Quantum repeaters: fundamental and future Yue Li∗a,b , Sha Huac, Yu Liuc, Jun Yec, Quan Zhoud a School of Computer Science & Technology, Huazhong University of Science & Technology, Wuhan, China 430074 b School of Computing, National University of Singapore, Kent Ridge, Singapore 117543 c Department of Electronics & Information Engineering, Huazhong University of Science & Technology, Wuhan, China 430074 d National Key Lab of Space Microwave Technology, Xi’an, China 710000

ABSTRACT An overview of the Quantum Repeater techniques based on Entanglement Distillation and Swapping is provided. Beginning with a brief history and the basic concepts of the quantum repeaters, the article primarily focuses on the communication model based on the quantum repeater techniques, which mainly consists of two fundamental modules --- the Entanglement Distillation module and the Swapping module. The realizations of Entanglement Distillation are discussed, including the Bernstein's Procrustean method, the Entanglement Concentration and the CNOT-purification method, etc. The schemes of implementing Swapping, which include the Swapping based on Bell-state measurement and the Swapping in Cavity QED, are also introduced. Then a comparison between these realizations and evaluations on them are presented. At last, the article discusses the experimental schemes of quantum repeaters at present, documents some remaining problems and emerging trends in this field. Keywords: Quantum Information, Quantum Repeater, Entanglement Distillation, Swapping, Experimental Scheme

1. INTRODUCTION Quantum Communication (QC) holds promise for absolutely secure transmission of secret messages and the faithful transfer of unknown quantum states. The destination of QC is to transmit quantum states between different parties. Such transmission has potential applications in the secret transfer of

∗

[email protected] or [email protected] Quantum Information and Computation V, edited by Eric J. Donkor, Andrew R. Pirich, Howard E. Brandt, Proc. of SPIE Vol. 6573, 65730X, (2007) · 0277-786X/07/$18 · doi: 10.1117/12.717206

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classical messages by means of quantum cryptography, and is also an essential element in the construction of Quantum Networks.1 The basic problem of QC is to generate nearly perfect entangled states between different parties. Such states can be used, for example, to implement secure quantum cryptography using the Ekert protocol, and to faithfully transfer quantum states via Quantum Teleportation (QT).1, 2 At present, all realistic schemes for QC are quantum channel based, however, QC only becomes achievable for moderate distances, and beyond this distance scale, serious problems will occur. One of the problems is caused by the photon loss and the detector noise, which will reduce the signal-to-noise-ratio (SNR) even to 0 and induce the failure of the communication; another issue is the decoherence in the quantum channel, which degrades the quality of entanglement between two particles when they propagate further. The problems mentioned above have become the most two imminency drawbacks to the unlimited extension on achievable distance of QC. To solve these problems, M. Zukowski et al. put forward the Entanglement Swapping (ES) to decrease the distance of distributing photons thus to reduce the photon loss;3 C. H. Bennett et al. brought forward Entanglement Distillation (ED) to overcome decoherence, then to obtain photon pairs with much higher entanglement degree;4~6 Based on these two schemes, H. J. Briegel proposed Quantum Repeaters (QR) this paper.

techniques based on ED and ES, which is the type of QR we will discuss in

7

We will first introduce the concepts and the QC model using QR, based on this model, the two key techniques --- the ED and the ES will then be highly focused; we will also cover the present researches including advanced experimental schemes which try to implement the two key techniques and the whole structures of QR; at the end of this paper, we will discuss some remaining issues and emerging trends in this field.

2. CONCEPTUAL MODEL 2. 1 Conceptual QR According to the concepts from H. J. Briegel, QR based on ED and ES were viewed as “the role of imperfect local operations and the basic techniques for extending the communication distance”, which indicates the necessity of using quantum repeaters techniques in the implementation of long distance QC.7, 8 The QR techniques we introduced here are mainly based on the two following key techniques --- the ED and the ES, which will be discussed in Sec. 3 and Sec. 4 respectively. 2. 2 Model of QC with QR The Model of QC using QR has been shown in Fig.1. This model mainly contains: the both parties who are involved in the communication, entanglement sources, classical channels, quantum channels, ED modules and ES module.

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3

J.

