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Discussion

Entanglement swapping with pure orbital angular momentum Bell-state analysis Ying Guo a,b, Zhiwen Zeng a,n, Guihua Zeng b a b

School of Information Science & Engineering, Central South University, Changsha 410083, China State Key Laboratory of Advanced Optical Communication Systems and Networks, Department of Electronic Engineering, Shanghai Jiaotong University, Shanghai 200240, China

a r t i c l e i n f o

abstract

Article history: Received 16 March 2012 Received in revised form 18 May 2012 Accepted 25 June 2012 Available online 13 July 2012

We investigate a framework of an orbital angular momentum (OAM) entanglement swapping in a multi-dimensional Hilbert space with the spin angular momentum (SAM)-based orbital angular momentum (OAM) Bell-sate analysis. By the implementations of entanglement swapping with the SAM and OAM Bell-state measurements subsequently, the OAM entanglement states (qudits) are generated and then transferred between photons in multi-dimensional Hilbert space in a point-to-point fashion. In the proposed scheme, two pairs of the SAM-based OAM hybrid entanglement photons are deployed to conduct the successive SAM and OAM Bell-state measurements. It provides an alternative technique to transfer pure OAM Bell-states in qudits, which illustrates a possible experimental approach for devising a full repeater in a complex quantum computation network where entanglement swapping serves as a critical constituent. & 2012 Elsevier B.V. All rights reserved.

Keywords: Quantum information Spin angular momentum Orbital angular momentum Entanglement swapping Quantum repeater

1. Introduction Fundamental laws of quantum mechanics, which prevent information from being eavesdropped [1], attract researchers’ eyes in quantum computation networks [2]. For example, Goldenberg and Vaidman presented a scheme based on entanglement of photons that are represented with the superposition of several localized wave packets [3]. Actually, entanglement produced by processing of spontaneous parametric down-conversion (SPDC) can be executed practically for quantum dense coding, teleportation, and hence information processing [4–8]. Entanglement swapping, an ingenious application of entanglement, was experimentally demonstrated to transfer unknown states in the pulsed or continuous-wave source [8–11]. To establish the multi-photon entanglement, Bose et al. [12] proposed the generalized entanglement swapping through a prior distribution of entangled singlets. Pan et al. [13] demonstrated experimentally entanglement swapping with entangled photons. Currently, entanglement swapping has become an essential ingredient in a quantum repeater, which plays a signiﬁcant role in a quantum networking system with multiple entangled photons [14]. It is shown that for an arbitrary photon it may have at least two different degrees of freedom, e.g., SAM and OAM [15]. SAM is

n

Corresponding author. E-mail address: [email protected] (Z. Zeng).

0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.06.064

associated with a polarization state that produces a qubit in twodimensional Hilbert space [16], while OAM is with the helical phase front eilf utilizing an additional degree of freedom which allows for encoding information in a qudit (i.e., state in multi-dimensional OAM Hilbert space) [17,18]. For example, Mair et al. demonstrated OAM entanglement of twin photons in three-dimensional Hilbert space [19]. Oemrawsingh et al. proposed the possible inﬁnitedimensional OAM entanglement in laboratory [20,21]. In order to transfer information with a high-rate for some practical applications, it is much convenient to deploy qudits instead of qubits. Actually, OAM has offered a powerful approach for transmitting information in the large-capacity communication since it can encode information in high-dimensional Hilbert space [22–24]. In addition, the OAM generator can be transmitted with multiple photons in the hybrid entanglement [25,26]. Unfortunately, the conventional processor cannot always work for transferring qudits in a certain way, and thus we are faced with a new challenge that extends to OAM entanglement swapping of qudits via the pure OAM state analysis. Motivated by Marrucci’s suggestion [22,23] that SAM carried by a circularly polarized light beam can be efﬁciently converted into OAM, we exclusively consider an implementation of OAM entanglement swapping on the basis of the SAM-based OAM Bell-state analysis, where OAM is transferred without transmitting the carrier itself [27,28]. Instead of performing the conventional SAM Bell-state measurement, we further perform the pure OAM Bell-state analysis on the corresponding photons, which

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Y. Guo et al. / Optics Communications 285 (2012) 4621–4625

may offer the high information-density coding while packing much data in the twisted photons. This paper is organized as follows. In Section 2, some notations and basic properties of the SAM-based OAM states are presented for the simple description of OAM entanglement swapping. In Section 3, a simple OAM entanglement swapping scheme is suggested in two phases on the basis of the SAM-based OAM Bell-state analysis. Finally, conclusions are drawn in Section 4.

