Abstract

We study the wave and particle nature in a symmetric Mach-Zehnder interferometer from the viewpoint of quantum information theory. By introducing either the von Neumann or Zurek’s model of quantum measurement, we find that the classical correlation can be used to quantify the particle nature since its monotonicity is similar to the path distinguishability. The environment in Zurek’s model induces the emergence of the optimal measuring basis, and reduces the classical and quantum correlation comparing to the von Neumann’s model. A way is presented analytically to calculate the quantum correlation of a two-qubit separable state other than X-type.

© 2017 Optical Society of America

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References

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  1. N. Bohr, “The Quantum Postulate and the Recent Development of Atomic Theory,” Nature (London) 121, 580 (1928).
    [Crossref]
  2. P. Ghose and D. Home, “Wave-Particle Duality of Single-Photon States,” Foundations of Physics 22, 1435 (1992).
    [Crossref]
  3. 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 (1979).
    [Crossref]
  4. D. M. Greenberger and A. Yasin, “Simultaneous wave and particle knowledge in a neutron interferometer,” Phys. Lett. A 128, 391 (1988).
    [Crossref]
  5. G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54 (1995).
    [Crossref] [PubMed]
  6. B. G. Englert, “Fringe Visibility and Which-Way Information: An Inequality,” Phys. Rev. Lett. 77, 2154 (1996).
    [Crossref] [PubMed]
  7. K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
    [Crossref]
  8. S. Dürr, T. Nonn, and G. Rempe, “Fringe Visibility and Which-Way Information in an Atom Interferometer,” Phys. Rev. Lett. 81, 5705 (1998).
    [Crossref]
  9. P. D. D. Schwindt, P. G. Kwiat, and B. G. Englert, “Quantitative wave-particle duality and nonerasing quantum erasure,” Phys. Rev. A 60, 4285 (1999).
    [Crossref]
  10. X. H. Peng, X. W. Zhu, D. Suter, J. F. Du, M. L. Liu, and K. L. Gao, “Quantification of complementarity in multiqubit systems,” Phys. Rev. A 72, 052109 (2005).
    [Crossref]
  11. T. Xin, H. Li, B. X. Wang, and G. L. Long, “Realization of an entanglement-assisted quantum delayed-choice experiment,” Phys. Rev. A 92, 022126 (2015).
    [Crossref]
  12. V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J. F. Roch, “Experimental Realization of Wheeler’s Delayed-Choice Gedanken Experiment,” Science 315, 966 (2007).
    [Crossref] [PubMed]
  13. 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]
  14. S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer,” Science 332, 1170 (2011).
    [Crossref] [PubMed]
  15. G. L. Long, W. Qin, Z. Yang, and J. L. Li, “Realistic Interpretation of Quantum Mechanics and Encounter-Delayed-Choice Experiment,” https://arxiv.org/abs/1410.4129 (2014).
  16. Z. Z. Qin, A. S. Prasad, T. Brannan, A. MacRae, A. Lezama, and A. Lvovsky, “Complete temporal characterization of a single photon,” Light: Science & Applications 4, e298 (2015).
    [Crossref]
  17. H. Yan, K. Y. Liao, Z. T. Deng, J. Y. He, Z. Y. Xue, Z. M. Zhang, and S. L. Zhu, “Experimental observation of simultaneous wave and particle behavior in a narrowband single-photon wave packet,” Phys. Rev. A 91, 042132 (2015).
    [Crossref]
  18. W. J. Guo, D. H. Fan, and L. F. Wei, “Experimentally testing Bell’s theorem based on Hardy’s nonlocal ladder proofs,” Science China Physics, Mechanics & Astronomy,  58, 1–5 (2015).
    [Crossref]
  19. I. Silveiro, J. M. P. Ortega, and F. J. G. De Abajo, “Quantum nonlocal effects in individual and interacting graphene nanoribbons,” Light: Science & Applications,  4, e241 (2015).
    [Crossref]
  20. T. C. Li and Z. Q. Yin, “Quantum superposition, entanglement, and state teleportation of a microorganism on an electromechanical oscillator,” Science Bulletin 61, 163–171 (2016).
    [Crossref]
  21. G. L. Long, “General quantum interference principle and duality computer,” Communications in Theoretical Physics 45, 825–844 (2006).
    [Crossref]
  22. W. H. Zurek, “Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?” Phys. Rev. D 24, 1516 (1981).
    [Crossref]
  23. C. A. Brasil and L. A. D. de Castro, “Understanding the pointer states,” Eur. J. Phys. 36, 065024 (2015).
    [Crossref]
  24. M. O. Scully, B. G. Englert, and H. Walther, “Quantum optical tests of complementarity,” Nature (London) 351, 111 (1991).
    [Crossref]
  25. E. P. Storey, S. M. Tan, M. J. Collett, and D. F. Walls, “Complementarity and uncertainty,” Nature (London) 375, 368 (1995).
    [Crossref]
  26. W. H. Zurek, “Environment-induced superselection rules,” Phys. Rev. D 26, 1862 (1982).
    [Crossref]
  27. L. Henderson and V. Vedral, “Classical, quantum and total correlations,” J. Phys. A 34, 6899 (2001).
    [Crossref]
  28. S. Hamieh, R. Kobes, and H. Zaraket, “Positive-operator-valued measure optimization of classical correlations,” Phys. Rev. A 70, 052325 (2004).
    [Crossref]
  29. V. Vedral, “Classical Correlations and Entanglement in Quantum Measurements,” Phys. Rev. Lett. 90, 050401 (2003).
    [Crossref] [PubMed]
  30. H. Ollivier and W. H. Zurek, “Quantum Discord: A Measure of the Quantumness of Correlations,” Phys. Rev. Lett. 88, 017901 (2001).
    [Crossref]
  31. K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: Discord and related measures,” Rev. Mod. Phys. 84, 1655 (2012).
    [Crossref]
  32. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).
  33. A. Chefles, “Quantum state discrimination,” Contemp. Phys. 41, 401 (2000).
    [Crossref]
  34. S. L. Luo, “Quantum discord for two-qubit systems,” Phys. Rev. A 77, 042303 (2008).
    [Crossref]
  35. S. L. Luo, “Using measurement-induced disturbance to characterize correlations as classical or quantum,” Phys. Rev. A 77, 022301 (2008).
    [Crossref]
  36. N. Li and S. L. Luo, “Classical states versus separable states,” Phys. Rev. A 78, 024303 (2008).
    [Crossref]
  37. R. F. Werner, “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277 (1989).
    [Crossref]

