Abstract

Entangled coherent states can be prepared remotely by subtracting nonlocally a single photon from two quantum superpositions of coherent states, the so-called “Schrödinger’s cat” state. Such entanglement can further be distributed over longer distances by successive entanglement swapping operations using linear optics and photon-number resolving detectors. The aim of this paper is to evaluate the performance of this approach to quantum repeaters for long-distance quantum communications. Despite many attractive features at first sight, we show that, when using state-of-the-art photon counters and quantum memories, they do not achieve higher entanglement generation rates than repeaters based on single-photon entanglement. We discuss potential developments, which may take better advantage of the richness of entanglement based on continuous variables, including in particular efficient parity measurements.

© 2010 Optical Society of America

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2009

N. Sangouard, R. Dubessy, and C. Simon, “Quantum repeaters based on single trapped ions,” Phys. Rev. A 79, 042340 (2009).
[CrossRef]

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[CrossRef]

2008

M. Riebe, T. Monz, K. Kim, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Deterministic entanglement swapping with an ion-trap quantum computer,” Nat. Phys. 4, 839–842 (2008).
[CrossRef]

Z.-S. Yuan, Y.-A. Chen, B. Zhao, S. Chen, J. Schmiedmayer, and J.-W. Pan, “Experimental demonstration of BDCZ quantum repeater node,” Nature 454, 1098–1101 (2008).
[CrossRef] [PubMed]

N. Sangouard, C. Simon, B. Zhao, Y.-A. Chen, H. De Riedmatten, J.-W. Pan, and N. Gisin, “Robust and efficient quantum repeaters with atomic ensembles and linear optics,” Phys. Rev. A 77, 062301 (2008).
[CrossRef]

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101, 233605 (2008).
[CrossRef] [PubMed]

A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16, 3032–3040 (2008).
[CrossRef] [PubMed]

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzkan, S. Gobler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

2007

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef] [PubMed]

C. Simon, H. De Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, “Quantum repeaters with photon pair sources and multimode memeries,” Phys. Rev. Lett. 98, 190503 (2007).
[CrossRef] [PubMed]

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical Schroedinger cats from photon number states,” Nature 448, 784–786 (2007).
[CrossRef] [PubMed]

N. Sangouard, C. Simon, J. Minar, H. Zbinden, H. De Riedmatten, and N. Gisin, “Long-distance entanglement distribution with single-photon sources,” Phys. Rev. A 76, 050301(R) (2007).
[CrossRef]

C.-W. Chou, J. Laurat, H. Deng, K. S. Choi, H. De Riedmattenn, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science 316, 1316–1320 (2007).
[CrossRef] [PubMed]

J. Laurat, C.-W. Chou, H. Deng, K.-S. Choi, D. Felinto, H. De Riedmatten, and H. J. Kimble, “Towards experimental entanglement connection with atomic ensembles in the single excitation regime,” New J. Phys. 9, 207 (2007).
[CrossRef]

L. Jiang, M. Taylor, and M. Lukin, “Fast and robust approach to long-distance quantum communication with atomic ensembles,” Phys. Rev. A 76, 012301 (2007).
[CrossRef]

Z.-B. Chen, B. Zhao, Y.-A. Chen, J. Schmiedmayer, and J.-W. Pan, “Fault tolerant quantum repeater with atomic ensembles and linear optics,” Phys. Rev. A 76, 022329 (2007).
[CrossRef]

B. Zhao, Z.-B. Chen, Y.-A. Chen, J. Schmiedmayer, and J.-W. Pan, “Robust creation of entanglement between remote memory qubits,” Phys. Rev. Lett. 98, 240502 (2007).
[CrossRef] [PubMed]

C. Simon, Y.-M. Niquet, X. Caillet, J. Eymery, J.-P. Poizat, and J.-M. Gerard, “Quantum communication with quantum dot spins,” Phys. Rev. B 75, 081302(R) (2007).
[CrossRef]

2006

L. Childress, J. M. Taylor, A. S. Sørensen, and M. D. Lukin, “Fault-tolerant quantum communication based on solid-state photon emiiters,” Phys. Rev. Lett. 96, 070504 (2006).
[CrossRef] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schroedinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef] [PubMed]

J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Molmer, and E. S. Polzik, “Generation of superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef] [PubMed]

2005

J. Laurat, T. Coudreau, G. Keller, N. Treps, and C. Fabre, “Effects of mode coupling on the generation of quadrature Einstein-Podolsky-Rosen entanglement in a type-II optical parametric oscillator below threshold,” Phys. Rev. A 71, 022313 (2005).
[CrossRef]

N. Takei, H. Yonezawa, T. Aoki, and A. Furusawa, “High-fidelity teleportation beyond the no-cloning limit and entanglement swapping for continuous variables,” Phys. Rev. Lett. 94, 220502 (2005).
[CrossRef] [PubMed]

