Abstract

Light is neither wave nor particle, but both, according to Bohr’s complementarity principle, which was first devised to qualitatively characterize quantum phenomena. Later, quantification was achieved through inequalities such as V2+D21, which engage visibility V and distinguishability D. Recently, equality V2+D2=P2—the polarization coherence theorem (PCT)—was established, incorporating polarization P and addressing both quantum and classical coherences. This shows that Bohr’s complementarity is not restricted to quantum phenomena. We derive an extension of the PCT that also applies to quantum and classical light fields carrying intertwined, dichotomic observables, such as polarization and two-path alternative. We discuss how constraints critically depend on the chosen measurement strategy. This may prompt various experiments to exhibit complementary features that possibly lurk behind hidden coherences.

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

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References

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  1. J. H. Eberly, X.-F. Qian, and A. N. Vamivakas, “Polarization coherence theorem,” Optica 4, 1113–1114 (2017).
    [Crossref]
  2. A. F. Abouraddy, “What is the maximum attainable visibility by a partially coherent electromagnetic field in Young’s double-slit interference?” Opt. Express 15, 18320–18331 (2017).
    [Crossref]
  3. R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
    [Crossref]
  4. O. Gamel and D. F. V. James, “Causality and the complete positivity of classical polarization maps,” Opt. Lett. 36, 2821–2823 (2011).
    [Crossref]
  5. A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherence matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
    [Crossref]
  6. B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
    [Crossref]
  7. K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and E. A. Saleh, “Optical coherence matrix tomography,” Sci. Rep. 5, 15333 (2015).
    [Crossref]
  8. D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
    [Crossref]
  9. C. Okoro, H. E. Kondakci, A. F. Abouraddy, and K. C. Toussaint, “Demonstration of an optical-coherence converter,” Optica 4, 1052–1058 (2017).
    [Crossref]
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    [Crossref]
  11. A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
    [Crossref]
  12. 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]
  13. X.-F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011).
    [Crossref]
  14. K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
    [Crossref]
  15. X.-F. Qian, B. Little, J. C. Howell, and J. H. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica 2, 611–615 (2015).
    [Crossref]
  16. M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
    [Crossref]
  17. A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
    [Crossref]
  18. N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
    [Crossref]
  19. A. Luis, “Coherence, polarization, and entanglement for classical fields,” Opt. Commun. 282, 3665–3670 (2009).
    [Crossref]
  20. 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).
    [Crossref]
  21. H. Chen, T. Peng, S. Karmakar, and Y. Shih, “Simulation of Bell states with incoherent thermal light,” New J. Phys. 13, 083018 (2011).
    [Crossref]
  22. J. H. Eberly, “Shimony-Wolf states and hidden coherences in classical light,” Contemp. Phys. 56, 407–416 (2015).
    [Crossref]
  23. J. H. Eberly, “Correlation, coherence and context,” Laser Phys. 26, 084004 (2016).
    [Crossref]
  24. J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
    [Crossref]
  25. T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
    [Crossref]
  26. B.-G. Englert, “Fringe visibility and which-way information: an inequality,” Phys. Rev. Lett. 77, 2154–2157 (1996).
    [Crossref]
  27. S. Dürr and G. Rempe, “Can wave-particle duality be based on the uncertainty relation?” Am. J. Phys. 68, 1021–1024 (2000).
    [Crossref]
  28. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).
  29. G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
    [Crossref]
  30. B.-G. Englert, “Remarks on some basic issues in quantum mechanics,” Z. Naturforsch. 54a, 11–32 (1999).
  31. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2007).
  32. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, 1980).
  33. F. Gori, M. Santarsiero, and R. Borghi, “Vector mode analysis of a Young interferometer,” Opt. Lett. 31, 858–860 (2006).
    [Crossref]

2017 (3)

2016 (4)

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
[Crossref]

J. H. Eberly, “Correlation, coherence and context,” Laser Phys. 26, 084004 (2016).
[Crossref]

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

2015 (6)

M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
[Crossref]

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
[Crossref]

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and E. A. Saleh, “Optical coherence matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

X.-F. Qian, B. Little, J. C. Howell, and J. H. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica 2, 611–615 (2015).
[Crossref]

J. H. Eberly, “Shimony-Wolf states and hidden coherences in classical light,” Contemp. Phys. 56, 407–416 (2015).
[Crossref]

2014 (2)

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherence matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref]

T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
[Crossref]

2013 (1)

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

2011 (3)

2010 (2)

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).
[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]

2009 (1)

A. Luis, “Coherence, polarization, and entanglement for classical fields,” Opt. Commun. 282, 3665–3670 (2009).
[Crossref]

2007 (1)

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
[Crossref]

2006 (1)

2001 (1)

R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
[Crossref]

2000 (1)

S. Dürr and G. Rempe, “Can wave-particle duality be based on the uncertainty relation?” Am. J. Phys. 68, 1021–1024 (2000).
[Crossref]

1999 (1)

B.-G. Englert, “Remarks on some basic issues in quantum mechanics,” Z. Naturforsch. 54a, 11–32 (1999).