2. Generation of SAM-based OAM entanglement As an ideal candidate to represent quantum data in multidimensional Hilbert space, OAM has drawn much attention since it creates an opportunity for increasing bandwidth in quantum computation networks. Prior to proposing a scheme of OAM entanglement swapping, we brief the generation of OAM entanglement through the Pancharatnam–Berry optical phase, known as the q plate (QP) [22,23]. It is the planar slab of a uniaxial birefringent medium. The orientation of an optical axis can be represented as aðfÞ ¼ qf þ a0 , where a0 and q represent two constants. It shows that input polarization can reshape output orbit wavefront. For example, Nagali et al. [29] proposed an approach in experiment to transfer entanglement from SAM to OAM with several q plates [30,31]. What we are concerned is the OAM generation in a case where only a single photon is considered with a q plate given by Q^ ðqÞ ¼ 9R,l þ 2qS/L,l9 þ 9L,l2qS/R,l9,

3. Implementation of OAM entanglement swapping The violation of the Bell inequality has been demonstrated with entanglement of photons in multi-dimensional OAM Hilbert space, where the OAM Bell-states can be created as follows: 1 9cS ¼ pﬃﬃﬃ ð9lSs 9lSi þ 9lSs 9lSi Þ, 2

where the subscripts s and i denote the signal and idler photons, respectively [34]. What we are interested is to perform OAM entanglement swapping via the SAM-based OAM Bell-state analysis, where the signal OAM states 9 7lSs and idler OAM states 97 kSi are not necessarily equal for two values l and k. A schematic diagram of the setup is sketched in Fig. 1. The ultraviolet (uv) pulse is weakly passed through two barium oxide crystals, where the type-I non-collinear spontaneous parametric down-conversions (SPDC) are utilized. The degenerated signal and idler beams are, respectively, separated by the non-polarizing beam splitters (BS), where two pairs of photons ðt 1 ,t 01 Þ, 8t 1 A f1; 2g, are produced in OAM states in multi-dimensional Hilbert space, i.e.,

9FSð0Þ t 1 ,t 0 ¼ 1

X C l,l 9lSt1 9lSt0 9HSt1 9HSt0 , 1

l

ð3Þ

1

D

ð1Þ

where 9lS represents an OAM eigenstate for encoding qudits associated with helical wavefront of expðilfÞ, l ¼ 2q [15,32], while 9RS and 9LS denote the spin eigenstates, i.e., left-handed and righthanded circular polarizations, respectively. We note that SAM results in the conventional qubit while OAM allows for a qudit represented in multi-dimensional OAM Hilbert space [33,34]. The generation of single-qudit photons in multi-dimensional OAM Hilbert space was demonstrated in experiments [34]. It plays a vital role in increasing the security of communications since much information can be packed in the yielded photons [35]. In what follows, we will derive an approach for transferring qudits to qudits in the SAM-based OAM hybrid entanglement system, which is a bit different from the previous case where entanglement is transferred from qubits to qubits or from qubits to qudits [22,27]. It is known that entanglement swapping is performed on the basis of independent pairs of entangled photons [36]. When photons from each pair are both subjected to a Bell-state (or other multi-qubit entanglement states) measurement, the remaining photons fall into another entanglement state [37]. Obviously, the entangled degree of freedom before and after swapping remains the same, i.e., both in polarization states (SAM states, qubits). Currently, there is an improved entanglement swapping, where degrees of freedom before and after entanglement swapping are sometimes different in SAM and OAM [22–24,27,28]. In our entanglement swapping, however, the degrees of freedom before and after entanglement swapping are both same in OAM. To achieve this goal, it is necessary to employ the SAM-based OAM entanglement instead of the pure SAM entanglement. We note that the SAM-based OAM entanglements where multiple photons are entangled in different degrees of freedom are actually not new as for its experimental generations and realizations. Based these entanglements we demonstrate how to transfer qudits via the SAM-based OAM Bell-state analysis. We implement OAM entanglement swapping on independent OAM entanglement photons via the SAM-based OAM Bell-state measurement while transferring the OAM entanglement state in multi-dimensional OAM Hilbert space.

