## High-fidelity transmission of polarization encoded qubits from an entangled source over 100 km of fiber

Optics Express, Vol. 15, Issue 12, pp. 7853-7862 (2007)

http://dx.doi.org/10.1364/OE.15.007853

Acrobat PDF (213 KB)

### Abstract

We demonstrate non-degenerate down-conversion at 810 and 1550 nm for long-distance fiber based quantum communication using polarization entangled photon pairs. Measurements of the two-photon visibility, without dark count subtraction, have shown that the quantum correlations (raw visibility 89%) allow secure quantum cryptography after 100 km of non-zero dispersion shifted fiber using commercially available single photon detectors. In addition, quantum state tomography has revealed little degradation of state negativity, decreasing from 0.99 at the source to 0.93 after 100 km, indicating minimal loss in fidelity during the transmission.

© 2007 Optical Society of America

## 1. Introduction

04. R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Oemer, M. Fuerst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Free-Space distribution of entanglement and single photons over 144 km,” http://www.arxiv.org/abs/quant-ph/0607182.

05. I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legré, and N. Gisin, “Distribution of Time-Bin Entangled Qubits over 50 km of Optical Fiber,” Phys. Rev. Lett. **93**, 180502 (2004). [CrossRef] [PubMed]

06. H. Takesue, “Long-distance distribution of time-bin entanglement generated in a cooled fiber,” Opt. Express **14**, 3453–3460 (2006). [CrossRef] [PubMed]

08. S. Sauge, M. Swillo, S. Albert-Seifried, G. B. Xavier, J. Waldebäck, M. Tengner, D. Ljunggren, and A. Karlsson, “Narrowband polarization-entangled photon pairs distributed over a WDM link for qubit networks,” Opt. Express **15**, 6926–6933 (2007). [CrossRef] [PubMed]

09. D.N. Matsukevich, T. Chaneliere, S.D. Jenkins, S.Y. Lan, T.A.B. Kennedy, and A. Kuzmich, “Entanglement of Remote Atomic Qubits,” Phys. Rev. Lett **96**, 030405 (2006). [CrossRef] [PubMed]

## 2. Source of polarization entangled photons

11. G. Ribordy, J. Brendel, J.D. Gautier, N. Gisin, and H. Zbinden, “Long-distance entanglement-based quantum key distribution,” Phys. Rev. A. **63**, 012309 (2000). [CrossRef]

12. M. Pelton, P. Marsden, D. Ljunggren, M. Tenger, A. Karlsson, A. Fragemann, C. Canalias, and F. Laurell, “Bright, single-spatial-mode source of frequency non-degenerate, polarization-entangled photon pairs using perioically pole KTP,” Opt. Express **12**, 3573–3580 (2004). [CrossRef] [PubMed]

13. F. Konig, E.J. Mason, F.N.C. Wong, and M.A. Albota, “Efficient and spectrally bright source of polarization-entangled photons,” Phys. Rev. A. **71**, 033805 (2005). [CrossRef]

14. P.G. Kwiat, E. Waks, A.G. White, I. Appelbaum, and P.H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A. **60**, R773-6 (1999). [CrossRef]

*z*

_{0}, is comparable to the individual crystal length

*L*(

*L*= 4 mm for both crystals) [15

15. D. Ljunggren and M. Tengner, “Optimal focusing for maximal collection of narrow-band photon pairs into single-mode fibers,” Phys. Rev. A. **72**, 062301 (2005). [CrossRef]

*L*/

*z*

_{0}ratio of 0.9.

_{4}(ppKTP), with a grating spacing of 9.7 μm, which has been tailored for type-I collinear generation of an asymmetric photon pair at 810 and 1550 nm from a 532 nm pump. These wavelengths were selected because of the efficient detection at 810 nm, low fiber absorption at 1550 nm and readily available stable radiation sources at 532 nm. The crystals are housed in a temperature controlled copper mount, heated to approximately 65 °C. Varying the temperature allows wavelength tuning for collinear emission. Each ppKTP crystal produces a pair with an intrinsic bandwidth of 800 GHz, which was reduced to 400 GHz with filters BP 810 and BP 1550 to minimize chromatic dispersion.