2016 (1)

T. C. Li and Z. Q. Yin, “Quantum superposition, entanglement, and state teleportation of a microorganism on an electromechanical oscillator,” Science Bulletin 61, 163–171 (2016).
[Crossref]

2015 (6)

C. A. Brasil and L. A. D. de Castro, “Understanding the pointer states,” Eur. J. Phys. 36, 065024 (2015).
[Crossref]

T. Xin, H. Li, B. X. Wang, and G. L. Long, “Realization of an entanglement-assisted quantum delayed-choice experiment,” Phys. Rev. A 92, 022126 (2015).
[Crossref]

Z. Z. Qin, A. S. Prasad, T. Brannan, A. MacRae, A. Lezama, and A. Lvovsky, “Complete temporal characterization of a single photon,” Light: Science & Applications 4, e298 (2015).
[Crossref]

H. Yan, K. Y. Liao, Z. T. Deng, J. Y. He, Z. Y. Xue, Z. M. Zhang, and S. L. Zhu, “Experimental observation of simultaneous wave and particle behavior in a narrowband single-photon wave packet,” Phys. Rev. A 91, 042132 (2015).
[Crossref]

W. J. Guo, D. H. Fan, and L. F. Wei, “Experimentally testing Bell’s theorem based on Hardy’s nonlocal ladder proofs,” Science China Physics, Mechanics & Astronomy,  58, 1–5 (2015).
[Crossref]

I. Silveiro, J. M. P. Ortega, and F. J. G. De Abajo, “Quantum nonlocal effects in individual and interacting graphene nanoribbons,” Light: Science & Applications,  4, e241 (2015).
[Crossref]

2013 (1)

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

2012 (1)

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: Discord and related measures,” Rev. Mod. Phys. 84, 1655 (2012).
[Crossref]

2011 (1)

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

2008 (4)

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]

S. L. Luo, “Quantum discord for two-qubit systems,” Phys. Rev. A 77, 042303 (2008).
[Crossref]

S. L. