Abstract

We develop a model for practical, entanglement-based long-distance quantum key distribution employing entanglement swapping as a key building block. Relying only on existing off-the-shelf technology, we show how to optimize resources so as to maximize secret key distribution rates. The tools comprise lossy transmission links, such as telecom optical fibers or free space, parametric down-conversion sources of entangled photon pairs, and threshold detectors that are inefficient and have dark counts. Our analysis provides the optimal trade-off between detector efficiency and dark counts, which are usually competing, as well as the optimal source brightness that maximizes the secret key rate for specified distances (i.e. loss) between sender and receiver.

© 2011 Optical Society of America

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  1. http://www.idquantique.com, http://www.magiqtech.com.
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    [CrossRef] [PubMed]
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    [CrossRef]
  27. 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).
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  29. J. Amirloo, M. Razavi, and A. H. Majedi, "Quantum key distribution over probabilistic quantum repeaters," Phys. Rev. A 82, 032304 (2010).
    [CrossRef]
  30. A. J. Miller, S. W. Nam, J. M. Martinis, and A. V. Sergienko, "Demonstration of a low-noise near-infrared photon counter with multiphoton discrimination," Appl. Phys. Lett. 83, 791-793 (2003).
    [CrossRef]
  31. D. Rosenberg, A. E. Lita, A. J. Miller, S. W. Nam, and R. E. Schwall, "Performance of photon-number resolving transition-edge sensors with integrated 1550 nm resonant cavities," IEEE Trans. Appl. Supercond. 15(2), 575-578 (2005).
    [CrossRef]
  32. D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, "Noise-free high-efficiency photon-number-resolving detectors," Phys. Rev. A 71, 061803 (2005).
    [CrossRef]
  33. G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, "Picosecond superconducting single-photon optical detector," Appl. Phys. Lett. 79, 705-707 (2001).
    [CrossRef]
  34. K. M. Rosfjord, J. K. W. Yang, E. A. Dauler, A. J. Kerman, V. Anant, B. M. Voronov, G. N. Gol’tsman, and K. K. Berggren, "Nanowire single-photon detector with an integrated optical cavity and anti-reflection coating," Opt. Express 14, 527-534 (2006).
    [CrossRef] [PubMed]
  35. A. J. Kerman, E. A. Dauler, W. E. Keicher, J. K. W. Yang, K. K. Berggren, G. Gol’tsman, and B. Voronov, "Kinetic-inductance-limited reset time of superconducting nanowire photon counters," Appl. Phys. Lett. 88, 111116 (2006).
    [CrossRef]
  36. A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, "Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths," Nat. Photonics 2, 302-306 (2008).
    [CrossRef]
  37. H. Hübel, M. R. Vanner, T. Lederer, B. Blauensteiner, T. Lorünser, A. Poppe, and A. Zeilinger, "High-fidelity transmission of polarization encoded qubits from an entangled source over 100 km of fiber," Opt. Express 15, 7853-7862 (2007).
    [CrossRef] [PubMed]
  38. R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Entanglement-based quantum communication over 144 km," Nat. Phys. 3, 481-486 (2007).
    [CrossRef]
  39. M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, "Entangling independent photons by time measurement," Nat. Phys. 3, 692-695 (2007).
    [CrossRef]
  40. E. Saglamyurek, N. Sinclair, J. Jin, J. A. Slater, D. Oblak, F. Bussires, M. George, R. Ricken, W. Sohler, and W. Tittel, "Broadband waveguide quantum memory for entangled photons," Nature 469, 512-515 (2011).
    [CrossRef] [PubMed]
  41. G. B. Xavier, G. Vilela de Faria, G. P. Temporão, and J. P. von der Weid, "Full polarization control for fiber optical quantum communication systems using polarization encoding," Opt. Express 16, 1867-1873 (2008).
    [CrossRef] [PubMed]
  42. I. Lucio-Martinez, P. Chan, X. Mo, S. Hosier, and W. Tittel, "Proof-of-concept of real-world quantum key distribution with quantum frames," N. J. Phys. 11, 095001 (2009).
    [CrossRef]
  43. H. de Riedmatten, I. Marcikic, J. A. W. van Houwelingen, W. Tittel, H. Zbinden, and N. Gisin, "Long-distance entanglement swapping with photons from separated sources," Phys. Rev. A 71, 050302 (2005).
    [CrossRef]
  44. J. G. Rarity, and P. R. Tapster, "Experimental violation of Bell’s inequality based on phase and momentum," Phys. Rev. Lett. 64, 2495-2498 (1990).
    [CrossRef] [PubMed]
  45. P. W. Shor, and J. Preskill, "Simple Proof of Security of the BB84 Quantum Key Distribution Protocol," Phys. Rev. Lett. 85, 441-444 (2000).
    [CrossRef] [PubMed]
  46. J. Calsamiglia, and N. Lütkenhaus, "Maximum efficiency of a linear-optical Bell-state analyzer," Appl. Phys. B 72, 67-71 (2001).
  47. G. Brassard, and L. Salvail, "Secret-Key Reconciliation by Public Discussion," in Advances in Cryptology -EUROCRYPT ’93 (Lecture Notes in Computer Science, Vol.765) (Springer, Berlin, 1994), pp. 410-423.
    [CrossRef]
  48. M. Koashi, and J. Preskill, "Secure Quantum Key Distribution with an Uncharacterized Source," Phys. Rev. Lett. 90, 057902 (2003).
    [CrossRef] [PubMed]
  49. N. J. Beaudry, T. Moroder, and N. Lütkenhaus, "Squashing Models for Optical Measurements in Quantum Communication," Phys. Rev. Lett. 101, 093601 (2008).
    [CrossRef] [PubMed]
  50. T. Moroder, O. Gühne, N. J. Beaudry, M. Piani, and N. Lütkenhaus, "Entanglement verification with realistic measurement devices via squashing operations," Phys. Rev. A 81, 052342 (2010).
    [CrossRef]