16. D. Ljunggren, M. Tengner, P. Marsden, and M. Pelton, “Theory and experiment of entanglement in a quasi-phase-matched two-crystal source,” Phys. Rev. A. **73**, 032326 (2006). [CrossRef]

^{±}).

*QE*) at this wavelength is in the range of 10-15%. Locally, with 32 m SMF fiber to connect Bob, we achieved the following rates of polarization entangled photon pairs:

## 3. Long-distance fiber transmission

*QE*= 6% with a dark count rate of ~ 3 × 10

^{-6}within each gate opening time. To further reduce the influence of dark counts, all measurements were performed at the smallest possible gate width of 1.5 ns on the InGaAs detector. It was therefore important to minimize chromatic dispersion (CD) so that the temporal broadening of the photon (τ

_{p}) was smaller than the applied gate window. The temporal broadening (τ

_{P}) due to chromatic dispersion (CD) is given by: τ

_{P}=CD λ

_{1550}∙

*L*, where λ

_{1550}is the spectral width of the 1550 photon in nm and

*L*is the fiber length in km.

_{1550}= 3.2 nm, CD would broaden the photon wavepacket to τ

_{P}= 5 and 1.5 ns after

*L*= 100 km for standard and NZDS fibers, respectively. Standard fibers are therefore inadequate to be used for long distance transmission. The first set (set I) of our investigations consisted of NZDS fibers with individual spool lengths of 6.3 and 12.6 km which could be concatenated (using FC/PC connectors) to give a total length of 63 km and a τ

_{P}of 1 ns. To reach longer distances, we added two spools of standard fiber (12.6 km each) totaling 88.2 km and resulting in an additional CD spread of 1.5 ns (total τ

_{P}of 2.5 ns). For measurements over even longer distances we had a second set (set II), consisting of two 50.4 km spools of NZDS fiber spliced together giving a total transmission distance of 100.8 km and a τ

_{P}of 1.5 ns, the size of the gate window.

*V*) and 90°(

*H*) respectively, which insured compensation in the

*HV*basis. In a second step the birefringent wedges, see Fig. 2, were adjusted to yield a minimal coincidence rate at 45°(+) and 135°(-) settings and obtain a |Φ

^{+}) state.

_{PMD}, after transmission. If τ

_{PMD}is larger than the coherence time of the pulse (τ

_{coh}) then the two components will temporally not overlap and the initial polarization state is destroyed. If the situation is reversed the pulse will stay together and no depolarization is caused. The DGD of a fiber is normally stated in ps/

_{PMD}=

*DGD*∙ √

*L*,

*L*being the length of the fiber. Since we had no means to measure the value of τ

_{PMD}directly at the time of our experiment, we relied upon the manufacturers specifications: For set I the average DGD, of our spools, was around 0.07 ps/

_{PMD}for each 6.3 km spool varied from 0.1 to 0.2 ps and between 0.2 to 0.4 ps for the 12.6 km spools. In set II, the maximum value is 0.1 ps/

_{PMD}. So the above values can only be seen as an estimate. Nevertheless, these parameters can be used to determine the significance of PMD. The 400 GHz bandwidth of the 1550 nm photon corresponds to a coherence time of τ

_{coh}≈ 1.6 ps. So for 50 km of fiber, τ

_{PMD}is around 0.6 ps, smaller but still in the range of τ

_{coh}. Therefore, PMD effects are expected to play a role in long distance transmission.

