Implementation of quantum state tomography for time-bin entangled photon pairs

Hiroki Takesue; Yuita Noguchi

doi:10.1364/OE.17.010976

Implementation of quantum state tomography for time-bin entangled photon pairs

Hiroki Takesue and Yuita Noguchi

Optics Express
Vol. 17,
Issue 13,
pp. 10976-10989
(2009)
•https://doi.org/10.1364/OE.17.010976

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Hiroki Takesue and Yuita Noguchi, "Implementation of quantum state tomography for time-bin entangled photon pairs," Opt. Express 17, 10976-10989 (2009)

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Related Topics
Table of Contents Category
- Quantum Optics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Nonlinear optics, fibers (190.4370)
- Quantum communications (270.5565)

About this Article
History
- Original Manuscript: May 7, 2009
- Revised Manuscript: June 15, 2009
- Manuscript Accepted: June 15, 2009
- Published: June 16, 2009

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Open Access

Abstract

Quantum state tomography (QST) is an important method for evaluating the quality of entangled photon pairs, and has been widely used to measure polarization entanglement. However, QST has not been applied to time-bin entanglement, which is a type of entanglement suitable for fiber transmission. In this paper, we clarify the way to implement QST on time-bin entangled photon pairs using a 1-bit delayed interferometer. We also provide experimental results for a demonstration of QST for time-bin entangled photon pairs generated using spontaneous four-wave mixing in a dispersion shifted fiber.

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References

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|
Author
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Publication

A. Aspect, P. Grangier, and G. Roger, “Experimental realization of Einstein-Podolsky-Rosen-Bohm gedanken-experiment: a new violation of Bell’s inequalities,” Phys. Rev. Lett. 49, 91–94 (1982).
[Crossref]
P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “High-visibility interference in a Bell-inequality experiment for energy and time,” Phys. Rev. A 47, R2472–R2475 (1993).
[Crossref] [PubMed]
D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]
Y. Nambu, K. Usami, Y. Tsuda, K. Matsumoto, and K. Nakamura, “Generation of polarization-entangled photon pairs in a cascade of two type-I crystals pumped by femtosecond pulses,” Phys. Rev. A 66, 033816 (2002)
[Crossref]
K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh, “Generation of ultraviolet entangled photons in a semiconductor,” Nature 431, 167–170 (2004).
[Crossref] [PubMed]
N. A. Peters, J. B. Altepeter, D. Branning, E. R. Jeffrey, T.-C. Wei, and P. G. Kwiat, “Maximally entangled mixed states: creation and concentration,” Phys. Rev. Lett. 92, 133601 (2004).
[Crossref] [PubMed]
R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, “A semiconductor source of triggered entangled photon pairs,” Nature 439, 179–182 (2006)
[Crossref] [PubMed]
S. Odate, A. Yoshizawa, and H. Tsuchida, “Polarisation-entangled photon-pair source at 1550 nm using 1 mm-long PPLN waveguide in fibre-loop configuration,” Electron. Lett. 43, 1376–1377 (2007).
[Crossref]
J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1999).
[Crossref]
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]
T. Honjo, H. Takesue, H. Kamada, Y. Nishida, O. Tadanaga, M. Asobe, and K. Inoue, “Long-distance distribution of time-bin entangled photon pairs over 100 km using frequency up-conversion detectors,” Opt. Express 15, 13957–13964 (2007).
[Crossref] [PubMed]
J. Dynes, H. Takesue, Z. L. Yuan, A. W. Sharpe, K. Harada, T. Honjo, H. Kamada, O. Tadanaga, Y. Nishida, M. Asobe, and A. J. Shields, “Ultra-long distance and efficient entanglement distribution over 200 km,” submitted to Opt. Express.
T. Honjo, S. W. Nam, H. Takesue, Q. Zhang, H. Kamada, Y. Nishida, O. Tadanaga, M. Asobe, B. Baek, R Hadfield, S. Miki, M. Fujiwara, M. Sasaki, Z. Wang, K. Inoue, and Y. Yamamoto, “Long-distance entanglement-based quantum key distribution over optical fiber,” Opt. Express 16, 19118–19126 (2008).
[Crossref]
H. Takesue, K. Inoue, O. Tadanaga, Y. Nishida, and M. Asobe, “Generation of pulsed polarization-entangled photon pairs in a 1.55-µm band with a periodically poled lithium niobate waveguide and an orthogonal polarization delay circuit,” Opt. Lett. 30, 293–295 (2005).
[Crossref] [PubMed]
G. Berlin, G. Brassard, F. Bussieres, N. Godbout, J. A. Slater, and W. Tittel, “Flipping quantum coins,” arXiv:0904.3946.
M. Fiorentino, P. L. Voss, J. E. Kumar, and P. Sharping, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]
W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[Crossref] [PubMed]
H. Takesue and K. Inoue, “Generation of 1.5-µm band time-bin entanglement using spontaneous fiber four-wave mixing and planar lightwave circuit interferometers,” Phys. Rev. A 72, 041804(R) (2005).
[Crossref]
N. Namekata, S. Sasamori, and S. Inoue, “800 MHz single-photon detection at 1550-nm using an InGaAs/InP avalanche photodiode operated with a sine wave gating,” Opt. Express 14, 10043–10049 (2006).
[Crossref] [PubMed]
B. Miquel and H. Takesue, “Observation of 1.5 µm band entanglement using single photon detectors based on sinusoidally gated InGaAs/InP avalanche photoidodes,” New J. Phys. 11, 045006 (2009).
[Crossref]
C. Langrock, E. Diamanti, R. V. Roussev, Y. Yamamoto, M. M. Fejer, and H. Takesue, “Highly efficient single-photon detection at communication wavelengths by use of upconversion in reverse-proton-exchanged periodically poled LiNbO3 waveguides,” Opt. Lett. 30, 1725–1727 (2005).
[Crossref] [PubMed]
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]

