Abstract

We present here a classical optics device based on an imaging architecture as an analogy of a quantum system where the violation of the Bell inequality can be evidenced. Quantum states are encoded using an electromagnetic wave modulated in amplitude and phase. Unitary operations involved in the measurement of the observables are simulated with the use of a coherent optical processor. The images obtained in the output of the process contain all the information about the possible outcomes of the joint measurement. By measuring the intensity distribution in the image plane we evaluate the mean values of the simulated observables. The obtained experimental results show how some correlations of Clauser–Horne–Shimony–Holt-type exceed the upper bound imposed by the local realism hypothesis as a consequence of the joint effect of entanglement and two-particle interference.

© 2010 Optical Society of America

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  1. N. J. Cerf, C. Adami, and P. G. Kwiat, “Optical simulation of quantum logic,” Phys. Rev. A 57, R1477-R1480 (1998).
    [CrossRef]
  2. R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361-374 (1998).
    [CrossRef]
  3. R. J. C. Spreeuw, Phys. Rev. A 63, 062302 (2001).
    [CrossRef]
  4. N. Bhattacharya, H. B. van Linden vanden Heuvell, and R. J. C. Spreeuw, “Implementation of quantum search algorithm using classical Fourier optics,” Phys. Rev. Lett. 88, 137901 (2002).
    [CrossRef] [PubMed]
  5. G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
    [CrossRef]
  6. D. Francisco, C. Iemmi, J. P. Paz, and S. Ledesma, “Optical simulation of the quantum Hadamard operator,” Opt. Commun. 268, 340-345 (2006).
    [CrossRef]
  7. D. Francisco, C. Iemmi, J. P. Paz, and S. Ledesma, “Simulating a quantum walk with classical optics,” Phys. Rev. A 74, 052327 (2006).
    [CrossRef]
  8. D. Francisco and S. Ledesma, “Classical optics analogy of quantum teleportation,” J. Opt. Soc. Am. B 25, 383-390 (2008).
    [CrossRef]
  9. J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A 70, 042313 (2004).
    [CrossRef]
  10. K. F. Lee and J. E. Thomas, “Experimental simulation of two-particle quantum entanglement using classical fields,” Phys. Rev. Lett. 88, 097902 (2002).
    [CrossRef] [PubMed]
  11. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
    [CrossRef]
  12. J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” Physics (Long Island City, N.Y.) 1, 195-200 (1964); reprinted in J. S. Bell, Speakable and Unispeakable in Quantum Mechanics (Cambridge U. Press, 1987).
  13. A. Aspect, J. Dalibard, and G. Roger, “Experimental test of Bell's inequalities using time-varying analyzers,” Phys. Rev. Lett. 49, 1804-1807 (1982).
    [CrossRef]
  14. W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart,” Phys. Rev. Lett. 81, 3563-3566 (1998).
    [CrossRef]
  15. G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell's inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039-5043 (1998).
    [CrossRef]
  16. A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell's theorem,” Phys. Rev. Lett. 47, 460-463 (1981).
    [CrossRef]
  17. A. Aspect, P. Grangier, and G. Roger, “Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: a new violation of Bell's inequalities,” Phys. Rev. Lett. 49, 91-94 (1982).
    [CrossRef]
  18. Z. Y. Ou and L. Mandel, “Violation of Bell's inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50-53 (1988).
    [CrossRef] [PubMed]
  19. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photons pair,” Phys. Rev. Lett. 75, 4337-4341 (1995).
    [CrossRef] [PubMed]
  20. D. M. Greenberger, M. Horne, and A. Zeilinger, “Going beyond Bell's theorem,” in Bell's Theorem, Quantum Theory and Conceptions of the Universe (Kluwer Academic, 1989), pp. 73-76.
  21. J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of The Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
    [CrossRef] [PubMed]
  22. A. Beige, W. J. Munro, and P. L. Knight, “Bell's inequality test with entangled atoms,” Phys. Rev. A 62, 052102 (2000).
    [CrossRef]
  23. A. M. Souza, A. Magalhaes, J. Teles, E. R. deAzevedo, T. J. Bonagamba, I. S. Oliveira, and R. S. Sarthour, “NMR analog of Bell's inequalities violation test,” New J. Phys. 10, 033020 (2008).
    [CrossRef]
  24. M. Genovese, “Research on hidden variable theories: a review of recent progress,” Phys. Rep. 413, 319-396 (2005).
    [CrossRef]
  25. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880-884 (1969).
    [CrossRef]
  26. N. Bohr, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 48, 696-702 (1935).
    [CrossRef]
  27. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  28. M. Nielsen and I. Chuang, Quantum Information and Computation (Cambridge U. Press, 2000).
  29. A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. (Bellingham) 40, 2558-2564 (2001).
    [CrossRef]

