Abstract

Fundamentally contradictory but inescapably joined dual attributes, wave and particle, remain a conceptually unsettling element at the heart of quantum mechanics. It was a career-long unanswered challenge for Bohr to rationalize quantum duality. The conceptual dilemma it presents has remained an open issue, a topic of continued discussion, ever since. Here we report the discovery of an experimentally manageable route to control the weirdness of duality. Ironically, entanglement, the other conceptually challenging weirdness of quantum theory, will be shown to be in control of duality. We establish a simple identity through which entanglement prescribes quantitatively the degree of duality, of combined waveness and particleness, that can be recorded in any one-quantum two-path coherence experiment.

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

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    [Crossref]
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  8. 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).
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  9. F. De Zela, “Relationship between the degree of polarization, indistinguishability, and entanglement,” Phys. Rev. A 89, 013845 (2014).
    [Crossref]
  10. J. H. Eberly, “Shimony-Wolf states and hidden coherences in classical light,” Contem. Phys. 56, 407–416 (2015).
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  11. J. Svozilík, A. Vallés, J. Peřina, and J. P. Torres, “Revealing hidden coherence in partially coherent light,” Phys. Rev. Lett. 115, 220501 (2015).
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  12. W. F. Balthazar, C. E. R. Souza, D. P. Caetano, E. F. Galvão, J. A. O. Huguenin, and A. Z. Khoury, “Tripartite nonseparability in classical optics,” Opt. Lett. 41, 5797–5800 (2016).
    [Crossref]
  13. X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (2016).
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  19. B.-G. Englert, M. O. Scully, and H. Walther, “The duality in matter and light,” Sci. Am. 271, 86–92 (1994).
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  20. G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
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  21. B.-G. Englert, “Fringe visibility and which-way information: an inequality,” Phys. Rev. Lett. 77, 2154–2157 (1996).
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  22. B.-G. Englert, M. O. Scully, and H. Walther, “Complementarity and uncertainty,” Nature 375, 367–368 (1995).
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  23. S. Dürr, T. Nonn, and G. Rempe, “Origin of quantum-mechanical complementarity probed by a ‘which-way’ experiment in an atom interferometer,” Nature 395, 33–37 (1998).
    [Crossref]
  24. S. Dürr, T. Nonn, and G. Rempe, “Fringe visibility and which-way information in an atom interferometer,” Phys. Rev. Lett. 81, 5705–5709 (1998).
    [Crossref]
  25. B.-G. Englert, M. O. Scully, and H. Walther, “On mechanisms that enforce complementarity,” J. Mod. Opt. 47, 2213–2220 (2000).
    [Crossref]
  26. F. de Melo, S. P. Walborn, J. A. Bergou, and L. Davidovich, “Quantum nondemolition circuit for testing bipartite complementarity,” Phys. Rev. Lett. 98, 250501 (2007).
    [Crossref]
  27. M. Jakob and J. A. Bergou, “Quantitative complementarity relations in bipartite systems: entanglement as a physical reality,” Opt. Commun. 283, 827–830 (2010).
    [Crossref]
  28. M. Lahiri, “Wave-particle duality and polarization properties of light in single-photon interference experiments,” Phys. Rev. A 83, 045803 (2011).
    [Crossref]
  29. R. Menzel, D. Puhlmann, A. Heuer, and W. P. Schleich, “Wave-particle dualism and complementarity unraveled by a different mode,” Proc. Natl. Acad. Sci. USA 109, 9314–9319 (2012).
    [Crossref]
  30. H.-Y. Liu, J.-H. Huang, J.-R. Gao, M. S. Zubairy, and S.-Y. Zhu, “Relation between wave-particle duality and quantum uncertainty,” Phys. Rev. A 85, 022106 (2012).
    [Crossref]
  31. A. Norrman, K. Blomstedt, T. Setälä, and A. Friberg, “Complementarity and polarization modulation in photon interference,” Phys. Rev. Lett. 119, 040401 (2017).
    [Crossref]
  32. W. K. Wootters and W. H. Zurek, “Complementarity in the double-slit experiment: quantum nonseparability and a quantitative statement of Bohr’s principle,” Phys. Rev. D 19, 473–484 (1979).
    [Crossref]
  33. The formulation we have followed is closely modeled on Chap. 3 of Theory of Coherence and Polarization of Light, by E. Wolf (Cambridge University, 2007).
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    [Crossref]
  35. R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998), for a very early recognition that entanglement (nonseparability of superpositions of tensor products) also belongs to classical wave theory as well as to quantum physics.
    [Crossref]
  36. P. L. Knight, “Quantum mechanics: where the weirdness comes from,” Nature 395, 12–13 (1998).
    [Crossref]
  37. 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]
  38. X.-F. Qian, A. N. Vamivakas, and J. H. Eberly, “Emerging connections, classical and quantum optics,” Opt. Photon. News 28(10), 34–41 (2017).
    [Crossref]
  39. X.-F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011).
    [Crossref]
  40. 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]
  41. T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
    [Crossref]
  42. A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, “Measuring quantum coherence with entanglement,” Phys. Rev. Lett. 115, 020403 (2015).
    [Crossref]
  43. A. Streltsov, G. Adesso, and M. B. Plenio, “Colloquium: quantum coherence as a resource,” Rev. Mod. Phys. 89, 041003 (2017).
    [Crossref]
  44. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998). Concurrence is designed for two-qubit entanglement, and so is particularly appropriate in qubit-analog contexts such as two-slit interference configurations. It quantifies entanglement (inseparability) and is strictly bounded, 0 ≤ C ≤ 1.
    [Crossref]
  45. F. De Zela, “Optical approach to concurrence and polarization,” Opt. Lett. 43, 2603–2606 (2018).
    [Crossref]
  46. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “On the measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
    [Crossref]
  47. A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
    [Crossref]
  48. A. Luis, “Coherence, polarization and entanglement for classical light fields,” Opt. Commun. 282, 3665–3670 (2009).
    [Crossref]
  49. M. A. Alonso, X.-F. Qian, and J. H. Eberly, “Center-of-mass interpretation for bipartite purity analysis of N-party entanglement,” Phys. Rev. A 94, 030303 (2016).
    [Crossref]
  50. J. P. Torres, X.-F. Qian, S. K. Manikandan, A. N. Vamivakas, and J. H. Eberly, “Distinguishability, visibility and quantum entanglement,” (in preparation).