2001

J. Calsamiglia and N. Lütkenhaus, “Maximum efficiency of a linear-optical Bell-state analyzer,” Appl. Phys. B 72, 67–71 (2001).
[CrossRef]

S. J. van Enk and O. Hirota, “Entangled coherent states: Teleportation and decoherence,” Phys. Rev. A 64, 022313 (2001).
[CrossRef]

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
[CrossRef] [PubMed]

2000

T. Opatrny, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
[CrossRef]

1998

A. Furusawa, J. L. Sorensen, S. L. Braustein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: The role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
[CrossRef]

1992

B. C. Sanders, “Entangled coherent states,” Phys. Rev. A 45, 6811–6815 (1992).
[CrossRef] [PubMed]

Afzelius, M.

C. Simon, H. De Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, “Quantum repeaters with photon pair sources and multimode memeries,” Phys. Rev. Lett. 98, 190503 (2007).
[CrossRef] [PubMed]

Aoki, T.

N. Takei, H. Yonezawa, T. Aoki, and A. Furusawa, “High-fidelity teleportation beyond the no-cloning limit and entanglement swapping for continuous variables,” Phys. Rev. Lett. 94, 220502 (2005).
[CrossRef] [PubMed]

Blatt, R.

M. Riebe, T. Monz, K. Kim, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Deterministic entanglement swapping with an ion-trap quantum computer,” Nat. Phys. 4, 839–842 (2008).
[CrossRef]

Braustein, S. L.

A. Furusawa, J. L. Sorensen, S. L. Braustein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

Briegel, H.-J.

H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: The role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
[CrossRef]

Caillet, X.

C. Simon, Y.-M. Niquet, X. Caillet, J. Eymery, J.-P. Poizat, and J.-M. Gerard, “Quantum communication with quantum dot spins,” Phys. Rev. B 75, 081302(R) (2007).
[CrossRef]

Calsamiglia, J.

J. Calsamiglia and N. Lütkenhaus, “Maximum efficiency of a linear-optical Bell-state analyzer,” Appl. Phys. B 72, 67–71 (2001).
[CrossRef]

Chelkowski, S.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzkan, S. Gobler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

Chen, S.

Z.-S. Yuan, Y.-A. Chen, B. Zhao, S. Chen, J. Schmiedmayer, and J.-W. Pan, “Experimental demonstration of BDCZ quantum repeater node,” Nature 454, 1098–1101 (2008).
[CrossRef] [PubMed]

Chen, Y.-A.

Z.-S. Yuan, Y.-A. Chen, B. Zhao, S. Chen, J. Schmiedmayer, and J.-W. Pan, “Experimental demonstration of BDCZ quantum repeater node,” Nature 454, 1098–1101 (2008).
[CrossRef] [PubMed]

N. Sangouard, C. Simon, B. Zhao, Y.-A. Chen, H. De Riedmatten, J.-W. Pan, and N. Gisin, “Robust and efficient quantum repeaters with atomic ensembles and linear optics,” Phys. Rev. A 77, 062301 (2008).
[CrossRef]

Z.-B. Chen, B. Zhao, Y.-A. Chen, J. Schmiedmayer, and J.-W. Pan, “Fault tolerant quantum repeater with atomic ensembles and linear optics,” Phys. Rev. A 76, 022329 (2007).
[CrossRef]

B. Zhao, Z.-B. Chen, Y.-A. Chen, J. Schmiedmayer, and J.-W. Pan, “Robust creation of entanglement between remote memory qubits,” Phys. Rev. Lett. 98, 240502 (2007).
[CrossRef] [PubMed]

Chen, Z.-B.

B. Zhao, Z.-B. Chen, Y.-A. Chen, J. Schmiedmayer, and J.-W. Pan, “Robust creation of entanglement between remote memory qubits,” Phys. Rev. Lett. 98, 240502 (2007).
[CrossRef] [PubMed]

Z.-B. Chen, B. Zhao, Y.-A. Chen, J. Schmiedmayer, and J.-W. Pan, “Fault tolerant quantum repeater with atomic ensembles and linear optics,” Phys. Rev. A 76, 022329 (2007).
[CrossRef]

Childress, L.

L. Childress, J. M. Taylor, A. S. Sørensen, and M. D. Lukin, “Fault-tolerant quantum communication based on solid-state photon emiiters,” Phys. Rev. Lett. 96, 070504 (2006).
[CrossRef] [PubMed]

Choi, K. S.

C.-W. Chou, J. Laurat, H. Deng, K. S. Choi, H. De Riedmattenn, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science 316, 1316–1320 (2007).
[CrossRef] [PubMed]

Choi, K.-S.