2008 (2)

D. Francisco and S. Ledesma, “Classical optics analogy of quantum teleportation,” J. Opt. Soc. Am. B 25, 383-390 (2008).
[CrossRef]

A. M. Souza, A. Magalhaes, J. Teles, E. R. deAzevedo, T. J. Bonagamba, I. S. Oliveira, and R. S. Sarthour, “NMR analog of Bell's inequalities violation test,” New J. Phys. 10, 033020 (2008).
[CrossRef]

2006 (2)

D. Francisco, C. Iemmi, J. P. Paz, and S. Ledesma, “Optical simulation of the quantum Hadamard operator,” Opt. Commun. 268, 340-345 (2006).
[CrossRef]

D. Francisco, C. Iemmi, J. P. Paz, and S. Ledesma, “Simulating a quantum walk with classical optics,” Phys. Rev. A 74, 052327 (2006).
[CrossRef]

2005 (1)

M. Genovese, “Research on hidden variable theories: a review of recent progress,” Phys. Rep. 413, 319-396 (2005).
[CrossRef]

2004 (3)

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of The Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[CrossRef] [PubMed]

G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
[CrossRef]

J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A 70, 042313 (2004).
[CrossRef]

2002 (2)

K. F. Lee and J. E. Thomas, “Experimental simulation of two-particle quantum entanglement using classical fields,” Phys. Rev. Lett. 88, 097902 (2002).
[CrossRef] [PubMed]

N. Bhattacharya, H. B. van Linden vanden Heuvell, and R. J. C. Spreeuw, “Implementation of quantum search algorithm using classical Fourier optics,” Phys. Rev. Lett. 88, 137901 (2002).
[CrossRef] [PubMed]

2001 (2)

R. J. C. Spreeuw, Phys. Rev. A 63, 062302 (2001).
[CrossRef]

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. (Bellingham) 40, 2558-2564 (2001).
[CrossRef]

2000 (1)

A. Beige, W. J. Munro, and P. L. Knight, “Bell's inequality test with entangled atoms,” Phys. Rev. A 62, 052102 (2000).
[CrossRef]

1998 (4)

N. J. Cerf, C. Adami, and P. G. Kwiat, “Optical simulation of quantum logic,” Phys. Rev. A 57, R1477-R1480 (1998).
[CrossRef]

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361-374 (1998).
[CrossRef]

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart,” Phys. Rev. Lett. 81, 3563-3566 (1998).
[CrossRef]

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell's inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039-5043 (1998).
[CrossRef]

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photons pair,” Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

1988 (1)

Z. Y. Ou and L. Mandel, “Violation of Bell's inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50-53 (1988).
[CrossRef] [PubMed]

1982 (2)

A. Aspect, J. Dalibard, and G. Roger, “Experimental test of Bell's inequalities using time-varying analyzers,” Phys. Rev. Lett. 49, 1804-1807 (1982).
[CrossRef]

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

1981 (1)

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell's theorem,” Phys. Rev. Lett. 47, 460-463 (1981).
[CrossRef]

1969 (1)

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880-884 (1969).
[CrossRef]

1964 (1)

J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” Physics (Long Island City, N.Y.) 1, 195-200 (1964); reprinted in J. S. Bell, Speakable and Unispeakable in Quantum Mechanics (Cambridge U. Press, 1987).

J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” Physics (Long Island City, N.Y.) 1, 195-200 (1964); reprinted in J. S. Bell, Speakable and Unispeakable in Quantum Mechanics (Cambridge U. Press, 1987).

1935 (2)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
[CrossRef]

N. Bohr, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 48, 696-702 (1935).
[CrossRef]

Adami, C.

N. J. Cerf, C. Adami, and P. G. Kwiat, “Optical simulation of quantum logic,” Phys. Rev. A 57, R1477-R1480 (1998).
[CrossRef]

Aspect, A.

A. Aspect, J. Dalibard, and G. Roger, “Experimental test of Bell's inequalities using time-varying analyzers,” Phys. Rev. Lett. 49, 1804-1807 (1982).
[CrossRef]

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

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell's theorem,” Phys. Rev. Lett. 47, 460-463 (1981).
[CrossRef]

Beige, A.

A. Beige, W. J. Munro, and P. L. Knight, “Bell's inequality test with entangled atoms,” Phys. Rev. A 62, 052102 (2000).
[CrossRef]

Bell, J. S.

J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” Physics (Long Island City, N.Y.) 1, 195-200 (1964); reprinted in J. S. Bell, Speakable and Unispeakable in Quantum Mechanics (Cambridge U. Press, 1987).

J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” Physics (Long Island City, N.Y.) 1, 195-200 (1964); reprinted in J. S. Bell, Speakable and Unispeakable in Quantum Mechanics (Cambridge U. Press, 1987).

Bennink, R. S.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of The Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[CrossRef] [PubMed]

Bentley, S. J.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of The Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[CrossRef] [PubMed]

Bhattacharya, N.

N. Bhattacharya, H. B. van Linden vanden Heuvell, and R. J. C. Spreeuw, “Implementation of quantum search algorithm using classical Fourier optics,” Phys. Rev. Lett. 88, 137901 (2002).
[CrossRef] [PubMed]

Bohr, N.

N. Bohr, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 48, 696-702 (1935).
[CrossRef]

Bonagamba, T. J.

A. M. Souza, A. Magalhaes, J. Teles, E. R. deAzevedo, T. J. Bonagamba, I. S. Oliveira, and R. S. Sarthour, “NMR analog of Bell's inequalities violation test,” New J. Phys. 10, 033020 (2008).
[CrossRef]

Boyd, R. W.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of The Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[CrossRef] [PubMed]

Brendel, J.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart,” Phys. Rev. Lett. 81, 3563-3566 (1998).
[CrossRef]

Campos, J.

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. (Bellingham) 40, 2558-2564 (2001).
[CrossRef]

Cerf, N. J.

N. J. Cerf, C. Adami, and P. G. Kwiat, “Optical simulation of quantum logic,” Phys. Rev. A 57, R1477-R1480 (1998).
[CrossRef]

Chuang, I.

M. Nielsen and I. Chuang, Quantum Information and Computation (Cambridge U. Press, 2000).

Clauser, J. F.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880-884 (1969).
[CrossRef]

Dalibard, J.

A. Aspect, J. Dalibard, and G. Roger, “Experimental test of Bell's inequalities using time-varying analyzers,” Phys. Rev. Lett. 49, 1804-1807 (1982).
[CrossRef]

Davis, J. A.

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. (Bellingham) 40, 2558-2564 (2001).
[CrossRef]

deAzevedo, E. R.

A. M. Souza, A. Magalhaes, J. Teles, E. R. deAzevedo, T. J. Bonagamba, I. S. Oliveira, and R. S. Sarthour, “NMR analog of Bell's inequalities violation test,” New J. Phys. 10, 033020 (2008).
[CrossRef]

Deng, J.

J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A 70, 042313 (2004).
[CrossRef]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
[CrossRef]

Francisco, D.

D. Francisco and S. Ledesma, “Classical optics analogy of quantum teleportation,” J. Opt. Soc. Am. B 25, 383-390 (2008).
[CrossRef]

D. Francisco, C. Iemmi, J. P. Paz, and S. Ledesma, “Optical simulation of the quantum Hadamard operator,” Opt. Commun. 268, 340-345 (2006).
[CrossRef]

D. Francisco, C. Iemmi, J. P. Paz, and S. Ledesma, “Simulating a quantum walk with classical optics,” Phys. Rev. A 74, 052327 (2006).
[CrossRef]

Fu, J.

