6 November 2000

Physics Letters A 276 (2000) 209–212www.elsevier.nl/locate/pla

Teleportation of a two-particle entangled state viaentanglement swapping

Hong Lua,b,∗, Guang-can Guoa

a Laboratory of Quantum Communication and Quantum Computation, University of Science and Technology of China,Hefei 230026, PR China

b Department of Physics of Foshan University, Foshan 528000, PR China

Received 5 September 2000; received in revised form 5 October 2000; accepted 9 October 2000Communicated by P.R. Holland

Abstract

A scheme of teleporting a two-particle entangled state is proposed. In this scheme, two pairs of entangled particles are usedas quantum channel. It is shown that if the two pairs of particles are not-maximally entangled, the probability of successfulteleportation is determined only by the smallest superposition coefficient. 2000 Elsevier Science B.V. All rights reserved.

PACS:03.67.Hk; 03.65.Bz

1. Introduction

Recently, teleportation [1] has drawn much at-tention because its fresh notion and latent appliedprospects in quantum communication and quantumcalculation [2–4]. In the process of teleportation, asender Alice sends an unknown quantum state throughthe quantum channel to a receiver Bob (correspondedsupplement information must still be delivered byclassical channel), while quantum entanglement [5]is used as the quantum channel in the process. If thestate vector|Ψ (A,B)〉 of a quantum system com-posed of two (or more) subsystems cannot be writ-ten as product of the state vectors of its subsystems|Ψ (A)〉 ⊗ |Ψ(B)〉, these subsystems in the quantumsystem are known as entangled. The state|Ψ (A,B)〉 isknown as entangled state. Entanglement reveals nonlo-

* Corresponding author.E-mail address:hlu@163.net (H. Lu).

cal character of quantum mechanics and violates Bellinequality.

There is entanglement phenomenon in many quan-tum systems [2,6–8]. Especially, in the process ofprocessing and transmitting quantum information, en-tanglement lies in extremely important position. Forteleporting a single particle quantum state, the quan-tum channel is usually an entangled Einstein–Podol-sky–Rosen (EPR) pair (i.e., two spin-1/2 particles inthe maximum entangled state).

Recently, Li et al. [9] proposed that by using onepair of entangled (but not-maximally entangled) two-state particles to teleport the quantum state of a sin-gle particle and extended the scheme to a multipar-ticle system. Another interesting problem is how toteleport the state of a pair of entangled particles. Theauthors of Refs. [10,11] have studied this problem.Ref. [10] indicates that probabilistic teleportation oftwo-particle entangled state can be realized by using athree-particle entangled state as quantum channel.

0375-9601/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0375-9601(00)00666-6

210 H. Lu, G.-c. Guo / Physics Letters A 276 (2000) 209–212

In this paper, we propose another scheme to realizethe teleportation of two-particle entangled state. Con-sidering two-particle entanglement is more easily gen-erated than three-particle entanglement, different fromRef. [10], we use two pairs of entangled particles toreplace three entangled particles as quantum channel,namely, we first construct two pairs of distant entan-gled particles between Alice and Bob [6], then utilizeentanglement swapping [7,12] to realize the teleporta-tion.

2. Teleportation of a two-particle entangled stateby two EPR pairs

We describe in present section how to teleport atwo-particle entangled state by using two EPR pairs.Suppose Alice has an entangled particle pair, whichconsists of particle 1 and particle 2. She wants toteleport the unknown state|ψ〉12 of the particle pairto receiver Bob. The state|ψ〉12 may be expressed as

(1)|ψ〉12= x|00〉12+ y|11〉12.

First, Alice sets up two distant EPR pairs betweenherself and Bob, which can be done by the methodsuggested in Ref. [6]. An EPR pair may be located inone of the four Bell base states below:

(2)∣∣Φ±⟩12=

1√2

(|00〉12± |11〉12),

(3)∣∣Ψ±⟩12=

1√2

(|01〉12± |10〉12).

Without losing generality, we suppose the two EPRpairs shared by Alice and Bob are in following states,respectively:

(4)∣∣Φ+⟩34=

1√2

(|00〉34+ |11〉34),

(5)∣∣Φ+⟩56=

1√2

(|00〉56+ |11〉56).