ð2Þ

D F

F

P

P

CBS D F P

P

F

CBS

BS

QP

QP

D

BS

QP

QP

2 UV

4

1

2

3

SPDC

SPDC 0

0

D

0

D

OBS

0

D

D

Fig. 1. Schematic diagram for OAM entanglement swapping with the SAM-based OAM Bell-state analysis. After emission, the ultraviolet pulse is passed through two barium oxide crystals, where two pairs of photons ð1; 10 Þ and ð2; 20 Þ are produced. After being reﬂected by a mirror, other two pairs ð3; 30 Þ and ð4; 40 Þ are generated subsequently. After that each photon is implemented with a suitable q plate. For the simplicity, the abbreviation CBS stands for a circular beam splitter that transmits 9LS while reﬂecting 9RS. OBS denotes an orbital-angular beam splitter that transmits 9lS while reﬂecting 9lS. Through the single-mode ﬁber (SMF), it produces four entanglement states 9FSt,t0 , 8t A f1; 2,3; 4g, which compose the SAM-based OAM hybrid entanglement states 9GS110 440 and 9GS220 330 . Superposing photons ð1; 4Þ and ð2; 3Þ at respective BSs yield the OAM entanglement state 9XS10 40 20 30 . Finally, the measurement of one OAM Bell-state on photons ð10 ,30 Þ results in projecting the remaining photons ð40 ,20 Þ onto another OAM Bell-state.

Y. Guo et al. / Optics Communications 285 (2012) 4621–4625

where 9HS represents the horizontal polarization; C l,l denotes the probability amplitude of detecting a signal photon with the OAM of l_ and an idler one with the OAM of l_. Since an interaction with the environment projects the signal and idler photons into superpositions of OAM states with amplitudes Am and Bm, i.e., X X 9lSt1 - Am 9m þlSt1 ,9lSt0 - Bm 9n þlSt 01 , ð4Þ 1

m

1

m,n

1

l

1

2

where ll denotes an eigenvalue calculated as ll ¼ 9C l,l Aml Bnl 9 . Then an OAM entanglement state can be selected with parameters (m,n), where m and n are two arbitrary values [33,38]. After being reﬂected by a mirror, the type-I SPDCs are done again. Then other two pairs of photons ðt 2 ,t 02 Þ, 8t 2 A f3; 4g, are similarly produced in OAM states X C l,l 9lSt2 9lSt0 9HSt2 9HSt0 : ð6Þ 9FStð0Þ 0 ¼ 2 ,t 2

2

l

which can be rewritten as 9GS1410 40 ¼

9GS2320 30 ¼

m

one modiﬁes the entanglement in Eq. (3) in the decomposing form XXpﬃﬃﬃﬃ ll 9m þ lSt1 9nlSt0 9HSt1 9HSt0 , ð5Þ 9FStð0Þ 0 ¼ 1 ,t

2

It is obvious that states in Eqs. (3) and (6) show OAM entanglement but no SAM entanglement between photons. For each photon t, 8t A f1; 2,3; 4g, one implements entanglement transferring with a suitable q plate in Eq. (1), i.e., ð0Þ ^ 9FSð1Þ t,t 0 ¼ Q ðqt Þ9FSt,t0 X ¼ C l,l ð9R,l þ kt St 9L,lkt St Þ9l,HSt0 ,

ð7Þ

l

where kt ¼ 2qt denotes an OAM generated by the q plate QPt . The subsequent single-mode ﬁber (SMF) is used for selecting the fundamental Gaussian mode with the zero OAM. According to the symmetry of the SPDC, one obtains C l,l ¼ C l,l , and hence 1 9FSt,t0 ¼ pﬃﬃﬃ ð9RSt 9kt St0 þ9LSt 9kt St0 Þ: 2