## 4. Evolution of the polarization state during transmission

### 4.1. Quantum state tomography

*HH, HV,VH, VV*,

*H*+,

*V*+,

*HR, VR*, +

*H*, +

*V*, ++, +

*R*,

*RH, RV*,

*R*+ and

*RR*, where

*R*is the right-circular polarization) [18

18. D.F.V. James, P.G. Kwiat, W.J. Munro, and A.G. White, “Measurement of qubits,” Phys. Rev. A. **64**, 052312 (2001). [CrossRef]

19. Z. Hradil, “Quantum-state estimation,” Phys. Rev. A. **55**, R1561–R1564 (1997). [CrossRef]

^{+}) state (four columns at the corners of the matrix) is large, even with the matrix obtained at 101 km. To quantitatively characterize our density matrices and to determine any depolarization of the state with increasing transmission distance we chose to use the negativity (

*N*) [20

20. G. Vidal and R.F. Werner,“Computable measure of entanglement,” Phys. Rev. A. **65**, 032314 (2002). [CrossRef]

^{T}) is taken. The higher the negativity, the more entanglement is present in the system. It is defined as the absolute value of the sum of negative eigenvalues of ρ

^{T}

_{i}are the eigenvalues of ρ

^{T}.

*E*

_{N}defined as:

21. P.G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, “Experimental entanglement distillation and ‘hidden’ non-locality,” Nature **409**, 1014–1017 (2001). [CrossRef] [PubMed]

*E*for increasing fiber lengths.

_{N}*E*is calculated from the raw data and also from corrected data, where the background has been removed. The actual background was measured for each individual data point by adding an additional 10 ns delay thereby temporally displacing the gate window from the coincidence peak. The recorded coincidences are then composed of detector dark counts and multi pair contributions only and were subtracted from the measured coincidences to calculate a corrected density matrix. The uncorrected values decrease from 0.94 at 0 km to 0.83 at 75 km. At 101 km, the negativity is higher at 0.88 due to lower losses in set II. When background counts are removed

_{N}*E*decreases from 0.99 to 0.93 in the first 50 km then remaining constant up to 101 km.

_{N}### 4.2. Two-photon visibility

*HV*and the +- basis for fiber lengths up to 101 km. The two-photon visibility (

*V*) was calculated using the definition

*Max*is the maximal coincidence rate as obtained for parallel polarizer settings at Alice and Bob and Min is the coincidence rate at orthogonal settings.

*HV*and the +- basis as a function of fiber length. We display the average of the two visibilities since any negative effect on the polarization induced by the fiber should affect both polarizations basis equally. Indeed, we found the difference of the two visibilities (

*HV*, +-) to be less than 3%.

*V*), the same data as in Fig. 4(a) but with background counts removed. At the source (0 km),

_{cor}*V*was measured to be 99.3%. For all distances measured with fiber set I, Vcor remains around 95%. Using the 101 km fiber we see an increase from 89% (uncorrected) to

_{cor}*V*= 98%.

_{cor}*V*over the transmission length

*l*based on dark counts, multi-pair emission and chromatic dispersion:

*Max*

_{0}and

*Min*

_{0}are the coincidence rates at 0 km,

*T*(

*l*) is the measured transmission after length

*l*,

*R*is the accidental coincidence count rate at 0 km (due to multi-pairs) and

_{acc}*R*is the detector dark count rate.

_{dark}*T*(

*l*)

^{2}terms for multi-pairs were omitted since their effect is less than 0.5% on the visibility. The dark count rate was measured by closing the fiber input port of the InGaAs detector but still triggering it with the Si-APD. Since

*R*is independent with transmission distance it will be the dominant factor in decreasing the visibility at large distances.

_{dark}*F*(

*l*) describes the effect of chromatic dispersion causing a broadening of the photon wavepacket which may eventually become wider than the detector gate window, leading to a decrease in

*Max*and

*Min*.

*F*(

*l*) is calculated from ∫

*D*(

*t*)

*G*(

*t,l*)

*dt*, the overlap between the detector response (gate window)

*D*(

*t*) and the distance-dependent width of the gaussian wavepacket

*G*(

*t,l*). Note that

*R*is not affected by chromatic dispersion. The parameters

_{acc}*Max*

_{0},

*Min*

_{0},

*R*and

_{acc}*R*were experimentally determined from measurements of the source.

_{dark}*T*(

*l*) and

*F*(

*l*) were found from experimental characterization of the fibers.

*T*(

*l*)) and lower chromatic dispersion (

*F*(

*l*)).