2018 (2)

2017 (4)

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

A. Norrman, K. Blomstedt, T. Setälä, and A. Friberg, “Complementarity and polarization modulation in photon interference,” Phys. Rev. Lett. 119, 040401 (2017).
[Crossref]

X.-F. Qian, A. N. Vamivakas, and J. H. Eberly, “Emerging connections, classical and quantum optics,” Opt. Photon. News 28(10), 34–41 (2017).
[Crossref]

A. Streltsov, G. Adesso, and M. B. Plenio, “Colloquium: quantum coherence as a resource,” Rev. Mod. Phys. 89, 041003 (2017).
[Crossref]

2016 (5)

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]

M. A. Alonso, X.-F. Qian, and J. H. Eberly, “Center-of-mass interpretation for bipartite purity analysis of N-party entanglement,” Phys. Rev. A 94, 030303 (2016).
[Crossref]

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

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (2016).
[Crossref]

W. F. Balthazar, C. E. R. Souza, D. P. Caetano, E. F. Galvão, J. A. O. Huguenin, and A. Z. Khoury, “Tripartite nonseparability in classical optics,” Opt. Lett. 41, 5797–5800 (2016).
[Crossref]

2015 (4)

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,” Contem. Phys. 56, 407–416 (2015).
[Crossref]

J. Svozilík, A. Vallés, J. Peřina, and J. P. Torres, “Revealing hidden coherence in partially coherent light,” Phys. Rev. Lett. 115, 220501 (2015).
[Crossref]

A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, “Measuring quantum coherence with entanglement,” Phys. Rev. Lett. 115, 020403 (2015).
[Crossref]

2014 (3)

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

F. De Zela, “Relationship between the degree of polarization, indistinguishability, and entanglement,” Phys. Rev. A 89, 013845 (2014).
[Crossref]

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

2012 (2)

R. Menzel, D. Puhlmann, A. Heuer, and W. P. Schleich, “Wave-particle dualism and complementarity unraveled by a different mode,” Proc. Natl. Acad. Sci. USA 109, 9314–9319 (2012).
[Crossref]

H.-Y. Liu, J.-H. Huang, J.-R. Gao, M. S. Zubairy, and S.-Y. Zhu, “Relation between wave-particle duality and quantum uncertainty,” Phys. Rev. A 85, 022106 (2012).
[Crossref]

2011 (2)

M. Lahiri, “Wave-particle duality and polarization properties of light in single-photon interference experiments,” Phys. Rev. A 83, 045803 (2011).
[Crossref]

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

2010 (1)

M. Jakob and J. A. Bergou, “Quantitative complementarity relations in bipartite systems: entanglement as a physical reality,” Opt. Commun. 283, 827–830 (2010).
[Crossref]

2009 (1)

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

2007 (1)

