Abstract

Wave-particle duality is a typical example of Bohr’s complementarity principle that plays a significant role in quantum mechanics. Previous studies used the visibility of an interference pattern to quantify the wave property and used path information to quantify the particle property. However, coherence is the core and basis of the interference phenomenon. If we could use coherence to characterize the wave property, the understanding of wave-particle duality would be strengthened. A recent theoretical work [ Phys. Rev. Lett. 116, 160406 (2016)] found two relations between quantum coherence and path information. Here, we demonstrate the new measure of wave-particle duality based on two kinds of coherence measures quantitatively for the first time. The wave property, quantified by the coherence in the l1-norm measure and the relative entropy measure, can be obtained via tomography of the target state, which is encoded in the path degree of freedom of the photons. The particle property, quantified by the path information, can be obtained via the discrimination of detector states, which is encoded in the polarization degree of freedom of the photons. Our work may deepen people’s understanding of coherence and provide a new perspective regarding wave-particle duality.

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

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References

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    [Crossref]
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    [Crossref]
  3. H. Rauch and J. Summhammer, “Static versus time-dependent absorption in neutron interferometry,” Phys. Lett. A 104, 44–46 (1984).
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  4. J. Summhammer, H. Rauch, and D. Tuppinger, “Stochastic and deterministic absorption in neutron-interference experiments,” Phys. Rev. A 36, 4447–4455 (1987).
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  5. E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, “Dephasing in electron interference by a ’which-path’ detector,” Nature 391, 871–874 (1998).
    [Crossref]
  6. 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]
  7. 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]
  8. D. M. Greenberger and A. Yasin, “Simultaneous wave and particle knowledge in a neutron interferometer,” Phys. Lett. A 128, 391–394 (1988).
    [Crossref]
  9. B.-G. Englert, “Fringe Visibility and Which-Way Information: An Inequality,” Phys. Rev. Lett. 77, 2154–2157 (1996).
    [Crossref] [PubMed]
  10. 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]
  11. X. Peng, X. Zhu, X. Fang, M. Feng, M. Liu, and K. Gao, “An interferometric complementarity experiment in a bulk nuclear magnetic resonance ensemble,” J. Phys. A: Math. Gen. 36, 2555–2563 (2003).
    [Crossref]
  12. X. Peng, X. Zhu, D. Suter, J. Du, M. Liu, and K. Gao, “Quantification of complementarity in multiqubit systems,” Phys. Rev. A 72, 052109 (2005).
    [Crossref]
  13. P. D. D. Schwindt, P. G. Kwiat, and B.-G. Englert, “Quantitative wave-particle duality and nonerasing quantum erasure,” Phys. Rev. A 60, 4285–4290 (1999).
    [Crossref]
  14. V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-Choice Test of Quantum Complementarity with Interfering Single Photons,” Phys. Rev. Lett. 100, 220402 (2008).
    [Crossref] [PubMed]
  15. G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
    [Crossref] [PubMed]
  16. S. Dürr, “Quantitative wave-particle duality in multibeam interferometers,” Phys. Rev. A 64, 042113 (2001).
    [Crossref]
  17. G. Bimonte and R. Musto, “On interferometric duality in multibeam experiments,” J. Phys. A: Math. Gen. 36, 11481–11502 (2003).
    [Crossref]
  18. B.-G. Englert, D. Kaszlikowski, L. C. Kwek, and W. H. Chee, “Wave-particle duality in multi-path interferometers: general concepts and three-path interferometers,” Int. J. Quantum. Inform. 6, 129–157 (2008).
    [Crossref]
  19. M. Mei and M. Weitz, “Controlled Decoherence in Multiple Beam Ramsey Interference,” Phys. Rev. Lett. 86, 559–563 (2001).
    [Crossref] [PubMed]
  20. T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying Coherence,” Phys. Rev. Lett. 113, 140401 (2014).
    [Crossref] [PubMed]
  21. A. Winter and D. Yang, “Operational Resource Theory of Coherence,” Phys. Rev. Lett. 116, 120404 (2016).
    [Crossref] [PubMed]
  22. S. Cheng and M. J. Hall, “Complementarity relations for quantum coherence,” Phys. Rev. A 92, 042101 (2015).
    [Crossref]
  23. E. Chitambar, A. Streltsov, S. Rana, M. N. Bera, G. Adesso, and M. Lewenstein, “Assisted Distillation of Quantum Coherence,” Phys. Rev. Lett. 116, 070402 (2016).
    [Crossref] [PubMed]
  24. K.-D. Wu, Z. Hou, H.-S. Zhong, Y. Yuan, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, “Experimentally obtaining maximal coherence via assisted distillation process,” Optica 4, 454–459 (2017).
    [Crossref]
  25. T. R. Bromley, M. Cianciaruso, and G. Adesso, “Frozen Quantum Coherence,” Phys. Rev. Lett. 114, 210401 (2015).
    [Crossref] [PubMed]
  26. X.-D. Yu, D.-J. Zhang, C. L. Liu, and D. M. Tong, “Measure-independent freezing of quantum coherence,” Phys. Rev. A 93, 060303 (2016).
    [Crossref]
  27. M. N. Bera, T. Qureshi, M. A. Siddiqui, and A. K. Pati, “Duality of quantum coherence and path distinguishability,” Phys. Rev. A 92, 012118 (2015).
    [Crossref]
  28. E. Bagan, J.A. Bergou, S.S. Cottrell, and M. Hillery, “Relations between Coherence and Path Information,” Phys. Rev. Lett. 116, 160406 (2016).
    [Crossref] [PubMed]
  29. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).
  30. S. M. Barnett and S. Croke, “Quantum state discrimination,” Adv. Opt. Photonics 1, 238–278 (2009).
    [Crossref]