D

uu•w,u4 4J4UU

iiq

V

9I-flI'

drnnJflw cpII

qiucc iuq qcc

F

I

11°p

iiq

cpiicJj cIflhI

IJGLCIJCC

Fig. 1. Model of QC using quantum repeaters.

In Fig. 1, Entanglement source E1 prepares entanglement pair 1-2 and distributes to distillators D1 and D2 respectively, in which the two distillators cooperate perfectly and it is the same with distillators D3 and D4 in the right part of this symmetric model. After these distillators take appropriate local operations and classical communications (LOCC), Maximal entanglement photon pairs will be produced, from which Alice will share pairs 1-2 with the swapper S and Bob will share pairs 3-4 with S. After the swapping procedure is fulfilled by the swapper, the entanglement in the two pairs will be swapped and the two parties, Alice and Bob, will finally obtain a maximal entanglement photon pairs 1-4, which can satisfy the prerequisites for photons to establish a quantum communication. As shown in this model, Maximal entanglement pairs which can only be successfully established between A and S (B and S) without QR now are able to be obtained between A and B, which means the distance of an efficiency QC is prolonged by one time. This model also implies some extensibility, since we can add series of QR between the two parties to get a much longer distance of distribution. To make it further, we will look into this model and find more detail about the two key modules --- ED and ES will be given in the following Sec. 3 and Sec. 4. The model is followed frequently in experimental schemes of QR, and we will see some important experimental work in Sec. 5.

3. ENTANGLEMENT DISTILLATION 3 The ED As shown in the model above, The ED is a significant module for constructing QR. The idea of ED is to extract from an ensemble of pairs of non-maximally entangled qubits a smaller number of pairs with higher entanglement degree, thus it’s an considerable mean to overcome the decoherence problem and other noise in the quantum channel.6 For the ED as a general conception, there are three main schemes for its further realization: the so called Bernstein’s Procrustean method, the Schmidt Projection method and the CNOT-purification method. These three methods will be focused on in the following three subsections.

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3. 1 The Procrustean method The Procrustean method6, 9 first given by H. J. Bernstein, is to cut off the extra probability of the larger term in the partly entangled pure states such as

cos θ ↑ A ⊗ ↓ B − sin θ ↓ A ⊗ ↑ B

(1)

Then turns out to be a perfectly entangled state. It requires that the photon pairs are all in a pure, non-maximally entangled state, say, α H V + β V H , here coefficients which satisfy

2

2

α + β = 1 , and H

α

and

β

are two known

( V ) denotes horizontal (vertical) polarization

of the photons. In this case, the scheme involves only local filtering operations on single pairs. This treatment does not correspond to any von Neumann measurement in the original two-dimensional spin space, but rather to POVM (positive-operator-valued measurement). Main procedure of the Procrustean method is given here: Step A. Assume we have ensemble which contains pure EPR pairs:

ρ AB = ψ −

AB

ψ−

(2)

Here let all the photons A, B in each pairs belong to Alice and Bob respectively, and due to decoherence in the quantum channel, states 11

are mixed into the ensembles, then it turns to the mix

state

ρ AB = (1 − x) ψ − Note that if

AB

ψ − + x 11

AB

11 , x ∈ [ 0,1)

(3)

0 ≤ x < (2 − 2) , CHSH inequality10 is not satisfied, and this violation makes

purification operation available. Step B. Perform on the photons A and B respectively the same POVM

A1i = α 2 0 Here

i

0 +β2 1 i 1 ,

α 2 + β 2 = 1, α , β ∈ (0,1)

and

A2i = β 2 0 Aki = m 0

i

k

0 + α 2 1 i 1 , i = A, B

(4)

0 + n 1 k 1 . After the measurements,

Alice and Bob use a classical channel to compare their results, if both of them get A1 , they will reserve this pair, otherwise, they will discard. After these two steps, the ensemble turns to

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(s) ρ AB =

= Note that In Eq. (5), if increases with

A1A A1B ρ AB A1B A1A tr ( A1A A1B ρ AB A1B A1A )

{(1 − x)α 2 ψ −

AB

ψ − + xβ 2 11

(1 − x)α + xβ 2

(s) α → 1 , ρ AB → ψ−

AB

11 }

2

(5)

, which means the efficiency of purification

α.

