Abstract

The growing recognition that entanglement is not exclusively a quantum property, and does not even originate with Schrödinger’s famous remark about it [Proc. Cambridge Philos. Soc. 31, 555 (1935) [CrossRef]  ], prompts the examination of its role in marking the quantum-classical boundary. We have done this by subjecting correlations of classical optical fields to new Bell-analysis experiments and report here values of the Bell parameter greater than B=2.54. This is many standard deviations outside the limit B=2 established by the Clauser–Horne–Shimony–Holt Bell inequality [Phys. Rev. Lett. 23, 880 (1969) [CrossRef]  ], in agreement with our theoretical classical prediction, and not far from the Tsirelson limit B=2.828…. These results cast a new light on the standard quantum-classical boundary description, and suggest a reinterpretation of it.

© 2015 Optical Society of America

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  1. In the paper where he introduced the English word entanglement [E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935)], Schrödinger referred to the math-physics textbook by R. Courant and D. Hilbert, Methoden der mathematischen Physik, 2nd ed. (specifically p. 134) where the mathematical character of entanglement appears in connection with the Schmidt theorem of analytic function theory.
    [Crossref]
  2. R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
    [Crossref]
  3. P. Ghose and M. K. Samal, “EPR type nonlocality in classical electrodynamics,” arXiv:quant-ph/0111119 (2001).
  4. K. F. Lee and J. E. Thomas, “Entanglement with classical fields,” Phys. Rev. Lett. 88, 097902 (2002).
    [Crossref]
  5. C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
    [Crossref]
  6. M. A. Goldin, D. Francisco, and S. Ledesma, “Simulating Bell inequality violations with classical optics encoded qubits,” J. Opt. Soc. Am. A 27, 779–786 (2010).
    [Crossref]
  7. A recent example is A. Tiranov, J. Lavoie, A. Ferrier, P. Goldner, V. B. Verma, S. W. Nam, R. P. Mirin, A. E. Lita, F. Marsili, H. Herrmann, C. Silberhorn, N. Gisin, M. Afzelius, and F. Bussieres, “Storage of hyperentanglement in a solid-state quantum memory,” Optica 2, 279–287 (2015).
    [Crossref]
  8. B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
    [Crossref]
  9. X. F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011). See also T. Setälä, A. Shevchenko, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002), and J. Ellis and A. Dogariu, “Optical polarimetry of random fields,” Phys. Rev. Lett. 95, 203905 (2005).
    [Crossref]
  10. K. H. Kagalwala, G. DiGiuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
    [Crossref]
  11. F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
    [Crossref]
  12. 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]
  13. A. Shimony, “Contextual hidden variables theories and Bell’s inequalities,” Br. J. Philos. Sci. 35, 25–45 (1984).See also A. Shimony, “Bell’s Theorem,” in Stanford Encyclopedia of Philosophy, http:/plato.stanford.edu/entries/bell-theorem (2009).
  14. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).
  15. See X.-F. Qian and J. H. Eberly, “Entanglement is sometimes enough,” arXiv:1307.3772 (2013).
  16. See C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998) and E. Wolf, ed. Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  17. Elliptical polarization needs different notation, but the results are the same.
  18. The Schmidt theorem is a continuous-space version of the singular-value decomposition theorem for matrices. For background, see M. V. Fedorov and N. I. Miklin, “Schmidt modes and entanglement,” Contemp. Phys. 55, 94–109 (2014). The original paper is E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. 1. Entwicklung willküriger Funktionen nach Systeme vorgeschriebener,” Math. Ann. 63, 433–476 (1907). See also A. Ekert and P. L. Knight, “Entangled quantum-systems and the Schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995).
    [Crossref]
  19. N. Gisin, “Bell’s inequality holds for all non-product states,” Phys. Lett. A 154, 201–202 (1991).
    [Crossref]
  20. Classical fields are often thought of as the many-photon macroscopic limit of quantum coherent states. It is interesting to propose connections between our work, and the CHSH Bell violations discussed theoretically for maximally entangled macroscopic coherent states in C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009).Following the number-state treatments of C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, “Strong violations of Bell-type inequalities for path-entangled number states,” Phys. Rev. A 76, 052101 (2007).
    [Crossref]
  21. J. S. Bell, “Quantum mechanical ideas,” Science 177, 880–881 (1972).
    [Crossref]

2015 (1)

2014 (2)