J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A 70, 042313 (2004).
[CrossRef]

Genovese, M.

M. Genovese, “Research on hidden variable theories: a review of recent progress,” Phys. Rep. 413, 319-396 (2005).
[CrossRef]

Gisin, N.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart,” Phys. Rev. Lett. 81, 3563-3566 (1998).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Grangier, P.

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

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell's theorem,” Phys. Rev. Lett. 47, 460-463 (1981).
[CrossRef]

Greenberger, D. M.

D. M. Greenberger, M. Horne, and A. Zeilinger, “Going beyond Bell's theorem,” in Bell's Theorem, Quantum Theory and Conceptions of the Universe (Kluwer Academic, 1989), pp. 73-76.

Holt, R. A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880-884 (1969).
[CrossRef]

Horne, M.

D. M. Greenberger, M. Horne, and A. Zeilinger, “Going beyond Bell's theorem,” in Bell's Theorem, Quantum Theory and Conceptions of the Universe (Kluwer Academic, 1989), pp. 73-76.

Horne, M. A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880-884 (1969).
[CrossRef]

Howell, J. C.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of The Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[CrossRef] [PubMed]

Iemmi, C.

D. Francisco, C. Iemmi, J. P. Paz, and S. Ledesma, “Simulating a quantum walk with classical optics,” Phys. Rev. A 74, 052327 (2006).
[CrossRef]

D. Francisco, C. Iemmi, J. P. Paz, and S. Ledesma, “Optical simulation of the quantum Hadamard operator,” Opt. Commun. 268, 340-345 (2006).
[CrossRef]

G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
[CrossRef]

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. (Bellingham) 40, 2558-2564 (2001).
[CrossRef]

Jennewein, T.

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell's inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039-5043 (1998).
[CrossRef]

Knight, P. L.

A. Beige, W. J. Munro, and P. L. Knight, “Bell's inequality test with entangled atoms,” Phys. Rev. A 62, 052102 (2000).
[CrossRef]

Kwiat, P. G.

N. J. Cerf, C. Adami, and P. G. Kwiat, “Optical simulation of quantum logic,” Phys. Rev. A 57, R1477-R1480 (1998).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photons pair,” Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

La Mela, C.

G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
[CrossRef]

Ledesma, S.

D. Francisco and S. Ledesma, “Classical optics analogy of quantum teleportation,” J. Opt. Soc. Am. B 25, 383-390 (2008).
[CrossRef]

D. Francisco, C. Iemmi, J. P. Paz, and S. Ledesma, “Optical simulation of the quantum Hadamard operator,” Opt. Commun. 268, 340-345 (2006).
[CrossRef]

D. Francisco, C. Iemmi, J. P. Paz, and S. Ledesma, “Simulating a quantum walk with classical optics,” Phys. Rev. A 74, 052327 (2006).
[CrossRef]

G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
[CrossRef]

Lee, K. F.

K. F. Lee and J. E. Thomas, “Experimental simulation of two-particle quantum entanglement using classical fields,” Phys. Rev. Lett. 88, 097902 (2002).
[CrossRef] [PubMed]

Magalhaes, A.

A. M. Souza, A. Magalhaes, J. Teles, E. R. deAzevedo, T. J. Bonagamba, I. S. Oliveira, and R. S. Sarthour, “NMR analog of Bell's inequalities violation test,” New J. Phys. 10, 033020 (2008).
[CrossRef]

Mandel, L.

Z. Y. Ou and L. Mandel, “Violation of Bell's inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50-53 (1988).
[CrossRef] [PubMed]

Marquez, A.

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. (Bellingham) 40, 2558-2564 (2001).
[CrossRef]

Mattle, K.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photons pair,” Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Moreno, I.

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. (Bellingham) 40, 2558-2564 (2001).
[CrossRef]

Munro, W. J.

A. Beige, W. J. Munro, and P. L. Knight, “Bell's inequality test with entangled atoms,” Phys. Rev. A 62, 052102 (2000).
[CrossRef]

Nielsen, M.