2011

E. Saglamyurek, N. Sinclair, J. Jin, J. A. Slater, D. Oblak, F. Bussires, M. George, R. Ricken, W. Sohler, and W. Tittel, "Broadband waveguide quantum memory for entangled photons," Nature 469, 512-515 (2011).
[CrossRef] [PubMed]

2010

J. B. Brask, L. Jiang, A. V. Gorshkov, V. Vuletic, A. S. Sørensen, and M. D. Lukin, "Fast entanglement distribution with atomic ensembles and fluorescent detection," Phys. Rev. A 81, 020303 (2010).
[CrossRef]

J. Amirloo, M. Razavi, and A. H. Majedi, "Quantum key distribution over probabilistic quantum repeaters," Phys. Rev. A 82, 032304 (2010).
[CrossRef]

T. Moroder, O. Gühne, N. J. Beaudry, M. Piani, and N. Lütkenhaus, "Entanglement verification with realistic measurement devices via squashing operations," Phys. Rev. A 81, 052342 (2010).
[CrossRef]

2009

I. Lucio-Martinez, P. Chan, X. Mo, S. Hosier, and W. Tittel, "Proof-of-concept of real-world quantum key distribution with quantum frames," N. J. Phys. 11, 095001 (2009).
[CrossRef]

A. Scherer, G. Howard, B. C. Sanders, and W. Tittel, "Quantum states prepared by realistic entanglement swapping," Phys. Rev. A 80, 062310 (2009).
[CrossRef]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, "The security of practical quantum key distribution," Rev. Mod. Phys. 81, 1301-1350 (2009).
[CrossRef]

2008

J. B. Brask, and A. S. Sørensen, "Memory imperfections in atomic-ensemble-based quantum repeaters," Phys. Rev. A 78, 012350 (2008).
[CrossRef]

A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, "Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths," Nat. Photonics 2, 302-306 (2008).
[CrossRef]

N. J. Beaudry, T. Moroder, and N. Lütkenhaus, "Squashing Models for Optical Measurements in Quantum Communication," Phys. Rev. Lett. 101, 093601 (2008).
[CrossRef] [PubMed]

G. B. Xavier, G. Vilela de Faria, G. P. Temporão, and J. P. von der Weid, "Full polarization control for fiber optical quantum communication systems using polarization encoding," Opt. Express 16, 1867-1873 (2008).
[CrossRef] [PubMed]

2007

H. Hübel, M. R. Vanner, T. Lederer, B. Blauensteiner, T. Lorünser, A. Poppe, and A. Zeilinger, "High-fidelity transmission of polarization encoded qubits from an entangled source over 100 km of fiber," Opt. Express 15, 7853-7862 (2007).
[CrossRef] [PubMed]

L. Jiang, J. M. Taylor, and M. D. Lukin, "Fast and robust approach to long-distance quantum communication with atomic ensembles," Phys. Rev. A 76, 012301 (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]

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Entanglement-based quantum communication over 144 km," Nat. Phys. 3, 481-486 (2007).
[CrossRef]

M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, "Entangling independent photons by time measurement," Nat. Phys. 3, 692-695 (2007).
[CrossRef]

X. Ma, C.-H. F. Fung, and H.-K. Lo, "Quantum key distribution with entangled photon sources," Phys. Rev. A 76, 012307 (2007).
[CrossRef]

2006

Y. Zhao, B. Qi, X. Ma, H.-K. Lo, and L. Qian, "Experimental Quantum Key Distribution with Decoy States," Phys. Rev. Lett. 96, 070502 (2006).
[CrossRef] [PubMed]

K. M. Rosfjord, J. K. W. Yang, E. A. Dauler, A. J. Kerman, V. Anant, B. M. Voronov, G. N. Gol’tsman, and K. K. Berggren, "Nanowire single-photon detector with an integrated optical cavity and anti-reflection coating," Opt. Express 14, 527-534 (2006).
[CrossRef] [PubMed]

A. J. Kerman, E. A. Dauler, W. E. Keicher, J. K. W. Yang, K. K. Berggren, G. Gol’tsman, and B. Voronov, "Kinetic-inductance-limited reset time of superconducting nanowire photon counters," Appl. Phys. Lett. 88, 111116 (2006).
[CrossRef]

2005

D. Rosenberg, A. E. Lita, A. J. Miller, S. W. Nam, and R. E. Schwall, "Performance of photon-number resolving transition-edge sensors with integrated 1550 nm resonant cavities," IEEE Trans. Appl. Supercond. 15(2), 575-578 (2005).
[CrossRef]

D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, "Noise-free high-efficiency photon-number-resolving detectors," Phys. Rev. A 71, 061803 (2005).
[CrossRef]

H. de Riedmatten, I. Marcikic, J. A. W. van Houwelingen, W. Tittel, H. Zbinden, and N. Gisin, "Long-distance entanglement swapping with photons from separated sources," Phys. Rev. A 71, 050302 (2005).
[CrossRef]

D. Collins, N. Gisin, and H. de Riedmatten, "Quantum relays for long distance quantum cryptography," J. Mod. Opt. 52, 735-753 (2005).
[CrossRef]

X.-B. Wang, "Beating the Photon-Number-Splitting Attack in Practical Quantum Cryptography," Phys. Rev. Lett. 94, 230503 (2005).
[CrossRef] [PubMed]

H.-K. Lo, X. Ma, and K. Chen, "Decoy State Quantum Key Distribution," Phys. Rev. Lett. 94, 230504 (2005).
[CrossRef] [PubMed]

X. Ma, B. Qi, Y. Zhao, and H.-K. Lo, "Practical decoy state for quantum key distribution," Phys. Rev. A 72, 012326 (2005).
[CrossRef]

X.-B. Wang, "Decoy-state protocol for quantum cryptography with four different intensities of coherent light," Phys. Rev. A 72, 012322 (2005).
[CrossRef]

2004

H. de Riedmatten, V. Scarani, I. Marcikic, A. Acín, W. Tittel, H. Zbinden, and N. Gisin, "Two independent photon pairs versus four-photon entangled states in parametric down conversion," J. Mod. Opt. 51, 1637-1649 (2004).

H. de Riedmatten, I. Marcikic, W. Tittel, H. Zbinden, D. Collins, and N. Gisin, "Long Distance Quantum Teleportation in a Quantum Relay Configuration," Phys. Rev. Lett. 92, 047904 (2004).
[CrossRef] [PubMed]

2003

W.-Y. Hwang, "Quantum Key Distribution with High Loss: Toward Global Secure Communication," Phys. Rev. Lett. 91, 057901 (2003).
[CrossRef] [PubMed]

M. Koashi, and J. Preskill, "Secure Quantum Key Distribution with an Uncharacterized Source," Phys. Rev. Lett. 90, 057902 (2003).
[CrossRef] [PubMed]

A. J. Miller, S. W. Nam, J. M. Martinis, and A. V. Sergienko, "Demonstration of a low-noise near-infrared photon counter with multiphoton discrimination," Appl. Phys. Lett. 83, 791-793 (2003).
[CrossRef]

2002

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

E. Waks, A. Zeevi, and Y. Yamamoto, "Security of quantum key distribution with entangled photons against individual attacks," Phys. Rev. A 65, 052310 (2002).
[CrossRef]

B. C. Jacobs, T. B. Pittman, and D. Franson, "Quantum relays and noise suppression using linear optics," Phys. Rev. A 66, 052307 (2002).
[CrossRef]

I. Marcikic, H. de Riedmatten, W. Tittel, V. Scarani, H. Zbinden, and N. Gisin, "Time-bin entangled qubits for quantum communication created by femtosecond pulses," Phys. Rev. A 66, 062308 (2002).
[CrossRef]

2001

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, "Picosecond superconducting single-photon optical detector," Appl. Phys. Lett. 79, 705-707 (2001).
[CrossRef]

J. Calsamiglia, and N. Lütkenhaus, "Maximum efficiency of a linear-optical Bell-state analyzer," Appl. Phys. B 72, 67-71 (2001).

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

P. W. Shor, and J. Preskill, "Simple Proof of Security of the BB84 Quantum Key Distribution Protocol," Phys. Rev. Lett. 85, 441-444 (2000).
[CrossRef] [PubMed]

N. Lütkenhaus, "Security against individual attacks for realistic quantum key distribution," Phys. Rev. A 61, 052304 (2000).
[CrossRef]

G. Brassard, N. Lütkenhaus, T. Mor, and B. C. Sanders, "Limitations on Practical Quantum Cryptography," Phys. Rev. Lett. 85, 1330-1333 (2000).
[CrossRef] [PubMed]

1998

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]

1993

M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, "‘Event-ready-detectors’ Bell experiment via entanglement swapping," Phys. Rev. Lett. 71, 4287-4290 (1993).
[CrossRef] [PubMed]

1992

C. H. Bennett, G. Brassard, and N. D. Mermin, "Quantum cryptography without Bell’s theorem," Phys. Rev. Lett. 68, 557-559 (1992).
[CrossRef] [PubMed]

1990

J. G. Rarity, and P. R. Tapster, "Experimental violation of Bell’s inequality based on phase and momentum," Phys. Rev. Lett. 64, 2495-2498 (1990).
[CrossRef] [PubMed]

Acín, A.

H. de Riedmatten, V. Scarani, I. Marcikic, A. Acín, W. Tittel, H. Zbinden, and N. Gisin, "Two independent photon pairs versus four-photon entangled states in parametric down conversion," J. Mod. Opt. 51, 1637-1649 (2004).

Amirloo, J.

J. Amirloo, M. Razavi, and A. H. Majedi, "Quantum key distribution over probabilistic quantum repeaters," Phys. Rev. A 82, 032304 (2010).
[CrossRef]

Anant, V.

Barbieri, C.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Entanglement-based quantum communication over 144 km," Nat. Phys. 3, 481-486 (2007).
[CrossRef]

Beaudry, N. J.

T. Moroder, O. Gühne, N. J. Beaudry, M. Piani, and N. Lütkenhaus, "Entanglement verification with realistic measurement devices via squashing operations," Phys. Rev. A 81, 052342 (2010).
[CrossRef]

N. J. Beaudry, T. Moroder, and N. Lütkenhaus, "Squashing Models for Optical Measurements in Quantum Communication," Phys. Rev. Lett. 101, 093601 (2008).
[CrossRef] [PubMed]

Bechmann-Pasquinucci, H.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, "The security of practical quantum key distribution," Rev. Mod. Phys. 81, 1301-1350 (2009).
[CrossRef]

Benkhaoul, M.

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J. B. Brask, L. Jiang, A. V. Gorshkov, V. Vuletic, A. S. Sørensen, and M. D. Lukin, "Fast entanglement distribution with atomic ensembles and fluorescent detection," Phys. Rev. A 81, 020303 (2010).
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V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, "The security of practical quantum key distribution," Rev. Mod. Phys. 81, 1301-1350 (2009).
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I. Lucio-Martinez, P. Chan, X. Mo, S. Hosier, and W. Tittel, "Proof-of-concept of real-world quantum key distribution with quantum frames," N. J. Phys. 11, 095001 (2009).
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G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, "Picosecond superconducting single-photon optical detector," Appl. Phys. Lett. 79, 705-707 (2001).
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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).
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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).
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H. de Riedmatten, I. Marcikic, W. Tittel, H. Zbinden, D. Collins, and N. Gisin, "Long Distance Quantum Teleportation in a Quantum Relay Configuration," Phys. Rev. Lett. 92, 047904 (2004).
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A. J. Kerman, E. A. Dauler, W. E. Keicher, J. K. W. Yang, K. K. Berggren, G. Gol’tsman, and B. Voronov, "Kinetic-inductance-limited reset time of superconducting nanowire photon counters," Appl. Phys. Lett. 88, 111116 (2006).
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K. M. Rosfjord, J. K. W. Yang, E. A. Dauler, A. J. Kerman, V. Anant, B. M. Voronov, G. N. Gol’tsman, and K. K. Berggren, "Nanowire single-photon detector with an integrated optical cavity and anti-reflection coating," Opt. Express 14, 527-534 (2006).
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H. de Riedmatten, I. Marcikic, J. A. W. van Houwelingen, W. Tittel, H. Zbinden, and N. Gisin, "Long-distance entanglement swapping with photons from separated sources," Phys. Rev. A 71, 050302 (2005).
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D. Collins, N. Gisin, and H. de Riedmatten, "Quantum relays for long distance quantum cryptography," J. Mod. Opt. 52, 735-753 (2005).
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H. de Riedmatten, I. Marcikic, W. Tittel, H. Zbinden, D. Collins, and N. Gisin, "Long Distance Quantum Teleportation in a Quantum Relay Configuration," Phys. Rev. Lett. 92, 047904 (2004).
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H. de Riedmatten, V. Scarani, I. Marcikic, A. Acín, W. Tittel, H. Zbinden, and N. Gisin, "Two independent photon pairs versus four-photon entangled states in parametric down conversion," J. Mod. Opt. 51, 1637-1649 (2004).

I. Marcikic, H. de Riedmatten, W. Tittel, V. Scarani, H. Zbinden, and N. Gisin, "Time-bin entangled qubits for quantum communication created by femtosecond pulses," Phys. Rev. A 66, 062308 (2002).
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A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, "Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths," Nat. Photonics 2, 302-306 (2008).
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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).
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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).
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Dušek, M.

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, "The security of practical quantum key distribution," Rev. Mod. Phys. 81, 1301-1350 (2009).
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Dzardanov, A.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, "Picosecond superconducting single-photon optical detector," Appl. Phys. Lett. 79, 705-707 (2001).
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M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, "‘Event-ready-detectors’ Bell experiment via entanglement swapping," Phys. Rev. Lett. 71, 4287-4290 (1993).
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A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, "Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths," Nat. Photonics 2, 302-306 (2008).
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R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Entanglement-based quantum communication over 144 km," Nat. Phys. 3, 481-486 (2007).
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A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, "Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths," Nat. Photonics 2, 302-306 (2008).
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E. Saglamyurek, N. Sinclair, J. Jin, J. A. Slater, D. Oblak, F. Bussires, M. George, R. Ricken, W. Sohler, and W. Tittel, "Broadband waveguide quantum memory for entangled photons," Nature 469, 512-515 (2011).
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M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, "Entangling independent photons by time measurement," Nat. Phys. 3, 692-695 (2007).
[CrossRef]

