- Email: [email protected]

Teleportation of two-photon entangled state via linear optical elements Liu Ye a,b,*, Jin Zhang b, Guang-Can Guo a a

Key Laboratory of Quantum Information, University of Science and Technology of China, Chinese Academy of Sciences, Hefei 230026, China b Department of Physics, Anhui University, Hefei 230039, China Received 5 February 2002; received in revised form 14 January 2003; accepted 14 January 2003

Abstract A scheme for teleporting a two-photon entangled state is proposed by using linear optical elements and postselection based on the output of single-photon detectors. It is shown that the probability of successful teleportation is 50%. By generalizing the scheme, the teleportation of multi-photon entangled state can also be realized. Ó 2003 Published by Elsevier Science B.V. PACS: 03.67.)a; 03.65.Ta; 42.50.)p Keywords: Teleportation; Two-photon entangled state; Polarizing beam splitters

In quantum teleportation [1], Alice and Bob share an entangled state (Bell state) as quantum channel. With the help of some classical information, an unknown state of a two-state particle can be teleported from Alice to Bob. Based on the quantum teleportation protocol, the teleportation of a photon in an unknown polarization state has been demonstrated experimentally by Bouwmeester et al. [2] and Boschi et al. [3] using parametric down-conversion. Apart from two-particle entangled states, Greenberger–Horne–Zeilinger [4] (GHZ) states also can be used as quantum chan-

*

Corresponding author. E-mail address: [email protected] (L. Ye).

nel. Karlsson and Bourennane [5] investigate the teleportation of a quantum state using three-particle entanglement. Gorbachev and Trubiko [6] considered the quantum teleportation of two-particle entangled state by three-particle GHZ state. Shi et al. [7] proposed the scheme of probabilistic teleportation of a two-particle entangled state by using a three-particle entangled state. The threephoton GHZ state has been realized experimentally [8]. From an experimental point of view linear optical elements might play an important role in development of quantum information applications [9]. Recently Lee et al. [10] and Lombardi et al. [11], respectively, propose a scheme to realize quantum teleportation of entangled states of a vacuum and one-photon qubit only requiring

0030-4018/03/$ - see front matter Ó 2003 Published by Elsevier Science B.V. doi:10.1016/S0030-4018(03)01168-4

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L. Ye et al. / Optics Communications 218 (2003) 333–336

linear optical devices such as beam splitters and phase shifter. These quantum teleportation protocols involve single-qubit state. Multi-particle quantum teleportation may be extremely useful in quantum computation and quantum communication [12]. It is therefore an important question to teleport a multi-particle state. In this paper, we propose an experimentally feasible method to teleport two polarized photon entangled state using linear optical devices (polarization beam splitter and quarter wave plate). In the proposal, a three-photon GHZ state is used as the quantum channel, which is needed for the teleportation of a two-photon entangled state. Meanwhile we show that the probability of successful teleportation for this scheme is 50%, which is optimal [13]. By a straightforward generalization of the scheme, teleportation of a multi-photon entangled state can be performed. The polarization conventions [14] that will be used in the paper are shown in Fig. 1. The horizontally polarized photon and vertically polarized photon will be represented by jH i and jV i, respectively, but measurements will also be made in jH 0 i and jV 0 i basis shown in the ﬁgure. Polarizing beam splitters (PBS) oriented in the HV basis will always transmit H-polarized photons and reﬂect V-polarized photons, while polarizing beam splitters oriented in the H 0 V 0 basis will transmit H 0 -polarized photons and reﬂect V 0 -polarized photons. Consider the experiment shown schematically in Fig. 2. A sender Alice has an unknown state j/i12 , which she wants to teleport to a receiver Bob. Alice and Bob share a three-photon entan-

Fig. 1. Orientations of the HV and H 0 V 0 polarization bases used in the paper. The H 0 V 0 basis is rotated 45° with respect to the HV basis.