- The model predicts the measured point very accurately for set II, Fig. 4(a), implying that with this fiber there are no additional effects leading to a reduction of the visibility.
- For set I, the measured raw visibility, lies on average 3%-5% lower than the model curve. This difference indicates that depolarization effects are present in the fiber which are not covered by our model. However, the dashed guide-to-the-eye (offset with the model curve) implies almost no additional loss of coherence at longer lengths.
- This tendency can also be seen when studying the corrected visibilities in Fig. 4(b). After an initial drop in the corrected visibility to 96% (3% lower than at the source), only a small additional decrease of about 2% for the remaining 75 km of set I is observed. Similarly, for set II, a decrease of about 1.3% is observed for the whole 101 km transmission, giving an estimate of how much the fiber disturbs the polarization state.
- The stronger decrease for set I at shorter distances could be because the first spool of fiber might have been slightly damaged or these fibers have an exceptionally high PMD. Since PMD could not be directly measured we did not include its effects in our model. We do however think that this point still deserves further investigations and are planning further experiments using short polarization maintaining fibers to study the effects of PMD alone.

*QE*and a lossy polarization measurement at Bob, the local coincidence rate was decreased to ~12 kcounts/s. The measured coincidence rates are displayed in Fig. 4(b). The exponential decrease corresponds to a loss of 0.23 dB/km on average for set I and a smaller loss of 0.21 dB/km for set II. With the latter, a coincidence rate of 104 counts/s could still be detected after 101 km. The same sifted key rate is expected for QKD [2

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

10. A. Poppe, A. Fedrizzi, R. Ursin, H. R. Böhm, T. Lorünser, O. Maurhardt, M. Peev, M. Suda, C. Kurtsiefer, H. Weinfurter, T. Jennewein, and A. Zeilinger, “Practical quantum key distribution with polarization entangled photons,” Opt. Express **12**, 3865–3871 (2004). [CrossRef] [PubMed]

22. 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]

23. G. Brassard and L. Salvail, “Secret key reconciliation by public discussion,” Lecture Notes in Computer Science **765**, 410423 (1994). [CrossRef]

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

*F*(

*l*) = 1) since smaller bandwidths can be achieved using longer crystals [8

08. S. Sauge, M. Swillo, S. Albert-Seifried, G. B. Xavier, J. Waldebäck, M. Tengner, D. Ljunggren, and A. Karlsson, “Narrowband polarization-entangled photon pairs distributed over a WDM link for qubit networks,” Opt. Express **15**, 6926–6933 (2007). [CrossRef] [PubMed]

*QE*of 0.5% [25]. This would correspond to a total dark count rate of 0.0015 c/s for our experimental set-up and would allow the detection of quantum correlations without background subtraction up to 200 km. The total rate at this distance would however be in the order of one count in 100 seconds.

## 5. Conclusion

## Acknowledgments

## References and links

01. | M. Dusek, N. Lutkenhaus, and M. Hendrych, “Quantum Cryptography,” Progress in Optics |

02. | C.H. Bennett, G. Brassard, and N.D. Mermin, “Quantum cryptography without Bell’s theorem,” Phys. Rev. Lett. |

03. | H. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. |

04. | R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Oemer, M. Fuerst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Free-Space distribution of entanglement and single photons over 144 km,” http://www.arxiv.org/abs/quant-ph/0607182. |

05. | I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legré, and N. Gisin, “Distribution of Time-Bin Entangled Qubits over 50 km of Optical Fiber,” Phys. Rev. Lett. |

06. | H. Takesue, “Long-distance distribution of time-bin entanglement generated in a cooled fiber,” Opt. Express |

07. | C. Liang, K. F. Lee, J. Chen, and P. Kumar, “Distribution of fiber-generated polarization entangled photon-pairs over 100 km of standard fiber in OC-192 WDM environment,” postdeadline paper, Optical Fiber Communications Conference (OFC2006), paper PDP35. |

08. | S. Sauge, M. Swillo, S. Albert-Seifried, G. B. Xavier, J. Waldebäck, M. Tengner, D. Ljunggren, and A. Karlsson, “Narrowband polarization-entangled photon pairs distributed over a WDM link for qubit networks,” Opt. Express |