2009 (1)

B. Miquel and H. Takesue, “Observation of 1.5 µm band entanglement using single photon detectors based on sinusoidally gated InGaAs/InP avalanche photoidodes,” New J. Phys. 11, 045006 (2009).
[Crossref]

2008 (1)

T. Honjo, S. W. Nam, H. Takesue, Q. Zhang, H. Kamada, Y. Nishida, O. Tadanaga, M. Asobe, B. Baek, R Hadfield, S. Miki, M. Fujiwara, M. Sasaki, Z. Wang, K. Inoue, and Y. Yamamoto, “Long-distance entanglement-based quantum key distribution over optical fiber,” Opt. Express 16, 19118–19126 (2008).
[Crossref]

2007 (2)

T. Honjo, H. Takesue, H. Kamada, Y. Nishida, O. Tadanaga, M. Asobe, and K. Inoue, “Long-distance distribution of time-bin entangled photon pairs over 100 km using frequency up-conversion detectors,” Opt. Express 15, 13957–13964 (2007).
[Crossref] [PubMed]

S. Odate, A. Yoshizawa, and H. Tsuchida, “Polarisation-entangled photon-pair source at 1550 nm using 1 mm-long PPLN waveguide in fibre-loop configuration,” Electron. Lett. 43, 1376–1377 (2007).
[Crossref]

2006 (3)

R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, “A semiconductor source of triggered entangled photon pairs,” Nature 439, 179–182 (2006)
[Crossref] [PubMed]

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

N. Namekata, S. Sasamori, and S. Inoue, “800 MHz single-photon detection at 1550-nm using an InGaAs/InP avalanche photodiode operated with a sine wave gating,” Opt. Express 14, 10043–10049 (2006).
[Crossref] [PubMed]

2005 (3)

H. Takesue and K. Inoue, “Generation of 1.5-µm band time-bin entanglement using spontaneous fiber four-wave mixing and planar lightwave circuit interferometers,” Phys. Rev. A 72, 041804(R) (2005).
[Crossref]

H. Takesue, K. Inoue, O. Tadanaga, Y. Nishida, and M. Asobe, “Generation of pulsed polarization-entangled photon pairs in a 1.55-µm band with a periodically poled lithium niobate waveguide and an orthogonal polarization delay circuit,” Opt. Lett. 30, 293–295 (2005).
[Crossref] [PubMed]