J. Laurat, C.-W. Chou, H. Deng, K.-S. Choi, D. Felinto, H. De Riedmatten, and H. J. Kimble, “Towards experimental entanglement connection with atomic ensembles in the single excitation regime,” New J. Phys. 9, 207 (2007).
[CrossRef]

Chou, C.-W.

C.-W. Chou, J. Laurat, H. Deng, K. S. Choi, H. De Riedmattenn, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science 316, 1316–1320 (2007).
[CrossRef] [PubMed]

J. Laurat, C.-W. Chou, H. Deng, K.-S. Choi, D. Felinto, H. De Riedmatten, and H. J. Kimble, “Towards experimental entanglement connection with atomic ensembles in the single excitation regime,” New J. Phys. 9, 207 (2007).
[CrossRef]

Chwalla, M.

M. Riebe, T. Monz, K. Kim, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Deterministic entanglement swapping with an ion-trap quantum computer,” Nat. Phys. 4, 839–842 (2008).
[CrossRef]

Cirac, J. I.

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
[CrossRef] [PubMed]

H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: The role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
[CrossRef]

Coudreau, T.

J. Laurat, T. Coudreau, G. Keller, N. Treps, and C. Fabre, “Effects of mode coupling on the generation of quadrature Einstein-Podolsky-Rosen entanglement in a type-II optical parametric oscillator below threshold,” Phys. Rev. A 71, 022313 (2005).
[CrossRef]

Dantan, A.

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef] [PubMed]

Danzmann, K.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzkan, S. Gobler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

De Riedmatten, H.

N. Sangouard, C. Simon, B. Zhao, Y.-A. Chen, H. De Riedmatten, J.-W. Pan, and N. Gisin, “Robust and efficient quantum repeaters with atomic ensembles and linear optics,” Phys. Rev. A 77, 062301 (2008).
[CrossRef]

J. Laurat, C.-W. Chou, H. Deng, K.-S. Choi, D. Felinto, H. De Riedmatten, and H. J. Kimble, “Towards experimental entanglement connection with atomic ensembles in the single excitation regime,” New J. Phys. 9, 207 (2007).
[CrossRef]

N. Sangouard, C. Simon, J. Minar, H. Zbinden, H. De Riedmatten, and N. Gisin, “Long-distance entanglement distribution with single-photon sources,” Phys. Rev. A 76, 050301(R) (2007).
[CrossRef]

C. Simon, H. De Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, “Quantum repeaters with photon pair sources and multimode memeries,” Phys. Rev. Lett. 98, 190503 (2007).
[CrossRef] [PubMed]

N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” arXiv:0906.2699

De Riedmattenn, H.

C.-W. Chou, J. Laurat, H. Deng, K. S. Choi, H. De Riedmattenn, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science 316, 1316–1320 (2007).
[CrossRef] [PubMed]

Deng, H.

J. Laurat, C.-W. Chou, H. Deng, K.-S. Choi, D. Felinto, H. De Riedmatten, and H. J. Kimble, “Towards experimental entanglement connection with atomic ensembles in the single excitation regime,” New J. Phys. 9, 207 (2007).
[CrossRef]

C.-W. Chou, J. Laurat, H. Deng, K. S. Choi, H. De Riedmattenn, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science 316, 1316–1320 (2007).
[CrossRef] [PubMed]

Duan, L.-M.

L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
[CrossRef] [PubMed]

Dubessy, R.

N. Sangouard, R. Dubessy, and C. Simon, “Quantum repeaters based on single trapped ions,” Phys. Rev. A 79, 042340 (2009).
[CrossRef]

Dür, W.

H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: The role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
[CrossRef]

Eymery, J.

C. Simon, Y.-M. Niquet, X. Caillet, J. Eymery, J.-P. Poizat, and J.-M. Gerard, “Quantum communication with quantum dot spins,” Phys. Rev. B 75, 081302(R) (2007).
[CrossRef]

Fabre, C.

J. Laurat, T. Coudreau, G. Keller, N. Treps, and C. Fabre, “Effects of mode coupling on the generation of quadrature Einstein-Podolsky-Rosen entanglement in a type-II optical parametric oscillator below threshold,” Phys. Rev. A 71, 022313 (2005).
[CrossRef]

Felinto, D.

J. Laurat, C.-W. Chou, H. Deng, K.-S. Choi, D. Felinto, H. De Riedmatten, and H. J. Kimble, “Towards experimental entanglement connection with atomic ensembles in the single excitation regime,” New J. Phys. 9, 207 (2007).
[CrossRef]

C.-W. Chou, J. Laurat, H. Deng, K. S. Choi, H. De Riedmattenn, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science 316, 1316–1320 (2007).
[CrossRef] [PubMed]

Ferreyrol, F.