As for purifying success rate:

P = tr ( A1A A1B ρ AB A1B A1A )

= tr{(α 2 0

A

×[(1 − x) ψ −

0 +β2 1

AB

A

1 )(α 2 0

ψ − + x 11

AB

B

0 +β2 1

B

11 ]}

= β 2 {(1 − x)α 2 + xβ 2 } In Eq. (6), if

α → 1, P → 0 .

1)

(6)

Thus it can be concluded that the higher the efficiency of the

Procrustean method pursues, the lower success rate it will reach. In its practical application, it is necessary to select a suitable α for a specific situation. This method has first been experimental realized by P. G. Kwiat et al. .11 3. 2 The Schmidt Projection method The second implementation of ED, which is also referred as Quantum Concentration, is the Schmidt Projection method proposed by C. H. Bennett et al. .6 In this method, the photon pairs are required in an unknown, pure, non-maximally entangled state α H V + β V H , and the joint state of n pairs of particles will be projected onto a subspace spanned by states having a common Schmidt coefficient. The main procedure is described in the following: Step A. Let n partly entangled pairs of two-state particles be shared between Alice and Bob, so the initial state is n

Ψ ( A, B) = ∏ [cos θ α1 (i ) β1 (i ) + sin θ α 2 (i) β 2 (i) ]

(7)

i =1

When binomially expanded, this state has 2n terms, with only n + 1 distinct coefficients,

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cos n θ , cos n −1 θ sin θ ,..., sin n θ . Let one of the parties, say Alice, performs an incomplete von Neumann measurement projecting the initial state onto one of the n+ 1 orthogonal subspace. Let Alice perform the measurement, obtaining some outcome k. She then tells Bob the result k (k = 0, 1, 2…n) using classical channel. The probability of outcomes is binomially distributed, with outcome k having probability

pk =

( ) (cos θ ) n k

2

n−k

(sin 2 θ ) k .

After some outcome k has been obtained, Alice and Bob will be left with a maximally entangled state in a known

2

( ) -dimensional subspace of the original n k

be used without further ado for faithful teleportation in a

22 n -dimensional space. Such states can

( ) -dimensional or smaller Hilbert space; or n k

they can be transformed, as described in step B, into a standard form such as singlets. Step B. To efficiently transformed these residual states Fix some small positive ε , with

ε

Ψ k into a standard form such as singlets,

= 0 corresponding to perfect efficiency of transformation. Let the

above measurement of k be performed independently on a sequence of batches of n pairs each. Each performance yields another k value; let the resulting sequence of k values be

Dm =

k1 , k2 ,..., km , and let

( )( ) ...( ) n k1

n k2

n km

(8)

be the product of the

( ) values for the first m batches. The sequence is continued until the

accumulated product of

Dm lies between 2l and 2l (1 + ε ) for some power l . For any single -

n k

pair entanglement E and any positive ε , the probability of

Dm = 2l will close to 1. Once a suitable

Dm is found, a local measurement will be performed by Alice (Bob or both) to project the joint system onto one large space of dimension

2 × 2l with probability greater than 1- ε , the residual state

l

is a maximally entangled state of 2 - dimensional subsystems, one is held by Alice and the other one is held by Bob. Using the Schmidt Decomposition, this can be converted by local unitary operations into a product of l standard singlets.12 Since the Schmidt Projection method requires simultaneous collective measurement and will only obtain an expect efficiency when the value of n is big enough, it is difficult to be implemented in practice. Z. Zhao et al. first reported the experimental demonstration of this method.13

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3. 3 Compare As shown in Sec. 3.1 and Sec. 3.2, both the Procrustean method and the Schmidt Projection method can be generalized to work on larger Hilbert spaces. The Procrustean method requires the bias to be known in advance, the Schmidt Projection method, on the other hand, works even when Alice and Bob do not know how entangled their partly entangled pairs are, provided all n pairs have equal biases Fig. 2 plots the yield of perfectly entangled pairs as a function of

θ.

cos 2 θ obtained by the

Schmidt Projection method with n = 2, 4, 8, and 32 (lower four curves), and the yield from the Procrustean method (inverted-V-shaped curve). Note that for n < 5 Schmidt projection is absolutely less efficient than the procrustean method.