2017 (1)

2016 (4)

E. Chitambar, A. Streltsov, S. Rana, M. N. Bera, G. Adesso, and M. Lewenstein, “Assisted Distillation of Quantum Coherence,” Phys. Rev. Lett. 116, 070402 (2016).
[Crossref] [PubMed]

A. Winter and D. Yang, “Operational Resource Theory of Coherence,” Phys. Rev. Lett. 116, 120404 (2016).
[Crossref] [PubMed]

X.-D. Yu, D.-J. Zhang, C. L. Liu, and D. M. Tong, “Measure-independent freezing of quantum coherence,” Phys. Rev. A 93, 060303 (2016).
[Crossref]

E. Bagan, J.A. Bergou, S.S. Cottrell, and M. Hillery, “Relations between Coherence and Path Information,” Phys. Rev. Lett. 116, 160406 (2016).
[Crossref] [PubMed]

2015 (3)

M. N. Bera, T. Qureshi, M. A. Siddiqui, and A. K. Pati, “Duality of quantum coherence and path distinguishability,” Phys. Rev. A 92, 012118 (2015).
[Crossref]

S. Cheng and M. J. Hall, “Complementarity relations for quantum coherence,” Phys. Rev. A 92, 042101 (2015).
[Crossref]

T. R. Bromley, M. Cianciaruso, and G. Adesso, “Frozen Quantum Coherence,” Phys. Rev. Lett. 114, 210401 (2015).
[Crossref] [PubMed]

2014 (1)

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

2009 (1)

S. M. Barnett and S. Croke, “Quantum state discrimination,” Adv. Opt. Photonics 1, 238–278 (2009).
[Crossref]

2008 (2)

B.-G. Englert, D. Kaszlikowski, L. C. Kwek, and W. H. Chee, “Wave-particle duality in multi-path interferometers: general concepts and three-path interferometers,” Int. J. Quantum. Inform. 6, 129–157 (2008).
[Crossref]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-Choice Test of Quantum Complementarity with Interfering Single Photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

2005 (1)

X. Peng, X. Zhu, D. Suter, J. Du, M. Liu, and K. Gao, “Quantification of complementarity in multiqubit systems,” Phys. Rev. A 72, 052109 (2005).
[Crossref]

2003 (2)

X. Peng, X. Zhu, X. Fang, M. Feng, M. Liu, and K. Gao, “An interferometric complementarity experiment in a bulk nuclear magnetic resonance ensemble,” J. Phys. A: Math. Gen. 36, 2555–2563 (2003).
[Crossref]

G. Bimonte and R. Musto, “On interferometric duality in multibeam experiments,” J. Phys. A: Math. Gen. 36, 11481–11502 (2003).
[Crossref]