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[CrossRef]

Franzen, A.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzkan, S. Gobler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

Fuchs, C. A.

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H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101, 233605 (2008).
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Gerard, J.-M.

C. Simon, Y.-M. Niquet, X. Caillet, J. Eymery, J.-P. Poizat, and J.-M. Gerard, “Quantum communication with quantum dot spins,” Phys. Rev. B 75, 081302(R) (2007).
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Gisin, N.

N. Sangouard, C. Simon, B. Zhao, Y.-A. Chen, H. De Riedmatten, J.-W. Pan, and N. Gisin, “Robust and efficient quantum repeaters with atomic ensembles and linear optics,” Phys. Rev. A 77, 062301 (2008).
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N. Sangouard, C. Simon, J. Minar, H. Zbinden, H. De Riedmatten, and N. Gisin, “Long-distance entanglement distribution with single-photon sources,” Phys. Rev. A 76, 050301(R) (2007).
[CrossRef]

C. Simon, H. De Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, “Quantum repeaters with photon pair sources and multimode memeries,” Phys. Rev. Lett. 98, 190503 (2007).
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N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” arXiv:0906.2699

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H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzkan, S. Gobler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
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A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
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A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical Schroedinger cats from photon number states,” Nature 448, 784–786 (2007).
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A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
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A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schroedinger kittens for quantum information processing,” Science 312, 83–86 (2006).
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H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzkan, S. Gobler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
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H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101, 233605 (2008).
[CrossRef] [PubMed]

H. Takahashi, S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Non-gaussian entanglement distillation for continuous variables,” arXiv:0907.2159

Hennrich, M.

M. Riebe, T. Monz, K. Kim, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Deterministic entanglement swapping with an ion-trap quantum computer,” Nat. Phys. 4, 839–842 (2008).
[CrossRef]

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J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Molmer, and E. S. Polzik, “Generation of superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
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S. J. van Enk and O. Hirota, “Entangled coherent states: Teleportation and decoherence,” Phys. Rev. A 64, 022313 (2001).
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A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical Schroedinger cats from photon number states,” Nature 448, 784–786 (2007).
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L. Jiang, M. Taylor, and M. Lukin, “Fast and robust approach to long-distance quantum communication with atomic ensembles,” Phys. Rev. A 76, 012301 (2007).
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J. Laurat, T. Coudreau, G. Keller, N. Treps, and C. Fabre, “Effects of mode coupling on the generation of quadrature Einstein-Podolsky-Rosen entanglement in a type-II optical parametric oscillator below threshold,” Phys. Rev. A 71, 022313 (2005).
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M. Riebe, T. Monz, K. Kim, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Deterministic entanglement swapping with an ion-trap quantum computer,” Nat. Phys. 4, 839–842 (2008).
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A. Furusawa, J. L. Sorensen, S. L. Braustein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

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T. Opatrny, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
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H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzkan, S. Gobler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
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C.-W. Chou, J. Laurat, H. Deng, K. S. Choi, H. De Riedmattenn, D. Felinto, and H. J. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science 316, 1316–1320 (2007).
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J. Laurat, C.-W. Chou, H. Deng, K.-S. Choi, D. Felinto, H. De Riedmatten, and H. J. Kimble, “Towards experimental entanglement connection with atomic ensembles in the single excitation regime,” New J. Phys. 9, 207 (2007).
[CrossRef]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schroedinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef] [PubMed]

J. Laurat, T. Coudreau, G. Keller, N. Treps, and C. Fabre, “Effects of mode coupling on the generation of quadrature Einstein-Podolsky-Rosen entanglement in a type-II optical parametric oscillator below threshold,” Phys. Rev. A 71, 022313 (2005).
[CrossRef]

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Lukin, M.

L. Jiang, M. Taylor, and M. Lukin, “Fast and robust approach to long-distance quantum communication with atomic ensembles,” Phys. Rev. A 76, 012301 (2007).
[CrossRef]

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L. Childress, J. M. Taylor, A. S. Sørensen, and M. D. Lukin, “Fault-tolerant quantum communication based on solid-state photon emiiters,” Phys. Rev. Lett. 96, 070504 (2006).
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P. Van Loock, N. Lütkenhaus, W. J. Munro, and K. Nemoto, “Quantum repeaters using coherent-state communication,” arXiv:0806.1153

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H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzkan, S. Gobler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

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J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Molmer, and E. S. Polzik, “Generation of superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef] [PubMed]

Miller, A. J.

Minar, J.

N. Sangouard, C. Simon, J. Minar, H. Zbinden, H. De Riedmatten, and N. Gisin, “Long-distance entanglement distribution with single-photon sources,” Phys. Rev. A 76, 050301(R) (2007).
[CrossRef]

Molmer, K.