Namely, particles 3 and 4 are in state|Φ+〉34, par-ticles 5 and 6 are in state|Φ+〉56. Suppose parti-cles 3 and 5 belong to Alice, while particles 4 and 6belonging to Bob, the state of the system is|Ψ 〉 =|ψ〉12|Φ+〉34|Φ+〉56 at this moment. Then, Alice op-erates a Bell measurement of particles 1 and 3. Thestate of particles 2, 4, 5 and 6 will collapse into one of

following states:⟨Φ±

∣∣13Ψ

⟩= 1

2√

2

(x|00〉24± y|11〉24

)(6)× (|00〉56+ |11〉56

),⟨

Ψ±∣∣13Ψ

⟩= 1

2√

2

(x|01〉24± y|10〉24

)(7)× (|00〉56+ |11〉56

).

For instance, if measurement result is|Φ+〉13, thestate of the system collapses into〈Φ+|13Ψ 〉 and theentanglement is erected between particles 2 and 4.Alice then measures Bell state of particle 2 and 5, ifthe measure result is|Ψ−〉25, the state of particles 4and 6 collapses into the following state:

(8)⟨Ψ−

∣∣25

⟨Φ+

∣∣13Ψ

⟩= 1

4

(x|01〉46− y|10〉46

).

Namely, after two separate measuring, the entangle-ment between particles 1 and 2 disappears, and thenew entanglement between particles 4 and 6 is setup. The entanglement swapping happens. Next, Al-ice informs Bob of her measurement results by aclassical channel. Bob, according to Alice’s measure-ment results, operates a unitary transformationI4 ⊗(|0〉〈1| − |1〉〈0|)6 on Eq. (8), thus reconstructs entan-gled state (1). Therefore, Alice successfully sends theunknown state (1) to Bob. It is evident that Bob mustoperate relevant unitary transformation against Alice’sdifferent measurement results. Table 1 gives Alice’s alldifferent measurement results and Bob’s relevant uni-tary transformations.

Table 1

Alice’s measurement Bob’s unitary transformationresult

〈Φ+|25〈Φ±|13Ψ 〉 I4⊗ (|0〉〈0| ± |1〉〈1|)6〈Φ−|25〈Φ±|13Ψ 〉 I4⊗ (|0〉〈0| ∓ |1〉〈1|)6〈Ψ+|25〈Φ±|13Ψ 〉 I4⊗ (|0〉〈1| ± |1〉〈0|)6〈Ψ−|25〈Φ±|13Ψ 〉 I4⊗ (|0〉〈1| ∓ |1〉〈0|)6〈Φ+|25〈Ψ±|13Ψ 〉 (|0〉〈1| ± |1〉〈0|)4⊗ I6〈Φ−|25〈Ψ±|13Ψ 〉 (|0〉〈1| ∓ |1〉〈0|)4⊗ I6〈Ψ+|25〈Ψ±|13Ψ 〉 (|0〉〈1| + |1〉〈0|)4⊗ (|0〉〈1| ± |1〉〈0|)6〈Ψ−|25〈Ψ±|13Ψ 〉 (|0〉〈1| + |1〉〈0|)4⊗ (|0〉〈1| ∓ |1〉〈0|)6

H. Lu, G.-c. Guo / Physics Letters A 276 (2000) 209–212 211

3. Teleportation of a two-particle entangled stateby two pairs of entangled particles

Suppose the pair of entangled particles (particles 1and 2) that Alice needs to send is still in state (1),but the two pairs entangled particles shared by Alice(particles 3 and 5) and Bob (particles 4 and 6) arenot-maximally entangled, but located in the followingstates, respectively:

(9)|ψ〉34= a|00〉34+ b|11〉34

and

(10)|ψ〉56= c|00〉56+ d|11〉56,

where |a| > |b|, |a|2 + |b|2 = 1, |c| > |d|, |c|2 +|d|2 = 1. We demonstrate that, in this case, by usingentanglement swapping, Alice can still successfullytransmit state (1) to Bob with certain probability.

At first, the state of the system is|Ψ 〉 = |ψ〉12|ψ〉34·|ψ〉56. After Alice measures the Bell states of particles1 and 3, particles 2 and 5, similar to Section 2, there arefollowing probable outcomes of the states of particles4 and 6:

(11)⟨Φ+

∣∣25

⟨Φ±

∣∣13Ψ

⟩= 1

2

(acx|00〉46± bdy|11〉46

),

(12)⟨Φ−

∣∣25

⟨Φ±

∣∣13Ψ

⟩= 1

2

(acx|00〉46∓ bdy|11〉46

),

(13)⟨Ψ+

∣∣25

⟨Φ±

∣∣13Ψ

⟩= 1

2

(adx|01〉46± bcy|10〉46

),

(14)⟨Ψ−

∣∣25

⟨Φ±

∣∣13Ψ

⟩= 1

2

(adx|01〉46∓ bcy|10〉46

),

(15)⟨Φ+

∣∣25

⟨Ψ±

∣∣13Ψ

⟩= 1

2

(bcx|10〉46± ady|01〉46

),

(16)⟨Φ−

∣∣25

⟨Ψ±

∣∣13Ψ

⟩= 1

2

(bcx|10〉46∓ ady|01〉46

),

(17)⟨Ψ+

∣∣25

⟨Ψ±

∣∣13Ψ

⟩= 1

2

(bdx|11〉46± acy|00〉46

),

(18)⟨Ψ−

∣∣25

⟨Ψ±

∣∣13Ψ

⟩= 1

2

(bdx|11〉46∓ acy|00〉46

).

If Alice’s two measurement outcomes of the Bellstates of particles 1 and 3, particles 2 and 5 arerespectively|Φ+〉13 and|Φ+〉25. She informs Bob ofthe results by the classical channel. From this, Bobconcludes the state of particles 4 and 6 is

(19)⟨Φ+

∣∣25

⟨Φ+

∣∣13Ψ

⟩= 1

2

(acx|00〉46+ bdy|11〉46

).

If |a| > |c| > |d| > |b|, then we have|ac| > |bd|,|ad| > |bc|. To obtain state (1), Bob introduces anauxiliary qubit with the original state|0〉A and oper-ates a unitary transformation under the basis{|0〉4|0〉A,|1〉4|0〉A, |0〉4|1〉A, |1〉4|1〉A}:

(20)

bdac

0√

1− ( bdac

)20

0 1 0 0

0 0 0 −1√1− ( bd

ac

)2 0 − bdac

0

.Under this transformation, the state of particles 4, 6and particleA becomes

1

2bd(x|00〉46+ y|11〉46

)|0〉A(21)+ 1

2ac

√1−

(bd

ac

)2

|10〉46|1〉A.

Then Bob measures the state of particleA. If the resultis |1〉A, the teleportation fail; if the result is|0〉A, thestate of particles 4 and 6 collapses to

(22)1

2bd(x|00〉46+ y|11〉46

).

The teleportation is successfully realized. So, forEq. (19), the probability of successful teleportationis 1/4 |bd|2.

In a similar way, we can obtain the probabilityof successful teleportation for Eqs. (11)–(18), respec-tively. For instance, if Alice’s two measurement resultsare respectively|Ψ−〉13 and |Φ+〉25, and then Bobknows the state of particles 4 and 6 is

(23)1

2

(bcx|10〉46− ady|01〉46

).

As a result, he operates a unitary transformation(|0〉〈1| + |1〉〈0|)4 ⊗ (|0〉〈0| − |1〉〈1|)6 on Eq. (23).Eq. (23) then changes to

(24)1

2

(bcx|00〉46+ ady|11〉46

).

After that, a collective unitary transformation is per-formed on the particles 4 andA. Because of|bc| 6|ad|, the transformational matrix now is

(25)

1 0 0 0

0 bcad

0√

1− ( bcad

)20 0 −1 0

0√

1− ( bcad

)2 0 − bcad

.

212 H. Lu, G.-c. Guo / Physics Letters A 276 (2000) 209–212

The state of particles 4 and 6 reduces to

1

2bc(x|00〉46+ y|11〉46

)|0〉A(26)+ 1

2ad

√1−

(bc

ad

)2

|11〉46|1〉A.

Admittedly, Bob has 1/4 |bc|2 probability to obtainstate (1).

Synthesizing all cases (sixteen kinds in all), we ob-tain the total probability of successful teleportation be-ing 2|b|2. If relationship between coefficientsa, b, c, dis |a|> |d|> |c|> |b|, we still obtain the total prob-ability. This means that the probability of success-ful teleportation is determined only by the smaller|a|, |b|, |c|, |d|. If |a| = |b| = |c| = |d|, and|b|2= 0.5,the total probability equals one. This is just the resultin Section 2.

In conclusion, we have proposed a scheme of tele-porting a two-particle entangled state via entangle-ment swapping. In this scheme, the quantum chan-nel is composed of two entangled particle pairs sharedby sender (Alice) and receiver (Bob). We show that ifthese two entangled particle pairs are in not-maximallyentangled states, there is still a certain probabilityof successful teleportation. The higher the entangle-

ment degree of the two pairs entangled particles is, thegreater the probability of successful teleportation. Be-cause two-particle entanglement is more easily gen-erated than three-particle entanglement is, the presentscheme may be easier to realize than using three-particle entanglement state as quantum channel.

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