1998 (1)

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

1996 (1)

B.-G. Englert, “Fringe visibility and which-way information: an inequality,” Phys. Rev. Lett. 77, 2154–2157 (1996).
[Crossref]

1995 (1)

G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
[Crossref]

Abouraddy, A. F.

A. F. Abouraddy, “What is the maximum attainable visibility by a partially coherent electromagnetic field in Young’s double-slit interference?” Opt. Express 15, 18320–18331 (2017).
[Crossref]

C. Okoro, H. E. Kondakci, A. F. Abouraddy, and K. C. Toussaint, “Demonstration of an optical-coherence converter,” Optica 4, 1052–1058 (2017).
[Crossref]

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and E. A. Saleh, “Optical coherence matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherence matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref]

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

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
[Crossref]

Agarwal, G. S.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
[Crossref]

Aiello, A.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Akhouayri, H.

N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
[Crossref]

Al Qasimi, A.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

Ali, H.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

Alonso, M. A.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

Baltanás, J. P.

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Baumgratz, T.

T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
[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).
[Crossref]

F. Gori, M. Santarsiero, and R. Borghi, “Vector mode analysis of a Young interferometer,” Opt. Lett. 31, 858–860 (2006).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

Cabello, A.

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Chen, H.

H. Chen, T. Peng, S. Karmakar, and Y. Shih, “Simulation of Bell states with incoherent thermal light,” New J. Phys. 13, 083018 (2011).
[Crossref]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2007).

Cramer, M.

T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
[Crossref]

Di Giuseppe, G.

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

Dürr, S.

S. Dürr and G. Rempe, “Can wave-particle duality be based on the uncertainty relation?” Am. J. Phys. 68, 1021–1024 (2000).
[Crossref]

Durt, T.

N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
[Crossref]

Eberly, J. H.

J. H. Eberly, X.-F. Qian, and A. N. Vamivakas, “Polarization coherence theorem,” Optica 4, 1113–1114 (2017).
[Crossref]

J. H. Eberly, “Correlation, coherence and context,” Laser Phys. 26, 084004 (2016).
[Crossref]

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

J. H. Eberly, “Shimony-Wolf states and hidden coherences in classical light,” Contemp. Phys. 56, 407–416 (2015).
[Crossref]

X.-F. Qian, B. Little, J. C. Howell, and J. H. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica 2, 611–615 (2015).
[Crossref]

X.-F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011).
[Crossref]

Englert, B.-G.

B.-G. Englert, “Remarks on some basic issues in quantum mechanics,” Z. Naturforsch. 54a, 11–32 (1999).

B.-G. Englert, “Fringe visibility and which-way information: an inequality,” Phys. Rev. Lett. 77, 2154–2157 (1996).
[Crossref]

Fernández-Prieto, A.

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Forbes, A.

M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
[Crossref]

Freire, M. J.

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Frustaglia, D.

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Gamel, O.

Giacobino, E.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Goldstein, H.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, 1980).

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).
[Crossref]

F. Gori, M. Santarsiero, and R. Borghi, “Vector mode analysis of a Young interferometer,” Opt. Lett. 31, 858–860 (2006).
[Crossref]

Gutiérrez-Cuevas, R.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[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]

Howell, J. C.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

X.-F. Qian, B. Little, J. C. Howell, and J. H. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica 2, 611–615 (2015).
[Crossref]

Hradil, Z.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
[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]

Jaeger, G.

G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
[Crossref]

James, D. F. V.

Kagalwala, K. H.

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and E. A. Saleh, “Optical coherence matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherence matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref]

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

Karmakar, S.

H. Chen, T. Peng, S. Karmakar, and Y. Shih, “Simulation of Bell states with incoherent thermal light,” New J. Phys. 13, 083018 (2011).
[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]

Kondakci, H. E.

C. Okoro, H. E. Kondakci, A. F. Abouraddy, and K. C. Toussaint, “Demonstration of an optical-coherence converter,” Optica 4, 1052–1058 (2017).
[Crossref]

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and E. A. Saleh, “Optical coherence matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

Konrad, T.

M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
[Crossref]

Leuchs, G.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Little, B.

Little, B. J.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

Losada, V.

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Luis, A.

A. Luis, “Coherence, polarization, and entanglement for classical fields,” Opt. Commun. 282, 3665–3670 (2009).
[Crossref]

Lujambio, A.