C. Langrock, E. Diamanti, R. V. Roussev, Y. Yamamoto, M. M. Fejer, and H. Takesue, “Highly efficient single-photon detection at communication wavelengths by use of upconversion in reverse-proton-exchanged periodically poled LiNbO3 waveguides,” Opt. Lett. 30, 1725–1727 (2005).
[Crossref] [PubMed]

2004 (3)

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]

K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh, “Generation of ultraviolet entangled photons in a semiconductor,” Nature 431, 167–170 (2004).
[Crossref] [PubMed]

N. A. Peters, J. B. Altepeter, D. Branning, E. R. Jeffrey, T.-C. Wei, and P. G. Kwiat, “Maximally entangled mixed states: creation and concentration,” Phys. Rev. Lett. 92, 133601 (2004).
[Crossref] [PubMed]

2002 (2)

Y. Nambu, K. Usami, Y. Tsuda, K. Matsumoto, and K. Nakamura, “Generation of polarization-entangled photon pairs in a cascade of two type-I crystals pumped by femtosecond pulses,” Phys. Rev. A 66, 033816 (2002)
[Crossref]

M. Fiorentino, P. L. Voss, J. E. Kumar, and P. Sharping, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

2001 (2)

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]

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

2000 (1)

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[Crossref] [PubMed]

1999 (1)

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1999).
[Crossref]

1993 (1)

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “High-visibility interference in a Bell-inequality experiment for energy and time,” Phys. Rev. A 47, R2472–R2475 (1993).
[Crossref] [PubMed]

1982 (1)

A. Aspect, P. Grangier, and G. Roger, “Experimental realization of Einstein-Podolsky-Rosen-Bohm gedanken-experiment: a new violation of Bell’s inequalities,” Phys. Rev. Lett. 49, 91–94 (1982).
[Crossref]

Altepeter, J. B.

Asobe, M.

J. Dynes, H. Takesue, Z. L. Yuan, A. W. Sharpe, K. Harada, T. Honjo, H. Kamada, O. Tadanaga, Y. Nishida, M. Asobe, and A. J. Shields, “Ultra-long distance and efficient entanglement distribution over 200 km,” submitted to Opt. Express.

Aspect, A.

Atkinson, P.

Baek, B.

Berlin, G.

G. Berlin, G. Brassard, F. Bussieres, N. Godbout, J. A. Slater, and W. Tittel, “Flipping quantum coins,” arXiv:0904.3946.

Branning, D.

Brassard, G.

G. Berlin, G. Brassard, F. Bussieres, N. Godbout, J. A. Slater, and W. Tittel, “Flipping quantum coins,” arXiv:0904.3946.

Brendel, J.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[Crossref] [PubMed]

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1999).
[Crossref]

Bussieres, F.

G. Berlin, G. Brassard, F. Bussieres, N. Godbout, J. A. Slater, and W. Tittel, “Flipping quantum coins,” arXiv:0904.3946.

Chiao, R. Y.

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “High-visibility interference in a Bell-inequality experiment for energy and time,” Phys. Rev. A 47, R2472–R2475 (1993).
[Crossref] [PubMed]

Chulkova, G.

Cooper, K.

de Riedmatten, H.

Diamanti, E.

Dynes, J.

Dzardanov, A.

Edamatsu, K.

K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh, “Generation of ultraviolet entangled photons in a semiconductor,” Nature 431, 167–170 (2004).
[Crossref] [PubMed]

Fejer, M. M.

Fiorentino, M.

M. Fiorentino, P. L. Voss, J. E. Kumar, and P. Sharping, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

Fujiwara, M.

Gisin, N.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[Crossref] [PubMed]

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1999).
[Crossref]

Godbout, N.

G. Berlin, G. Brassard, F. Bussieres, N. Godbout, J. A. Slater, and W. Tittel, “Flipping quantum coins,” arXiv:0904.3946.

Gol’tsman, G. N.

Grangier, P.

Hadfield, R

Harada, K.

Honjo, T.

Inoue, K.

Inoue, S.

Itoh, T.

K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh, “Generation of ultraviolet entangled photons in a semiconductor,” Nature 431, 167–170 (2004).
[Crossref] [PubMed]

James, D. F. V.

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

Jeffrey, E. R.

Kamada, H.

Kumar, J. E.

M. Fiorentino, P. L. Voss, J. E. Kumar, and P. Sharping, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

Kwiat, P. G.