09. | D.N. Matsukevich, T. Chaneliere, S.D. Jenkins, S.Y. Lan, T.A.B. Kennedy, and A. Kuzmich, “Entanglement of Remote Atomic Qubits,” Phys. Rev. Lett |

10. | A. Poppe, A. Fedrizzi, R. Ursin, H. R. Böhm, T. Lorünser, O. Maurhardt, M. Peev, M. Suda, C. Kurtsiefer, H. Weinfurter, T. Jennewein, and A. Zeilinger, “Practical quantum key distribution with polarization entangled photons,” Opt. Express |

11. | G. Ribordy, J. Brendel, J.D. Gautier, N. Gisin, and H. Zbinden, “Long-distance entanglement-based quantum key distribution,” Phys. Rev. A. |

12. | M. Pelton, P. Marsden, D. Ljunggren, M. Tenger, A. Karlsson, A. Fragemann, C. Canalias, and F. Laurell, “Bright, single-spatial-mode source of frequency non-degenerate, polarization-entangled photon pairs using perioically pole KTP,” Opt. Express |

13. | F. Konig, E.J. Mason, F.N.C. Wong, and M.A. Albota, “Efficient and spectrally bright source of polarization-entangled photons,” Phys. Rev. A. |

14. | P.G. Kwiat, E. Waks, A.G. White, I. Appelbaum, and P.H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A. |

15. | D. Ljunggren and M. Tengner, “Optimal focusing for maximal collection of narrow-band photon pairs into single-mode fibers,” Phys. Rev. A. |

16. | D. Ljunggren, M. Tengner, P. Marsden, and M. Pelton, “Theory and experiment of entanglement in a quasi-phase-matched two-crystal source,” Phys. Rev. A. |

17. | J. N. Damask, |

18. | D.F.V. James, P.G. Kwiat, W.J. Munro, and A.G. White, “Measurement of qubits,” Phys. Rev. A. |

19. | Z. Hradil, “Quantum-state estimation,” Phys. Rev. A. |

20. | G. Vidal and R.F. Werner,“Computable measure of entanglement,” Phys. Rev. A. |

21. | P.G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, “Experimental entanglement distillation and ‘hidden’ non-locality,” Nature |

22. | E. Waks, A. Zeevi, and Y. Yamamoto, “Security of quantum key distribution with entangled photons against individual attacks,” Phys. Rev. A. |

23. | G. Brassard and L. Salvail, “Secret key reconciliation by public discussion,” Lecture Notes in Computer Science |

24. | N. Lütkenhaus, “Security against individual attacks for realistic quantum key distribution,” Phys. Rev. A |

25. | C. Liang, K. F. Lee, M. Medic, and P. Kumar, “Characterization of fiber-generated entangled photon pairs with superconducting single-photon detectors,” Opt. Express |

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: March 26, 2007

Revised Manuscript: June 1, 2007

Manuscript Accepted: June 5, 2007

Published: June 8, 2007

**Citation**

Hannes Hübel, Michael R. Vanner, Thomas Lederer, Bibiane Blauensteiner, Thomas Lorünser, Andreas Poppe, and Anton Zeilinger, "High-fidelity transmission of polarization encoded qubits from an entangled source over 100 km of fiber," Opt. Express **15**, 7853-7862 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-12-7853