H. de Riedmatten, I. Marcikic, J. A. W. van Houwelingen, W. Tittel, H. Zbinden, and N. Gisin, "Long-distance entanglement swapping with photons from separated sources," Phys. Rev. A 71, 050302 (2005).
[CrossRef]

D. Collins, N. Gisin, and H. de Riedmatten, "Quantum relays for long distance quantum cryptography," J. Mod. Opt. 52, 735-753 (2005).
[CrossRef]

H. de Riedmatten, I. Marcikic, W. Tittel, H. Zbinden, D. Collins, and N. Gisin, "Long Distance Quantum Teleportation in a Quantum Relay Configuration," Phys. Rev. Lett. 92, 047904 (2004).
[CrossRef] [PubMed]

H. de Riedmatten, V. Scarani, I. Marcikic, A. Acín, W. Tittel, H. Zbinden, and N. Gisin, "Two independent photon pairs versus four-photon entangled states in parametric down conversion," J. Mod. Opt. 51, 1637-1649 (2004).

I. Marcikic, H. de Riedmatten, W. Tittel, V. Scarani, H. Zbinden, and N. Gisin, "Time-bin entangled qubits for quantum communication created by femtosecond pulses," Phys. Rev. A 66, 062308 (2002).
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N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
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A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, "Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths," Nat. Photonics 2, 302-306 (2008).
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A. J. Kerman, E. A. Dauler, W. E. Keicher, J. K. W. Yang, K. K. Berggren, G. Gol’tsman, and B. Voronov, "Kinetic-inductance-limited reset time of superconducting nanowire photon counters," Appl. Phys. Lett. 88, 111116 (2006).
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Gol’tsman, G. N.

K. M. Rosfjord, J. K. W. Yang, E. A. Dauler, A. J. Kerman, V. Anant, B. M. Voronov, G. N. Gol’tsman, and K. K. Berggren, "Nanowire single-photon detector with an integrated optical cavity and anti-reflection coating," Opt. Express 14, 527-534 (2006).
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G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, "Picosecond superconducting single-photon optical detector," Appl. Phys. Lett. 79, 705-707 (2001).
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J. B. Brask, L. Jiang, A. V. Gorshkov, V. Vuletic, A. S. Sørensen, and M. D. Lukin, "Fast entanglement distribution with atomic ensembles and fluorescent detection," Phys. Rev. A 81, 020303 (2010).
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T. Moroder, O. Gühne, N. J. Beaudry, M. Piani, and N. Lütkenhaus, "Entanglement verification with realistic measurement devices via squashing operations," Phys. Rev. A 81, 052342 (2010).
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M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, "Entangling independent photons by time measurement," Nat. Phys. 3, 692-695 (2007).
[CrossRef]

Horne, M. A.

M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, "‘Event-ready-detectors’ Bell experiment via entanglement swapping," Phys. Rev. Lett. 71, 4287-4290 (1993).
[CrossRef] [PubMed]

Hosier, S.

I. Lucio-Martinez, P. Chan, X. Mo, S. Hosier, and W. Tittel, "Proof-of-concept of real-world quantum key distribution with quantum frames," N. J. Phys. 11, 095001 (2009).
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A. Scherer, G. Howard, B. C. Sanders, and W. Tittel, "Quantum states prepared by realistic entanglement swapping," Phys. Rev. A 80, 062310 (2009).
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Hwang, W.-Y.

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B. C. Jacobs, T. B. Pittman, and D. Franson, "Quantum relays and noise suppression using linear optics," Phys. Rev. A 66, 052307 (2002).
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Jennewein, T.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Entanglement-based quantum communication over 144 km," Nat. Phys. 3, 481-486 (2007).
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Jiang, L.

J. B. Brask, L. Jiang, A. V. Gorshkov, V. Vuletic, A. S. Sørensen, and M. D. Lukin, "Fast entanglement distribution with atomic ensembles and fluorescent detection," Phys. Rev. A 81, 020303 (2010).
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L. Jiang, J. M. Taylor, and M. D. Lukin, "Fast and robust approach to long-distance quantum communication with atomic ensembles," Phys. Rev. A 76, 012301 (2007).
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Jin, J.

E. Saglamyurek, N. Sinclair, J. Jin, J. A. Slater, D. Oblak, F. Bussires, M. George, R. Ricken, W. Sohler, and W. Tittel, "Broadband waveguide quantum memory for entangled photons," Nature 469, 512-515 (2011).
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Kaurova, N.

A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, "Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths," Nat. Photonics 2, 302-306 (2008).
[CrossRef]

Keicher, W. E.

A. J. Kerman, E. A. Dauler, W. E. Keicher, J. K. W. Yang, K. K. Berggren, G. Gol’tsman, and B. Voronov, "Kinetic-inductance-limited reset time of superconducting nanowire photon counters," Appl. Phys. Lett. 88, 111116 (2006).
[CrossRef]

Kerman, A. J.

A. J. Kerman, E. A. Dauler, W. E. Keicher, J. K. W. Yang, K. K. Berggren, G. Gol’tsman, and B. Voronov, "Kinetic-inductance-limited reset time of superconducting nanowire photon counters," Appl. Phys. Lett. 88, 111116 (2006).
[CrossRef]

K. M. Rosfjord, J. K. W. Yang, E. A. Dauler, A. J. Kerman, V. Anant, B. M. Voronov, G. N. Gol’tsman, and K. K. Berggren, "Nanowire single-photon detector with an integrated optical cavity and anti-reflection coating," Opt. Express 14, 527-534 (2006).
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Koashi, M.