2001 (2)

S. Dürr, “Quantitative wave-particle duality in multibeam interferometers,” Phys. Rev. A 64, 042113 (2001).
[Crossref]

M. Mei and M. Weitz, “Controlled Decoherence in Multiple Beam Ramsey Interference,” Phys. Rev. Lett. 86, 559–563 (2001).
[Crossref] [PubMed]

1999 (1)

P. D. D. Schwindt, P. G. Kwiat, and B.-G. Englert, “Quantitative wave-particle duality and nonerasing quantum erasure,” Phys. Rev. A 60, 4285–4290 (1999).
[Crossref]

1998 (3)

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]

E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, “Dephasing in electron interference by a ’which-path’ detector,” Nature 391, 871–874 (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]

1996 (1)

B.-G. Englert, “Fringe Visibility and Which-Way Information: An Inequality,” Phys. Rev. Lett. 77, 2154–2157 (1996).
[Crossref] [PubMed]

1995 (1)

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

1988 (1)

D. M. Greenberger and A. Yasin, “Simultaneous wave and particle knowledge in a neutron interferometer,” Phys. Lett. A 128, 391–394 (1988).
[Crossref]

1987 (1)

J. Summhammer, H. Rauch, and D. Tuppinger, “Stochastic and deterministic absorption in neutron-interference experiments,” Phys. Rev. A 36, 4447–4455 (1987).
[Crossref]

1984 (1)

H. Rauch and J. Summhammer, “Static versus time-dependent absorption in neutron interferometry,” Phys. Lett. A 104, 44–46 (1984).
[Crossref]

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]

1928 (2)

N. Bohr, “Das Quantenpostulat und die neuere Entwicklung der Atomistik,” Naturwissenschaften 16, 245–257 (1928).
[Crossref]

N. Bohr, “The quantum postulate and the recent development of atomic theory,” Nature 121, 580–590 (1928).
[Crossref]

Adesso, G.

E. Chitambar, A. Streltsov, S. Rana, M. N. Bera, G. Adesso, and M. Lewenstein, “Assisted Distillation of Quantum Coherence,” Phys. Rev. Lett. 116, 070402 (2016).
[Crossref] [PubMed]

T. R. Bromley, M. Cianciaruso, and G. Adesso, “Frozen Quantum Coherence,” Phys. Rev. Lett. 114, 210401 (2015).
[Crossref] [PubMed]

Aspect, A.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-Choice Test of Quantum Complementarity with Interfering Single Photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

Bagan, E.

E. Bagan, J.A. Bergou, S.S. Cottrell, and M. Hillery, “Relations between Coherence and Path Information,” Phys. Rev. Lett. 116, 160406 (2016).
[Crossref] [PubMed]

Barnett, S. M.

S. M. Barnett and S. Croke, “Quantum state discrimination,” Adv. Opt. Photonics 1, 238–278 (2009).
[Crossref]

Baumgratz, T.

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

Bera, M. N.

E. Chitambar, A. Streltsov, S. Rana, M. N. Bera, G. Adesso, and M. Lewenstein, “Assisted Distillation of Quantum Coherence,” Phys. Rev. Lett. 116, 070402 (2016).
[Crossref] [PubMed]

M. N. Bera, T. Qureshi, M. A. Siddiqui, and A. K. Pati, “Duality of quantum coherence and path distinguishability,” Phys. Rev. A 92, 012118 (2015).
[Crossref]

Bergou, J.A.

E. Bagan, J.A. Bergou, S.S. Cottrell, and M. Hillery, “Relations between Coherence and Path Information,” Phys. Rev. Lett. 116, 160406 (2016).
[Crossref] [PubMed]

Bimonte, G.

G. Bimonte and R. Musto, “On interferometric duality in multibeam experiments,” J. Phys. A: Math. Gen. 36, 11481–11502 (2003).
[Crossref]

Bohr, N.

N. Bohr, “The quantum postulate and the recent development of atomic theory,” Nature 121, 580–590 (1928).
[Crossref]

N. Bohr, “Das Quantenpostulat und die neuere Entwicklung der Atomistik,” Naturwissenschaften 16, 245–257 (1928).
[Crossref]

Bromley, T. R.