F. de Melo, S. P. Walborn, J. A. Bergou, and L. Davidovich, “Quantum nondemolition circuit for testing bipartite complementarity,” Phys. Rev. Lett. 98, 250501 (2007).
[Crossref]

2006 (1)

2001 (1)

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “On the measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

2000 (1)

B.-G. Englert, M. O. Scully, and H. Walther, “On mechanisms that enforce complementarity,” J. Mod. Opt. 47, 2213–2220 (2000).
[Crossref]

1998 (5)

S. Dürr, T. Nonn, and G. Rempe, “Origin of quantum-mechanical complementarity probed by a ‘which-way’ experiment in an atom interferometer,” Nature 395, 33–37 (1998).
[Crossref]

S. Dürr, T. Nonn, and G. Rempe, “Fringe visibility and which-way information in an atom interferometer,” Phys. Rev. Lett. 81, 5705–5709 (1998).
[Crossref]

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998). Concurrence is designed for two-qubit entanglement, and so is particularly appropriate in qubit-analog contexts such as two-slit interference configurations. It quantifies entanglement (inseparability) and is strictly bounded, 0 ≤ C ≤ 1.
[Crossref]

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998), for a very early recognition that entanglement (nonseparability of superpositions of tensor products) also belongs to classical wave theory as well as to quantum physics.
[Crossref]

P. L. Knight, “Quantum mechanics: where the weirdness comes from,” Nature 395, 12–13 (1998).
[Crossref]

1996 (1)

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

1995 (2)

B.-G. Englert, M. O. Scully, and H. Walther, “Complementarity and uncertainty,” Nature 375, 367–368 (1995).
[Crossref]

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

1994 (1)

B.-G. Englert, M. O. Scully, and H. Walther, “The duality in matter and light,” Sci. Am. 271, 86–92 (1994).
[Crossref]

1993 (1)

G. Jaeger, M. A. Horne, and A. Shimony, “Complementarity of one-particle and two-particle interference,” Phys. Rev. A 48, 1023–1027 (1993).
[Crossref]

1991 (1)

1979 (1)

W. K. Wootters and W. H. Zurek, “Complementarity in the double-slit experiment: quantum nonseparability and a quantitative statement of Bohr’s principle,” Phys. Rev. D 19, 473–484 (1979).
[Crossref]

1925 (1)

L. de Broglie, “Recherches sur la théorie des quanta,” Ann. Phys. 10, 22–128 (1925). English translation by A. F. Kracklauer, “On the Theory of Quanta,” available from aflb.ensmp.fr/LDB-oeuvres/De_Broglie_Kracklauer.pdf.
[Crossref]

1802 (1)

Th. Young, “An account of some cases of the production of colours, not hitherto described,” Philos. Trans. R. Soc. London 92, 387–397 (1802).
[Crossref]

Abouraddy, A. F.

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

Adesso, G.

A. Streltsov, G. Adesso, and M. B. Plenio, “Colloquium: quantum coherence as a resource,” Rev. Mod. Phys. 89, 041003 (2017).
[Crossref]

A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, “Measuring quantum coherence with entanglement,” Phys. Rev. Lett. 115, 020403 (2015).
[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]

M. A. Alonso, X.-F. Qian, and J. H. Eberly, “Center-of-mass interpretation for bipartite purity analysis of N-party entanglement,” Phys. Rev. A 94, 030303 (2016).
[Crossref]

Balthazar, W. F.

Baumgratz, T.

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

Bera, M. N.

A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, “Measuring quantum coherence with entanglement,” Phys. Rev. Lett. 115, 020403 (2015).
[Crossref]

Bergou, J. A.

M. Jakob and J. A. Bergou, “Quantitative complementarity relations in bipartite systems: entanglement as a physical reality,” Opt. Commun. 283, 827–830 (2010).
[Crossref]

F. de Melo, S. P. Walborn, J. A. Bergou, and L. Davidovich, “Quantum nondemolition circuit for testing bipartite complementarity,” Phys. Rev. Lett. 98, 250501 (2007).
[Crossref]

Blomstedt, K.

A. Norrman, K. Blomstedt, T. Setälä, and A. Friberg, “Complementarity and polarization modulation in photon interference,” Phys. Rev. Lett. 119, 040401 (2017).
[Crossref]

Borghi, R.

Caetano, D. P.

Cramer, M.

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

Davidovich, L.

F. de Melo, S. P. Walborn, J. A. Bergou, and L. Davidovich, “Quantum nondemolition circuit for testing bipartite complementarity,” Phys. Rev. Lett. 98, 250501 (2007).
[Crossref]

de Broglie, L.