Fig. 2. Schematic representation of the experimental setup for the teleportation of an entangled polarized photon state.

gled state jUi345 . Photon 3 belongs to Alice, and photons 4 and 5 belong to Bob. AliceÕs photon 3 and photon 2 of the state j/i12 are incident on the polarizing beam splitter (PBS) 1. After this PBS1, the photons on the output modes a, b proceed to detectors Da , Db , at the same time, photon 1 on the mode 1 is directed to detector D1 . The details of Di (i ¼ a; b; 1) is shown in the dashed-box of Fig. 2. The dashed-box insert consists of a polarizing beam splitter in the H 0 V 0 basis followed by two ordinary single-photon detectors. Alice makes a measurement on Da , Db , D1 , and tells Bob her measurement results via a classical channel (shown by a wavy line in Fig. 2). Bob performs a relevant operation. Then he can transform his photons 3 and 4 into his wanted state. So the teleportation is achieved. Now we will describe how to realize this teleportation scheme in more detail. First, we set up a three-photon GHZ state for photons 3–5 to be used as quantum channel between Alice and Bob, which is in the following state: 1 jUi345 ¼ pﬃﬃﬃ ðjH i3 jH i4 jH i5 þ jV i3 jV i4 jV i5 Þ: 2

ð1Þ

L. Ye et al. / Optics Communications 218 (2003) 333–336

The above state can be generated in the laboratory using entanglement swapping starting from three down converters [15] or as recently demonstrated experimentally using two pairs of entangled photons. We suppose Alice has an entangled photon pair, which consists of photons 1 and 2. She wants to teleport the unknown state j/i12 of the photon pair to Bob. The state j/i12 may be expressed as j/i12 ¼ ajH i1 jH i2 þ bjV i1 jV i2 ;

ð2Þ

where jaj2 þ jbj2 ¼ 1. Such a state can be prepared experimentally using a spontaneous parametric down conversion photon source [16] for an unknown pump polarization angle. The state of the entire photon ﬁeld incident on PBS 1 is j/iin ¼ j/i12 jUi345 1 ¼ pﬃﬃﬃ ðajH i1 jH i2 þ bjV i1 jV i2 ÞðjH i3 jH i4 jH i5 2 þ jV i3 jV i4 jV i5 Þ: ð3Þ After passing through PBS1, the incident state will evolve into 1 j/iin ! pﬃﬃﬃ ½ðajH i4 jH i5 bjV i4 jV i5 Þ 2

ðjH i1 jH ia jH ib jV i1 jV ia jV ib Þ þ ðajV i4 jV i5 bjH i4 jH i5 ÞðjH i1 jH ib jV ib jV i1 jV ia jH ia Þ: ð4Þ Using the polarization conventions shown in Fig. 1 to write the mode i (i ¼ a; b; 1) amplitudes in the H 0 V 0 basis leads to j/iin !

1 ðajH i4 jH i5 þ bjV i4 jV i5 Þ½ðjH 0 ia jH 0 ib 2 þ jV 0 ia jV 0 ib ÞjH 0 i1 þ ðjH 0 ia jV 0 ib þ jV 0 ia jH 0 ib ÞjV 0 i1 ðajH i4 jH i5 bjV i4 jV i5 Þ½ðjH 0 ia jH 0 ib þ jV 0 ia jV 0 ib ÞjV 0 i1 þ ðjH 0 ia jV 0 ib þ jV 0 ia jH 0 ib ÞjH 0 i1 ð5Þ þ ðajV i4 jV i5 bjH i4 jH i5 ÞjwiP ;

where jwiP ¼ ðjH 0 i1 jV 0 i1 ÞðjH 0 ib jH 0 ib jV 0 ib j V 0 ib Þ ðjH 0 i1 þ jV 0 i1 ÞðjH 0 ia jH 0 ia jV 0 ia jV 0 ia Þ. If Alice accepts the outcomes in which detectors 0 0 0 DH1 , DHa , DHb simultaneously receive one photon or 0 0 H0 detectors D1 , DVa , DVb simultaneously receive one