M. Nielsen and I. Chuang, Quantum Information and Computation (Cambridge U. Press, 2000).

Oliveira, I. S.

A. M. Souza, A. Magalhaes, J. Teles, E. R. deAzevedo, T. J. Bonagamba, I. S. Oliveira, and R. S. Sarthour, “NMR analog of Bell's inequalities violation test,” New J. Phys. 10, 033020 (2008).
[CrossRef]

Ou, Z. Y.

Z. Y. Ou and L. Mandel, “Violation of Bell's inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50-53 (1988).
[CrossRef] [PubMed]

Paz, J. P.

D. Francisco, C. Iemmi, J. P. Paz, and S. Ledesma, “Optical simulation of the quantum Hadamard operator,” Opt. Commun. 268, 340-345 (2006).
[CrossRef]

D. Francisco, C. Iemmi, J. P. Paz, and S. Ledesma, “Simulating a quantum walk with classical optics,” Phys. Rev. A 74, 052327 (2006).
[CrossRef]

G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
[CrossRef]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
[CrossRef]

Puentes, G.

G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
[CrossRef]

Roger, G.

A. Aspect, J. Dalibard, and G. Roger, “Experimental test of Bell's inequalities using time-varying analyzers,” Phys. Rev. Lett. 49, 1804-1807 (1982).
[CrossRef]

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

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell's theorem,” Phys. Rev. Lett. 47, 460-463 (1981).
[CrossRef]

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
[CrossRef]

Saraceno, M.

G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
[CrossRef]

Sarthour, R. S.

A. M. Souza, A. Magalhaes, J. Teles, E. R. deAzevedo, T. J. Bonagamba, I. S. Oliveira, and R. S. Sarthour, “NMR analog of Bell's inequalities violation test,” New J. Phys. 10, 033020 (2008).
[CrossRef]

Sergienko, A. V.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photons pair,” Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Shih, Y. H.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photons pair,” Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Shimony, A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880-884 (1969).
[CrossRef]

Si, Z.

J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A 70, 042313 (2004).
[CrossRef]

Simon, C.

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell's inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039-5043 (1998).
[CrossRef]

Souza, A. M.

A. M. Souza, A. Magalhaes, J. Teles, E. R. deAzevedo, T. J. Bonagamba, I. S. Oliveira, and R. S. Sarthour, “NMR analog of Bell's inequalities violation test,” New J. Phys. 10, 033020 (2008).
[CrossRef]

Spreeuw, R. J. C.

N. Bhattacharya, H. B. van Linden vanden Heuvell, and R. J. C. Spreeuw, “Implementation of quantum search algorithm using classical Fourier optics,” Phys. Rev. Lett. 88, 137901 (2002).
[CrossRef] [PubMed]

R. J. C. Spreeuw, Phys. Rev. A 63, 062302 (2001).
[CrossRef]

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361-374 (1998).
[CrossRef]

Tang, S.

J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A 70, 042313 (2004).
[CrossRef]

Teles, J.

A. M. Souza, A. Magalhaes, J. Teles, E. R. deAzevedo, T. J. Bonagamba, I. S. Oliveira, and R. S. Sarthour, “NMR analog of Bell's inequalities violation test,” New J. Phys. 10, 033020 (2008).
[CrossRef]

Thomas, J. E.

K. F. Lee and J. E. Thomas, “Experimental simulation of two-particle quantum entanglement using classical fields,” Phys. Rev. Lett. 88, 097902 (2002).
[CrossRef] [PubMed]

Tittel, W.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart,” Phys. Rev. Lett. 81, 3563-3566 (1998).
[CrossRef]

van Linden vanden Heuvell, H. B.

N. Bhattacharya, H. B. van Linden vanden Heuvell, and R. J. C. Spreeuw, “Implementation of quantum search algorithm using classical Fourier optics,” Phys. Rev. Lett. 88, 137901 (2002).
[CrossRef] [PubMed]

Weihs, G.

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell's inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039-5043 (1998).
[CrossRef]

Weinfurter, H.

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell's inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039-5043 (1998).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photons pair,” Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Yzuel, M. J.