1•0

o. OQ

0•1

O5

00

03

00

01

o.

10

C02

Fig. 2. Yield of maximally entangled output states from partly entangled input states ( cos θ

↑↓ − sin θ ↓↑ ),

as a function, equals to the asymptotic yield of the Schmidt Projection method. Successively lower smooth curves give yields of Schmidt projection applied to n = 32, 8, 4, and 2 input pairs. The inverted-V-shaped curve gives yield by the procrustean method applied to one input pair.

3. 4 The General Purification method C. H. Bennett et al. reported the General Purification method in 1996, as it is called, this general method works for arbitrary mixed states.4 It requires controlled-NOT (CNOT) operations9 between different photons to increase the fidelity of its subensemble to some definite state, thereby to process a distillation. The main steps are described below: In general, assume the mixed state is

Ψ−

AB

ρ AB

, and the target states of the purification is

= (↑↓ − ↓↑) / 2 . Then the purity of ρ AB can be conveniently expressed by its fidelity

F=

AB

Ψ − ρ AB Ψ −

AB

>

1 2

(9)

Step A. According to the hypothesis we made, Alice and Bob perform the same specific local

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rotation on each shared pair which stays on the Ψ

Ψ−

AB

− AB

state, but to each pair which is different from

, random rotation should be performed on.

Step B. In this 4-dimensional space, impose unilateral and bilateral random rotations to state

Ψ−

with fidelity F, then we can obtain a mixed ensemble

AB

mix ρ AB = F Ψ−

AB

Ψ− +

1− F + Ψ 3

AB

Ψ+ +

1− F + φ 3

AB

φ+ +

1− F − φ 3

AB

φ−

(10)

which is so called Werner state. Note that this state can always be produced and the value of F determines whether such state is separable. Step C. Alice (Bob) performs transformation

σy

on her (his) own particle, which turns the state

in Eq. (10) into a rotational symmetry mixed state +

mixφ ρ AB = F φ+

AB

φ+ +

1− F − φ 3

AB

φ− +

1− F + Ψ 3

AB

Ψ+ +

1− F − Ψ 3

AB

Ψ − (11)

Step D. The controlled-NOT operations are performed bilaterally by both Alice and Bob on corresponding members of two shared pairs: If Alice holds spins 1 and 3, and Bob holds spins 2 and 4, a CNOT, with spins 1 and 2 as source and spins 3 and 4 as target, would conditionally flip spin 3 if and only if spin 1 was up, while conditionally flipping spin 4 if and only if spin 2 was up. The results of applying CNOT to other combinations of Bell states have been shown in Tab. 1, omitting phases: Tab. 1. CNOT operations

Before Source ±

Φ Ψ± Ψ± Φ± Φ± Ψ± ± `Ψ Φ±

After (n. c. = no change) Target +

Φ Φ+ Ψ+ Ψ+ Φ− Φ− Ψ− Ψ−

Source

Target

n. c.

n. c.

n. c. n. c.

Ψ+ Φ+

n. c.

n. c.

Φ Ψm Ψm Φm

n. c.

m

Ψ− Φ− n. c.

Step E. Alice and Bob perform a kind of specific measurement: measuring both spins in a given

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pair along the Z spin axis. Then they compare their results, if the results are same, they preserve the source spins; if not, they discard them. According to Tab. 1, this selection equals to projecting the initial state to a subspace before CNOT operation, in this subspace, the states will be only in Ψ or

φ. After the purification, the fidelity is

F'= Note that if F >

10 F 2 − 2 F + 1 8F 2 − 4 F + 5

(12)