ð8Þ

It is necessary to note that the above-mentioned SAM-based OAM hybrid entanglement state plays an important role for our OAM entanglement swapping. Actually, the hybrid entanglement was elegantly employed for teleportation of an OAM generator based on the SAM Bell-state measurement (BSM) [25]. Since this hybrid entanglement can be generated from entangled photon pairs emitted by SPDC with a q plate coupling the spin and orbital degrees of freedom of the corresponding photon [26], it can be deployed in practical experimentations for creating the pure OAM Bell-states or other OAM entanglement states in multi-dimensional Hilbert space [27]. Based on four pairs of the hybrid entanglement states in Eq. (8), one combines the crossing hybrid entanglement photons in different modes as follows 9GS110 440 ¼ 9FS110 9FS440 ,

4623

1 ð9F þ S14 9Y þ S10 40 þ 9F S14 9O þ S10 40 2 þi9C þ S14 9O S10 40 i9C S14 9Y S10 40 Þ,

ð10Þ

1 ð9F þ S23 9Y þ S20 30 þ 9F S23 9O þ S20 30 2 þi9C þ S23 9O S20 30 i9C S23 9Y S20 30 Þ:

ð11Þ

7

7

The notations 9F S and 9C S denote the SAM Bell-states 1 9F 7 S ¼ pﬃﬃﬃ ð9HS9HS 79VS9VSÞ, 2 1 9C 7 S ¼ pﬃﬃﬃ ð9HS9VS 79VS9HSÞ, 2

ð12Þ

while 9Y 7 S and 9O 7 S represent the OAM Bell-states 1 9O 7 S ¼ pﬃﬃﬃ ð9mS9nS 79mS9nSÞ, 2 1 9Y 7 S ¼ pﬃﬃﬃ ð9mS9nS 79mS9nSÞ, 2

ð13Þ

where ðm,nÞ A fðk1 ,k4 Þ,ðk2 ,k3 Þg. The key process in the afore-mentioned transformation is that quantum mechanics allows for the relations of the horizontal and circular polarizations given by [27,28,38] 1 i 9HS ¼ pﬃﬃﬃ ð9LS þ 9RSÞ,9VS ¼ pﬃﬃﬃ ð9LS9RSÞ: 2 2

ð14Þ

According to the hybrid entanglement states in Eqs. (10) and (11), it shows the principle of OAM entanglement swapping in two successive phases via the SAM-based OAM Bell-state analysis. In the ﬁrst phase, the SAM Bell-state measurements of photons (1, 4) and (2, 3) result in two pure OAM entanglements on ð10 ,40 Þ and ð20 ,30 Þ, respectively. Namely, photons (1, 4) and (2, 3) are, respectively, projected onto one of four SAM Bell-states. In essence, it is an entanglement transferring from SAM-entanglement to OAM-entanglement. The practical difﬁculty of this process is to identify unambiguously four SAM Bell-states 9F 7 S and 9C 7 S. However, it is sufﬁcient to project photons (1, 4) and (2, 3) while superposing them at the non-polarizing beam splitters (BS) and registering coincidence counts between outputs of these BSs. After performing the SAM Bell-state analysis, one obtains four-qudit OAM states on photons ð10 ,40 Þ and ð20 ,30 Þ, as illustrated in Table 1. In the second phase, we consider a case where photons (1, 4) collapse into 9F þ S14 while photons (2, 3) collapse into 9F þ S23 as results of the SAM Bell-state measurements. The remaining photons are correspondingly projected to the combined OAM state 9XS10 40 20 30 given by 9XS10 40 20 30 ¼ 9Y þ S10 40 9Y þ S20 30 1 ¼ ð9mS9nS þ9mS9nSÞð9mS9nSþ 9mS9nSÞ: 2 ð15Þ 0

9GS220 330 ¼ 9FS220 9FS330 ,

ð9Þ

0

It also shows that the emerging state of photons ð1 ,3 Þ has an intimate relationship with photons ð40 ,20 Þ due to the fact that

Table 1 Relations of the SAM Bell-state measurement on two pairs of photons and the resulting OAM Bell-states. Measurement Basis