L. de Broglie, “Recherches sur la théorie des quanta,” Ann. Phys. 10, 22–128 (1925). English translation by A. F. Kracklauer, “On the Theory of Quanta,” available from aflb.ensmp.fr/LDB-oeuvres/De_Broglie_Kracklauer.pdf.
[Crossref]

de Melo, F.

F. de Melo, S. P. Walborn, J. A. Bergou, and L. Davidovich, “Quantum nondemolition circuit for testing bipartite complementarity,” Phys. Rev. Lett. 98, 250501 (2007).
[Crossref]

De Zela, F.

Dhar, H. S.

A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, “Measuring quantum coherence with entanglement,” Phys. Rev. Lett. 115, 020403 (2015).
[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]

Dürr, S.

S. Dürr, T. Nonn, and G. Rempe, “Origin of quantum-mechanical complementarity probed by a ‘which-way’ experiment in an atom interferometer,” Nature 395, 33–37 (1998).
[Crossref]

S. Dürr, T. Nonn, and G. Rempe, “Fringe visibility and which-way information in an atom interferometer,” Phys. Rev. Lett. 81, 5705–5709 (1998).
[Crossref]

Eberly, J. H.

X.-F. Qian, A. N. Vamivakas, and J. H. Eberly, “Emerging connections, classical and quantum optics,” Opt. Photon. News 28(10), 34–41 (2017).
[Crossref]

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

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (2016).
[Crossref]

M. A. Alonso, X.-F. Qian, and J. H. Eberly, “Center-of-mass interpretation for bipartite purity analysis of N-party entanglement,” Phys. Rev. A 94, 030303 (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]

J. H. Eberly, “Shimony-Wolf states and hidden coherences in classical light,” Contem. 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]

J. P. Torres, X.-F. Qian, S. K. Manikandan, A. N. Vamivakas, and J. H. Eberly, “Distinguishability, visibility and quantum entanglement,” (in preparation).

Englert, B.-G.

B.-G. Englert, M. O. Scully, and H. Walther, “On mechanisms that enforce complementarity,” J. Mod. Opt. 47, 2213–2220 (2000).
[Crossref]

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

B.-G. Englert, M. O. Scully, and H. Walther, “Complementarity and uncertainty,” Nature 375, 367–368 (1995).
[Crossref]

B.-G. Englert, M. O. Scully, and H. Walther, “The duality in matter and light,” Sci. Am. 271, 86–92 (1994).
[Crossref]

Friberg, A.

A. Norrman, K. Blomstedt, T. Setälä, and A. Friberg, “Complementarity and polarization modulation in photon interference,” Phys. Rev. Lett. 119, 040401 (2017).
[Crossref]

Galvão, E. F.

Gao, J.-R.

H.-Y. Liu, J.-H. Huang, J.-R. Gao, M. S. Zubairy, and S.-Y. Zhu, “Relation between wave-particle duality and quantum uncertainty,” Phys. Rev. A 85, 022106 (2012).
[Crossref]

Gori, F.

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]

Heuer, A.

R. Menzel, D. Puhlmann, A. Heuer, and W. P. Schleich, “Wave-particle dualism and complementarity unraveled by a different mode,” Proc. Natl. Acad. Sci. USA 109, 9314–9319 (2012).
[Crossref]

Horne, M. A.

G. Jaeger, M. A. Horne, and A. Shimony, “Complementarity of one-particle and two-particle interference,” Phys. Rev. A 48, 1023–1027 (1993).
[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]

Huang, J.-H.

H.-Y. Liu, J.-H. Huang, J.-R. Gao, M. S. Zubairy, and S.-Y. Zhu, “Relation between wave-particle duality and quantum uncertainty,” Phys. Rev. A 85, 022106 (2012).
[Crossref]

Huguenin, J. A. O.

Jaeger, G.

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

G. Jaeger, M. A. Horne, and A. Shimony, “Complementarity of one-particle and two-particle interference,” Phys. Rev. A 48, 1023–1027 (1993).
[Crossref]

Jakob, M.

M. Jakob and J. A. Bergou, “Quantitative complementarity relations in bipartite systems: entanglement as a physical reality,” Opt. Commun. 283, 827–830 (2010).
[Crossref]

James, D. F. V.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “On the measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Kagalwala, K. H.

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

Khoury, A. Z.

Knight, P. L.

P. L. Knight, “Quantum mechanics: where the weirdness comes from,” Nature 395, 12–13 (1998).
[Crossref]

Kwiat, P. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “On the measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Lahiri, M.

M. Lahiri, “Wave-particle duality and polarization properties of light in single-photon interference experiments,” Phys. Rev. A 83, 045803 (2011).
[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]

Liu, H.-Y.

H.-Y. Liu, J.-H. Huang, J.-R. Gao, M. S. Zubairy, and S.-Y. Zhu, “Relation between wave-particle duality and quantum uncertainty,” Phys. Rev. A 85, 022106 (2012).
[Crossref]

Luis, A.

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

Malhotra, T.

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (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]

Mandel, L.

Manikandan, S. K.

J. P. Torres, X.-F. Qian, S. K. Manikandan, A. N. Vamivakas, and J. H. Eberly, “Distinguishability, visibility and quantum entanglement,” (in preparation).

Menzel, R.

R. Menzel, D. Puhlmann, A. Heuer, and W. P. Schleich, “Wave-particle dualism and complementarity unraveled by a different mode,” Proc. Natl. Acad. Sci. USA 109, 9314–9319 (2012).
[Crossref]

Munro, W. J.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “On the measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Murdoch, D.

D. Murdoch, Niels Bohr’s Philosophy of Physics (Cambridge University, 1987).

Nonn, T.

S. Dürr, T. Nonn, and G. Rempe, “Origin of quantum-mechanical complementarity probed by a ‘which-way’ experiment in an atom interferometer,” Nature 395, 33–37 (1998).
[Crossref]

S. Dürr, T. Nonn, and G. Rempe, “Fringe visibility and which-way information in an atom interferometer,” Phys. Rev. Lett. 81, 5705–5709 (1998).
[Crossref]

Norrman, A.

A. Norrman, K. Blomstedt, T. Setälä, and A. Friberg, “Complementarity and polarization modulation in photon interference,” Phys. Rev. Lett. 119, 040401 (2017).
[Crossref]

Perina, J.

J. Svozilík, A. Vallés, J. Peřina, and J. P. Torres, “Revealing hidden coherence in partially coherent light,” Phys. Rev. Lett. 115, 220501 (2015).
[Crossref]

Plenio, M. B.

A. Streltsov, G. Adesso, and M. B. Plenio, “Colloquium: quantum coherence as a resource,” Rev. Mod. Phys. 89, 041003 (2017).
[Crossref]

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

Puhlmann, D.

R. Menzel, D. Puhlmann, A. Heuer, and W. P. Schleich, “Wave-particle dualism and complementarity unraveled by a different mode,” Proc. Natl. Acad. Sci. USA 109, 9314–9319 (2012).
[Crossref]

Qian, X.-F.

X.-F. Qian, A. N. Vamivakas, and J. H. Eberly, “Emerging connections, classical and quantum optics,” Opt. Photon. News 28(10), 34–41 (2017).
[Crossref]

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

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (2016).
[Crossref]

M. A. Alonso, X.-F. Qian, and J. H. Eberly, “Center-of-mass interpretation for bipartite purity analysis of N-party entanglement,” Phys. Rev. A 94, 030303 (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]

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]

J. P. Torres, X.-F. Qian, S. K. Manikandan, A. N. Vamivakas, and J. H. Eberly, “Distinguishability, visibility and quantum entanglement,” (in preparation).

Rempe, G.

S. Dürr, T. Nonn, and G. Rempe, “Fringe visibility and which-way information in an atom interferometer,” Phys. Rev. Lett. 81, 5705–5709 (1998).
[Crossref]

S. Dürr, T. Nonn, and G. Rempe, “Origin of quantum-mechanical complementarity probed by a ‘which-way’ experiment in an atom interferometer,” Nature 395, 33–37 (1998).
[Crossref]

Saleh, B. E. A.

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

Santarsiero, M.

Schleich, W. P.

R. Menzel, D. Puhlmann, A. Heuer, and W. P. Schleich, “Wave-particle dualism and complementarity unraveled by a different mode,” Proc. Natl. Acad. Sci. USA 109, 9314–9319 (2012).
[Crossref]

Scully, M. O.

B.-G. Englert, M. O. Scully, and H. Walther, “On mechanisms that enforce complementarity,” J. Mod. Opt. 47, 2213–2220 (2000).
[Crossref]

B.-G. Englert, M. O. Scully, and H. Walther, “Complementarity and uncertainty,” Nature 375, 367–368 (1995).
[Crossref]

B.-G. Englert, M. O. Scully, and H. Walther, “The duality in matter and light,” Sci. Am. 271, 86–92 (1994).
[Crossref]

Setälä, T.

A. Norrman, K. Blomstedt, T. Setälä, and A. Friberg, “Complementarity and polarization modulation in photon interference,” Phys. Rev. Lett. 119, 040401 (2017).
[Crossref]

Shimony, A.