1 , F ' > F , thus the purification is obtained. 2

The General Purification method is based on CNOT or other similar logical computational operations, however, logic CNOT gate for long distance QC has not been realized yet, which makes the realization of this scheme more difficult than the Schmidt Projection method, and there is no implementation for the General Purification method, which dues to the hard realizations of the simultaneous collective measurements and the CNOT operations. 3. 5 Summary of ED Now we have showed the three main methods of ED with analysis of prerequisites and efficiency, operations that involved, we have also briefly listed drawbacks in their experimental realizations. We emphasize that, though only the General Purification method can be suitable to purify arbitrary mixed states, both the Procrustean method (local filtering) and the Schmidt Projection method (entanglement concentration) are of interest in their own rights. On the one hand, both methods provide a way to generate scheme where only highly entangled states are generated. On the other hand, with the help of local filtering, any inseparable states can be purified, while the general scheme alone works only for the cases where the entanglement fidelity F is larger than 1/2. So far, except the General Purification method which has not been demonstrated due to the difficulties caused by collective measurement and CNOT operation, both the Procrustean method and the Schmidt Projection method have been successfully implemented, which has made ED experimental available. However, to further complete the QR, we still need to master the principle of another essential module --- the ES, which will be extended in the following section.

4. ENTANGLEMENT SWAPPING As we have shown in Fig. 1, the ES is another key module for the whole structure of QR, which aims to establish the entanglement between two photons from different entanglement pairs using some necessary operations, it means that the ES generalizes or recovers non-local correlations between different independent particles. Here we will introduce two types of methods, the classical ES method

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which is based on Bell-state measurement and a new type without joint measurement but cavity QED. 4. 1 ES based on Bell-state joint measurement The ES based on Bell-state joint measurement (Fig. 3) was first published by M. Zukowski et al. .3 The procedures ask for two maximal entanglement pairs and the necessary operation we mentioned in the beginning is the Bell-state joint measurement. CIJ1LIIG 11611 41G

VI!"

11°!

/jGIflLGUJ6114

Eb1{011LCG

CG fl

Fig. 3. Principle of ES based on Bell-state joint measurement. Two EPR sources produce two pairs of entangled photons, pair 1-2 and pair 3-4. One photon from each pair (photons 2 and 3) is subjected to a Bell-state measurement. This results in projecting the other two outgoing photons 1 and 4 onto an entangled state. Change of the shading of the lines indicates the change in the set of possible predictions that can be made.

The core steps of this Swapping are described below: Step A. Assume the polarization entangled photons in the state

Ψ− =

1 (H V −V H ) 2

(13)

Alice and Charlie share entangled photon pair 1-2, Bob and Charlie share 3-4. Then these polarization entangled photons are in the state:

Ψ

1234

1 ( H 1 V 2 − V 1 H 2)× ( H 3 V 4 − V 3 H 4) 2 1 = ( Ψ+ Ψ+ + Ψ− Ψ− + Φ+ Φ+ + Φ− 14 23 14 23 14 23 2 =

14

Φ−

23

)

(14)

Where

1 (H V ±V H ) 2 1 = (H H ±V V ) 2

Ψ± = Φ

±

(15)

Step B. Charlie perform a joint Bell-state measurement on photons 2 and 3; that is, photons 2 and 3 are projected onto one of the four Bell states which form the complete basis for the combined state

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Ψ±

23

or Φ

±

23

. According to this combined state and Eq. (15), the entangled state that photons 1

and 4 are swapped onto can be determined, thus the ES has been fulfilled. This method is widely used and many experimental realizations14, 15 have been reported, the first implementation was reported by J. W. Pan et al. .16 Because the Bell-state or other types of joint measurement is the core of the previous entanglement swapping schemes, the realization of the Bell-state measurement is of the importance for the entanglement swapping. Hitherto, we still can not discriminate the four Bell states totally and conclusively. The realization of the Bell-state measurement is still difficult in experiments, especially for the atomic Bell states. In the following Sec. 4.2, we will introduce a new type of ES based on cavity QED, which can overcome some of the difficulties.17 4. 2 The Cavity QED Based ES M. Yang et al. have recently proposed a new type of Swapping using cavity QED.18 In this scheme, only single measurement rather than a joint measurement is needed, thus it overcomes the difficulty of realization brought by Bell-state joint measurement. Here we will describe the scheme in steps: Step A. Suppose there are three spatially separate parties Alice (A), Bob (B), and Charlie (C). A and C have shared one pair of atoms 1-2 with atom 1 belonging to A and 2 belonging to C. These two atoms have been previously prepared in the following entangled state:

Φ

12

=a e

1

e 2 +b g

Note that a and b are the normalization coefficients, e

1

g

(16)