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Malhotra, T.

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M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
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B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
[Crossref]

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).
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T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
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Qian, X.-F.

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B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
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A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherence matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
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A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
[Crossref]

Saleh, E. A.

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and E. A. Saleh, “Optical coherence matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
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B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
[Crossref]

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N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
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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).
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F. Gori, M. Santarsiero, and R. Borghi, “Vector mode analysis of a Young interferometer,” Opt. Lett. 31, 858–860 (2006).
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H. Chen, T. Peng, S. Karmakar, and Y. Shih, “Simulation of Bell states with incoherent thermal light,” New J. Phys. 13, 083018 (2011).
[Crossref]

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G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
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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).
[Crossref]

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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).
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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).
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B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
[Crossref]

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A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
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A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

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G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
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J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

Velázquez-Ahumada, M. C.

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
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Yarnall, T.

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
[Crossref]

Am. J. Phys. (1)

S. Dürr and G. Rempe, “Can wave-particle duality be based on the uncertainty relation?” Am. J. Phys. 68, 1021–1024 (2000).
[Crossref]

Contemp. Phys. (1)

J. H. Eberly, “Shimony-Wolf states and hidden coherences in classical light,” Contemp. Phys. 56, 407–416 (2015).
[Crossref]

Found. Phys. (1)

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
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J. H. Eberly, “Correlation, coherence and context,” Laser Phys. 26, 084004 (2016).
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Nat. Photonics (1)

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

New J. Phys. (3)

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
[Crossref]

H. Chen, T. Peng, S. Karmakar, and Y. Shih, “Simulation of Bell states with incoherent thermal light,” New J. Phys. 13, 083018 (2011).
[Crossref]

Opt. Commun. (1)

A. Luis, “Coherence, polarization, and entanglement for classical fields,” Opt. Commun. 282, 3665–3670 (2009).
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Opt. Express (1)

A. F. Abouraddy, “What is the maximum attainable visibility by a partially coherent electromagnetic field in Young’s double-slit interference?” Opt. Express 15, 18320–18331 (2017).
[Crossref]

Opt. Lett. (4)

Optica (3)

Phys. Rev. A (6)

R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
[Crossref]

M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
[Crossref]

N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
[Crossref]

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (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]

G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
[Crossref]

Phys. Rev. Lett. (4)

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).
[Crossref]

T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
[Crossref]

B.-G. Englert, “Fringe visibility and which-way information: an inequality,” Phys. Rev. Lett. 77, 2154–2157 (1996).
[Crossref]

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Phys. Scripta (1)

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

Sci. Rep. (1)

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and E. A. Saleh, “Optical coherence matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

Z. Naturforsch. (1)

B.-G. Englert, “Remarks on some basic issues in quantum mechanics,” Z. Naturforsch. 54a, 11–32 (1999).

Other (3)

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2007).

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, 1980).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

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Figures (2)

Fig. 1.
Fig. 1. Young-type array. By addressing multiple DoFs, e.g., polarization and path, one may have access to a richer domain of phenomena than in the standard, two-slit Young setup.
Fig. 2.
Fig. 2. Mach–Zehnder-type array that serves to forge one qubit as a two-way superposition, while a second qubit (polarization) can be manipulated with optical elements (e.g., wave plates) that realize unitaries Uj=1,2. The input beam is prepared in a polarization state ρM(0) and acts as a “marker” for the path-qubit, which is prepared in state ρS(0) by means of the BS and the first mirror (M). The phase shifter (PS) allows us to generate interference patterns at the output detectors, in which intensities I(1) and I(2) can be recorded. While unitaries Uj=1,2 act only on the polarization qubit, they are activated by the path qubit.

Equations (54)

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Ic=Ia+Ib+2|ϕa*ϕb|cos[arg(ϕa*ϕb)].
PF=14DetW(F)(TrW(F))2,
VF=IcmaxIcminIcmax+Icmin=2|Wab(F)|Waa(F)+Wbb(F),DF=|IaIb|Ia+Ib=|Waa(F)Wbb(F)|Waa(F)+Wbb(F).
PF2=DF2+VF2.
λ±=12[(ρ11+ρ22)±(ρ11ρ22)2+4|ρ12|2],
P2=(ρ11ρ22)2(ρ11+ρ22)2+4|ρ12|2(ρ11+ρ22)2.
ρ=12k=03Tr(ρ·σk)σk12k=03Skσk.
P2=S12+S22+S32S02=4|ρ12|2+(ρ11ρ22)2(ρ11+ρ22)2.
D2=(S3S0)2,V2=(S1S0)2+(S2S0)2,
P2D2+V2.
ρ(f)=12k=03Sk(f)σk,
Iϕ=Tr(|++|·ρ(f))=12(1S2S0sinϕ+S3S0cosϕ)12(1+Vcos(ϕα)),
V=[(S2S0)2+(S3S0)2]1/2.
Dw=|I+I|=|S1|.
Dw2+V21,
|+3UBS|+1Uϕ12(|+3+eiϕ|3)UBS12(|+1+eiϕ|1)=12[(1+eiϕ)|+3+(1eiϕ)|3]|ψϕ.
Iϕ,3±|ψϕ|±3|2=12(1±cosϕ),Iϕ,1±|ψϕ|±1|2=12.
Iϕ=Tr(|+1+|·ρ(f))=12(1+S1S0cosϕ+S2S0sinϕ)12(1+Vcos(ϕα)),
V=IϕmaxIϕmin=[(S1S0)2+(S2S0)2]1/2,
I(θ,ϵ)=Jxxcos2θ+Jyysin2θ+2JxxJyycosθsinθ|jxy|cos(βxyϵ).
J=(Ex*ExEy*ExEx*EyEy*Ey).
I(θ,ϵ)=Ix+Iy+2IxIy|jxy|cos(βxyϵ),
Imax(ϵ)Imin(ϵ)Imax(ϵ)+Imin(ϵ)=2|Jxy|sinθcosθJxxcos2θ+Jyysin2θ.
Vϵ=Imax(ϵ)Imin(ϵ)Imax(ϵ)+Imin(ϵ),Dϵ=|IxIy|Ix+Iy.
Vϵ2+Dϵ2=14(sin2θcos2θ)DetJ(Jxxcos2θ+Jyysin2θ)2.
Jϵ=(Jxxcos2θeiϵJxysinθcosθeiϵJyxsinθcosθJyysin2θ).
14DetJϵ(TrJϵ)2Pϵ2,
Dϵ2+Vϵ2=Pϵ2.
Imax(θ,ϵ)Imin(θ,ϵ)Imax(θ,ϵ)+Imin(θ,ϵ)=14DetJ(Jxx+Jyy)2.
V(θ,ϵ)=P,
G=(GaaHHGaaHVGabHHGabHVGaaVHGaaVVGabVHGabVVGbaHHGbaHVGbbHHGbbHVGbaVHGbaVVGbbHHGbbHV),
GS=TrP(G)=(GaaHH+GaaVVGabHH+GabVVGbaHH+GbaVVGbbHH+GbbVV),GP=TrS(G)=(GaaHH+GbbHHGaaHV+GbbHVGaaVH+GbbVHGaaVV+GbbVV).
(λaλb)2=(GaaGbb)2+4|Gab|2DS2+VS2,
VmaxS=DmaxS=|λaλb|.
VmaxS=λ1+λ2λ3λ4=12(λ3+λ4).
VmaxP=λ1+λ3λ2λ4=12(λ2+λ4).
G=12(λ1+λ200λ1λ20λ3+λ4λ3λ400λ3λ4λ3+λ40λ1λ200λ1+λ2).
G=λ1|ϕ+ϕ+|+λ2|ϕϕ|+λ3|Ψ+Ψ+|+λ4|ΨΨ|.
U=|aHϕ+|+|aVϕ|+|bHΨ+|+|bVΨ|.
GD=UGU=λ1|aHaH|+λ2|aVaV|+λ3|bHbH|+λ4|bVbV|.
ρS(0)=|ψψ|=|α|2σσ+|β|2σσ+αβ*σ+α*βσ,
USM=σσU1+σσU2eiϕ.
ρSM=USM(ρS(0)ρM(0))USM=|α|2σσρM(1)+|β|2σσρM(2)+α*βσeiϕρ˜M+β*ασeiϕρ˜M.
ρS=TrM(ρSM)=|α|2σσ+|β|2σσ+α*βeiϕCσ+αβ*eiϕC*σ,
ρSρSout=UBSρSUBS.
I(1)=Tr(σσρSout)=12[|α|2+|β|2+2R(αβ*eiϕC)],
ρM=TrS(ρSM)=|α|2ρM(1)+|β|2ρM(2).
V=|TrMρ˜M|,D=12Tr|ρM(1)ρM(2)|,
ρM(0)=12(1+S·σ),
D12=12Tr|ρ1ρ2|=12|S1S2|.
(R11R2)S=(e02e2)S+2(e·S)e+2e0(S×e).
D2=e2S2(e·S)2.
V=|TrM(U1U2ρM(0))|=12|Tr[(cos(γ2)1+isin(γ2)n^·σ)(1+S^·σ)]|=|cos(γ2)+isin(γ2)n^·S|=|e0+ie·S|.
D2+V2=e02+e2S2=cos2(γ2)+P2sin2(γ2).

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