Abstract

High-dimensional entanglement offers promising perspectives in quantum information science. However, how to generate high-quality high-dimensional entanglement and control it efficiently is still a challenge. Here, we experimentally demonstrate a polarization-path hybrid high-dimensional entangled two-photon source with extremely high quality. Based on stable interferometers, we measured fidelities exceeding 0.99 for both three-dimensional and four-dimensional maximal entanglement. The experimental setup can also be used to prepare arbitrary high-dimensional pure state and can be efficiently extended to even higher dimensional systems. Our new source will shed new light on high-dimensional quantum information processes.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (3)

X. M. Hu, B. H. Liu, Y. Guo, G. Y. Xiang, Y. F. Huang, C. F. Li, G. C. Guo, M. Kleinmann, T. Vértesi, and A. Cabello, “Observation of stronger-than-binary correlations with entangled photonic qutrits,” Phys. Rev. Lett. 120, 180402 (2018).
[Crossref] [PubMed]

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Maninska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acin, K. Rottwitt, L. K. Oxenlowe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref] [PubMed]

X. M. Hu, Y. Ga, B. H. Liu, Y. F. Huang, C. F. Li, and G. C. Guo, “Beating the channel capacity limit for superdense coding with entangled ququarts,” Sci. Adv. 4, eaat9304 (2018).
[Crossref] [PubMed]

2017 (4)

Anthony Martin, Thiago Guerreiro, Alexey Tiranov, Sébastien Designolle, Florian Fröwis, Nicolas Brunner, Marcus Huber, and Nicolas Gisin, “Quantifying Photonic High-Dimensional Entanglement,” Phys. Rev. Lett. 118, 110501 (2017).
[Crossref] [PubMed]

A. Babazadeh, M. Erhard, F. Wang, M. Malik, R. Nouroozi, M. Krenn, and A. Zeilinger, “High-Dimensional Single-Photon Quantum Gates: Concepts and Experiments,” Phys. Rev. Lett. 119180510 (2017).
[Crossref] [PubMed]

F. Wang, M. Erhard, A. Babazadeh, M. Malik, M. Krenn, and A. Zeilinger, “Generation of the complete four-dimensional Bell basis,” Optica 4, 1462–1467 (2017).
[Crossref]

M. Kues, C. Remer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref] [PubMed]

2016 (5)

R. B. Jin, R. Shimizu, M. Fujiwara, M. Takeoka, R. Wakabayashi, T. Yamashita, S. Miki, H. Terai, T. Gerrits, and M. Sasaki, “Simple method of generating and distributing frequency-entangled qudits,” Quantum Sci. Technol.  1, 015004 (2016).
[Crossref]

T. Ikuta and H. Takesue, “Enhanced violation of the Collins-Gisin-Linden-Massar-Popescu inequality with optimized time-bin-entangled ququarts,” Phys. Rev. A 93, 022307 (2016).
[Crossref]

B. H. Liu, X. M. Hu, J. S. Chen, Y. F. Huang, Y. J. Han, C. F. Li, G. C. Guo, and A. Cabello, “Nonlocality from Local Contextuality,” Phys. Rev. Lett. 117, 220402 (2016).
[Crossref] [PubMed]

M. Kleinmann and A. Cabello, “Quantum Correlations are Stronger than All Nonsignaling Correlations Produced by n-Outcome Measurements,” Phys. Rev. Lett. 117, 150401 (2016).
[Crossref] [PubMed]

X. M. Hu, J. S. Chen, B. H. Liu, Y. Guo, Y. F. Huang, Z. Q. Zhou, Y. J. Han, C. F. Li, and G. C. Guo, “Experimental Test of Compatibility-Loophole-Free Contextuality with Spatially Separated Entangled Qutrits,” Phys. Rev. Lett. 117, 170403 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (1)

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. U.S.A.  111, 6243 (2014).
[Crossref]

2013 (1)

C. Bernhard, B. Bessire, T. Feurer, and A. Stefanov, “Shaping frequency-entangled qudits,” Phys. Rev. A 88, 032322 (2013).
[Crossref]