J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Molmer, and E. S. Polzik, “Generation of superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef] [PubMed]

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M. Riebe, T. Monz, K. Kim, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Deterministic entanglement swapping with an ion-trap quantum computer,” Nat. Phys. 4, 839–842 (2008).
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P. Van Loock, N. Lütkenhaus, W. J. Munro, and K. Nemoto, “Quantum repeaters using coherent-state communication,” arXiv:0806.1153

Nam, S. W.

Nam, Sae Woo

Sae Woo Nam, private communication.

Neergaard-Nielsen, J. S.

J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Molmer, and E. S. Polzik, “Generation of superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef] [PubMed]

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H. Takahashi, S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Non-gaussian entanglement distillation for continuous variables,” arXiv:0907.2159

Nemoto, K.

P. Van Loock, N. Lütkenhaus, W. J. Munro, and K. Nemoto, “Quantum repeaters using coherent-state communication,” arXiv:0806.1153

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C. Simon, Y.-M. Niquet, X. Caillet, J. Eymery, J.-P. Poizat, and J.-M. Gerard, “Quantum communication with quantum dot spins,” Phys. Rev. B 75, 081302(R) (2007).
[CrossRef]

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T. Opatrny, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
[CrossRef]

Ourjoumtsev, A.

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[CrossRef]

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical Schroedinger cats from photon number states,” Nature 448, 784–786 (2007).
[CrossRef] [PubMed]

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schroedinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef] [PubMed]

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Z.-S. Yuan, Y.-A. Chen, B. Zhao, S. Chen, J. Schmiedmayer, and J.-W. Pan, “Experimental demonstration of BDCZ quantum repeater node,” Nature 454, 1098–1101 (2008).
[CrossRef] [PubMed]

N. Sangouard, C. Simon, B. Zhao, Y.-A. Chen, H. De Riedmatten, J.-W. Pan, and N. Gisin, “Robust and efficient quantum repeaters with atomic ensembles and linear optics,” Phys. Rev. A 77, 062301 (2008).
[CrossRef]

B. Zhao, Z.-B. Chen, Y.-A. Chen, J. Schmiedmayer, and J.-W. Pan, “Robust creation of entanglement between remote memory qubits,” Phys. Rev. Lett. 98, 240502 (2007).
[CrossRef] [PubMed]

Z.-B. Chen, B. Zhao, Y.-A. Chen, J. Schmiedmayer, and J.-W. Pan, “Fault tolerant quantum repeater with atomic ensembles and linear optics,” Phys. Rev. A 76, 022329 (2007).
[CrossRef]

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C. Simon, Y.-M. Niquet, X. Caillet, J. Eymery, J.-P. Poizat, and J.-M. Gerard, “Quantum communication with quantum dot spins,” Phys. Rev. B 75, 081302(R) (2007).
[CrossRef]

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J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Molmer, and E. S. Polzik, “Generation of superposition of odd photon number states for quantum information networks,” Phys. Rev. Lett. 97, 083604 (2006).
[CrossRef] [PubMed]

A. Furusawa, J. L. Sorensen, S. L. Braustein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

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H. Jeong and T. C. Ralph, Quantum information with continuous variables of atoms and light, Chapter 9, N.Cerf, G.Leuchs, and E.Polzik, Imperial College Press.

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M. Riebe, T. Monz, K. Kim, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Deterministic entanglement swapping with an ion-trap quantum computer,” Nat. Phys. 4, 839–842 (2008).
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N. Sangouard, R. Dubessy, and C. Simon, “Quantum repeaters based on single trapped ions,” Phys. Rev. A 79, 042340 (2009).
[CrossRef]

N. Sangouard, C. Simon, B. Zhao, Y.-A. Chen, H. De Riedmatten, J.-W. Pan, and N. Gisin, “Robust and efficient quantum repeaters with atomic ensembles and linear optics,” Phys. Rev. A 77, 062301 (2008).
[CrossRef]

N. Sangouard, C. Simon, J. Minar, H. Zbinden, H. De Riedmatten, and N. Gisin, “Long-distance entanglement distribution with single-photon sources,” Phys. Rev. A 76, 050301(R) (2007).
[CrossRef]

C. Simon, H. De Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, “Quantum repeaters with photon pair sources and multimode memeries,” Phys. Rev. Lett. 98, 190503 (2007).
[CrossRef] [PubMed]

N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” arXiv:0906.2699

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H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101, 233605 (2008).
[CrossRef] [PubMed]

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O. Hirota and M. Sasaki, “Entangled state based on nonorthogonal state,” arXiv:quant-ph/0101018

Schindler, P.

M. Riebe, T. Monz, K. Kim, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Deterministic entanglement swapping with an ion-trap quantum computer,” Nat. Phys. 4, 839–842 (2008).
[CrossRef]

Schmiedmayer, J.

