Abstract

Encoding information in the position of single photons has no known limits, given infinite resources. Using a heralded single-photon source and a spatial light modulator (SLM), we steer single photons to specific positions in a virtual grid on a large-area spatially resolving photon-counting detector (ICCD). We experimentally demonstrate selective addressing any location (symbol) in a 9072 size grid (alphabet) to achieve 10.5 bit of mutual information per detected photon between the sender and receiver. Our results can be useful for very-high-dimensional quantum information processing.

© 2017 Optical Society of America

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References

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2016 (2)

D. J. Lum, J. C. Howell, M. S. Allman, T. Gerrits, V. B. Verma, S. W. Nam, C. Lupo, and S. Lloyd, “Quantum enigma machine: experimentally demonstrating quantum data locking,” Phys. Rev. A 94, 022315 (2016).
[Crossref]

T. A. W. Wolterink, R. Uppu, G. Ctistis, W. L. Vos, K.-J. Boller, and P. W. H. Pinkse, “Programmable two-photon quantum interference in 103 channels in opaque scattering media,” Phys. Rev. A 93, 053817 (2016).
[Crossref]

2015 (4)

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

T. Zhong, H. Zhou, R. D. Horansky, C. Lee, V. B. Verma, A. E. Lita, A. Restelli, J. C. Bienfang, R. P. Mirin, and T. Gerrits, “Photon-efficient quantum key distribution using time–energy entanglement with high-dimensional encoding,” New J. Phys. 17, 022002 (2015).
[Crossref]

N. Zhao, X. Li, G. Li, and J. M. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photonics 9, 822–826 (2015).
[Crossref]

L. V. Amitonova, A. P. Mosk, and P. W. H. Pinkse, “Rotational memory effect of a multimode fiber,” Opt. Express 23, 20569–20575 (2015).
[Crossref] [PubMed]

2014 (1)

2013 (3)

R. S. Aspden, D. S. Tasca, R. W. Boyd, and M. J. Padgett, “Epr-based ghost imaging using a single-photon-sensitive camera,” New J. Phys. 15, 073032 (2013).
[Crossref]

R. Fickler, M. Krenn, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Real-time imaging of quantum entanglement,” Sci. Rep. 3, 1914 (2013).
[Crossref] [PubMed]

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lütkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

2012 (2)

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett. 108, 143603 (2012).
[Crossref] [PubMed]

J. Wang, J-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

2011 (1)

2009 (2)

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 (2009).
[Crossref]

H. D. L. Pires, C. H. Monken, and M. P. van Exter, “Direct measurement of transverse-mode entanglement in two-photon states,” Phys. Rev. A 80, 022307 (2009).
[Crossref]

2008 (1)

S. P. Walborn, D. S. Lemelle, D. S. Tasca, and P. H. S. Ribeiro, “Schemes for quantum key distribution with higher-order alphabets using single-photon fractional fourier optics,” Phys. Rev. A 77, 062323 (2008).
[Crossref]

2007 (2)

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
[Crossref] [PubMed]

2006 (2)

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
[Crossref]

S. P. Walborn, D. S. Lemelle, M. P. Almeida, and P. H. S. Ribeiro, “Quantum key distribution with higher-order alphabets using spatially encoded qudits,” Phys. Rev. Lett. 96, 090501 (2006).
[Crossref] [PubMed]

2005 (1)

O. Haderka, J. Peřina, M. Hamar, and J. Peřina, “Direct measurement and reconstruction of nonclassical features of twin beams generated in spontaneous parametric down-conversion,” Phys. Rev. A 71, 033815 (2005).
[Crossref]

2004 (2)

O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93, 243601 (2004).
[Crossref]

D. P. DiVincenzo, M. Horodecki, D. W. Leung, J. A. Smolin, and B. M. Terhal, “Locking classical correlations in quantum states,” Phys. Rev. Lett. 92, 067902 (2004).
[Crossref] [PubMed]