2

denotes the excited state and

g denotes

the ground state. In addition to the atom 2, C also possesses one single mode cavity 3, which is entangled with another single mode cavity 4, and the cavity 4 belongs to B. Similarly, we also suppose the two cavities have been prepared in the following entangled state:

Φ where 0

34

= a 1 3 1 4 +b 0

3

0

(17)

4

and 1 denote the vacuum state and one photon state of the cavity mode, respectively,

and the normalization coefficients are same with the atomic entangled state in Eq. (16). Before swapping, the total system becomes

Ψ

1234

= (a e

1

e 2 +b g

1

g 2 ) ⊗ (a 1 3 1 4 + b 0

3

0 4)

(18)

Step B. To realize the ES, C will let the atom 2 fly through the cavity 3. After interaction time t, the state of the whole system will evolve into the following state

Ψ

' 1234

= a 2 e 1 1 4 [cos( 2 gt ) e

2

1 3 − i sin( 2 gt ) g

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2

2 3]

+ ab e

1

0 4 [cos( gt ) e

+ ab g 1 1 4 [cos( gt ) g + ab g 1 1 4 [cos( gt ) g +b g 2

1

0

4

g

2

0

0 3 − i sin( gt ) g

2

1 3]

2

1 3 − i sin( gt ) e

2

0 3]

2

1 3 − i sin( gt ) e

2

0 3]

2

(19)

3

After the atom flying out of the cavity, C will detect the atom 2. If the atom 2 is detected in excited state, the atom 1 and cavities 3, 4 will collapse into

Ψ

' 134

= N {[ab[cos( gt ) e

1

0 4 − i sin( gt ) g 1 1 4 ] 0

3

+ a 2 cos( 2 gt ) e 1 1 3 1 4 } where N is the normalization factor. If C chooses the interaction time to satisfy gt

(20)

= 7π / 4 , we get

cos 2 gt = 0.079 ≈ 0 . So the third term in Eq. (20) can be eliminated. Then the atom 1 and cavity 4 collapsed approximately into a maximally entangled state without detection on the cavity 3

Ψ

14

=

1 (e 2

1

0 4 + i g 1 1 4)

(21)

by some probability. After a rotation is performed, the entangled state will be transformed into the standard form with a zero relative phase factor. After swapping, Bob sets the interaction time appropriately, the interaction can swap the atom and cavity excitations. Then the two atoms, which respectively belong to Alice and Bob, and never interacted before, are in a maximally entangled state. The most distinct advantage of this scheme is that it does not need any joint measurement needed by the previous ES. It only needs a resonant interaction between an atom and a cavity mode and a measurement on cavity (or atom). Its proposal is for non-post-selection, after the ES is done, the swapped entanglement still exits. This indicates a potential value for real application in QR. 4. 3 Summary of ES So far we have showed two types of ES with specific operations and analyzed their merits and obstacles. The prominent difference between the two schemes is whether the scheme needs a joint measurement which also causes the different difficulties of their realizations. Although the ES module is not so intricate in principles as the ED module, it is always the core which should works well with the Distillation module to enable a QR. ED provides ES maximally entangled photon pairs as the source, and Swapping made a further establishment of entanglement available. Such two key modules consist most of the QR. In the following Sec. 5, we will document QR realizations by quickly reviewing some typical

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experimental schemes for QR, most of which are based on the ED and ES mode while some other advanced repeaters will also be mentioned.

5.

EXPERIMENTAL SCHEMES FOR QUANTUM REPEATERS

Comparing to other branches in the QC research fields, the development of QR techniques is only a very limited period. Due to the difficulties of implementing its module technologies, main stream of the research mainly focuses on proposing schemes and theories for experimental implementation. At present, more specific experimental schemes have been proposed, and some of them have successfully been realized. In the following, we will briefly review some forefront experimental schemes of QR. After the report of their innovative scheme19 for ED using Schmidt Projection method in 2001, J. W. Pan’s research group reported a successful experimental realization of Entanglement Concentration using two polarization entangled photon pairs produced by pulsed parametric down-conversion in 2003.13 In the meantime, their setups also provided a proof-in-principle demonstration of a QR. The quality of their procedure is verified by observing a violation of Bell’s inequality. The high experimental accuracy achieved in the experiment implied that the requirements of tolerable error rate in multistage realization of quantum repeaters can be fulfilled. To further their advanced research, in the same year, a QR based on ED and ES was used and played an important role in their great achievement --- Distribution of entangled photon pairs over a noisy ground atmosphere of 13 km.20 Moreover, in 2006, Based on the same model with Schmidt Projection method, T. Chaneliere et