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

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “High-visibility interference in a Bell-inequality experiment for energy and time,” Phys. Rev. A 47, R2472–R2475 (1993).
[Crossref] [PubMed]

Langrock, C.

Legre, M.

Lipatov, A.

Marcikic, I.

Matsumoto, K.

Miki, S.

Miquel, B.

Munro, W. J.

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

Nakamura, K.

Nam, S. W.

Nambu, Y.

Namekata, N.

Nishida, Y.

Odate, S.

Okunev, O.

Oohata, G.

K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh, “Generation of ultraviolet entangled photons in a semiconductor,” Nature 431, 167–170 (2004).
[Crossref] [PubMed]

Peters, N. A.

Ritchie, D. A.

Roger, G.

Roussev, R. V.

Sasaki, M.

Sasamori, S.

Semenov, A.

Sharpe, A. W.

Sharping, P.

M. Fiorentino, P. L. Voss, J. E. Kumar, and P. Sharping, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

Shields, A. J.

Shimizu, R.

K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh, “Generation of ultraviolet entangled photons in a semiconductor,” Nature 431, 167–170 (2004).
[Crossref] [PubMed]

Slater, J. A.

G. Berlin, G. Brassard, F. Bussieres, N. Godbout, J. A. Slater, and W. Tittel, “Flipping quantum coins,” arXiv:0904.3946.

Smirnov, K.

Sobolewski, R.

Steinberg, A. M.

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “High-visibility interference in a Bell-inequality experiment for energy and time,” Phys. Rev. A 47, R2472–R2475 (1993).
[Crossref] [PubMed]

Stevenson, R. M.

Tadanaga, O.

Takesue, H.

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

Tittel, W.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[Crossref] [PubMed]

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1999).
[Crossref]

G. Berlin, G. Brassard, F. Bussieres, N. Godbout, J. A. Slater, and W. Tittel, “Flipping quantum coins,” arXiv:0904.3946.

Tsuchida, H.

Tsuda, Y.

Usami, K.

Voronov, B.

Voss, P. L.

M. Fiorentino, P. L. Voss, J. E. Kumar, and P. Sharping, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

Wang, Z.

Wei, T.-C.

White, A. G.

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

Williams, C.

Yamamoto, Y.

Yoshizawa, A.

Young, R. J.

Yuan, Z. L.

Zbinden, H.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[Crossref] [PubMed]

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1999).
[Crossref]

Zhang, Q.

Appl. Phys. Lett. (1)

Electron. Lett. (1)

IEEE Photon. Technol. Lett. (1)

M. Fiorentino, P. L. Voss, J. E. Kumar, and P. Sharping, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

Nature (2)

K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh, “Generation of ultraviolet entangled photons in a semiconductor,” Nature 431, 167–170 (2004).
[Crossref] [PubMed]

New J. Phys. (1)

Opt. Express (4)

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

Opt. Lett. (2)

Phys. Rev. A (4)

P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, “High-visibility interference in a Bell-inequality experiment for energy and time,” Phys. Rev. A 47, R2472–R2475 (1993).
[Crossref] [PubMed]

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

Phys. Rev. Lett. (5)

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1999).
[Crossref]

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84, 4737–4740 (2000).
[Crossref] [PubMed]

Other (2)

G. Berlin, G. Brassard, F. Bussieres, N. Godbout, J. A. Slater, and W. Tittel, “Flipping quantum coins,” arXiv:0904.3946.

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Fig. 1.

Poincaré spheres for (a) polarization and (b) time-bin qubits.

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Fig. 2.

Projection measurement using 1-bit delayed interferometer.

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Fig. 3.

Experimental setup.

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Fig. 4.

Typical histogram of photon arrival time for D1.

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Fig. 5.

Density matrix of time-bin entangled photon pairs obtained by QST. (a) and (b) show the real and the imaginary parts of the matrix obtained by linear tomography, while (c) and (d) are those obtained by maximum likelihood estimation.

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Tables (3)

Table 1. Tomographic analysis states used in the experimentDThe fourth column refers to the two-photon states to which photons 1 and 2 are projected.