2012 (2)

L. Olislager, I. Mbodji, E. Woodhead, J. Cussey, L. Furfaro, P. Emplit, S. Massar, K. P. Huy, and J. M. Merolla, “Implementing two-photon interference in the frequency domain with electro-optic phase modulators,” New J. Phys.  14, 043015 (2012).
[Crossref]

D. Richart, Y. Fischer, and H. Weinfurter, “Experimental implementation of higher dimensional time-energy entanglement,” Appl. Phys. B 106, 543 (2012).
[Crossref]

2011 (1)

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys.  7, 677 (2011).
[Crossref]

2009 (1)

R. Inoue, T. Yonehara, Y. Miyamoto, M. Koashi, and M. Kozuma, “Measuring qutrit-qutrit entanglement of orbital angular momentum states of an atomic ensemble and a photon,” Phys. Rev. Lett. 103, 110503 (2009).
[Crossref] [PubMed]

2005 (1)

D. Stucki, H. Zbinden, and N. Gisin, “A Fabry-Perot-like two-photon interferometer for high-dimensional time-bin entanglement,” J. Mod. Opt. 52, 2637 (2005).
[Crossref]

2004 (2)

R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
[Crossref]

C. Brukner, M. Zukowski, and A. Zeilinger, “Quantum communication complexity protocol with two entangled qutrits,” Phys. Rev. Lett. 89, 197901 (2004).
[Crossref]

2002 (3)

R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66, 012303 (2002).
[Crossref]

D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, “Bell inequalities for arbitrarily high-dimensional systems,” Phys. Rev. Lett. 88, 040404 (2002).
[Crossref] [PubMed]

H. d. Riedmatten, I. Marcikic, H. Zbinden, and N. Gisin, “Creating high dimensional time-bin entanglement using mode-locked lasers,” Quantum Inf. Comput.  2, 425 (2002).

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

1999 (2)

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

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–R776 (1999).

1995 (1)

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gate for quantum computation,” Phys. Rev. A 52, 3457 (1995).
[Crossref] [PubMed]

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Proposal for Direct, Local Measurement of Entanglement for Pure Bipartite Systems of Arbitrary Dimension,” Phys. Rev. Lett. 73, 58 (1994).
[Crossref] [PubMed]

1989 (1)

J. D. Franson, “Bell inequality for position and time Phys,” Phys. Rev. Lett. 62, 2205 (1989).
[Crossref] [PubMed]

1982 (1)

W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature 299, 5886 (1982).
[Crossref]

Acin, A.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Maninska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acin, K. Rottwitt, L. K. Oxenlowe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref] [PubMed]

Acín, A.

R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
[Crossref]

Andersson, E.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys.  7, 677 (2011).
[Crossref]

Appelbaum, I.

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–R776 (1999).

Augusiak, R.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Maninska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acin, K. Rottwitt, L. K. Oxenlowe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref] [PubMed]

Azana, J.

M. Kues, C. Remer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref] [PubMed]

Babazadeh, A.

A. Babazadeh, M. Erhard, F. Wang, M. Malik, R. Nouroozi, M. Krenn, and A. Zeilinger, “High-Dimensional Single-Photon Quantum Gates: Concepts and Experiments,” Phys. Rev. Lett. 119180510 (2017).
[Crossref] [PubMed]

F. Wang, M. Erhard, A. Babazadeh, M. Malik, M. Krenn, and A. Zeilinger, “Generation of the complete four-dimensional Bell basis,” Optica 4, 1462–1467 (2017).
[Crossref]

Bacco, D.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Maninska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acin, K. Rottwitt, L. K. Oxenlowe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref] [PubMed]

Barenco, A.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gate for quantum computation,” Phys. Rev. A 52, 3457 (1995).
[Crossref] [PubMed]

Bennett, C. H.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gate for quantum computation,” Phys. Rev. A 52, 3457 (1995).
[Crossref] [PubMed]

Bernhard, C.