T. R. Bromley, M. Cianciaruso, and G. Adesso, “Frozen Quantum Coherence,” Phys. Rev. Lett. 114, 210401 (2015).
[Crossref] [PubMed]

Buks, E.

E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, “Dephasing in electron interference by a ’which-path’ detector,” Nature 391, 871–874 (1998).
[Crossref]

Chee, W. H.

B.-G. Englert, D. Kaszlikowski, L. C. Kwek, and W. H. Chee, “Wave-particle duality in multi-path interferometers: general concepts and three-path interferometers,” Int. J. Quantum. Inform. 6, 129–157 (2008).
[Crossref]

Cheng, S.

S. Cheng and M. J. Hall, “Complementarity relations for quantum coherence,” Phys. Rev. A 92, 042101 (2015).
[Crossref]

Chitambar, E.

E. Chitambar, A. Streltsov, S. Rana, M. N. Bera, G. Adesso, and M. Lewenstein, “Assisted Distillation of Quantum Coherence,” Phys. Rev. Lett. 116, 070402 (2016).
[Crossref] [PubMed]

Cianciaruso, M.

T. R. Bromley, M. Cianciaruso, and G. Adesso, “Frozen Quantum Coherence,” Phys. Rev. Lett. 114, 210401 (2015).
[Crossref] [PubMed]

Cottrell, S.S.

E. Bagan, J.A. Bergou, S.S. Cottrell, and M. Hillery, “Relations between Coherence and Path Information,” Phys. Rev. Lett. 116, 160406 (2016).
[Crossref] [PubMed]

Cramer, M.

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

Croke, S.

S. M. Barnett and S. Croke, “Quantum state discrimination,” Adv. Opt. Photonics 1, 238–278 (2009).
[Crossref]

Du, J.

X. Peng, X. Zhu, D. Suter, J. Du, M. Liu, and K. Gao, “Quantification of complementarity in multiqubit systems,” Phys. Rev. A 72, 052109 (2005).
[Crossref]

Dürr, S.

S. Dürr, “Quantitative wave-particle duality in multibeam interferometers,” Phys. Rev. A 64, 042113 (2001).
[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]

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]

Englert, B.-G.

B.-G. Englert, D. Kaszlikowski, L. C. Kwek, and W. H. Chee, “Wave-particle duality in multi-path interferometers: general concepts and three-path interferometers,” Int. J. Quantum. Inform. 6, 129–157 (2008).
[Crossref]

P. D. D. Schwindt, P. G. Kwiat, and B.-G. Englert, “Quantitative wave-particle duality and nonerasing quantum erasure,” Phys. Rev. A 60, 4285–4290 (1999).
[Crossref]

B.-G. Englert, “Fringe Visibility and Which-Way Information: An Inequality,” Phys. Rev. Lett. 77, 2154–2157 (1996).
[Crossref] [PubMed]

Fang, X.

X. Peng, X. Zhu, X. Fang, M. Feng, M. Liu, and K. Gao, “An interferometric complementarity experiment in a bulk nuclear magnetic resonance ensemble,” J. Phys. A: Math. Gen. 36, 2555–2563 (2003).
[Crossref]

Feng, M.

X. Peng, X. Zhu, X. Fang, M. Feng, M. Liu, and K. Gao, “An interferometric complementarity experiment in a bulk nuclear magnetic resonance ensemble,” J. Phys. A: Math. Gen. 36, 2555–2563 (2003).
[Crossref]

Gao, K.

X. Peng, X. Zhu, D. Suter, J. Du, M. Liu, and K. Gao, “Quantification of complementarity in multiqubit systems,” Phys. Rev. A 72, 052109 (2005).
[Crossref]

X. Peng, X. Zhu, X. Fang, M. Feng, M. Liu, and K. Gao, “An interferometric complementarity experiment in a bulk nuclear magnetic resonance ensemble,” J. Phys. A: Math. Gen. 36, 2555–2563 (2003).
[Crossref]

Grangier, P.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-Choice Test of Quantum Complementarity with Interfering Single Photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

Greenberger, D. M.