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Table 2. Experimentally obtained coincidence counts. A dash (-) indicates that coincidences were not obtained for the corresponding projection measurement. The 4th, 5th, 6th and 7th columns correspond to the obtained coincidence counts for four different energy-basis projection measurement settings, and the 8th column gives the total coincidence counts denoted as n_ν .

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Table 3. Quantities derived from the density matrix

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Equations (26)

Equations on this page are rendered with MathJax. Learn more.

∣ ϕ 〉 = α ∣ 1 〉 + β ∣ 2 〉,

\frac{∣ 1 〉 + e^{- i θ} ∣ 2 〉}{\sqrt{2}} .

\hat{ρ} = Σ_{μ = 1}^{16} {\hat{Γ}}_{μ} r_{μ}

r_{μ} = T r {{\hat{Γ}}_{μ} \hat{ρ}}

n_{v} = C 〈 ψ_{v} ∣ \hat{ρ} ∣ ψ_{v} 〉,

(\begin{array}{c} n_{1} \\ n_{2} \\ ⋮ \\ n_{16} \end{array}) = C (\begin{array}{c} B_{1,1} & B_{1,2} & \dots & B_{1,16} \\ B_{2,1} & \dots & \dots & B_{2,16} \\ ⋮ & ⋮ \\ B_{16,1} & \dots & \dots & B_{16,16} \end{array}) (\begin{array}{c} r_{1} \\ r_{2} \\ ⋮ \\ r_{16} \end{array}),

B_{x, y} = 〈 ψ_{x} ∣ Γ_{y} ∣ ψ_{x} 〉 .

(\begin{array}{c} r_{1} \\ r_{2} \\ ⋮ \\ r_{16} \end{array}) = C B^{- 1} (\begin{array}{c} n_{1} \\ n_{2} \\ ⋮ \\ n_{16} \end{array})

\hat{ρ} = C^{- 1} Σ_{v = 1}^{16} {\hat{M}}_{v} n_{v},

{\hat{M}}_{v} = \underset{x}{Σ} Γ_{x} {(B^{- 1})}_{x, ν} .

C = Σ_{ν = 1}^{4} n_{ν}

\hat{ρ} = (Σ_{ν = 1}^{16} {\hat{M}}_{ν} n_{ν}) ⁄ (Σ_{ν = 1}^{4} n_{ν}) .

∣ Ψ 〉 = \frac{1}{\sqrt{2}} ({∣ 1 〉}_{s} {∣ 1 〉}_{i} + {∣ 2 〉}_{s} {∣ 2 〉}_{i})

ρ = (\begin{array}{c} 0.445626 & 0.011447 + i 0.013083 & 0.012674 + i 0.010221 & 0.421504 - i 0.107931 \\ 0.011447 - i 0.013083 & 0.040065 & 0.059689 - i 0.012674 & - 0.05070 + i 0.048242 \\ 0.012674 - i 0.010221 & 0.059689 + i 0.012674 & 0.026165 & - 0.022486 + i 0.004497 \\ 0.421504 + i 0.107931 & - 0.05070 - i 0.048242 & - 0.022486 - i 0.004497 & 0.488144 \end{array})

ρ_{p} = (\begin{array}{c} 0.461266 & 0.006138 + i 0.014805 & - 0.002069 + i 0.021371 & 0.395186 - i 0.090409 \\ 0.006138 - i 0.014805 & 0.039398 & 0.019510 + i 0.013286 & - 0.034332 + i 0.028899 \\ - 0.002069 - i 0 . 021371 & 0.019510 - i 0.013286 & 0.027424 & - 0.000301 + i 0.001415 \\ 0.395186 + i 0.090409 & - 0.034332 - i 0.028899 & - 0.000301 - i 0.001415 & 0.471911 \end{array})

∣ Ψ 〉 = \frac{1}{\sqrt{2}} (∣ {1 〉}_{s} ∣ {1 〉}_{i} + e^{iϕ} ∣ {2 〉}_{s} {2 〉}_{i})

∣ Ψ 〉 \to ∣ {1 〉}_{s} ∣ {1 〉}_{i} + e^{i ϕ_{i}} ∣ {1 〉}_{s} ∣ {2 〉}_{i} + e^{i ϕ_{s}} ∣ {2 〉}_{s} ∣ {1 〉}_{i} + (e^{i (ϕ_{s} + ϕ_{i})} + e^{iϕ}) ∣ {2 〉}_{s} ∣ {2 〉}_{i}