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

The Schmidt theorem is a continuous-space version of the singular-value decomposition theorem for matrices. For background, see M. V. Fedorov and N. I. Miklin, “Schmidt modes and entanglement,” Contemp. Phys. 55, 94–109 (2014). The original paper is E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. 1. Entwicklung willküriger Funktionen nach Systeme vorgeschriebener,” Math. Ann. 63, 433–476 (1907). See also A. Ekert and P. L. Knight, “Entangled quantum-systems and the Schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995).
[Crossref]

2013 (1)

K. H. Kagalwala, G. DiGiuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

2011 (1)

2010 (3)

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

M. A. Goldin, D. Francisco, and S. Ledesma, “Simulating Bell inequality violations with classical optics encoded qubits,” J. Opt. Soc. Am. A 27, 779–786 (2010).
[Crossref]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

2009 (1)

Classical fields are often thought of as the many-photon macroscopic limit of quantum coherent states. It is interesting to propose connections between our work, and the CHSH Bell violations discussed theoretically for maximally entangled macroscopic coherent states in C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009).Following the number-state treatments of C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, “Strong violations of Bell-type inequalities for path-entangled number states,” Phys. Rev. A 76, 052101 (2007).
[Crossref]

2002 (1)

K. F. Lee and J. E. Thomas, “Entanglement with classical fields,” Phys. Rev. Lett. 88, 097902 (2002).
[Crossref]

1998 (1)

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

1991 (1)

N. Gisin, “Bell’s inequality holds for all non-product states,” Phys. Lett. A 154, 201–202 (1991).
[Crossref]

1984 (1)

A. Shimony, “Contextual hidden variables theories and Bell’s inequalities,” Br. J. Philos. Sci. 35, 25–45 (1984).See also A. Shimony, “Bell’s Theorem,” in Stanford Encyclopedia of Philosophy, http:/plato.stanford.edu/entries/bell-theorem (2009).

1972 (1)

J. S. Bell, “Quantum mechanical ideas,” Science 177, 880–881 (1972).
[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]

1959 (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).

1935 (1)

In the paper where he introduced the English word entanglement [E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935)], Schrödinger referred to the math-physics textbook by R. Courant and D. Hilbert, Methoden der mathematischen Physik, 2nd ed. (specifically p. 134) where the mathematical character of entanglement appears in connection with the Schmidt theorem of analytic function theory.
[Crossref]

Abouraddy, A. F.

K. H. Kagalwala, G. DiGiuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

Afzelius, M.

Aiello, A.

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Bell, J. S.

J. S. Bell, “Quantum mechanical ideas,” Science 177, 880–881 (1972).
[Crossref]

Benmoussa, A.

Classical fields are often thought of as the many-photon macroscopic limit of quantum coherent states. It is interesting to propose connections between our work, and the CHSH Bell violations discussed theoretically for maximally entangled macroscopic coherent states in C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009).Following the number-state treatments of C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, “Strong violations of Bell-type inequalities for path-entangled number states,” Phys. Rev. A 76, 052101 (2007).
[Crossref]

Borges, C. V. S.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

Borghi, R.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

Brosseau, C.

See C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998) and E. Wolf, ed. Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Bussieres, F.

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]

DiGiuseppe, G.

K. H. Kagalwala, G. DiGiuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

Eberly, J. H.

Fedorov, M. V.

The Schmidt theorem is a continuous-space version of the singular-value decomposition theorem for matrices. For background, see M. V. Fedorov and N. I. Miklin, “Schmidt modes and entanglement,” Contemp. Phys. 55, 94–109 (2014). The original paper is E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. 1. Entwicklung willküriger Funktionen nach Systeme vorgeschriebener,” Math. Ann. 63, 433–476 (1907). See also A. Ekert and P. L. Knight, “Entangled quantum-systems and the Schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995).
[Crossref]

Ferrier, A.

Francisco, D.

M. A. Goldin, D. Francisco, and S. Ledesma, “Simulating Bell inequality violations with classical optics encoded qubits,” J. Opt. Soc. Am. A 27, 779–786 (2010).
[Crossref]

Gerry, C. C.

Classical fields are often thought of as the many-photon macroscopic limit of quantum coherent states. It is interesting to propose connections between our work, and the CHSH Bell violations discussed theoretically for maximally entangled macroscopic coherent states in C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009).Following the number-state treatments of C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, “Strong violations of Bell-type inequalities for path-entangled number states,” Phys. Rev. A 76, 052101 (2007).
[Crossref]

Ghose, P.

P. Ghose and M. K. Samal, “EPR type nonlocality in classical electrodynamics,” arXiv:quant-ph/0111119 (2001).

Giacobino, E.

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Gisin, N.

Goldin, M. A.

M. A. Goldin, D. Francisco, and S. Ledesma, “Simulating Bell inequality violations with classical optics encoded qubits,” J. Opt. Soc. Am. A 27, 779–786 (2010).
[Crossref]

Goldner, P.

Gori, F.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

Herrmann, H.

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]

Hor-Meyll, M.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

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]

Huguenin, J. A. O.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

Kagalwala, K. H.

K. H. Kagalwala, G. DiGiuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

Khoury, A. Z.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

Lavoie, J.

Ledesma, S.

M. A. Goldin, D. Francisco, and S. Ledesma, “Simulating Bell inequality violations with classical optics encoded qubits,” J. Opt. Soc. Am. A 27, 779–786 (2010).
[Crossref]

Lee, K. F.

K. F. Lee and J. E. Thomas, “Entanglement with classical fields,” Phys. Rev. Lett. 88, 097902 (2002).
[Crossref]

Leuchs, G.

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Lita, A. E.

Marquardt, C.

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Marsili, F.

Miklin, N. I.

The Schmidt theorem is a continuous-space version of the singular-value decomposition theorem for matrices. For background, see M. V. Fedorov and N. I. Miklin, “Schmidt modes and entanglement,” Contemp. Phys. 55, 94–109 (2014). The original paper is E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. 1. Entwicklung willküriger Funktionen nach Systeme vorgeschriebener,” Math. Ann. 63, 433–476 (1907). See also A. Ekert and P. L. Knight, “Entangled quantum-systems and the Schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995).
[Crossref]

Mimih, J.

Classical fields are often thought of as the many-photon macroscopic limit of quantum coherent states. It is interesting to propose connections between our work, and the CHSH Bell violations discussed theoretically for maximally entangled macroscopic coherent states in C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009).Following the number-state treatments of C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, “Strong violations of Bell-type inequalities for path-entangled number states,” Phys. Rev. A 76, 052101 (2007).
[Crossref]

Mirin, R. P.

Mukunda, N.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

Nam, S. W.

Qian, X. F.

Qian, X.-F.

See X.-F. Qian and J. H. Eberly, “Entanglement is sometimes enough,” arXiv:1307.3772 (2013).

Saleh, B. E. A.

K. H. Kagalwala, G. DiGiuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

Samal, M. K.

P. Ghose and M. K. Samal, “EPR type nonlocality in classical electrodynamics,” arXiv:quant-ph/0111119 (2001).

Santarsiero, M.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

Schrödinger, E.

In the paper where he introduced the English word entanglement [E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935)], Schrödinger referred to the math-physics textbook by R. Courant and D. Hilbert, Methoden der mathematischen Physik, 2nd ed. (specifically p. 134) where the mathematical character of entanglement appears in connection with the Schmidt theorem of analytic function theory.
[Crossref]

Shimony, A.

A. Shimony, “Contextual hidden variables theories and Bell’s inequalities,” Br. J. Philos. Sci. 35, 25–45 (1984).See also A. Shimony, “Bell’s Theorem,” in Stanford Encyclopedia of Philosophy, http:/plato.stanford.edu/entries/bell-theorem (2009).

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]

Silberhorn, C.

Simon, B. N.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

Simon, R.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

Simon, S.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

Spreeuw, R. J. C.

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

Thomas, J. E.

K. F. Lee and J. E. Thomas, “Entanglement with classical fields,” Phys. Rev. Lett. 88, 097902 (2002).
[Crossref]

Tiranov, A.

Töppel, F.

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Verma, V. B.

Wolf, E.

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).

Br. J. Philos. Sci. (1)

A. Shimony, “Contextual hidden variables theories and Bell’s inequalities,” Br. J. Philos. Sci. 35, 25–45 (1984).See also A. Shimony, “Bell’s Theorem,” in Stanford Encyclopedia of Philosophy, http:/plato.stanford.edu/entries/bell-theorem (2009).