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

G. Jaeger, M. A. Horne, and A. Shimony, “Complementarity of one-particle and two-particle interference,” Phys. Rev. A 48, 1023–1027 (1993).
[Crossref]

Singh, U.

A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, “Measuring quantum coherence with entanglement,” Phys. Rev. Lett. 115, 020403 (2015).
[Crossref]

Souza, C. E. R.

Spreeuw, R. J. C.

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998), for a very early recognition that entanglement (nonseparability of superpositions of tensor products) also belongs to classical wave theory as well as to quantum physics.
[Crossref]

Streltsov, A.

A. Streltsov, G. Adesso, and M. B. Plenio, “Colloquium: quantum coherence as a resource,” Rev. Mod. Phys. 89, 041003 (2017).
[Crossref]

A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, “Measuring quantum coherence with entanglement,” Phys. Rev. Lett. 115, 020403 (2015).
[Crossref]

Svozilík, J.

J. Svozilík, A. Vallés, J. Peřina, and J. P. Torres, “Revealing hidden coherence in partially coherent light,” Phys. Rev. Lett. 115, 220501 (2015).
[Crossref]

Torres, J. P.

J. Svozilík, A. Vallés, J. Peřina, and J. P. Torres, “Revealing hidden coherence in partially coherent light,” Phys. Rev. Lett. 115, 220501 (2015).
[Crossref]

J. P. Torres, X.-F. Qian, S. K. Manikandan, A. N. Vamivakas, and J. H. Eberly, “Distinguishability, visibility and quantum entanglement,” (in preparation).

Vaidman, L.

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

Vallés, A.

J. Svozilík, A. Vallés, J. Peřina, and J. P. Torres, “Revealing hidden coherence in partially coherent light,” Phys. Rev. Lett. 115, 220501 (2015).
[Crossref]

Vamivakas, A. N.

X.-F. Qian, A. N. Vamivakas, and J. H. Eberly, “Emerging connections, classical and quantum optics,” Opt. Photon. News 28(10), 34–41 (2017).
[Crossref]

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

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (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. P. Torres, X.-F. Qian, S. K. Manikandan, A. N. Vamivakas, and J. H. Eberly, “Distinguishability, visibility and quantum entanglement,” (in preparation).

Walborn, S. P.

F. de Melo, S. P. Walborn, J. A. Bergou, and L. Davidovich, “Quantum nondemolition circuit for testing bipartite complementarity,” Phys. Rev. Lett. 98, 250501 (2007).
[Crossref]

Walther, H.

B.-G. Englert, M. O. Scully, and H. Walther, “On mechanisms that enforce complementarity,” J. Mod. Opt. 47, 2213–2220 (2000).
[Crossref]

B.-G. Englert, M. O. Scully, and H. Walther, “Complementarity and uncertainty,” Nature 375, 367–368 (1995).
[Crossref]

B.-G. Englert, M. O. Scully, and H. Walther, “The duality in matter and light,” Sci. Am. 271, 86–92 (1994).
[Crossref]

Whitaker, A.

A. Whitaker, Einstein, Bohr and the Quantum Dilemma, 2nd ed. (Cambridge University, 2006).

White, A. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “On the measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

Wolf, E.

The formulation we have followed is closely modeled on Chap. 3 of Theory of Coherence and Polarization of Light, by E. Wolf (Cambridge University, 2007).

Wootters, W. K.

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998). Concurrence is designed for two-qubit entanglement, and so is particularly appropriate in qubit-analog contexts such as two-slit interference configurations. It quantifies entanglement (inseparability) and is strictly bounded, 0 ≤ C ≤ 1.
[Crossref]

W. K. Wootters and W. H. Zurek, “Complementarity in the double-slit experiment: quantum nonseparability and a quantitative statement of Bohr’s principle,” Phys. Rev. D 19, 473–484 (1979).
[Crossref]

Young, Th.

Th. Young, “An account of some cases of the production of colours, not hitherto described,” Philos. Trans. R. Soc. London 92, 387–397 (1802).
[Crossref]

Zhu, S.-Y.

H.-Y. Liu, J.-H. Huang, J.-R. Gao, M. S. Zubairy, and S.-Y. Zhu, “Relation between wave-particle duality and quantum uncertainty,” Phys. Rev. A 85, 022106 (2012).
[Crossref]

Zubairy, M. S.

H.-Y. Liu, J.-H. Huang, J.-R. Gao, M. S. Zubairy, and S.-Y. Zhu, “Relation between wave-particle duality and quantum uncertainty,” Phys. Rev. A 85, 022106 (2012).
[Crossref]

Zurek, W. H.