M. Koashi, and J. Preskill, "Secure Quantum Key Distribution with an Uncharacterized Source," Phys. Rev. Lett. 90, 057902 (2003).
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Korneev, A.

A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, "Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths," Nat. Photonics 2, 302-306 (2008).
[CrossRef]

Lagoudakis, K. G.

A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, "Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths," Nat. Photonics 2, 302-306 (2008).
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Lederer, T.

Leoni, R.

A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, "Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths," Nat. Photonics 2, 302-306 (2008).
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Lévy, F.

A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, "Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths," Nat. Photonics 2, 302-306 (2008).
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Lindenthal, M.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Entanglement-based quantum communication over 144 km," Nat. Phys. 3, 481-486 (2007).
[CrossRef]

Lipatov, A.

G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, "Picosecond superconducting single-photon optical detector," Appl. Phys. Lett. 79, 705-707 (2001).
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Lita, A. E.

D. Rosenberg, A. E. Lita, A. J. Miller, S. W. Nam, and R. E. Schwall, "Performance of photon-number resolving transition-edge sensors with integrated 1550 nm resonant cavities," IEEE Trans. Appl. Supercond. 15(2), 575-578 (2005).
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D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, "Noise-free high-efficiency photon-number-resolving detectors," Phys. Rev. A 71, 061803 (2005).
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Lo, H.-K.

X. Ma, C.-H. F. Fung, and H.-K. Lo, "Quantum key distribution with entangled photon sources," Phys. Rev. A 76, 012307 (2007).
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Y. Zhao, B. Qi, X. Ma, H.-K. Lo, and L. Qian, "Experimental Quantum Key Distribution with Decoy States," Phys. Rev. Lett. 96, 070502 (2006).
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D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, "Noise-free high-efficiency photon-number-resolving detectors," Phys. Rev. A 71, 061803 (2005).
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A. Scherer, G. Howard, B. C. Sanders, and W. Tittel, "Quantum states prepared by realistic entanglement swapping," Phys. Rev. A 80, 062310 (2009).
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V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, "The security of practical quantum key distribution," Rev. Mod. Phys. 81, 1301-1350 (2009).
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I. Marcikic, H. de Riedmatten, W. Tittel, V. Scarani, H. Zbinden, and N. Gisin, "Time-bin entangled qubits for quantum communication created by femtosecond pulses," Phys. Rev. A 66, 062308 (2002).
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R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Entanglement-based quantum communication over 144 km," Nat. Phys. 3, 481-486 (2007).
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A. Scherer, G. Howard, B. C. Sanders, and W. Tittel, "Quantum states prepared by realistic entanglement swapping," Phys. Rev. A 80, 062310 (2009).
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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).
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R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Entanglement-based quantum communication over 144 km," Nat. Phys. 3, 481-486 (2007).
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D. Rosenberg, A. E. Lita, A. J. Miller, S. W. Nam, and R. E. Schwall, "Performance of photon-number resolving transition-edge sensors with integrated 1550 nm resonant cavities," IEEE Trans. Appl. Supercond. 15(2), 575-578 (2005).
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A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, "Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths," Nat. Photonics 2, 302-306 (2008).
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G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, "Picosecond superconducting single-photon optical detector," Appl. Phys. Lett. 79, 705-707 (2001).
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A. J. Miller, S. W. Nam, J. M. Martinis, and A. V. Sergienko, "Demonstration of a low-noise near-infrared photon counter with multiphoton discrimination," Appl. Phys. Lett. 83, 791-793 (2003).
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P. W. Shor, and J. Preskill, "Simple Proof of Security of the BB84 Quantum Key Distribution Protocol," Phys. Rev. Lett. 85, 441-444 (2000).
[CrossRef] [PubMed]