Contemp. Phys. (1)

The Schmidt theorem is a continuous-space version of the singular-value decomposition theorem for matrices. For background, see M. V. Fedorov and N. I. Miklin, “Schmidt modes and entanglement,” Contemp. Phys. 55, 94–109 (2014). The original paper is E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. 1. Entwicklung willküriger Funktionen nach Systeme vorgeschriebener,” Math. Ann. 63, 433–476 (1907). See also A. Ekert and P. L. Knight, “Entangled quantum-systems and the Schmidt decomposition,” Am. J. Phys. 63, 415–423 (1995).
[Crossref]

Found. Phys. (1)

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

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

M. A. Goldin, D. Francisco, and S. Ledesma, “Simulating Bell inequality violations with classical optics encoded qubits,” J. Opt. Soc. Am. A 27, 779–786 (2010).
[Crossref]

Nat. Photonics (1)

K. H. Kagalwala, G. DiGiuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

New J. Phys. (1)

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Nuovo Cimento (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165–1181 (1959).

Opt. Lett. (1)

Optica (1)

Phys. Lett. A (1)

N. Gisin, “Bell’s inequality holds for all non-product states,” Phys. Lett. A 154, 201–202 (1991).
[Crossref]

Phys. Rev. A (2)

Classical fields are often thought of as the many-photon macroscopic limit of quantum coherent states. It is interesting to propose connections between our work, and the CHSH Bell violations discussed theoretically for maximally entangled macroscopic coherent states in C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009).Following the number-state treatments of C. F. Wildfeuer, A. P. Lund, and J. P. Dowling, “Strong violations of Bell-type inequalities for path-entangled number states,” Phys. Rev. A 76, 052101 (2007).
[Crossref]

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

Phys. Rev. Lett. (3)

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]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).See also M. Sanjay Kumar and R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[Crossref]

K. F. Lee and J. E. Thomas, “Entanglement with classical fields,” Phys. Rev. Lett. 88, 097902 (2002).
[Crossref]

Proc. Cambridge Philos. Soc. (1)

In the paper where he introduced the English word entanglement [E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935)], Schrödinger referred to the math-physics textbook by R. Courant and D. Hilbert, Methoden der mathematischen Physik, 2nd ed. (specifically p. 134) where the mathematical character of entanglement appears in connection with the Schmidt theorem of analytic function theory.
[Crossref]

Science (1)

J. S. Bell, “Quantum mechanical ideas,” Science 177, 880–881 (1972).
[Crossref]

Other (4)

See X.-F. Qian and J. H. Eberly, “Entanglement is sometimes enough,” arXiv:1307.3772 (2013).

See C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998) and E. Wolf, ed. Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Elliptical polarization needs different notation, but the results are the same.

P. Ghose and M. K. Samal, “EPR type nonlocality in classical electrodynamics,” arXiv:quant-ph/0111119 (2001).

Supplementary Material (1)

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Figures (2)

Fig. 1.
Fig. 1. Experimental setup consists of a source of unpolarized light and a measurement using a modified MZ interferometer. HWP and a QWP control the polarization of the source. All beam splitters are 50:50 unless marked as a PBS. Intensities needed for obtaining the required joint projections are measured at detector D1. Shutters S independently block the arms of the interferometer in order to measure light through the arms separately. A removable mirror (RM) directs the light to a polarization tomography setup where the orthogonal components of the polarization in the basis determined by the wave plate are measured at detectors D2 and D3.

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Fig. 2.
Fig. 2. Plots of the correlation functions C(a,b) obtained by rotating polarizer a in the polarization space and keeping angle b in the function space constant. Curves 1–4 correspond to different fixed values of b separated by π/4. The invariant cosine function required to violate the Bell inequality is clearly present. Error bars are included but are scarcely visible.

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Equations (10)

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E⃗(t)=x^Ex(t)+y^Ey(t).
e⃗(t)≡E⃗(t)/I={x^ex(t)+y^ey(t)},
E⃗/I=x^ex+y^ey=|e〉=(|u1〉|f1〉+|u2〉|f2〉)/2.
|u1a〉=cos a|u1〉−sin a|u2〉and|u2a〉=sin a|u1〉+cos a|u2〉.
|f1b〉=cos b|f1〉−sin b|f2〉and|f2b〉=sin b|f1〉+cos b|f2〉,
C(a,b)=〈e|Zu(a)⊗Zf(b)|e〉,
P11(a,b)=〈e|(|u1a〉|f1b〉〈f1b|〈u1a|)|e〉=|〈f1b|〈u1a|e〉|2.
B=|C(a,b)−C(a′,b)+C(a,b′)+C(a′,b′)|.
|E1a〉=I1a|u1a〉(c11|f1b〉+c12|f2b〉),
P11(a,b)=(2I1T−I¯1a−I1a)2/4II¯1a.

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