C. Bernhard, B. Bessire, T. Feurer, and A. Stefanov, “Shaping frequency-entangled qudits,” Phys. Rev. A 88, 032322 (2013).
[Crossref]

Bernstein, H. J.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Proposal for Direct, Local Measurement of Entanglement for Pure Bipartite Systems of Arbitrary Dimension,” Phys. Rev. Lett. 73, 58 (1994).
[Crossref] [PubMed]

Bertani, P.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Proposal for Direct, Local Measurement of Entanglement for Pure Bipartite Systems of Arbitrary Dimension,” Phys. Rev. Lett. 73, 58 (1994).
[Crossref] [PubMed]

Bessire, B.

C. Bernhard, B. Bessire, T. Feurer, and A. Stefanov, “Shaping frequency-entangled qudits,” Phys. Rev. A 88, 032322 (2013).
[Crossref]

Bonneau, D.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Maninska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acin, K. Rottwitt, L. K. Oxenlowe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref] [PubMed]

Brendel, J.

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

Brukner, C.

C. Brukner, M. Zukowski, and A. Zeilinger, “Quantum communication complexity protocol with two entangled qutrits,” Phys. Rev. Lett. 89, 197901 (2004).
[Crossref]

Brunner, Nicolas

Anthony Martin, Thiago Guerreiro, Alexey Tiranov, Sébastien Designolle, Florian Fröwis, Nicolas Brunner, Marcus Huber, and Nicolas Gisin, “Quantifying Photonic High-Dimensional Entanglement,” Phys. Rev. Lett. 118, 110501 (2017).
[Crossref] [PubMed]

Buller, G. S.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys.  7, 677 (2011).
[Crossref]

Cabello, A.

X. M. Hu, B. H. Liu, Y. Guo, G. Y. Xiang, Y. F. Huang, C. F. Li, G. C. Guo, M. Kleinmann, T. Vértesi, and A. Cabello, “Observation of stronger-than-binary correlations with entangled photonic qutrits,” Phys. Rev. Lett. 120, 180402 (2018).
[Crossref] [PubMed]

B. H. Liu, X. M. Hu, J. S. Chen, Y. F. Huang, Y. J. Han, C. F. Li, G. C. Guo, and A. Cabello, “Nonlocality from Local Contextuality,” Phys. Rev. Lett. 117, 220402 (2016).
[Crossref] [PubMed]

M. Kleinmann and A. Cabello, “Quantum Correlations are Stronger than All Nonsignaling Correlations Produced by n-Outcome Measurements,” Phys. Rev. Lett. 117, 150401 (2016).
[Crossref] [PubMed]

Caspani, L.

M. Kues, C. Remer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref] [PubMed]

Chen, J. S.

X. M. Hu, J. S. Chen, B. H. Liu, Y. Guo, Y. F. Huang, Z. Q. Zhou, Y. J. Han, C. F. Li, and G. C. Guo, “Experimental Test of Compatibility-Loophole-Free Contextuality with Spatially Separated Entangled Qutrits,” Phys. Rev. Lett. 117, 170403 (2016).
[Crossref] [PubMed]

B. H. Liu, X. M. Hu, J. S. Chen, Y. F. Huang, Y. J. Han, C. F. Li, G. C. Guo, and A. Cabello, “Nonlocality from Local Contextuality,” Phys. Rev. Lett. 117, 220402 (2016).
[Crossref] [PubMed]

Chu, S. T.

M. Kues, C. Remer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref] [PubMed]

Cino, A.

M. Kues, C. Remer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref] [PubMed]

Cleve, R.

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[Crossref] [PubMed]

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[Crossref]

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H. d. Riedmatten, I. Marcikic, H. Zbinden, and N. Gisin, “Creating high dimensional time-bin entanglement using mode-locked lasers,” Quantum Inf. Comput.  2, 425 (2002).