D. M. Greenberger and A. Yasin, “Simultaneous wave and particle knowledge in a neutron interferometer,” Phys. Lett. A 128, 391–394 (1988).
[Crossref]

Grosshans, F.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-Choice Test of Quantum Complementarity with Interfering Single Photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

Guo, G.-C.

Hall, M. J.

S. Cheng and M. J. Hall, “Complementarity relations for quantum coherence,” Phys. Rev. A 92, 042101 (2015).
[Crossref]

Heiblum, M.

E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, “Dephasing in electron interference by a ’which-path’ detector,” Nature 391, 871–874 (1998).
[Crossref]

Helstrom, C. W.

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).

Hillery, M.

E. Bagan, J.A. Bergou, S.S. Cottrell, and M. Hillery, “Relations between Coherence and Path Information,” Phys. Rev. Lett. 116, 160406 (2016).
[Crossref] [PubMed]

Hou, Z.

Jacques, V.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-Choice Test of Quantum Complementarity with Interfering Single Photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

Jaeger, G.

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

Kaszlikowski, D.

B.-G. Englert, D. Kaszlikowski, L. C. Kwek, and W. H. Chee, “Wave-particle duality in multi-path interferometers: general concepts and three-path interferometers,” Int. J. Quantum. Inform. 6, 129–157 (2008).
[Crossref]

Kwek, L. C.

B.-G. Englert, D. Kaszlikowski, L. C. Kwek, and W. H. Chee, “Wave-particle duality in multi-path interferometers: general concepts and three-path interferometers,” Int. J. Quantum. Inform. 6, 129–157 (2008).
[Crossref]

Kwiat, P. G.

P. D. D. Schwindt, P. G. Kwiat, and B.-G. Englert, “Quantitative wave-particle duality and nonerasing quantum erasure,” Phys. Rev. A 60, 4285–4290 (1999).
[Crossref]

Lewenstein, M.

E. Chitambar, A. Streltsov, S. Rana, M. N. Bera, G. Adesso, and M. Lewenstein, “Assisted Distillation of Quantum Coherence,” Phys. Rev. Lett. 116, 070402 (2016).
[Crossref] [PubMed]

Li, C.-F.

Liu, C. L.

X.-D. Yu, D.-J. Zhang, C. L. Liu, and D. M. Tong, “Measure-independent freezing of quantum coherence,” Phys. Rev. A 93, 060303 (2016).
[Crossref]

Liu, M.

X. Peng, X. Zhu, D. Suter, J. Du, M. Liu, and K. Gao, “Quantification of complementarity in multiqubit systems,” Phys. Rev. A 72, 052109 (2005).
[Crossref]

X. Peng, X. Zhu, X. Fang, M. Feng, M. Liu, and K. Gao, “An interferometric complementarity experiment in a bulk nuclear magnetic resonance ensemble,” J. Phys. A: Math. Gen. 36, 2555–2563 (2003).
[Crossref]

Mahalu, D.

E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, “Dephasing in electron interference by a ’which-path’ detector,” Nature 391, 871–874 (1998).
[Crossref]

Mei, M.

M. Mei and M. Weitz, “Controlled Decoherence in Multiple Beam Ramsey Interference,” Phys. Rev. Lett. 86, 559–563 (2001).
[Crossref] [PubMed]

Musto, R.

G. Bimonte and R. Musto, “On interferometric duality in multibeam experiments,” J. Phys. A: Math. Gen. 36, 11481–11502 (2003).
[Crossref]

Nonn, T.

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]

Pati, A. K.

M. N. Bera, T. Qureshi, M. A. Siddiqui, and A. K. Pati, “Duality of quantum coherence and path distinguishability,” Phys. Rev. A 92, 012118 (2015).
[Crossref]

Peng, X.

X. Peng, X. Zhu, D. Suter, J. Du, M. Liu, and K. Gao, “Quantification of complementarity in multiqubit systems,” Phys. Rev. A 72, 052109 (2005).
[Crossref]

X. Peng, X. Zhu, X. Fang, M. Feng, M. Liu, and K. Gao, “An interferometric complementarity experiment in a bulk nuclear magnetic resonance ensemble,” J. Phys. A: Math. Gen. 36, 2555–2563 (2003).
[Crossref]

Plenio, M. B.