2002 (2)

N. Gisin, G. Ribody, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145 (2002).
[Crossref]

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[Crossref] [PubMed]

2001 (1)

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of the complementarity of one-and two-photon interference,” Phys. Rev. A 63, 063803 (2001).
[Crossref]

2000 (1)

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
[Crossref]

1998 (1)

1982 (1)

W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature 299, 802–803 (1982).
[Crossref]

1962 (1)

R. Gallager, “Low-density parity-check codes,” IRE Trans. Inf. Theory 8, 21–28 (1962).
[Crossref]

Abouraddy, A.

Abouraddy, A. F.

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of the complementarity of one-and two-photon interference,” Phys. Rev. A 63, 063803 (2001).
[Crossref]

Ahmed, N.

J. Wang, J-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Ali-Khan, I.

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

Allman, M. S.

D. J. Lum, J. C. Howell, M. S. Allman, T. Gerrits, V. B. Verma, S. W. Nam, C. Lupo, and S. Lloyd, “Quantum enigma machine: experimentally demonstrating quantum data locking,” Phys. Rev. A 94, 022315 (2016).
[Crossref]

Almeida, M. P.

S. P. Walborn, D. S. Lemelle, M. P. Almeida, and P. H. S. Ribeiro, “Quantum key distribution with higher-order alphabets using spatially encoded qudits,” Phys. Rev. Lett. 96, 090501 (2006).
[Crossref] [PubMed]

Amitonova, L. V.

Aspden, R. S.

R. S. Aspden, D. S. Tasca, R. W. Boyd, and M. J. Padgett, “Epr-based ghost imaging using a single-photon-sensitive camera,” New J. Phys. 15, 073032 (2013).
[Crossref]

Bache, M.

O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93, 243601 (2004).
[Crossref]

Banaszek, K.

Barnett, S. M.

D. J. Gauthier, C. F. Wildfeuer, H. Stipcević, B. Christensen, D. Kumor, P. Kwiat, K. McCusker, T. Brougham, and S. M. Barnett, “Quantum key distribution using hyperentangled time-bin states,” in “Proceedings of The Tenth Rochester Conference on Coherence on Quantum Optics (CQO10),” (2014), pp. 234–239.

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 (2009).
[Crossref]

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
[Crossref]

Bennett, C. H.

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in “International Conference on Computers, Systems & Signal Processing, Bangalore, India, Dec 9–12, 1984,” (1984), pp. 175–179.

Bienfang, J. C.

T. Zhong, H. Zhou, R. D. Horansky, C. Lee, V. B. Verma, A. E. Lita, A. Restelli, J. C. Bienfang, R. P. Mirin, and T. Gerrits, “Photon-efficient quantum key distribution using time–energy entanglement with high-dimensional encoding,” New J. Phys. 17, 022002 (2015).
[Crossref]

Boller, K.-J.

T. A. W. Wolterink, R. Uppu, G. Ctistis, W. L. Vos, K.-J. Boller, and P. W. H. Pinkse, “Programmable two-photon quantum interference in 103 channels in opaque scattering media,” Phys. Rev. A 93, 053817 (2016).
[Crossref]

Bourennane, M.

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[Crossref] [PubMed]

Boyd, R. W.

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

R. S. Aspden, D. S. Tasca, R. W. Boyd, and M. J. Padgett, “Epr-based ghost imaging using a single-photon-sensitive camera,” New J. Phys. 15, 073032 (2013).
[Crossref]

Brambilla, E.

O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93, 243601 (2004).
[Crossref]

Brassard, G.

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in “International Conference on Computers, Systems & Signal Processing, Bangalore, India, Dec 9–12, 1984,” (1984), pp. 175–179.

Broadbent, C. J.

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref] [PubMed]

Brougham, T.

D. J. Gauthier, C. F. Wildfeuer, H. Stipcević, B. Christensen, D. Kumor, P. Kwiat, K. McCusker, T. Brougham, and S. M. Barnett, “Quantum key distribution using hyperentangled time-bin states,” in “Proceedings of The Tenth Rochester Conference on Coherence on Quantum Optics (CQO10),” (2014), pp. 234–239.

Cerf, N. J.

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 (2009).
[Crossref]

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[Crossref] [PubMed]

Chrapkiewicz, R.

Christensen, B.

D. J. Gauthier, C. F. Wildfeuer, H. Stipcević, B. Christensen, D. Kumor, P. Kwiat, K. McCusker, T. Brougham, and S. M. Barnett, “Quantum key distribution using hyperentangled time-bin states,” in “Proceedings of The Tenth Rochester Conference on Coherence on Quantum Optics (CQO10),” (2014), pp. 234–239.

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2010).
[Crossref]

Cižmár, T.

Ctistis, G.

T. A. W. Wolterink, R. Uppu, G. Ctistis, W. L. Vos, K.-J. Boller, and P. W. H. Pinkse, “Programmable two-photon quantum interference in 103 channels in opaque scattering media,” Phys. Rev. A 93, 053817 (2016).
[Crossref]

Dholakia, K.

Di Trapani, P.

O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93, 243601 (2004).
[Crossref]

DiVincenzo, D. P.

D. P. DiVincenzo, M. Horodecki, D. W. Leung, J. A. Smolin, and B. M. Terhal, “Locking classical correlations in quantum states,” Phys. Rev. Lett. 92, 067902 (2004).
[Crossref] [PubMed]

Dixon, P. B.

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett. 108, 143603 (2012).
[Crossref] [PubMed]

Dolinar, S.

J. Wang, J-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Dudley, A.

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lütkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

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 (2009).
[Crossref]

Fazal, I. M.

J. Wang, J-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
[Crossref]

Fickler, R.

R. Fickler, M. Krenn, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Real-time imaging of quantum entanglement,” Sci. Rep. 3, 1914 (2013).
[Crossref] [PubMed]

Forbes, A.

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lütkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

Gallager, R.

R. Gallager, “Low-density parity-check codes,” IRE Trans. Inf. Theory 8, 21–28 (1962).
[Crossref]

Gatti, A.

O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. Lugiato, and P. Di Trapani, “Detection of sub-shot-noise spatial correlation in high-gain parametric down conversion,” Phys. Rev. Lett. 93, 243601 (2004).
[Crossref]

Gauthier, D. J.

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D. J. Lum, J. C. Howell, M. S. Allman, T. Gerrits, V. B. Verma, S. W. Nam, C. Lupo, and S. Lloyd, “Quantum enigma machine: experimentally demonstrating quantum data locking,” Phys. Rev. A 94, 022315 (2016).
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M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
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D. J. Gauthier, C. F. Wildfeuer, H. Stipcević, B. Christensen, D. Kumor, P. Kwiat, K. McCusker, T. Brougham, and S. M. Barnett, “Quantum key distribution using hyperentangled time-bin states,” in “Proceedings of The Tenth Rochester Conference on Coherence on Quantum Optics (CQO10),” (2014), pp. 234–239.

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M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lütkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
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M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lütkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
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R. Fickler, M. Krenn, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Real-time imaging of quantum entanglement,” Sci. Rep. 3, 1914 (2013).
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J. Wang, J-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
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T. Zhong, H. Zhou, R. D. Horansky, C. Lee, V. B. Verma, A. E. Lita, A. Restelli, J. C. Bienfang, R. P. Mirin, and T. Gerrits, “Photon-efficient quantum key distribution using time–energy entanglement with high-dimensional encoding,” New J. Phys. 17, 022002 (2015).
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S. P. Walborn, D. S. Lemelle, D. S. Tasca, and P. H. S. Ribeiro, “Schemes for quantum key distribution with higher-order alphabets using single-photon fractional fourier optics,” Phys. Rev. A 77, 062323 (2008).
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Saleh, B. E. A.