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A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gate for quantum computation,” Phys. Rev. A 52, 3457 (1995).
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Anthony Martin, Thiago Guerreiro, Alexey Tiranov, Sébastien Designolle, Florian Fröwis, Nicolas Brunner, Marcus Huber, and Nicolas Gisin, “Quantifying Photonic High-Dimensional Entanglement,” Phys. Rev. Lett. 118, 110501 (2017).
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L. Olislager, I. Mbodji, E. Woodhead, J. Cussey, L. Furfaro, P. Emplit, S. Massar, K. P. Huy, and J. M. Merolla, “Implementing two-photon interference in the frequency domain with electro-optic phase modulators,” New J. Phys.  14, 043015 (2012).
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R. B. Jin, R. Shimizu, M. Fujiwara, M. Takeoka, R. Wakabayashi, T. Yamashita, S. Miki, H. Terai, T. Gerrits, and M. Sasaki, “Simple method of generating and distributing frequency-entangled qudits,” Quantum Sci. Technol.  1, 015004 (2016).
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M. Kues, C. Remer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref] [PubMed]

Moss, D. J.

M. Kues, C. Remer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
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R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66, 012303 (2002).
[Crossref]

Nemoto, K.

R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66, 012303 (2002).
[Crossref]

Nouroozi, R.

A. Babazadeh, M. Erhard, F. Wang, M. Malik, R. Nouroozi, M. Krenn, and A. Zeilinger, “High-Dimensional Single-Photon Quantum Gates: Concepts and Experiments,” Phys. Rev. Lett. 119180510 (2017).
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O’Brien, J. L.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Maninska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acin, K. Rottwitt, L. K. Oxenlowe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
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Olislager, L.

L. Olislager, I. Mbodji, E. Woodhead, J. Cussey, L. Furfaro, P. Emplit, S. Massar, K. P. Huy, and J. M. Merolla, “Implementing two-photon interference in the frequency domain with electro-optic phase modulators,” New J. Phys.  14, 043015 (2012).
[Crossref]

Oxenlowe, L. K.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Maninska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acin, K. Rottwitt, L. K. Oxenlowe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref] [PubMed]

Padgett, M. J.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys.  7, 677 (2011).
[Crossref]

Paesani, S.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Maninska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acin, K. Rottwitt, L. K. Oxenlowe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref] [PubMed]

Polster, R.

Popescu, S.

D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, “Bell inequalities for arbitrarily high-dimensional systems,” Phys. Rev. Lett. 88, 040404 (2002).
[Crossref] [PubMed]

Ramelow, S.

Reck, M.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Proposal for Direct, Local Measurement of Entanglement for Pure Bipartite Systems of Arbitrary Dimension,” Phys. Rev. Lett. 73, 58 (1994).
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Remer, C.

M. Kues, C. Remer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref] [PubMed]

Richart, D.

D. Richart, Y. Fischer, and H. Weinfurter, “Experimental implementation of higher dimensional time-energy entanglement,” Appl. Phys. B 106, 543 (2012).
[Crossref]

Riedmatten, H. d.

H. d. Riedmatten, I. Marcikic, H. Zbinden, and N. Gisin, “Creating high dimensional time-bin entanglement using mode-locked lasers,” Quantum Inf. Comput.  2, 425 (2002).

Rottwitt, K.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Maninska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acin, K. Rottwitt, L. K. Oxenlowe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref] [PubMed]

Roztocki, P.

M. Kues, C. Remer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref] [PubMed]

Salavrakos, A.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Maninska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acin, K. Rottwitt, L. K. Oxenlowe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref] [PubMed]

Santagati, R.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Maninska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acin, K. Rottwitt, L. K. Oxenlowe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref] [PubMed]

Sasaki, M.

R. B. Jin, R. Shimizu, M. Fujiwara, M. Takeoka, R. Wakabayashi, T. Yamashita, S. Miki, H. Terai, T. Gerrits, and M. Sasaki, “Simple method of generating and distributing frequency-entangled qudits,” Quantum Sci. Technol.  1, 015004 (2016).
[Crossref]

Schaeff, C.

Sciara, S.

M. Kues, C. Remer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref] [PubMed]

Shimizu, R.