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### References

- M. Dusek, N. Lutkenhaus, and M. Hendrych, "Quantum Cryptography," Prog. Opt. 49, 381-454 (2006).
- C. H. Bennett, G. Brassard, and N. D. Mermin, "Quantum cryptography without Bell’s theorem," Phys. Rev. Lett. 68, 557-559 (1992). [CrossRef] [PubMed]
- H. Briegel, W. Dur, 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]
- R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Oemer, M. Fuerst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, "Free-Space distribution of entanglement and single photons over 144 km," http://www.arxiv.org/abs/quant-ph/0607182>.
- I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legre, and N. Gisin, "Distribution of time-bin entangled qubits over 50 km of optical fiber," Phys. Rev. Lett. 93, 180502 (2004). [CrossRef] [PubMed]
- H. Takesue, "Long-distance distribution of time-bin entanglement generated in a cooled fiber," Opt. Express 14, 3453-3460 (2006). [CrossRef] [PubMed]
- C. Liang, K. F. Lee, J. Chen, and P. Kumar, "Distribution of fiber-generated polarization entangled photon-pairs over 100 km of standard fiber in OC-192WDMenvironment," postdeadline paper, Optical Fiber Communications Conference (OFC2006), paper PDP35.
- S. Sauge, M. Swillo, S. Albert-Seifried, G. B. Xavier, J. Waldeback, M. Tengner, D. Ljunggren, A. Karlsson, "Narrowband polarization-entangled photon pairs distributed over aWDMlink for qubit networks," Opt. Express 15, 6926-6933 (2007). [CrossRef] [PubMed]
- D. N. Matsukevich, T. Chaneliere, S. D. Jenkins, S. Y. Lan, T. A. B. Kennedy, and A. Kuzmich, "Entanglement of remote atomic qubits," Phys. Rev. Lett 96, 030405 (2006). [CrossRef] [PubMed]
- A. Poppe, A. Fedrizzi, R. Ursin, H. R. Bohm, T. Lorunser, O. Maurhardt, M. Peev, M. Suda, C. Kurtsiefer, H. Weinfurter, T. Jennewein, and A. Zeilinger, "Practical quantum key distribution with polarization entangled photons," Opt. Express 12, 3865-3871 (2004). [CrossRef] [PubMed]
- G. Ribordy, J. Brendel, J. D. Gautier, N. Gisin, and H. Zbinden, "Long-distance entanglement-based quantum key distribution," Phys. Rev. A. 63, 012309 (2000). [CrossRef]
- M. Pelton, P. Marsden, D. Ljunggren, M. Tenger, A. Karlsson, A. Fragemann, C. Canalias, and F. Laurell, "Bright, single-spatial-mode source of frequency non-degenerate, polarization-entangled photon pairs using periodically pole KTP," Opt. Express 12, 3573-3580 (2004). [CrossRef] [PubMed]
- F. Konig, E.J . Mason, F. N. C. Wong, and M. A. Albota, "Efficient and spectrally bright source of polarizationentangled photons," Phys. Rev. A. 71, 033805 (2005). [CrossRef]
- P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A. 60, R773-6 (1999). [CrossRef]
- D. Ljunggren and M. Tengner, "Optimal focusing for maximal collection of narrow-band photon pairs into single-mode fibers," Phys. Rev. A. 72, 062301 (2005). [CrossRef]
- D. Ljunggren, M. Tengner, P. Marsden, and M. Pelton, "Theory and experiment of entanglement in a quasiphase- matched two-crystal source," Phys. Rev. A. 73, 032326 (2006). [CrossRef]
- J. N. Damask, Polarization Optics in Telecommunications (Springer 2005).
- D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, "Measurement of qubits," Phys. Rev. A. 64, 052312 (2001). [CrossRef]
- Z. Hradil, "Quantum-state estimation," Phys. Rev. A. 55, R1561-R1564 (1997). [CrossRef]
- G. Vidal and R. F. Werner,"Computable measure of entanglement," Phys. Rev. A. 65, 032314 (2002). [CrossRef]
- P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, "Experimental entanglement distillation and ‘hidden’ non-locality," Nature 409, 1014-1017 (2001). [CrossRef] [PubMed]
- 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]
- G. Brassard and L. Salvail, "Secret key reconciliation by public discussion," Lecture notes in Computer Science 765, 410423 (1994). [CrossRef]
- N. Lutkenhaus, "Security against individual attacks for realistic quantum key distribution," Phys. Rev. A 61, 052304 (2000). [CrossRef]
- C. Liang, K. F. Lee, M. Medic, and P. Kumar, "Characterization of fiber-generated entangled photon pairs with superconducting single-photon detectors," Opt. Express 12, 3573-3580 (2004).

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