1). The coefficients ki, k̄i (i = 2, 3, . . . ,m − 1) depend on the total number of Bell state measurements byagent f (f = i, i + 1, . . . ,m − 1). If the total number of |ψ±

⟩Af ,Bf−1(f = i, i + 1, . . . ,m − 1) is even, ki = αi, k̄i = βi.Otherwise, ki = βi, k̄i = αi.

Here we perform a numerical analysis to deduce Eqs. (37)–(40).We start from Eq. (37). If the total number of |ψ−

⟩AiBi−1 (i = 2, 3, . . . ,m) and |φ−⟩AiBi−1 as well as the total number of

|ψ−⟩AiBi−1 and |ψ+

⟩AiBi−1 is even, we can obtain the state of Eq. (37). Now assume the agentsm andm+1 share the followingchannel:

|φ′⟩BmAm = αm|00⟩BmAm + βm|11⟩BmAm . (41)

Then, agentm and m + 1 possess particles Am and Bm, respectively. The state of particles A1, Bm, Am and Bm−1 is

|ϕ⟩BmAmBm−1A1 = (αm|00⟩BmAm + βm|11⟩BmAm)

αm−1

m−2∏i=1

ki|00⟩Bn−1A1 + βm−1

m−2∏i=1

k̄i|11⟩Bn−1A1

. (42)

Now, agentm performs a Bell measurement on his particles Bm−1 and Am, the state of particles A1 and Bm will be writtenas one of the following states:

φ+|ϕBmAmBm−1A1

=1

(√2)

αmαm−1

m−2∏i=1

ki|00⟩BmA1 + βmβm−1

m−2∏i=1

k̄i|11⟩BmA1

=1

(√2)

αm

m−1∏i=1

ki|00⟩BmA1 + βm

m−1∏i=1

k̄i|11⟩BmA1

(43)

φ−

|ϕBmAmBm−1A1

=1

(√2)

αmαm−1

m−2∏i=1

ki|00⟩BmA1 − βmβm−1

m−2∏i=1

k̄i|11⟩BmA1

=1

(√2)

αm

m−1∏i=1

ki|00⟩BmA1 − βm

m−1∏i=1

k̄i|11⟩BmA1

(44)

ψ+

|ϕBmAmBm−1A1

=1

(√2)

αmβm−1

m−2∏i=1

k̄i|01⟩BmA1 + βmαm−1

m−2∏i=1

ki|10⟩BmA1

=1

(√2)

αm

m−1∏i=1

k̄i|01⟩BmA1 + βm

m−1∏i=1

ki|10⟩BmA1

(45)

ψ−

|ϕBmAmBm−1A1

=1

(√2)

αmβm−1

m−2∏i=1

ki|01⟩BmA1 − βmαm−1

m−2∏i=1

k̄i|10⟩BmA1

=1

(√2)

αm

m−1∏i=1

k̄i|01⟩BmA1 − βm

m−1∏i=1

ki|10⟩BmA1

. (46)

Here are four possible measurement outcomes obtained by agentm.Assume the measurement result is |ψ+

⟩AmBm−1 , the total number of |ψ−⟩AiBi−1 and |φ−

⟩AiBi−1 is even, while the totalnumber of |ψ−

⟩AiBi−1 and |ψ+⟩AiBi−1 obtained by agents i (i = 2, 3, . . . ,m) is odd. This form matches Eq. (39) as will as the

coefficients.Similarly, we can further analyze the other three cases |φ+

⟩AmBm−1 , |φ−⟩AmBm−1 and |ψ−

⟩AmBm−1 . All of these cases agreewith Eqs. (37), (38) and (40) respectively.

Using a similar method, we can analyze the other three states of Eqs. (38)–(40). The results are consistent with ourprevious one. In fact, the numerical calculation methods which will be programmed can be used to determine the directchannel between the sender and receiver.

According to both total numbers of |ψ−⟩AiBi−1 and |φ−

⟩AiBi−1 as well as the total number of |ψ−⟩AiBi−1 and |ψ+

⟩AiBi−1 , thesender or the receiver applies an appropriate operation onto her particle A1. The system will be transformed into the finalstate which is consistent with the state of Eq. (7).

Using this approach, the same probability as that in Section 5 is obtained. Since the direct construction consumes lesstime and less local operations, it is superior to the indirect method in Section 5.

768 M. Jiang et al. / Physica A 390 (2011) 760–768

7. Conclusion

In this paper, we have presented a new method to teleport entangled multi-particle quantum information via non-maximally entangled states involving multi spatially remote agents. Using our scheme, the integrity of unknown statescan be maintained even when the construction of an EPR channel between the sender and the receiver via the assistanceof agents fails. Besides that, the presented scheme needs only the sender or receiver to offer an ancillary particle, followedby implementing only one two-particle reduction unitary operation. As a result, the required auxiliary resources and thenumber of local operations and two-particle operations are greatly reduced. Hence our scheme is of considerable interest,especially due to its convenience for faithfully teleporting entangled multi-particle systems via probabilistic channels in anefficient and simple manner over quantum networks.

Acknowledgement

This research is supported by the National Natural Science Foundation of China under Grant Number 60904034 and61071214 and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant Number09KJD120003.

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