R. B. Jin, R. Shimizu, M. Fujiwara, M. Takeoka, R. Wakabayashi, T. Yamashita, S. Miki, H. Terai, T. Gerrits, and M. Sasaki, “Simple method of generating and distributing frequency-entangled qudits,” Quantum Sci. Technol.  1, 015004 (2016).
[Crossref]

Shor, P.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gate for quantum computation,” Phys. Rev. A 52, 3457 (1995).
[Crossref] [PubMed]

Silverstone, J. W.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Maninska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acin, K. Rottwitt, L. K. Oxenlowe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref] [PubMed]

Skrzypczyk, P.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Maninska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acin, K. Rottwitt, L. K. Oxenlowe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref] [PubMed]

Sleator, T.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gate for quantum computation,” Phys. Rev. A 52, 3457 (1995).
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Smolin, J.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gate for quantum computation,” Phys. Rev. A 52, 3457 (1995).
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Stefanov, A.

C. Bernhard, B. Bessire, T. Feurer, and A. Stefanov, “Shaping frequency-entangled qudits,” Phys. Rev. A 88, 032322 (2013).
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Stucki, D.

D. Stucki, H. Zbinden, and N. Gisin, “A Fabry-Perot-like two-photon interferometer for high-dimensional time-bin entanglement,” J. Mod. Opt. 52, 2637 (2005).
[Crossref]

Takeoka, M.

R. B. Jin, R. Shimizu, M. Fujiwara, M. Takeoka, R. Wakabayashi, T. Yamashita, S. Miki, H. Terai, T. Gerrits, and M. Sasaki, “Simple method of generating and distributing frequency-entangled qudits,” Quantum Sci. Technol.  1, 015004 (2016).
[Crossref]

Takesue, H.

T. Ikuta and H. Takesue, “Enhanced violation of the Collins-Gisin-Linden-Massar-Popescu inequality with optimized time-bin-entangled ququarts,” Phys. Rev. A 93, 022307 (2016).
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Terai, H.

R. B. Jin, R. Shimizu, M. Fujiwara, M. Takeoka, R. Wakabayashi, T. Yamashita, S. Miki, H. Terai, T. Gerrits, and M. Sasaki, “Simple method of generating and distributing frequency-entangled qudits,” Quantum Sci. Technol.  1, 015004 (2016).
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Thew, R. T.

R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
[Crossref]

R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66, 012303 (2002).
[Crossref]

Thompson, M. G.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Maninska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acin, K. Rottwitt, L. K. Oxenlowe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
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Tiranov, Alexey

Anthony Martin, Thiago Guerreiro, Alexey Tiranov, Sébastien Designolle, Florian Fröwis, Nicolas Brunner, Marcus Huber, and Nicolas Gisin, “Quantifying Photonic High-Dimensional Entanglement,” Phys. Rev. Lett. 118, 110501 (2017).
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J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594 (1999).
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Tura, J.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Maninska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acin, K. Rottwitt, L. K. Oxenlowe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
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Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
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Vértesi, T.

X. M. Hu, B. H. Liu, Y. Guo, G. Y. Xiang, Y. F. Huang, C. F. Li, G. C. Guo, M. Kleinmann, T. Vértesi, and A. Cabello, “Observation of stronger-than-binary correlations with entangled photonic qutrits,” Phys. Rev. Lett. 120, 180402 (2018).
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Wakabayashi, R.

R. B. Jin, R. Shimizu, M. Fujiwara, M. Takeoka, R. Wakabayashi, T. Yamashita, S. Miki, H. Terai, T. Gerrits, and M. Sasaki, “Simple method of generating and distributing frequency-entangled qudits,” Quantum Sci. Technol.  1, 015004 (2016).
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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–R776 (1999).

Wang, F.

A. Babazadeh, M. Erhard, F. Wang, M. Malik, R. Nouroozi, M. Krenn, and A. Zeilinger, “High-Dimensional Single-Photon Quantum Gates: Concepts and Experiments,” Phys. Rev. Lett. 119180510 (2017).
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F. Wang, M. Erhard, A. Babazadeh, M. Malik, M. Krenn, and A. Zeilinger, “Generation of the complete four-dimensional Bell basis,” Optica 4, 1462–1467 (2017).
[Crossref]

Wang, J.