W. K. Wootters and W. H. Zurek, “Complementarity in the double-slit experiment: quantum nonseparability and a quantitative statement of Bohr’s principle,” Phys. Rev. D 19, 473–484 (1979).
[Crossref]

Ann. Phys. (1)

L. de Broglie, “Recherches sur la théorie des quanta,” Ann. Phys. 10, 22–128 (1925). English translation by A. F. Kracklauer, “On the Theory of Quanta,” available from aflb.ensmp.fr/LDB-oeuvres/De_Broglie_Kracklauer.pdf.
[Crossref]

Contem. Phys. (1)

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

Found. Phys. (1)

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998), for a very early recognition that entanglement (nonseparability of superpositions of tensor products) also belongs to classical wave theory as well as to quantum physics.
[Crossref]

J. Mod. Opt. (1)

B.-G. Englert, M. O. Scully, and H. Walther, “On mechanisms that enforce complementarity,” J. Mod. Opt. 47, 2213–2220 (2000).
[Crossref]

Laser Phys. (1)

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

Nature (3)

B.-G. Englert, M. O. Scully, and H. Walther, “Complementarity and uncertainty,” Nature 375, 367–368 (1995).
[Crossref]

S. Dürr, T. Nonn, and G. Rempe, “Origin of quantum-mechanical complementarity probed by a ‘which-way’ experiment in an atom interferometer,” Nature 395, 33–37 (1998).
[Crossref]

P. L. Knight, “Quantum mechanics: where the weirdness comes from,” Nature 395, 12–13 (1998).
[Crossref]

Opt. Commun. (2)

M. Jakob and J. A. Bergou, “Quantitative complementarity relations in bipartite systems: entanglement as a physical reality,” Opt. Commun. 283, 827–830 (2010).
[Crossref]

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

Opt. Lett. (6)

Opt. Photon. News (1)

X.-F. Qian, A. N. Vamivakas, and J. H. Eberly, “Emerging connections, classical and quantum optics,” Opt. Photon. News 28(10), 34–41 (2017).
[Crossref]

Optica (3)

Philos. Trans. R. Soc. London (1)

Th. Young, “An account of some cases of the production of colours, not hitherto described,” Philos. Trans. R. Soc. London 92, 387–397 (1802).
[Crossref]

Phys. Rev. A (7)

F. De Zela, “Relationship between the degree of polarization, indistinguishability, and entanglement,” Phys. Rev. A 89, 013845 (2014).
[Crossref]

G. Jaeger, M. A. Horne, and A. Shimony, “Complementarity of one-particle and two-particle interference,” Phys. Rev. A 48, 1023–1027 (1993).
[Crossref]

H.-Y. Liu, J.-H. Huang, J.-R. Gao, M. S. Zubairy, and S.-Y. Zhu, “Relation between wave-particle duality and quantum uncertainty,” Phys. Rev. A 85, 022106 (2012).
[Crossref]

M. Lahiri, “Wave-particle duality and polarization properties of light in single-photon interference experiments,” Phys. Rev. A 83, 045803 (2011).
[Crossref]

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

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “On the measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[Crossref]

M. A. Alonso, X.-F. Qian, and J. H. Eberly, “Center-of-mass interpretation for bipartite purity analysis of N-party entanglement,” Phys. Rev. A 94, 030303 (2016).
[Crossref]

Phys. Rev. D (1)

W. K. Wootters and W. H. Zurek, “Complementarity in the double-slit experiment: quantum nonseparability and a quantitative statement of Bohr’s principle,” Phys. Rev. D 19, 473–484 (1979).
[Crossref]

Phys. Rev. Lett. (9)

A. Norrman, K. Blomstedt, T. Setälä, and A. Friberg, “Complementarity and polarization modulation in photon interference,” Phys. Rev. Lett. 119, 040401 (2017).
[Crossref]

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

S. Dürr, T. Nonn, and G. Rempe, “Fringe visibility and which-way information in an atom interferometer,” Phys. Rev. Lett. 81, 5705–5709 (1998).
[Crossref]

F. de Melo, S. P. Walborn, J. A. Bergou, and L. Davidovich, “Quantum nondemolition circuit for testing bipartite complementarity,” Phys. Rev. Lett. 98, 250501 (2007).
[Crossref]

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (2016).
[Crossref]

J. Svozilík, A. Vallés, J. Peřina, and J. P. Torres, “Revealing hidden coherence in partially coherent light,” Phys. Rev. Lett. 115, 220501 (2015).
[Crossref]

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998). Concurrence is designed for two-qubit entanglement, and so is particularly appropriate in qubit-analog contexts such as two-slit interference configurations. It quantifies entanglement (inseparability) and is strictly bounded, 0 ≤ C ≤ 1.
[Crossref]