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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 (2009).
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Sergienko, A. V.

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D. P. DiVincenzo, M. Horodecki, D. W. Leung, J. A. Smolin, and B. M. Terhal, “Locking classical correlations in quantum states,” Phys. Rev. Lett. 92, 067902 (2004).
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R. S. Aspden, D. S. Tasca, R. W. Boyd, and M. J. Padgett, “Epr-based ghost imaging using a single-photon-sensitive camera,” New J. Phys. 15, 073032 (2013).
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Teich, M. C.

A. F. Abouraddy, M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of the complementarity of one-and two-photon interference,” Phys. Rev. A 63, 063803 (2001).
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D. P. DiVincenzo, M. Horodecki, D. W. Leung, J. A. Smolin, and B. M. Terhal, “Locking classical correlations in quantum states,” Phys. Rev. Lett. 92, 067902 (2004).
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J. Wang, J-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012).
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T. A. W. Wolterink, R. Uppu, G. Ctistis, W. L. Vos, K.-J. Boller, and P. W. H. Pinkse, “Programmable two-photon quantum interference in 103 channels in opaque scattering media,” Phys. Rev. A 93, 053817 (2016).
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H. D. L. Pires, C. H. Monken, and M. P. van Exter, “Direct measurement of transverse-mode entanglement in two-photon states,” Phys. Rev. A 80, 022307 (2009).
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S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
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Verma, V. B.

D. J. Lum, J. C. Howell, M. S. Allman, T. Gerrits, V. B. Verma, S. W. Nam, C. Lupo, and S. Lloyd, “Quantum enigma machine: experimentally demonstrating quantum data locking,” Phys. Rev. A 94, 022315 (2016).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

Schematic representation of the setup. The Type-II Spontaneous Parametric Down-Conversion (SPDC) source produces photon pairs, which are split by a Polarising Beam Splitter (PBS). One of the photons is detected by a Single-Photon Counting Module (SPCM) acting as a herald for an Intensified CCD (ICCD). The other photon is fiber-coupled and is incident on a Spatial Light Modulator (SLM). Its Fourier image is projected on the ICCD. The position of the focus is scanned by adjusting the blazed grating on the SLM, indicated by the arrows. An accumulated focus is shown in a zoom-in of the ICCD image integrated over an average of 1000 photons. The red lines show the 8 × 8 pixel binning of the symbols.

Fig. 2
Fig. 2

Measured ICCD counts in each of the measured symbols as a function of the sent symbol with a binning size of 8 × 8 pixels. The exposure time was 0.6 s for each symbol. Graph (a) illustrates the correlation between all 9072 symbols in a log-log plot. The graph (b) shows the measured correlation between the first 200 symbols in a linear plot. The measurement samples the joint probability distribution P(X, Y).

Fig. 3
Fig. 3

Dependence of the mutual information and the average hit probability in the correct symbol on the binning size of the detection areas. The blue circles represent the theoretical limit Imax given no noise or crosstalk. The red dots correspond to the measured mutual information for 8 × 8 and 12 × 12 pixel bin size. The theoretical mutual information is shown as gray bars in the figure, corrected for a signal-to-dark-count photon ratio between 10 and 100. The green + markers illustrate the average hit probability in the correct area for a finite focal diameter with FWHM of 8 pixels as shown in Fig. 1.

Fig. 4
Fig. 4

The Bit Error Rate (BER) of the received bit string versus the BER of the bit string after performing error correction. The dashed diagonal line represents the result without error correction. The vertical bars indicate the estimated BER of our experiment in case of 8 × 8 (green) and 12 × 12 (blue) binning. Their left and right edges indicate a signal-to-dark-count photon ratio of 100 and 10, respectively.

Equations (1)

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I ( X : Y ) = x X , y Y p ( x , y ) log 2 ( p ( x , y ) p ( x ) p ( y ) ) ,

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