Z.-S. Yuan, Y.-A. Chen, B. Zhao, S. Chen, J. Schmiedmayer, and J.-W. Pan, “Experimental demonstration of BDCZ quantum repeater node,” Nature 454, 1098–1101 (2008).
[CrossRef] [PubMed]

Z.-B. Chen, B. Zhao, Y.-A. Chen, J. Schmiedmayer, and J.-W. Pan, “Fault tolerant quantum repeater with atomic ensembles and linear optics,” Phys. Rev. A 76, 022329 (2007).
[CrossRef]

B. Zhao, Z.-B. Chen, Y.-A. Chen, J. Schmiedmayer, and J.-W. Pan, “Robust creation of entanglement between remote memory qubits,” Phys. Rev. Lett. 98, 240502 (2007).
[CrossRef] [PubMed]

Schnabel, R.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzkan, S. Gobler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

Simon, C.

N. Sangouard, R. Dubessy, and C. Simon, “Quantum repeaters based on single trapped ions,” Phys. Rev. A 79, 042340 (2009).
[CrossRef]

N. Sangouard, C. Simon, B. Zhao, Y.-A. Chen, H. De Riedmatten, J.-W. Pan, and N. Gisin, “Robust and efficient quantum repeaters with atomic ensembles and linear optics,” Phys. Rev. A 77, 062301 (2008).
[CrossRef]

N. Sangouard, C. Simon, J. Minar, H. Zbinden, H. De Riedmatten, and N. Gisin, “Long-distance entanglement distribution with single-photon sources,” Phys. Rev. A 76, 050301(R) (2007).
[CrossRef]

C. Simon, Y.-M. Niquet, X. Caillet, J. Eymery, J.-P. Poizat, and J.-M. Gerard, “Quantum communication with quantum dot spins,” Phys. Rev. B 75, 081302(R) (2007).
[CrossRef]

C. Simon, H. De Riedmatten, M. Afzelius, N. Sangouard, H. Zbinden, and N. Gisin, “Quantum repeaters with photon pair sources and multimode memeries,” Phys. Rev. Lett. 98, 190503 (2007).
[CrossRef] [PubMed]

N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” arXiv:0906.2699

Sorensen, J. L.

A. Furusawa, J. L. Sorensen, S. L. Braustein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[CrossRef] [PubMed]

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L. Childress, J. M. Taylor, A. S. Sørensen, and M. D. Lukin, “Fault-tolerant quantum communication based on solid-state photon emiiters,” Phys. Rev. Lett. 96, 070504 (2006).
[CrossRef] [PubMed]

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H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101, 233605 (2008).
[CrossRef] [PubMed]

Takahashi, H.

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101, 233605 (2008).
[CrossRef] [PubMed]

H. Takahashi, S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Non-gaussian entanglement distillation for continuous variables,” arXiv:0907.2159

Takei, N.

N. Takei, H. Yonezawa, T. Aoki, and A. Furusawa, “High-fidelity teleportation beyond the no-cloning limit and entanglement swapping for continuous variables,” Phys. Rev. Lett. 94, 220502 (2005).
[CrossRef] [PubMed]

Takeoka, M.

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101, 233605 (2008).
[CrossRef] [PubMed]

H. Takahashi, S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Non-gaussian entanglement distillation for continuous variables,” arXiv:0907.2159

Takeuchi, M.

H. Takahashi, S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Non-gaussian entanglement distillation for continuous variables,” arXiv:0907.2159

Taylor, J. M.

L. Childress, J. M. Taylor, A. S. Sørensen, and M. D. Lukin, “Fault-tolerant quantum communication based on solid-state photon emiiters,” Phys. Rev. Lett. 96, 070504 (2006).
[CrossRef] [PubMed]

Taylor, M.

L. Jiang, M. Taylor, and M. Lukin, “Fast and robust approach to long-distance quantum communication with atomic ensembles,” Phys. Rev. A 76, 012301 (2007).
[CrossRef]

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J. Laurat, T. Coudreau, G. Keller, N. Treps, and C. Fabre, “Effects of mode coupling on the generation of quadrature Einstein-Podolsky-Rosen entanglement in a type-II optical parametric oscillator below threshold,” Phys. Rev. A 71, 022313 (2005).
[CrossRef]

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A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[CrossRef]

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical Schroedinger cats from photon number states,” Nature 448, 784–786 (2007).
[CrossRef] [PubMed]

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[CrossRef] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schroedinger kittens for quantum information processing,” Science 312, 83–86 (2006).
[CrossRef] [PubMed]

Vahlbruch, H.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzkan, S. Gobler, K. Danzmann, and R. Schnabel, “Observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[CrossRef] [PubMed]