+ e^{i (ϕ + ϕ_{i})} ∣ {2 〉}_{s} ∣ {3 〉}_{i} + e^{i (ϕ + ϕ_{s})} ∣ {3 〉}_{s} ∣ {2 〉}_{i} + e^{i (ϕ + ϕ_{s} + ϕ_{i})} ∣ {3 〉}_{s} ∣ {3 〉}_{i},

\overset{ˉ}{δ S_{μ} δ S_{ν}} = \frac{S_{ν} δ_{μ, ν}}{C} .

\overset{ˉ}{δ S_{μ} δ S_{μ}} = δ_{ν, μ} \underset{x = s, i ε,}{Σ} Σ_{λ = 1}^{16} f_{ν, ε}^{(x)} f_{ν, λ}^{(x)} S_{ε} S_{λ} {(Δ θ)}^{2}

f_{ν, μ}^{(x)} = \{\frac{\partial}{{\partial θ}_{ν, x}} 〈 ψ_{ν} ∣\} {\hat{M}}_{μ} ∣ ψ_{ν} 〉 + 〈 ψ_{ν} ∣ {\hat{M}}_{μ} {\frac{\partial}{\partial θ_{ν, x}} ∣ ψ_{ν} 〉},

\overset{ˉ}{δ S_{ν} δ S_{μ}} = δ_{ν, μ} Λ_{ν}

Λ_{ν} = [\frac{s_{v}}{C} + \underset{x = s, i}{Σ} Σ_{ε, λ = 1}^{16} f_{ν, ε}^{(x)} f_{ν, λ}^{(x)} S_{ε} S_{λ} {(Δ θ)}^{2}] .

F = 〈 ψ ∣ ρ ∣ ψ 〉 .

{(Δ F)}^{2} = Σ_{ν = 1}^{16} {(\frac{\partial F}{\partial s_{ν}})}^{2} Λ_{ν} .

\frac{\partial F}{\partial s_{v}} = \frac{1}{2} {〈 11 ∣ M_{v} ∣ 11 〉 + 〈 11 ∣ M_{v} ∣ 22 〉 + 〈 22 ∣ M_{v} ∣ 11 〉 + 〈 22 ∣ M_{v} ∣ 22 〉}

ν	Photon 1	Photon 2	\|ψ_ν 〉=(\|11〉, \|12〉, \|21〉, \|22〉)
1	\|1〉	\|1〉	(1,0,0,0)
2	\|1〉	\|2〉	(0,1,0,0)
3	\|2〉	\|1〉	(0,0,1,0)
4	\|2〉	\|2〉	(0,0,0,1)
5	\|2〉	\|+〉	(0,0 $\frac{1}{\sqrt{2}}$ , $\frac{1}{\sqrt{2}}$ )
6	\|1〉	\|+〉	( $\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$ ,0,0)
7	\|+〉	\|+〉	( $\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}$ )
8	\|L〉	\|+〉	( $\frac{1}{2}, \frac{1}{2}, \frac{i}{2}, \frac{i}{2}$ )
9	\|L〉	\|1〉	( $\frac{1}{\sqrt{2}}, 0, \frac{i}{\sqrt{2}}, 0$ )
10	\|L〉	\|2〉	( $0, \frac{1}{\sqrt{2}}, 0, \frac{i}{\sqrt{2}}$ )
11	\|L〉	\|L〉	( $\frac{1}{2}, \frac{i}{2}, \frac{i}{2}, - \frac{1}{2}$ )
12	\|1〉	\|L〉	( $\frac{1}{\sqrt{2}}, \frac{i}{\sqrt{2}}, 0, 0$ )
13	\|2〉	\|L〉	( $0, 0, \frac{1}{\sqrt{2}}, \frac{i}{\sqrt{2}}$ )
14	\|+〉	\|L〉	( $\frac{1}{2}, \frac{i}{2}, \frac{1}{2}, \frac{i}{2}$ )
15	\|+〉	\|1〉	( $\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}, 0$ )
16	\|+〉	\|2〉	( $0, \frac{1}{\sqrt{2}} 0, \frac{1}{\sqrt{2}}$ )