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

Qureshi, T.

M. N. Bera, T. Qureshi, M. A. Siddiqui, and A. K. Pati, “Duality of quantum coherence and path distinguishability,” Phys. Rev. A 92, 012118 (2015).
[Crossref]

Rana, S.

E. Chitambar, A. Streltsov, S. Rana, M. N. Bera, G. Adesso, and M. Lewenstein, “Assisted Distillation of Quantum Coherence,” Phys. Rev. Lett. 116, 070402 (2016).
[Crossref] [PubMed]

Rauch, H.

J. Summhammer, H. Rauch, and D. Tuppinger, “Stochastic and deterministic absorption in neutron-interference experiments,” Phys. Rev. A 36, 4447–4455 (1987).
[Crossref]

H. Rauch and J. Summhammer, “Static versus time-dependent absorption in neutron interferometry,” Phys. Lett. A 104, 44–46 (1984).
[Crossref]

Rempe, G.

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]

Roch, J.-F.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-Choice Test of Quantum Complementarity with Interfering Single Photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

Schuster, R.

E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, “Dephasing in electron interference by a ’which-path’ detector,” Nature 391, 871–874 (1998).
[Crossref]

Schwindt, P. D. D.

P. D. D. Schwindt, P. G. Kwiat, and B.-G. Englert, “Quantitative wave-particle duality and nonerasing quantum erasure,” Phys. Rev. A 60, 4285–4290 (1999).
[Crossref]

Shimony, A.

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

M. N. Bera, T. Qureshi, M. A. Siddiqui, and A. K. Pati, “Duality of quantum coherence and path distinguishability,” Phys. Rev. A 92, 012118 (2015).
[Crossref]

Streltsov, A.

E. Chitambar, A. Streltsov, S. Rana, M. N. Bera, G. Adesso, and M. Lewenstein, “Assisted Distillation of Quantum Coherence,” Phys. Rev. Lett. 116, 070402 (2016).
[Crossref] [PubMed]

Summhammer, J.

J. Summhammer, H. Rauch, and D. Tuppinger, “Stochastic and deterministic absorption in neutron-interference experiments,” Phys. Rev. A 36, 4447–4455 (1987).
[Crossref]

H. Rauch and J. Summhammer, “Static versus time-dependent absorption in neutron interferometry,” Phys. Lett. A 104, 44–46 (1984).
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Suter, D.

X. Peng, X. Zhu, D. Suter, J. Du, M. Liu, and K. Gao, “Quantification of complementarity in multiqubit systems,” Phys. Rev. A 72, 052109 (2005).
[Crossref]

Tong, D. M.

X.-D. Yu, D.-J. Zhang, C. L. Liu, and D. M. Tong, “Measure-independent freezing of quantum coherence,” Phys. Rev. A 93, 060303 (2016).
[Crossref]

Treussart, F.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-Choice Test of Quantum Complementarity with Interfering Single Photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

Tuppinger, D.

J. Summhammer, H. Rauch, and D. Tuppinger, “Stochastic and deterministic absorption in neutron-interference experiments,” Phys. Rev. A 36, 4447–4455 (1987).
[Crossref]

Umansky, V.

E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, “Dephasing in electron interference by a ’which-path’ detector,” Nature 391, 871–874 (1998).
[Crossref]

Vaidman, L.

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

Weitz, M.

M. Mei and M. Weitz, “Controlled Decoherence in Multiple Beam Ramsey Interference,” Phys. Rev. Lett. 86, 559–563 (2001).
[Crossref] [PubMed]

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A. Winter and D. Yang, “Operational Resource Theory of Coherence,” Phys. Rev. Lett. 116, 120404 (2016).
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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).
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V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-Choice Test of Quantum Complementarity with Interfering Single Photons,” Phys. Rev. Lett. 100, 220402 (2008).
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Xiang, G.-Y.

Yang, D.

A. Winter and D. Yang, “Operational Resource Theory of Coherence,” Phys. Rev. Lett. 116, 120404 (2016).
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D. M. Greenberger and A. Yasin, “Simultaneous wave and particle knowledge in a neutron interferometer,” Phys. Lett. A 128, 391–394 (1988).
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X.-D. Yu, D.-J. Zhang, C. L. Liu, and D. M. Tong, “Measure-independent freezing of quantum coherence,” Phys. Rev. A 93, 060303 (2016).
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Yuan, Y.