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M. Halder, A. Beveratos, N. Gisin, V. Scarani, C. Simon, and H. Zbinden, "Entangling independent photons by time measurement," Nat. Phys. 3, 692-695 (2007).
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E. Saglamyurek, N. Sinclair, J. Jin, J. A. Slater, D. Oblak, F. Bussires, M. George, R. Ricken, W. Sohler, and W. Tittel, "Broadband waveguide quantum memory for entangled photons," Nature 469, 512-515 (2011).
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E. Saglamyurek, N. Sinclair, J. Jin, J. A. Slater, D. Oblak, F. Bussires, M. George, R. Ricken, W. Sohler, and W. Tittel, "Broadband waveguide quantum memory for entangled photons," Nature 469, 512-515 (2011).
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G. N. Gol’tsman, O. Okunev, G. Chulkova, A. Lipatov, A. Semenov, K. Smirnov, B. Voronov, A. Dzardanov, C. Williams, and R. Sobolewski, "Picosecond superconducting single-photon optical detector," Appl. Phys. Lett. 79, 705-707 (2001).
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R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Entanglement-based quantum communication over 144 km," Nat. Phys. 3, 481-486 (2007).
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E. Saglamyurek, N. Sinclair, J. Jin, J. A. Slater, D. Oblak, F. Bussires, M. George, R. Ricken, W. Sohler, and W. Tittel, "Broadband waveguide quantum memory for entangled photons," Nature 469, 512-515 (2011).
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J. B. Brask, L. Jiang, A. V. Gorshkov, V. Vuletic, A. S. Sørensen, and M. D. Lukin, "Fast entanglement distribution with atomic ensembles and fluorescent detection," Phys. Rev. A 81, 020303 (2010).
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J. B. Brask, and A. S. Sørensen, "Memory imperfections in atomic-ensemble-based quantum repeaters," Phys. Rev. A 78, 012350 (2008).
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J. G. Rarity, and P. R. Tapster, "Experimental violation of Bell’s inequality based on phase and momentum," Phys. Rev. Lett. 64, 2495-2498 (1990).
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L. Jiang, J. M. Taylor, and M. D. Lukin, "Fast and robust approach to long-distance quantum communication with atomic ensembles," Phys. Rev. A 76, 012301 (2007).
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Tiefenbacher, F.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Entanglement-based quantum communication over 144 km," Nat. Phys. 3, 481-486 (2007).
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E. Saglamyurek, N. Sinclair, J. Jin, J. A. Slater, D. Oblak, F. Bussires, M. George, R. Ricken, W. Sohler, and W. Tittel, "Broadband waveguide quantum memory for entangled photons," Nature 469, 512-515 (2011).
[CrossRef] [PubMed]

I. Lucio-Martinez, P. Chan, X. Mo, S. Hosier, and W. Tittel, "Proof-of-concept of real-world quantum key distribution with quantum frames," N. J. Phys. 11, 095001 (2009).
[CrossRef]

A. Scherer, G. Howard, B. C. Sanders, and W. Tittel, "Quantum states prepared by realistic entanglement swapping," Phys. Rev. A 80, 062310 (2009).
[CrossRef]

H. de Riedmatten, I. Marcikic, J. A. W. van Houwelingen, W. Tittel, H. Zbinden, and N. Gisin, "Long-distance entanglement swapping with photons from separated sources," Phys. Rev. A 71, 050302 (2005).
[CrossRef]

H. de Riedmatten, V. Scarani, I. Marcikic, A. Acín, W. Tittel, H. Zbinden, and N. Gisin, "Two independent photon pairs versus four-photon entangled states in parametric down conversion," J. Mod. Opt. 51, 1637-1649 (2004).

H. de Riedmatten, I. Marcikic, W. Tittel, H. Zbinden, D. Collins, and N. Gisin, "Long Distance Quantum Teleportation in a Quantum Relay Configuration," Phys. Rev. Lett. 92, 047904 (2004).
[CrossRef] [PubMed]

I. Marcikic, H. de Riedmatten, W. Tittel, V. Scarani, H. Zbinden, and N. Gisin, "Time-bin entangled qubits for quantum communication created by femtosecond pulses," Phys. Rev. A 66, 062308 (2002).
[CrossRef]

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

Trojek, P.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Entanglement-based quantum communication over 144 km," Nat. Phys. 3, 481-486 (2007).
[CrossRef]

Ursin, R.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Entanglement-based quantum communication over 144 km," Nat. Phys. 3, 481-486 (2007).
[CrossRef]

van Houwelingen, J. A. W.

H. de Riedmatten, I. Marcikic, J. A. W. van Houwelingen, W. Tittel, H. Zbinden, and N. Gisin, "Long-distance entanglement swapping with photons from separated sources," Phys. Rev. A 71, 050302 (2005).
[CrossRef]

Vanner, M. R.

Vilela de Faria, G.

von der Weid, J. P.

Voronov, B.

A. J. Kerman, E. A. Dauler, W. E. Keicher, J. K. W. Yang, K. K. Berggren, G. Gol’tsman, and B. Voronov, "Kinetic-inductance-limited reset time of superconducting nanowire photon counters," Appl. Phys. Lett. 88, 111116 (2006).
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R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Entanglement-based quantum communication over 144 km," Nat. Phys. 3, 481-486 (2007).
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Figures (9)

Fig. 1
Fig. 1

Entanglement swapping based on two PDC sources and an interferometric BSM. Four spatial modes are involved: a, b, c and d. The modes b and c are combined at a balanced beamsplitter (BS). Outputs b′ and c′ are directed to polarizing beamsplitters (PBS) and then detected at four photon detectors: one for the H and one for the V polarization of each of the c′ and b′ modes. The detectors are inefficient and subject to dark counts. Their readout is denoted by {q1, q2, q3, q4}. In a QKD experiment (see Fig. 2), the polarization-entangled photons of the remaining modes a and d are distributed between Alice and Bob, respectively, and the BBM92 protocol [16] can be applied to make the secret key.

Fig. 2
Fig. 2

Illustration of ES-based QKD. The quantum channel between Alice and Bob is split into shorter segments, with two PDC sources placed 1/4 and 3/4 of the way along the channel and a joint BSM performed halfway. Given a successful BSM (with success probability equal to 1 2 η 0 2), the photons arriving at Alice and Bob are entangled despite never having interacted with one another, and the BBM92 protocol can be used to create the secret key.

