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]

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]

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]

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]

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]

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]

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

[Crossref]

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

[Crossref]

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

[Crossref]

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

[Crossref]

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]

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

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]

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]

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]

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

[Crossref]

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

[Crossref]

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]

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]

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).

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]

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]

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]

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]

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

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]

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]

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]

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]

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

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

[Crossref]

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]

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

[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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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

[Crossref]

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

[Crossref]

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]

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

[Crossref]

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]

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]

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]

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]

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]

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]

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]

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

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]

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

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]

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]

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]

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]

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]

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]

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]

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

[Crossref]

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

[Crossref]

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]

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

[Crossref]

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]

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

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).

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]

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

[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]

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]

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

[Crossref]

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

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]

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]

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

[Crossref]

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]

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]

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]

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]

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

[Crossref]

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).