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. (Bellingham) 40, 2558-2564 (2001).
[CrossRef]

Zbinden, H.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart,” Phys. Rev. Lett. 81, 3563-3566 (1998).
[CrossRef]

Zeilinger, A.

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell's inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039-5043 (1998).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photons pair,” Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

D. M. Greenberger, M. Horne, and A. Zeilinger, “Going beyond Bell's theorem,” in Bell's Theorem, Quantum Theory and Conceptions of the Universe (Kluwer Academic, 1989), pp. 73-76.

Found. Phys. (1)

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361-374 (1998).
[CrossRef]

J. Opt. Soc. Am. B (1)

New J. Phys. (1)

A. M. Souza, A. Magalhaes, J. Teles, E. R. deAzevedo, T. J. Bonagamba, I. S. Oliveira, and R. S. Sarthour, “NMR analog of Bell's inequalities violation test,” New J. Phys. 10, 033020 (2008).
[CrossRef]

Opt. Commun. (1)

D. Francisco, C. Iemmi, J. P. Paz, and S. Ledesma, “Optical simulation of the quantum Hadamard operator,” Opt. Commun. 268, 340-345 (2006).
[CrossRef]

Opt. Eng. (Bellingham) (1)

A. Marquez, C. Iemmi, I. Moreno, J. A. Davis, J. Campos, and M. J. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. (Bellingham) 40, 2558-2564 (2001).
[CrossRef]

Phys. Rep. (1)

M. Genovese, “Research on hidden variable theories: a review of recent progress,” Phys. Rep. 413, 319-396 (2005).
[CrossRef]

Phys. Rev. (2)

N. Bohr, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 48, 696-702 (1935).
[CrossRef]

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777-780 (1935).
[CrossRef]

Phys. Rev. A (6)

A. Beige, W. J. Munro, and P. L. Knight, “Bell's inequality test with entangled atoms,” Phys. Rev. A 62, 052102 (2000).
[CrossRef]

D. Francisco, C. Iemmi, J. P. Paz, and S. Ledesma, “Simulating a quantum walk with classical optics,” Phys. Rev. A 74, 052327 (2006).
[CrossRef]

J. Fu, Z. Si, S. Tang, and J. Deng, “Classical simulation of quantum entanglement using optical transverse modes in multimode waveguides,” Phys. Rev. A 70, 042313 (2004).
[CrossRef]

R. J. C. Spreeuw, Phys. Rev. A 63, 062302 (2001).
[CrossRef]

N. J. Cerf, C. Adami, and P. G. Kwiat, “Optical simulation of quantum logic,” Phys. Rev. A 57, R1477-R1480 (1998).
[CrossRef]

G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
[CrossRef]

Phys. Rev. Lett. (11)

N. Bhattacharya, H. B. van Linden vanden Heuvell, and R. J. C. Spreeuw, “Implementation of quantum search algorithm using classical Fourier optics,” Phys. Rev. Lett. 88, 137901 (2002).
[CrossRef] [PubMed]

K. F. Lee and J. E. Thomas, “Experimental simulation of two-particle quantum entanglement using classical fields,” Phys. Rev. Lett. 88, 097902 (2002).
[CrossRef] [PubMed]

A. Aspect, J. Dalibard, and G. Roger, “Experimental test of Bell's inequalities using time-varying analyzers,” Phys. Rev. Lett. 49, 1804-1807 (1982).
[CrossRef]

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart,” Phys. Rev. Lett. 81, 3563-3566 (1998).
[CrossRef]

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell's inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039-5043 (1998).
[CrossRef]

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell's theorem,” Phys. Rev. Lett. 47, 460-463 (1981).
[CrossRef]

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

Z. Y. Ou and L. Mandel, “Violation of Bell's inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50-53 (1988).
[CrossRef] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, “New high-intensity source of polarization-entangled photons pair,” Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of The Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[CrossRef] [PubMed]

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880-884 (1969).
[CrossRef]

Physics (Long Island City, N.Y.) (1)

J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” Physics (Long Island City, N.Y.) 1, 195-200 (1964); reprinted in J. S. Bell, Speakable and Unispeakable in Quantum Mechanics (Cambridge U. Press, 1987).