Fig. 3
Fig. 3

QBER as a function of χ for various values of the normalized distance αd and fixed η0 anddc. The function is plotted for αd = 0, 5, 10, 25 and 50, corresponding, respectively, to the curves of lowest- to highest-qber values in all diagrams, and η0 = 0.1 & dc ≈ 3 × 10−6 in figures (a) and (b), or η0 = 0.3 &dc ≈ 10−4 in figures (c) and (d). The values of η0 and dc are related to one another by constraint (2). Figures (a) and (c) display a higher resolution for very small χ values in both cases. To have Rsec > 0 the QBER must assume values less than approx. 0.094 if κ = 1.22 (0.11 if κ = 1.0).

Fig. 4
Fig. 4

QBER vs αd for various χ values and fixed η0 and dc. The function is plotted for χ = 10−4, 10−3, 10−2, 0.1, and 0.2, corresponding to the dotted, dot-dashed, dashed, gray solid and dark solid curves, respectively, in both diagrams, and (a) η0 = 0.1 & dc ≈ 3×10−6 or (b) η0 = 0.3 &dc ≈ 10−4. The values of η0 anddc are related to one another by constraint (2).

Fig. 5
Fig. 5

logRsec as a function of χ for various values of the product αd, and fixed η0 anddc. Plots are displayed for αd = 0, 5, 10, 25, and 50, corresponding, respectively, to the dark solid, gray solid, dashed, dot-dashed and dotted curves in both diagrams, and (a) η0 = 0.1 & dc ≈ 3 × 10−6 or (b) η0 = 0.3 & dc ≈ 10−4. Dark count probabilities are related to the values of η0 by constraint (2). The value of Rsec is the number of secure bits created per single pump-laser pulse.

Fig. 6
Fig. 6

Secret key rate Rsec as a function of χ and η0 for various exemplary values of the product αd. From left to right: (a) αd = 1, (b) αd = 5, (c) αd = 10, (d) αd = 25, (e) αd = 40 and (f) αd = 50. Dark count probabilities are related to the values of η0 by constraint (2), respectively. Here Rsec is given in terms of the number of secure bits created per single pump-laser pulse (precisely: for each attempt of ES, which requires two laser pulses, specifically with one per crystal).

Fig. 7
Fig. 7

(a) Optimal χ and (b) optimal η0 values for QKD as a function of the product αd. The dark count parameterdc is related to η0opt by constraint (2).

Fig. 8
Fig. 8

Comparison between decoy-BB84 and PDC-ES-BBM92 QKD performance for decreasing dark count noise. All three diagrams display the logarithm of Rsec vs αd for decoy-BB84 QKD (gray curve) and for PDC-ES-BBM92 QKD (dark curve), with optimal choice of source brightness for every value of αd, respectively, and the fixed detector efficiency η0 = 0.2. The detector dark count probabilities are (a) dc = 1.8 × 10−5 (complying with constraint (2)), (b) dc = 10−6 and (c) dc = 10−10 (ignoring constraint (2) in (b) and (c)).

Fig. 9
Fig. 9

(a) Effect of source brightness variation on key generation rate and distribution range for fixed η0 = 0.2 and negligible dark counts, dc = 10−12. In decoy-BB84 QKD, the key rate and range are fairly stable against a change of μ (as already pointed out in [20]): the curves corresponding to μ = 0.8 (dotted) and μ = 0.4 (gray solid) coincide. In PDC-ES-BBM92 QKD, the key rate and achievable range depend drastically on source brightness, as demonstrated by the curves corresponding to χ = 0.174 (dark solid), χ = 0.172 (dot-dashed) and χ = 0.12 (dashed), respectively. (b) Effect of detector efficiency variation on key generation rate and distribution range for fixed dc = 10−12 and fixed source brightness, μ = 0.7 and χ = 0.120, respectively. Predictions for two different η0 values are shown. For η0 = 0.9 yielding the dotted curve for decoy-BB84 and the dashed curve for PDC-ES-BBM92 QKD, higher key rates and longer distances are achieved than for η0 = 0.1 yielding the gray solid curve for decoy-BB84 and dark solid curve for PDC-ES-BBM92 QKD. The effect is substantially greater for PDC-ES-BBM92 than for decoy-BB84 QKD.

Equations (6)

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| χ a b = exp [ i χ ( a ^ H b ^ H + a ^ H b ^ H ) ] exp [ i χ ( a ^ V b ^ V + a ^ V b ^ V ) ] | vac
dc = A exp ( B η 0 )
R sec = R sift [ 1 κ H 2 ( QBER ) H 2 ( QBER ) ] .
R sift = 1 2 χ 2 χ 2 ( 10 α d / 40 ) 4 η 0 2 ( 1 2 η 0 2 ) = 1 4 χ 4 η 0 4 × 10 α d / 10 .
H 2 ( x ) x log 2 ( x ) ( 1 x ) log 2 ( 1 x ) for x [ 0 , 1 ]
R sec decoy 1 2 { Q μ f ( E μ ) H 2 ( E μ ) + Q 1 [ 1 H 2 ( e 1 ) ] } ,

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