Zhang, D.-J.

X.-D. Yu, D.-J. Zhang, C. L. Liu, and D. M. Tong, “Measure-independent freezing of quantum coherence,” Phys. Rev. A 93, 060303 (2016).
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Zhong, H.-S.

Zhu, X.

X. Peng, X. Zhu, D. Suter, J. Du, M. Liu, and K. Gao, “Quantification of complementarity in multiqubit systems,” Phys. Rev. A 72, 052109 (2005).
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X. Peng, X. Zhu, X. Fang, M. Feng, M. Liu, and K. Gao, “An interferometric complementarity experiment in a bulk nuclear magnetic resonance ensemble,” J. Phys. A: Math. Gen. 36, 2555–2563 (2003).
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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).
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Adv. Opt. Photonics (1)

S. M. Barnett and S. Croke, “Quantum state discrimination,” Adv. Opt. Photonics 1, 238–278 (2009).
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B.-G. Englert, D. Kaszlikowski, L. C. Kwek, and W. H. Chee, “Wave-particle duality in multi-path interferometers: general concepts and three-path interferometers,” Int. J. Quantum. Inform. 6, 129–157 (2008).
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J. Phys. A: Math. Gen. (2)

X. Peng, X. Zhu, X. Fang, M. Feng, M. Liu, and K. Gao, “An interferometric complementarity experiment in a bulk nuclear magnetic resonance ensemble,” J. Phys. A: Math. Gen. 36, 2555–2563 (2003).
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[Crossref]

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N. Bohr, “The quantum postulate and the recent development of atomic theory,” Nature 121, 580–590 (1928).
[Crossref]

E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, “Dephasing in electron interference by a ’which-path’ detector,” Nature 391, 871–874 (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]

Naturwissenschaften (1)

N. Bohr, “Das Quantenpostulat und die neuere Entwicklung der Atomistik,” Naturwissenschaften 16, 245–257 (1928).
[Crossref]

Optica (1)

Phys. Lett. A (2)

H. Rauch and J. Summhammer, “Static versus time-dependent absorption in neutron interferometry,” Phys. Lett. A 104, 44–46 (1984).
[Crossref]

D. M. Greenberger and A. Yasin, “Simultaneous wave and particle knowledge in a neutron interferometer,” Phys. Lett. A 128, 391–394 (1988).
[Crossref]

Phys. Rev. A (8)

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

S. Dürr, “Quantitative wave-particle duality in multibeam interferometers,” Phys. Rev. A 64, 042113 (2001).
[Crossref]

J. Summhammer, H. Rauch, and D. Tuppinger, “Stochastic and deterministic absorption in neutron-interference experiments,” Phys. Rev. A 36, 4447–4455 (1987).
[Crossref]

X. Peng, X. Zhu, D. Suter, J. Du, M. Liu, and K. Gao, “Quantification of complementarity in multiqubit systems,” Phys. Rev. A 72, 052109 (2005).
[Crossref]

P. D. D. Schwindt, P. G. Kwiat, and B.-G. Englert, “Quantitative wave-particle duality and nonerasing quantum erasure,” Phys. Rev. A 60, 4285–4290 (1999).
[Crossref]

S. Cheng and M. J. Hall, “Complementarity relations for quantum coherence,” Phys. Rev. A 92, 042101 (2015).
[Crossref]

X.-D. Yu, D.-J. Zhang, C. L. Liu, and D. M. Tong, “Measure-independent freezing of quantum coherence,” Phys. Rev. A 93, 060303 (2016).
[Crossref]

M. N. Bera, T. Qureshi, M. A. Siddiqui, and A. K. Pati, “Duality of quantum coherence and path distinguishability,” Phys. Rev. A 92, 012118 (2015).
[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)

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, “Delayed-Choice Test of Quantum Complementarity with Interfering Single Photons,” Phys. Rev. Lett. 100, 220402 (2008).
[Crossref] [PubMed]

B.-G. Englert, “Fringe Visibility and Which-Way Information: An Inequality,” Phys. Rev. Lett. 77, 2154–2157 (1996).
[Crossref] [PubMed]