9F þ S23

9F S23

9C þ S23

9C S23

9F þ S14

9Y þ S10 40 9Y þ S20 30

9Y þ S10 40 9O þ S20 30

i9Y þ S10 40 9O S20 30

i9Y þ S10 40 9Y S20 30

9F S14

9O þ S10 40 9Y þ S20 30

9O þ S10 40 9O þ S20 30

i9O þ S10 40 9O S20 30

i9O þ S10 40 9Y S20 30

9C þ S14

i9O S10 40 9Y þ S20 30

i9O S10 40 9O þ S20 30

9O S10 40 9O S20 30

9O S10 40 9Y S20 30

9C S14

i9Y S10 40 9Y þ S20 30

i9Y S10 40 9O þ S20 30

9Y S10 40 9O S20 30

9Y S10 40 9Y S20 30

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Y. Guo et al. / Optics Communications 285 (2012) 4621–4625

9XS10 40 20 30 in Eq. (15) can be rewritten as 9XS10 40 20 30 ¼

photons in spatial modes ð10 ,30 Þ given by

1 ð9Y þ S10 30 9Y þ S40 20 9Y S10 30 9Y S40 20 2 9O þ S10 30 9O þ S40 20 þ9O S10 30 9O S40 20 Þ:

ð16Þ

Next we perform the OAM Bell-state analysis, instead of the SAM Bell-state analysis, on photons ð10 ,30 Þ. According to the results of the SAM Bell-state measurements represented in Eq. (16), the emerging OAM state of photons ð40 ,20 Þ equals to that of photons ð10 ,30 Þ. As for the OAM Bell-state measurement on photons ð10 ,30 Þ, we demonstrate an experimental setup to realize the similar implementation of the relevant OAM state measurements using the cross-Kerr non-linearities [39]. The principle of the OAM Bell-state analysis is shown in Fig. 2 using the quantum nondemolition detectors (QND) with two steps. Without loss of generality, we consider the pure OAM Bell-state analysis performed on photons ð10 ,30 Þ, which results in the OAM Bell-state on the remaining photons ð40 ,20 Þ via OAM entanglement swapping. In order to show the OAM Bell-state measurement on photons ð10 ,30 Þ, one has to distinguish unambiguously four OAM Bell-states, i.e., 9Y 7 S10 30 and 9O 7 S10 30 . According to Fig. 2(a) in step 1, two groups of the OAM Bell-states f9Y 7 S10 30 g and f9O 7 S10 30 g on modes ð10 ,30 Þ with an auxiliary photon in coherent state 9aS evolve as follows: 9O 7 S10 30 9aS/9O 7 S10 30 9aS,

ð17Þ

1 9Y 7 S10 30 9aS/ ð9mS10 9nS30 9aeiy S7 9mS10 9nS30 9aeiy SÞ: 2 ð18Þ One can distinguish f9Y 7 S10 30 g and f9O 7 S10 30 g using the distinctive phase shift through the homodyne–heterodyne measurement on the auxiliary photon without destroying photons ð10 ,30 Þ. Specifying in rough, the initial state on photons ð10 ,30 Þ is one of the OAM Bellstates in an assemble f9Y 7 S10 30 g when 9mS10 9nS30 (or 9mS10 9nS30 ) picks up no phase shift; otherwise, it is one of the OAM Bellstates in f9O 7 S10 30 g picking up the phase shift y (or y). In this manner, one distinguishes two groups of the OAM Bell-states, i.e., f9Y 7 S10 30 g and f9O 7 S10 30 g,according to the phase shift achieved from the homodyne–heterodyne measurement on auxiliary photon. In step 2, one further distinguishes two OAM Bell-states of each group, as shown in Fig. 2(b), where the abbreviation BS denotes the 50:50 beam splitter that accomplishes the transformations on

m

BS

k l

m n

BS

k

n

l 2

1 1 9mS10 / pﬃﬃﬃ ð9kS10 þ 9lS10 Þ,9nS30 / pﬃﬃﬃ ð9kS30 þ 9lS30 Þ, 2 2

ð19Þ

1 1 9mS10 / pﬃﬃﬃ ð9kS10 9lS10 Þ,9nS30 / pﬃﬃﬃ ð9kS30 9lS30 Þ, 2 2

ð20Þ

where 9kS10 and 9lS10 are the transformed qudits on mode 10 while 9kS30 and 9lS30 represent the transformed qudits on mode 30 . These transformations show the correlations of OAM states in different computation bases. After passing through two BSs [39–41], the OAM Bell-states f9Y 7 S10 30 ,9O 7 S10 30 g evolve as follows: 1 9Y þ S10 30 / ð9kS10 9kS30 þ 9lS10 9lS30 , 2