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S. J. van Enk and O. Hirota, “Entangled coherent states: Teleportation and decoherence,” Phys. Rev. A 64, 022313 (2001).
[CrossRef]

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P. Van Loock, N. Lütkenhaus, W. J. Munro, and K. Nemoto, “Quantum repeaters using coherent-state communication,” arXiv:0806.1153

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M. Riebe, T. Monz, K. Kim, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Deterministic entanglement swapping with an ion-trap quantum computer,” Nat. Phys. 4, 839–842 (2008).
[CrossRef]

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H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101, 233605 (2008).
[CrossRef] [PubMed]

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T. Opatrny, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
[CrossRef]

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N. Takei, H. Yonezawa, T. Aoki, and A. Furusawa, “High-fidelity teleportation beyond the no-cloning limit and entanglement swapping for continuous variables,” Phys. Rev. Lett. 94, 220502 (2005).
[CrossRef] [PubMed]

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Z.-S. Yuan, Y.-A. Chen, B. Zhao, S. Chen, J. Schmiedmayer, and J.-W. Pan, “Experimental demonstration of BDCZ quantum repeater node,” Nature 454, 1098–1101 (2008).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

Schematic architecture of an elementary link connecting two distant locations A and B. Memories and detectors are represented by squares and half-circles respectively. Vertical bars denote beamsplitters. Odd superposition of Fock states are generated at each location and sent through an asymmetric beamsplitter with small transmission. The reflected light is stored in local memories whereas the transmitted part is combined on a 50 50 beamsplitter. The detection of a single photon at the central station heralds the storage of an entangled coherent state | ϕ θ A B defined in Eq. (6) with high probability, due to the asymmetry of the local beamsplitters.

Fig. 2
Fig. 2

Fidelity of the distributed state within an elementary link versus the tapped fraction sin 2 θ of local beamsplitters. Different amplitudes α have been considered: | α | 2 = 0.5 (solid curve), | α | 2 = 1 (dashed curve), | α | 2 = 2 (dashed-dotted curve). ( η d = 0.9 , L 0 = 100 km ).

Fig. 3
Fig. 3

Average time for the distribution of an entangled state within an elementary link versus the transmission coefficient sin 2 θ . The distance is L 0 = 100 km , and the photon-number resolving detector efficiency η d is 90%. Entangled coherent states with different amplitudes α have been considered: | α | 2 = 0.5 (solid curve), | α | 2 = 1 (dashed curve), | α | 2 = 2 (dashed-dotted curve).

Fig. 4
Fig. 4

Entanglement swapping: starting from entangled links A–B and B–C, entanglement between A and C is obtained if one retrieves the light fields stored in the memories at location B, combines them in a 50–50 beamsplitter and then counts the photon number in the output modes d b and d ̃ b . Memories and detectors are represented by squares and half-circles, respectively. The vertical bar symbolizes a beamsplitter.

Fig. 5
Fig. 5

Fidelity of the conditional state after one entanglement swapping operation as a function of the tapped fraction sin 2 θ of local beamsplitters in the case of large amplitude coherent states. For coherent states with an amplitude such that | α | 2 1 , the fidelity of the distributed state depends on the parity of the number of photons detected. Memory and photon-detector efficiencies are taken equal to η m = η d = 0.9 .

Fig. 6
Fig. 6

Fidelity of the conditional state after one entanglement swapping operation as a function of the tapped fraction of the local beamsplitters sin 2 θ in the case of small amplitude coherent states without postselection (three lower curves) and with (three upper curves) postselection.

Fig. 7
Fig. 7

Schematic representation of postselection protocol. Entanglement has been distributed independently within two chains (labelled by the subscripts 1 and 2) such that the ensembles A 1 C 1 and A 2 C 2 store entangled coherent states. Light fields stored at the same location are retrieved and combined into a beamsplitter. The detection of a single photon at each location postselects a two-photon entangled state. Memories and detectors are represented by squares and half-circles respectively. Horizontal bars denote beamsplitters.

Equations (38)