al. proposed a quantum repeater21 at telecommunications wavelengths with long-lived atomic memory, its critical elements have been experimentally demonstrated using a cold atomic ensemble. They used atomic cascade emission to generate entangled pairs of 1.53 µ m and 780 nm photons, the former is ideal for long-distance QC, and the latter naturally suites the mapping to a long-lived atomic memory. Together with their demonstration of photonic-to-atomic qubit conversion, both of the essential elements for the proposed telecommunications QR were realized. With the same ED and ES mode, in 2003, using the combination of the General Purification and Swapping, P. Kok et al. proposed another experimental scheme for QR.22 The architecture of their repeater was also provided in their repeater in the report; P. van Loock et al. published their scheme which involves the application of bright coherent light to propose a hybrid QR.23 Due to the difficulties of implementing general purification method, these two schemes are still waiting to be experimental demonstrated. On the other hand, there still preserves other schemes which are not based on the model we emphasized in this paper, they introduce new conceptual models of QR, which are also significant to push the research forward. It is clear that simpler but more efficient schemes will be proposed and implemented with the development of the research in quantum information processing.

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6.

THE FUTURE

The research of QR techniques will step much further with the development of the whole quantum information area, and here we will do some brief predictions about QR techniques on the trends of their architectures and potential applications. A challenge for the QR in theoretical architecture will still be the issue about finding a quantum source with a long quantum information storage time, or quantum memory. This aspect of design is recursive, since overcoming decoherence of the transmitted state implies performing an operation at some place in the channel which can copy the quantum information and maintain a coherent quantum state for a sufficient amount of time, time enough to generate additional qubits for the signal, swap over the copied state, and continue it along its path. If these are realizable in the future, QR architecture will be improved for much easier implementations and low complexity, for instance, the integration of QR based on ED and ES as a single module based repeater which implement the ED and ES in the same step. QR are of technical significance in quantum information processing due to the relevance in extending the distance of Quantum Key Distribution (QKD), and as resources of entanglement over long distances, which for example, may be used in QT or even in Quantum Secure Direct Communication (QSDC) and Quantum Computer Networks (QCN). With the successfully proving of the feasibility of QC based on satellites and the development of satellite data transmission technology, satellite-carried QR will become a promising research branch.24, 25 These applications of QR, no matter in national defense or in civilian areas, will show greater potential value and play an elementary role to create the new period of secure communication.

7.

CONCLUSIONS

The general purpose of QR is thus naively to perform a unitary transform: given an input state, can the entanglement properties of the state be preserved for output at another physical location. The point is that the signal can or become lost due to attenuation, scattering, or absorption or it can obtain errors for example, due to depolarization, dispersion, in an optical fiber. Given multiple resources, with imperfect quantum operations and imperfect quantum measurements ( e.g POVM), the computational task becomes one of distilling or purifying a set of maximally entangled quantum states at the ends of the channel with a scalable number of resources and operations along the channel. Note that the idea of quantum repeaters is analogous to the classical scheme for signal transmission: divide the channel into N segments through which a suitable threshold of SNR can be met so as to regenerate the information and send it along another segment of the channel. In summary, based on ED and ES, detail about QR techniques have been given in this paper, including concepts, QC model with QR, key techniques, experimental schemes; some remaining

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problems and emerging trends in this field have also been discussed. With the continuously increasing maturity of quantum information theory and the experimental level, we believe that the stage for practical QR will soon be achieved in the future.

ACKNOWLEDGMENTS Y. Li would like to thank the Quantum Information Technology Group (QITG), Department of Electronics & Information Engineering, Huazhong University of Science & Technology. This work is supported by the Innovation Foundation of Aerospace Science and Technology of the People’s Republic of China under Grant No. 20060110 and the Research Foundation of the National Innovative Training Program for College Students of the People’s Republic of China.

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