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Maninska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acin, K. Rottwitt, L. K. Oxenlowe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
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Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
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Weinfurter, H.

D. Richart, Y. Fischer, and H. Weinfurter, “Experimental implementation of higher dimensional time-energy entanglement,” Appl. Phys. B 106, 543 (2012).
[Crossref]

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gate for quantum computation,” Phys. Rev. A 52, 3457 (1995).
[Crossref] [PubMed]

Wetzel, B.

M. Kues, C. Remer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref] [PubMed]

White, A. G.

R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66, 012303 (2002).
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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–R776 (1999).

Woodhead, E.

L. Olislager, I. Mbodji, E. Woodhead, J. Cussey, L. Furfaro, P. Emplit, S. Massar, K. P. Huy, and J. M. Merolla, “Implementing two-photon interference in the frequency domain with electro-optic phase modulators,” New J. Phys.  14, 043015 (2012).
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Wootters, W. K.

W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature 299, 5886 (1982).
[Crossref]

Xiang, G. Y.

X. M. Hu, B. H. Liu, Y. Guo, G. Y. Xiang, Y. F. Huang, C. F. Li, G. C. Guo, M. Kleinmann, T. Vértesi, and A. Cabello, “Observation of stronger-than-binary correlations with entangled photonic qutrits,” Phys. Rev. Lett. 120, 180402 (2018).
[Crossref] [PubMed]

Yamashita, T.

R. B. Jin, R. Shimizu, M. Fujiwara, M. Takeoka, R. Wakabayashi, T. Yamashita, S. Miki, H. Terai, T. Gerrits, and M. Sasaki, “Simple method of generating and distributing frequency-entangled qudits,” Quantum Sci. Technol.  1, 015004 (2016).
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Yonehara, T.

R. Inoue, T. Yonehara, Y. Miyamoto, M. Koashi, and M. Kozuma, “Measuring qutrit-qutrit entanglement of orbital angular momentum states of an atomic ensemble and a photon,” Phys. Rev. Lett. 103, 110503 (2009).
[Crossref] [PubMed]

Zbinden, H.

D. Stucki, H. Zbinden, and N. Gisin, “A Fabry-Perot-like two-photon interferometer for high-dimensional time-bin entanglement,” J. Mod. Opt. 52, 2637 (2005).
[Crossref]

R. T. Thew, A. Acín, H. Zbinden, and N. Gisin, “Bell-type test of energy-time entangled qutrits,” Phys. Rev. Lett. 93, 010503 (2004).
[Crossref]

H. d. Riedmatten, I. Marcikic, H. Zbinden, and N. Gisin, “Creating high dimensional time-bin entanglement using mode-locked lasers,” Quantum Inf. Comput.  2, 425 (2002).

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

Zeilinger, A.

A. Babazadeh, M. Erhard, F. Wang, M. Malik, R. Nouroozi, M. Krenn, and A. Zeilinger, “High-Dimensional Single-Photon Quantum Gates: Concepts and Experiments,” Phys. Rev. Lett. 119180510 (2017).
[Crossref] [PubMed]

F. Wang, M. Erhard, A. Babazadeh, M. Malik, M. Krenn, and A. Zeilinger, “Generation of the complete four-dimensional Bell basis,” Optica 4, 1462–1467 (2017).
[Crossref]

C. Schaeff, R. Polster, M. Huber, S. Ramelow, and A. Zeilinger, “Experimental access to higher-dimensional entangled quantum systems using integrated optics,” Optica 2, 523–529 (2015).
[Crossref]

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. U.S.A.  111, 6243 (2014).
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C. Brukner, M. Zukowski, and A. Zeilinger, “Quantum communication complexity protocol with two entangled qutrits,” Phys. Rev. Lett. 89, 197901 (2004).
[Crossref]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Proposal for Direct, Local Measurement of Entanglement for Pure Bipartite Systems of Arbitrary Dimension,” Phys. Rev. Lett. 73, 58 (1994).
[Crossref] [PubMed]

Zhang, Y.