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]

M. Mei and M. Weitz, “Controlled Decoherence in Multiple Beam Ramsey Interference,” Phys. Rev. Lett. 86, 559–563 (2001).
[Crossref] [PubMed]

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

A. Winter and D. Yang, “Operational Resource Theory of Coherence,” Phys. Rev. Lett. 116, 120404 (2016).
[Crossref] [PubMed]

E. Bagan, J.A. Bergou, S.S. Cottrell, and M. Hillery, “Relations between Coherence and Path Information,” Phys. Rev. Lett. 116, 160406 (2016).
[Crossref] [PubMed]

E. Chitambar, A. Streltsov, S. Rana, M. N. Bera, G. Adesso, and M. Lewenstein, “Assisted Distillation of Quantum Coherence,” Phys. Rev. Lett. 116, 070402 (2016).
[Crossref] [PubMed]

T. R. Bromley, M. Cianciaruso, and G. Adesso, “Frozen Quantum Coherence,” Phys. Rev. Lett. 114, 210401 (2015).
[Crossref] [PubMed]

Other (1)

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).

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Figures (3)

Fig. 1
Fig. 1 Experimental setup. In the source module, the single photon source is generated by the spontaneous parametric down-conversion(SPDC) progress with a type-II beamlike phase-matching beta-barium-borate(BBO). In the state preparation module, the angle of HWP1 is 22.5°; thus, the photons pass through path-1 and path-2 equiprobably. The detector states can be changed by rotating HWP2 and HWP3. In the measurement module, we measure coherence and path information independently. The angle of HWP4 is 0° or 45°, depending on the observation of coherence via tomography of the target state or observation of the path information via discrimination of the detector states. HWP5 and HWP6 are both rotated by 45° in an effort to rotate V polarized to H polarized and rotate H polarized to V polarized, respectively. The phase φ = 180° is added only in the wave property measurement. Quarter-wave plate (QWP)1, QWP2, HWP7, HWP8 and PBS are used to perform tomography and optimal minimum-error state discrimination. Output photons are detected using avalanche photo-diode(APD).
Fig. 2
Fig. 2 Experimental results regarding the relation between coherence in the relative entropy measure and the path information. Figure 2(a) shows the coherence C (green) and the mutual information H (red) between the detector states labeling the paths and the results of probing them as a function of the detector states θ. The solid lines are the theoretical expectations Eqs. (4) and (6). Figure 2(b) shows the sum of coherence C and the mutual information H; the blue dashed line denotes the theoretical values of C + H. The black solid line represents the upper bound of inequality Eq. (7). The error is too small to identify using this coordinate system; thus, we provide partially enlarged drawings using a magnifying power of 5×.
Fig. 3
Fig. 3 Experimental results for the relation between coherence in the l1 measure and path information. Figure 3(a) shows coherence X (green) and path distinguishability P (red) as a function of the detector states θ. Here, we define P = Ps − 1/2. The solid lines are the theoretical expectations Eqs. (9) and (10). Figure 3(b) shows the sum of the squares of coherence and the squares of path distinguishability. The blue dashed line denotes the theoretical values. We provide partially enlarged drawings using a magnifying power of 5×.

Equations (10)

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V 2 + D 2 1 ,
| η 1 = cos θ | H + sin θ | V , | η 2 = cos θ | H sin θ | V ,
ρ = Tr det ( | Ψ Ψ | ) = ( 1 2 1 2 cos 2 θ 1 2 cos 2 θ 1 2 ) .
C ( ρ ) = 1 + ( cos 2 θ log 2 cos 2 θ + sin 2 θ log 2 sin 2 θ ) .
| ϕ 1 = 1 2 ( | H + | V ) , | ϕ 2 = 1 2 ( | H | V ) .
H ( M : D ) = 2 + 2 p 11 log 2 p 11 + 2 p 12 log 2 p 12 ,
C ( ρ ) + H ( M : D ) H ( p i ) .
( P s 1 N ) 2 + X 2 ( 1 1 N ) 2 ,
X = 1 2 cos 2 θ .
P s = i = 1 2 1 2 η i | Π i | η i = 1 2 + 1 2 sin 2 θ .

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