335 0

0

0

photon, or detectors DV1 , DHa , DVb simultaneously 0 0 0 receive one photon or detectors DV1 , DVa , DHb simultaneously receive one photon, the state of photons 4 and 5 collapses to ajH i4 jH i5 þ bjV i4 jV i5 , exactly the state that Alice wants to teleport to Bob. In this case, Bob needs to do nothing and teleportation is successfully achieved. If Alice ac0 0 0 cepts the outcomes in which detectors DV1 , DHa , DHb simultaneously receive one photon or detectors 0 0 0 DV1 , DVa , DVb simultaneously receive one photon, 0 0 0 or detectors DH1 , DHa , DVb simultaneously receive 0 0 0 one photon or detectors DH1 , DVa , DHb simultaneously receive one photon, the state of photons 4, 5 collapses to ajH i4 jH i5 bjV i4 jV i5 . If Bob is informed of such a measurement result from Alice through classical communication channels, he needs to apply a quarter-wave plate set in the output port of photon 4 to generate a p phase shift, and teleportation is then successfully achieved. For the states jwiQ , there are always two photons registered in the same detector. Alice cannot distinguish the two states jwiþ P and jwiP , therefore the probability of success for the scheme of two-photon entanglement teleportation is 50%. The teleportation described above can be generalized to the case of multi-photon entanglement in a straightforward way. Consider teleportation of an N-photon entangled state of the form ajH i1 jH i2 jH in þ bjV i1 jV i2 jV in :

ð6Þ

The quantum channel that links Alice and Bob is provided by an N þ 1-photon entangled state 1 pﬃﬃﬃ ðjH inþ1 jH inþ2 jH i2n jH i2nþ1 2 þ jV inþ1 jV inþ2 jV i2n jV i2nþ1 Þ:

ð7Þ

In Eqs. (6) and (7) the nth photon and the (n + 1)th photon are sent to PBS. Alice makes particular combinations of coincidence detection of a single photon among (N þ 1) detectors. An appropriate unitary transformation performed by Bob, then photons ðn þ 1Þ; . . . ; ð2nÞ, (2n + 1) will collapse to the state Alice that wants to teleport. The teleportation is completed. The probability of success for teleportation of an N-photon entangled state is 50%.

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Let us consider the case N ¼ 3. The case corresponds to teleportation of an unknown threephoton entangled state ajH i1 jH i2 jH i3 þ bjV i1 jV i2 jV i3 . The quantum channel in this case is provided by a four-photon entangled state p1ﬃﬃ2ðjH i4 jH i5 jH i6 j H i7 þ jV i4 jV i5 jV i6 jV i7 Þ, which can be generated using the method of Pan et al. [17]. Let the photons 3 and 4 pass through PBS and make a measurement in H 0 V 0 basis with four detectors. If Alice simultaneously receives the same polarized (horizontally or vertically) photon at detectors Da (D1 ) and Db (D2 ) (corresponding to the input state 0þ jU0þ ab ijU12 i), or receives contrary polarized photon at detectors Da (D1 ) and Db (D2 ) (corresponding to 0þ the input state jW0þ ab ijW12 i), then the state of the photons 5–7 reduces to (ajH i5 jH i6 jH i7 þ bjV i5 j V i6 jV i7 ). If Alice simultaneously obtains the same polarized photon at detectors Da , Db , and contrary polarized photon at detectors D1 , D2 (corre0þ sponding to the input state jU0þ ab ijW12 i) or Alice simultaneously obtains the contrary polarized photon at detectors Da , Db , and the same polarized photon at detectors D1 , D2 (corresponding to the 0þ input state jW0þ ab ijU12 i), then the state of the photons 5–7 reduces to (ajH i5 jH i6 jH i7 bjV i5 jV i6 j V i7 ). The total probability of the teleportation of the three-photon entangled state is 50%. In conclusion, we have proposed an experimentally feasible protocol for implementing quantum teleportation of the polarization-entangled photon using linear optical devices. The scheme utilizes two-photon entanglement and three-photon entanglement generalizations, and requires detection of photons at a single-photon level. In particular, the Bell measurements required in the scheme consist of identifying particular combinations of coincidence detection of a single photon among three detectors. By using linear optical elements, half of the four Bell states can be discriminated. So the probability of successful teleportation is 50%, which is optimal. Our

scheme can also be performed in the case of multiphoton entangled states. With the enhancement of the sensitivity of the detectors, so that the detectors can distinguish between no photon, one photon or two photons, the experimental realization of the scheme seems to be within the reach of the present technology.

Acknowledgements We thank the referees for helpful suggests and comments. This work was funded by National Fundamental Research Program (2001CB309300), the Chinese Natural Science Foundation, the innovation funds from Chinese Academy of Sciences.

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