M. Kues, C. Remer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref] [PubMed]

Zhou, Z. Q.

X. M. Hu, J. S. Chen, B. H. Liu, Y. Guo, Y. F. Huang, Z. Q. Zhou, Y. J. Han, C. F. Li, and G. C. Guo, “Experimental Test of Compatibility-Loophole-Free Contextuality with Spatially Separated Entangled Qutrits,” Phys. Rev. Lett. 117, 170403 (2016).
[Crossref] [PubMed]

Zukowski, M.

C. Brukner, M. Zukowski, and A. Zeilinger, “Quantum communication complexity protocol with two entangled qutrits,” Phys. Rev. Lett. 89, 197901 (2004).
[Crossref]

Zurek, W. H.

W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature 299, 5886 (1982).
[Crossref]

Appl. Phys. B (1)

D. Richart, Y. Fischer, and H. Weinfurter, “Experimental implementation of higher dimensional time-energy entanglement,” Appl. Phys. B 106, 543 (2012).
[Crossref]

J. Mod. Opt. (1)

D. Stucki, H. Zbinden, and N. Gisin, “A Fabry-Perot-like two-photon interferometer for high-dimensional time-bin entanglement,” J. Mod. Opt. 52, 2637 (2005).
[Crossref]

Nat. Phys (1)

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys.  7, 677 (2011).
[Crossref]

Nature (3)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

M. Kues, C. Remer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azana, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref] [PubMed]

W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature 299, 5886 (1982).
[Crossref]

New J. Phys (1)

L. Olislager, I. Mbodji, E. Woodhead, J. Cussey, L. Furfaro, P. Emplit, S. Massar, K. P. Huy, and J. M. Merolla, “Implementing two-photon interference in the frequency domain with electro-optic phase modulators,” New J. Phys.  14, 043015 (2012).
[Crossref]

Optica (2)

Phys. Rev. A (5)

T. Ikuta and H. Takesue, “Enhanced violation of the Collins-Gisin-Linden-Massar-Popescu inequality with optimized time-bin-entangled ququarts,” Phys. Rev. A 93, 022307 (2016).
[Crossref]

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gate for quantum computation,” Phys. Rev. A 52, 3457 (1995).
[Crossref] [PubMed]

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–R776 (1999).

R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66, 012303 (2002).
[Crossref]

C. Bernhard, B. Bessire, T. Feurer, and A. Stefanov, “Shaping frequency-entangled qudits,” Phys. Rev. A 88, 032322 (2013).
[Crossref]

Phys. Rev. Lett. (13)

X. M. Hu, J. S. Chen, B. H. Liu, Y. Guo, Y. F. Huang, Z. Q. Zhou, Y. J. Han, C. F. Li, and G. C. Guo, “Experimental Test of Compatibility-Loophole-Free Contextuality with Spatially Separated Entangled Qutrits,” Phys. Rev. Lett. 117, 170403 (2016).
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D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, “Bell inequalities for arbitrarily high-dimensional systems,” Phys. Rev. Lett. 88, 040404 (2002).
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X. M. Hu, B. H. Liu, Y. Guo, G. Y. Xiang, Y. F. Huang, C. F. Li, G. C. Guo, M. Kleinmann, T. Vértesi, and A. Cabello, “Observation of stronger-than-binary correlations with entangled photonic qutrits,” Phys. Rev. Lett. 120, 180402 (2018).
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Figures (6)