ν	Photon 1	Photon 2	++	+L	L+	LL	n_ν
1	\|1〉	\|1〉	139	142	130	134	545
2	\|1〉	\|2〉	7	14	14	14	49
3	\|2〉	\|1〉	9	7	9	7	32
4	\|2〉	\|2〉	149	144	153	151	597
5	\|2〉	\|+〉	146	-	141	-	287
6	\|1〉	\|+〉	151	-	160	-	311
7	\|+〉	\|+〉	570	-	-	-	570
8	\|L〉	\|+〉	-	-	337	-	337
9	\|L〉	\|1〉	-	-	128	148	276
10	\|L〉	\|2〉	-	-	144	120	264
11	\|L〉	\|L〉	-	-	-	38	38
12	\|1〉	\|L〉	-	156	-	125	281
13	\|2〉	\|L〉	-	160	-	149	309
14	\|+〉	\|L〉	-	330	-	-	330
15	\|+〉	\|1〉	154	150	-	-	304
16	\|+〉	\|2〉	135	126	-	-	261

Fidelity	0.862±0.044
Von Neumann entropy	0.648±0.247
Linear entropy	0.303±0.101
Concurrence	0.777±0.043

ν	Photon 1	Photon 2	\|ψ_ν 〉=(\|11〉, \|12〉, \|21〉, \|22〉)
1	\|1〉	\|1〉	(1,0,0,0)
2	\|1〉	\|2〉	(0,1,0,0)
3	\|2〉	\|1〉	(0,0,1,0)
4	\|2〉	\|2〉	(0,0,0,1)
5	\|2〉	\|+〉	(0,0 $\frac{1}{\sqrt{2}}$ , $\frac{1}{\sqrt{2}}$ )
6	\|1〉	\|+〉	( $\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$ ,0,0)
7	\|+〉	\|+〉	( $\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}$ )
8	\|L〉	\|+〉	( $\frac{1}{2}, \frac{1}{2}, \frac{i}{2}, \frac{i}{2}$ )
9	\|L〉	\|1〉	( $\frac{1}{\sqrt{2}}, 0, \frac{i}{\sqrt{2}}, 0$ )
10	\|L〉	\|2〉	( $0, \frac{1}{\sqrt{2}}, 0, \frac{i}{\sqrt{2}}$ )
11	\|L〉	\|L〉	( $\frac{1}{2}, \frac{i}{2}, \frac{i}{2}, - \frac{1}{2}$ )
12	\|1〉	\|L〉	( $\frac{1}{\sqrt{2}}, \frac{i}{\sqrt{2}}, 0, 0$ )
13	\|2〉	\|L〉	( $0, 0, \frac{1}{\sqrt{2}}, \frac{i}{\sqrt{2}}$ )
14	\|+〉	\|L〉	( $\frac{1}{2}, \frac{i}{2}, \frac{1}{2}, \frac{i}{2}$ )
15	\|+〉	\|1〉	( $\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}, 0$ )
16	\|+〉	\|2〉	( $0, \frac{1}{\sqrt{2}} 0, \frac{1}{\sqrt{2}}$ )

ν	Photon 1	Photon 2	++	+L	L+	LL	n_ν
1	\|1〉	\|1〉	139	142	130	134	545
2	\|1〉	\|2〉	7	14	14	14	49
3	\|2〉	\|1〉	9	7	9	7	32
4	\|2〉	\|2〉	149	144	153	151	597
5	\|2〉	\|+〉	146	-	141	-	287
6	\|1〉	\|+〉	151	-	160	-	311
7	\|+〉	\|+〉	570	-	-	-	570
8	\|L〉	\|+〉	-	-	337	-	337
9	\|L〉	\|1〉	-	-	128	148	276
10	\|L〉	\|2〉	-	-	144	120	264
11	\|L〉	\|L〉	-	-	-	38	38
12	\|1〉	\|L〉	-	156	-	125	281
13	\|2〉	\|L〉	-	160	-	149	309
14	\|+〉	\|L〉	-	330	-	-	330
15	\|+〉	\|1〉	154	150	-	-	304
16	\|+〉	\|2〉	135	126	-	-	261