ð21Þ

1 9Y S10 30 / ð9kS10 9lS30 þ9lS10 9kS30 , 2

ð22Þ

1 9O þ S10 30 / ð9kS10 9kS30 9lS10 9lS30 , 2

ð23Þ

1 9O S10 30 / ð9kS10 9lS30 9lS10 9kS30 : 2

ð24Þ

Let y denote the phase shift of the cross-Ker non-linear media [42], and then the transformed OAM states 9kS10 9kS30 , 9lS10 9lS30 , 9kS10 9lS30 and 9lS10 9kS30 achieve the phase shifts 3y, 3y, y and y, respectively. Based on these phase shifts, one distinguishes the OAM Bell-states in each group at different outports. That is, photons ð10 ,30 Þ are originally in f9Y þ S10 30 ,9O þ S10 30 g when they appear in state 9kS10 9kS30 (or 9lS10 9lS30 ) with the phase shift 3y (or 3y); otherwise they are in f9Y S10 30 ,9O S10 30 g when 9kS10 9lS30 (or 9lS10 9kS30 ) picks up the phase shift y (or y). In brief, one can in principle distinguish unambiguously four OAM Bell-states in the transformed spatial modes in the non-destructive way according to the measurement result of the auxiliary coherent beam without destroying photons ð10 ,30 Þ. Based on the afore-mentioned SAM-based OAM Bell-state analysis, we can show how the detection of an OAM Bell-state on photons ð10 ,30 Þ results in projecting another OAM Bell-state on the remaining photons ð40 ,20 Þ, which is the principle of the OAM entanglement swapping. According to the process of the pure OAM Bell-state analysis on photons ð10 ,30 Þ, the OAM Bell-states in an assemble f9Y 7 S10 30 ,9O 7 S10 30 g are orthogonal, which may serve as an OAM Bell-state measurement basis while implementing OAM entanglement swapping on the basis of the OAM entanglement relation illustrated in Eq. (16). In detail, the measurement of the OAM Bell-state on photons ð10 ,30 Þ can always lead to an OAM Bellstate on remaining photons ð40 ,20 Þ. As an example for the combined OAM entanglement state 9XS10 40 20 30 in Eq. (15) shown in Table 1, after performing the OAM Bell-state analysis on photons ð10 ,30 Þ with an OAM Bell-state 9Y þ S10 30 , we gain the same OAM Bell-state 9Y þ S40 20 on photons ð40 ,20 Þ. Namely, the OAM-entanglement state on photons ð40 ,20 Þ can be induced from the OAM Bell-state analysis on photons ð10 ,30 Þ, and vice versa.

2

Homodyne 4. Conclusion Fig. 2. The OAM Bell-state analysis of photons in modes ð10 ,30 Þ. The states f9mS,9mSg are in mode 10 while f9nS,9nSg are in mode 30 . (a) After the nonlinear interactions, the probe beam picks up no phase shift or the phase shift in f7 yg, from which two groups f9Y 7 S10 30 g and f9O 7 S10 30 g can be roughly distinguished. If two photons have no phase shift, they are originally in f9Y 7 S10 30 g. But if two photons have the phase shift in f 7 yg, they are originally in f9O 7 S10 30 g. (b) After photons pass through two BSs, they pick up the distinctive phase shifts 3y, 3y, y and y from the transformed states 9kS9kS, 9lS9lS, 9kS9lS and 9lS9kS, respectively. If the resulting photons have the phase shift in f7 3yg, they are initially in one of states in f9Y þ S10 30 ,9O þ S10 30 g. But if they have the phase shift in f7 yg, they are in f9Y S10 30 ,9O S10 30 g.

We have created a fashion of OAM entanglement swapping for transferring the pure OAM entanglement in multi-dimensional Hilbert space via the SAM-based OAM Bell-state analysis. We show how to transfer the pure two-qudit entanglement state in the combined multi-dimensional Hilbert space with OAM entanglement swapping while implementing the SAM-based OAM Bell-state measurements on the corresponding photons subsequently. It provides an alternative technique to transfer OAM sates with a

Y. Guo et al. / Optics Communications 285 (2012) 4621–4625

quantum repeater in multi-dimensional Hilbert space, where entanglement swapping serves as a critical constituent.

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