Fig. 1
Fig. 1 Experimental setup. (a) Entanglement source. A continuous wave laser at 404 nm serves as the pump source. To generate qutrit-qutrit entanglement, the pump laser is separated into two paths by three HWPs and a BD, where HWP1 is set at 17.63°, HWP2 is set at 22.5° and HWP3 is set at 0°. Then the pump laser is directed into two 0.5 mm thick type-I cut β-borate (BBO) crystal. After the BD and three HWPs, the pump state is prepared on ( | V u + | H u + | V l ) / 3 . The pump light is then focused on two spots of the crystals to generate spatial and polarizing hybrid entangled state. Thus the state ( | 00 + | 11 + | 22 ) / 3 is prepared if we encode the upper path |H〉 as |0〉, |V〉 as |1〉 and the lower path |H〉 as |2〉. (b) A typical single-observable measurement device. PBSs, HWPs, and BDs are used to construct the observable. The angles of HWPs are chosen to project the state to the eigenstates of the corresponding observable. (c) Three-dimensional gate. By applying different voltages, the liquid crystal phase plates (LCs) will introduce different phases between the fast axis and the slow axis. In our scheme, the optical axes of LCs are set at 0°. The setup realizes an I gate when we set HWP7-12 at 45°,, 45°, 45°,, 45°. The setup realizes an X gate when we set HWP7-12 at 0°,, 45°,,, 45°. The setup realize an X2 gate when we set HWP1-6 at 45°,, 45°,, 45°, 45°. Combining the gates and phase controller, we can operate locally on one photon of the maximally entangled state ( | 00 + | 11 + | 22 ) / 3 , producing all nine Bell-states. Notation for optical elements: half-wave plate (HWP), beam displacer (BD), polarizing beamsplitter (PBS).
Fig. 2
Fig. 2 Graphical representation of the reconstructed density matrix of the three-dimensional maximally entangled state. The density matrix of the two qutrits is reconstructed from a set of 81 measurements represented by the operators uiuj (with i, j=1, 2,…,9) and uk = |Ψ k 〉(Ψ k |. The kets |Ψ k 〉 for both the idler and the signal photons are selected from the following set: {|0〉, |1〉, |2〉, ( | 0 + | 1 ) / 2 , ( | 0 + i | 1 ) / 2 , ( | 1 + | 2 ) / 2 , ( | 1 + i | 2 ) / 2 , ( | 0 + | 2 ) / 2 , ( | 0 + i | 2 ) / 2 }. Detailed descriptions of the tomographic measurements are presented in Ref. [25, 26].
Fig. 3
Fig. 3 Graphical representation of the reconstructed density matrix of the four-dimensional maximally entangled state. The method of constructing four-dimensional density matrices is similar to that of the three-dimensional ones.
Fig. 4
Fig. 4 The coincidence results of varying proportion of Schmidt coefficients of state a|00〉+b|11〉+c|22〉 when both photons are measured with computational basis {|0〉, |1〉, |2〉}. (a–c) We have prepared three states with a : b : c = 1 : 1 : 2 , 1 : 2 : 1 , 2 : 1 : 1 respectively.
Fig. 5
Fig. 5 Coincidence fringes for state a|00〉 + b|11〉 + c|22〉. (a–c) show the phase variation of the three two-qubit subspaces of the state with a : b : c = 1 : 1 : 1. (a) For example, taking the subspace composed of |0〉 and |1〉, the two photons are subjected to the measurement base (|0〉 + e |1〉) ⊗ (|0〉 + |1〉) with varied φ. (d–f) present the fringe results of the state with a : b : c = 2 : 1 : 1 .
Fig. 6
Fig. 6 Design of experimental device for eight dimensional entanglement. After transmitting through the BDs, the state of pump light is ( | H 1 + e i φ 1 | V 1 + e i φ 2 | H 2 + e i φ 3 | V 2 + e i φ 4 | H 3 + e i φ 5 | V 3 + e i φ 6 | H 4 + e i φ 7 | V 4 ) / 8 . When all the phases φi(i = 1, 2, …, 7〉 equal 0, the eight-dimensional maximally entangled two-photon state ( | 00 + | 11 + | 22 + | 33 + | 44 + | 55 + | 66 + | 77 ) / 8 is prepared. Specially, the length of BD2 is half of BD1, thus introducing half displacement compared with BD1.

Equations (4)

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| Ψ = ( | H u + | V u ) + | V l ) / 3 ,
| Ψ = ( | H H u + | V V u + | H H l ) / 3 .
| Ψ = ( | 00 + | 11 + | 22 ) / 3 .
| Ψ 4 = ( | 00 + | 11 + | 22 + | 33 ) / 2

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