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| ϕ A B = 1 M ( | α A | α B | α A | α B ) ,
M = 2 ( 1 e 4 | α | 2 ) ,
| + = 1 N + ( | α + | α ) ,
N + = 2 ( 1 + e 2 | α | 2 ) ,
| = 1 N ( | α | α ) ,
N = 2 ( 1 e 2 | α | 2 ) ,
| ϕ ± A B = 1 M ± ( | α A | α B ± | α A | α B ) ,
| ψ ± A B = 1 M ± ( | α A | α B ± | α A | α B ) ,
M ± = 2 ( 1 ± e 4 | α | 2 )
| ϕ + | even out 1 | 0 out 2 ,
| ϕ | odd out 1 | 0 out 2 ,
| ψ + | 0 out 1 | even out 2 ,
| ψ | 0 out 1 | odd out 2 ,
| A = 1 N ( | α A | α A ) = e | α | 2 2 N ( e α a e α a ) | 0
| B = 1 N ( | α B | α B ) = e | α | 2 2 N ( e α b e α b ) | 0
| ϕ θ A B = 1 M θ ( | α cos θ A | α cos θ B | α cos θ A | α cos θ B ) ,
M θ = 2 ( 1 e 4 | α | 2 cos 2 θ )
| ψ θ A B = 1 M θ ( | α cos θ A | α cos θ B | α cos θ A | α cos θ B ) .
| ϕ + θ A B = 1 M + θ ( | α cos θ A | α cos θ B + | α cos θ A | α cos θ B ) ,
M + θ = 2 ( 1 + e 4 | α | 2 cos 2 θ ) .
ρ 0 A B = 1 M θ + 2 M + θ | α | 2 sin 2 θ ( M θ | ϕ θ A B ϕ θ | A B + 2 M + θ | α | 2 sin 2 θ | ϕ + θ A B ϕ + θ | A B ) ,
P 0 ϕ = 1 N 2 2 sin 2 θ | α | 2 e 2 sin 2 θ | α | 2 ( M θ + 2 M + θ | α | 2 sin 2 θ ) η t η d .
T 0 = L 0 c 1 P 0 .
F 0 = ϕ θ | A B ρ 0 A B | ϕ θ A B
P 0 = sin 2 θ | α | 2 e 2 sin 2 θ | α | 2 ( 2 + 4 | α | 2 sin 2 θ ) η t η d
F 0 = 1 1 + 2 | α | 2 sin 2 θ .
ρ 0 A B = F 0 | ϕ A B ϕ | A B + F + 0 | ϕ + A B ϕ + | A B ,
N odd = ( ( F 0 ) 2 + ( F + 0 M θ M + θ ) 2 ) k = 0 + 2 2 k ( 2 k ) ! ( 1 η ) 2 k | α | 4 k cos 4 k θ + 2 F 0 F + 0 M θ M + θ k = 0 + 2 2 k + 1 ( 2 k + 1 ) ! ( 1 η ) 2 k + 1 | α | 2 ( 2 k + 1 ) cos 2 ( 2 k + 1 ) θ = ( ( F 0 ) 2 + ( F + 0 M θ M + θ ) 2 ) cosh ( 2 ( 1 η ) | α | 2 cos 2 θ ) + 2 F 0 F + 0 M θ M + θ sinh ( 2 ( 1 η ) | α | 2 cos 2 θ ) ;
D odd = ( ( F 0 ) 2 + 2 F 0 F + 0 + ( F + 0 M θ M + θ ) 2 ) k = 0 + 2 2 k ( 2 k ) ! ( 1 η ) 2 k | α | 4 k cos 4 k θ + ( ( F + 0 ) 2 + 2 F 0 F + 0 + ( F 0 M + θ M θ ) 2 ) M θ M + θ k = 0 + 2 2 k + 1 ( 2 k + 1 ) ! ( 1 η ) 2 k + 1 | α | 2 ( 2 k + 1 ) cos 2 ( 2 k + 1 ) θ = ( ( F 0 ) 2 + 2 F 0 F + 0 + ( F + 0 M θ M + θ ) 2 ) cosh ( 2 ( 1 η ) | α | 2 cos 2 θ ) + ( ( F + 0 ) 2 + 2 F 0 F + 0 + ( F 0 M + θ M θ ) 2 ) M θ M + θ sinh ( 2 ( 1 η ) | α | 2 cos 2 θ ) .
P 1 n , odd = 2 M θ η n 2 n | α | 2 n cos 2 n θ n ! e 2 | α | 2 cos 2 θ × D odd .
P 1 n , even = 2 M + θ η n 2 n | α | 2 n cos 2 n θ n ! e 2 | α | 2 cos 2 θ × D even .
F 1 1 1 + tanh ( 2 ( 1 η ) | α | 2 cos 2 θ ) M + θ M θ + o ( F + 0 ) ,
G + 1 1 1 + tanh ( 2 ( 1 η ) | α | 2 cos 2 θ ) M θ M + θ + o ( F + 0 ) ,
F 1 G + 1 1 1 + tanh ( 2 ( 1 η ) | α | 2 cos 2 θ ) .
P 1 = n = 1 + ( P 1 2 n 1 , odd + P 1 2 n , even ) = 1 e 2 η | α | 2 cos 2 θ ,
F 1 1 1 + ( 1 η ) .
P 1 P 1 1 , odd 1 2 η ( 2 η ) .
T ( 3 2 ) 2 T 0 1 P 1 × P 2 × P ps .

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