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

M. Nielsen and I. Chuang, Quantum Information and Computation (Cambridge U. Press, 2000).

D. M. Greenberger, M. Horne, and A. Zeilinger, “Going beyond Bell's theorem,” in Bell's Theorem, Quantum Theory and Conceptions of the Universe (Kluwer Academic, 1989), pp. 73-76.

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Figures (8)

Fig. 1
Fig. 1

Optical representation of the single qubit state. (a) Optical representation of the states of the computational basis. (b) The input scene associated with the optical single qubit. The state α | 0 + β | 1 is represented by the “left” or “right” slices where the constants α and β are the complex amplitudes of the electromagnetic field in each slice.

Fig. 2
Fig. 2

Schematic picture of the representation of two qubit states by using optical scenes. (a) Spatial organization of the input plane in order to emulate two qubit states. (b) Optical representation of the | 0 A | 0 B state. (c) Optical representation of the general pure two qubit state. Gray level scale corresponds to different amplitudes and phase modulations of the classical wavefront.

Fig. 3
Fig. 3

Schematic picture of the optically simulated U ( 2 ) operation. Complex amplitudes α and β are mapping onto α and β by means of a 4 f coherent optical processor with an almenary phase grating in the Fourier plane.

Fig. 4
Fig. 4

Optical representation (a) of the maximally entangled state, ( | 0 A | 0 B + | 1 A | 1 B ) / 2 , and (b) of the maximally mixed state, 1 2 [ ( | 0 A | 0 B ) ( 0 | A 0 | B ) + ( | 1 A | 1 B ) ( 1 | A 1 | B ) ] , as optical scenes.

Fig. 5
Fig. 5

Experimental setup for simulating the Bell experiment as an imaging system.

Fig. 6
Fig. 6

Detail of the protocol of the full experiment.

Fig. 7
Fig. 7

Measuring a two qubit system in computational basis from the output distribution intensities.

Fig. 8
Fig. 8

Experimental results. (a) Theoretical predictions and experimental results of the simulation O versus θ plotted in the full range θ [ 0 , 2 π ) for the maximally entangled state q = 1 and for the mixed state q = 0 . (b) Theoretical and experimental results of the simulations corresponding to mixed states with density matrix defined in Eq. (7) for q = 1 , 2/3, 1/3, and 0 in the Bell inequality violation domain.

Tables (1)

Tables Icon

Table 1 Hermitian Observables, Unitary Change of Basis, and Parameters of the Optical Simulation

Equations (11)

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

A = α ̂ σ A ,     A = α ̂ σ A ,     B = β ̂ σ B ,     B = β ̂ σ B ,
| O | = | A B + A B + A B A B | 2.
O = A B + A C C B 1.
A = σ x ,     B = sin   θ σ x + cos   θ σ z ,     C = σ z .
ρ = q ρ pure + ( 1 q ) ρ mixed = 1 2 ( 1 0 0 q 0 0 0 0 0 0 0 0 q 0 0 1 ) ,
O ( θ ) = tr [ ( A B ) ρ ] + tr [ ( A C ) ρ ] tr [ ( C B ) ρ ] = q   sin   θ cos   θ .
x o | 0 Rect ( x o + a b ) ,     x o | x o Rect ( x o a b ) ,
H ( f x ) = { e i ϕ , if | f x f c | < p / 2 1 , in   other   case , }
h ( x ) = + H ( f x ) exp ( i 2 π f x x ) d f x = cos ϕ 2 δ ( x ) + 2 π sin ϕ 2 e i γ δ ( x + 1 2 p ) + 2 π sin ϕ 2 e i γ + δ ( x 1 2 p ) ,
( α β ) ( α β ) = ( cos ϕ 2 2 π sin ϕ 2 e i γ 2 π sin ϕ 2 e i γ + cos ϕ 2 ) ( α β ) ,
σ z σ z = P ( A + , B + ) + P ( A , B ) P ( A + , B ) P ( A , B + ) = | α 00 | 2 + | α 11 | 2 | α 01 | 2 | α 10 | 2 | α 00 | 2 + | α 01 | 2 + | α 10 | 2 + | α 11 | 2 .

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