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

A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, “Measuring quantum coherence with entanglement,” Phys. Rev. Lett. 115, 020403 (2015).
[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]

Proc. Natl. Acad. Sci. USA (1)

R. Menzel, D. Puhlmann, A. Heuer, and W. P. Schleich, “Wave-particle dualism and complementarity unraveled by a different mode,” Proc. Natl. Acad. Sci. USA 109, 9314–9319 (2012).
[Crossref]

Rev. Mod. Phys. (1)

A. Streltsov, G. Adesso, and M. B. Plenio, “Colloquium: quantum coherence as a resource,” Rev. Mod. Phys. 89, 041003 (2017).
[Crossref]

Sci. Am. (1)

B.-G. Englert, M. O. Scully, and H. Walther, “The duality in matter and light,” Sci. Am. 271, 86–92 (1994).
[Crossref]

Other (7)

P. A. Schilpp, ed., Albert Einstein: Philosopher-Scientist (Library of the Living Philosophers, 1949).

Bohr’s multiply revised and delayed report of his address to an international Physics Congress at Como in 1927 appeared as “The Quantum Postulate and the Recent Development of Atomic Theory,” Nature121, 580–590 (1928). See also Bohr’s book Atomic Theory and the Description of Nature (Cambridge University, 1934), p. 10.

Bohr was not the only one confused or frustrated. This is made clear in Jammer’s account of discussions following Bohr’s address at the Como conference, where he introduced complementarity to attendees including Born, Fermi, Heisenberg, Kramers, Pauli, Rosenfeld, von Neumann, and Wigner. See an overview by M. Jammer in The Conceptual Development of Quantum Mechanics (McGraw-Hill, 1966), p. 354. See also M. Jammer, The Philosophy of Quantum Mechanics (Wiley, 1974).

A. Whitaker, Einstein, Bohr and the Quantum Dilemma, 2nd ed. (Cambridge University, 2006).

D. Murdoch, Niels Bohr’s Philosophy of Physics (Cambridge University, 1987).

The formulation we have followed is closely modeled on Chap. 3 of Theory of Coherence and Polarization of Light, by E. Wolf (Cambridge University, 2007).

J. P. Torres, X.-F. Qian, S. K. Manikandan, A. N. Vamivakas, and J. H. Eberly, “Distinguishability, visibility and quantum entanglement,” (in preparation).

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Figures (4)

Fig. 1.
Fig. 1. Young two-slit experiment [6] and a range of visibilities. Depending on how the diagonal polarizer engages the polarization degrees of freedom of the two paths, the observed visibility, a proxy for spatial coherence, is suitably modified. This is a simple model for many more elaborate scenarios.
Fig. 2.
Fig. 2. Experimental setup. An analogous version of Fig. 1 with an entanglement tomography setup shown.
Fig. 3.
Fig. 3. Tomographic coherence matrices W of the joint spatial and polarization degrees of freedom for 13 representative beams numbered as in Table 1.
Fig. 4.
Fig. 4. Thirteen points locating the actually measured sets of (V,D,C) are identified by the red dots.

Tables (1)

Tables Icon

Table 1. Measured Values of Visibility, Distinguishability, and Entanglement, where SUM Refers to the Identity Combination V2+D2+C2

Equations (17)

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

V2+D21,
Eac(rc,t)=As^aEa(ra,tta),
Ebc(rc,t)=Bs^bEb(rb,ttb),
Ic(rc,t)=|A|2Ea*(ra,tta)Ea(ra,tta)+|B|2Eb*(rb,ttb)Eb(rb,ttb),+2|AB|Rs^a*Ea*(ra,t)·s^bEb(rb,t+τ),
Γab(ra,rb,τ)s^a*·s^bEa*(ra,t)Eb(rb,t+τ).
Ic=Ia+Ib+2RIaIbγ(ra,rb,τ),
γ(ra,rb,τ)=Γab(ra,rb,τ)Γaa(ra,ra,0)Γbb(rb,rb,0).
V=ImaxIminImax+Imin=2|γ|IaIbIa+Ib.
D=|IaIbIa+Ib|.
V2+D2=14IacIbc(1|γ|2)(Iac+Ibc)21.
P2=V2+D2.
P2=1C2,
V2+D2=1C2.
V2+D2+C2=1.
E(r,z)=As^aua(r,z)+Bs^bub(r,z),
γexp=s^a*·s^b=cosθeiξ.
W=[W1,1W1,2W1,3W1,4W2,1W2,2W2,3W2,4W3,1W3,2W3,3W3,4W4,1W4,2W4,3W4,4].

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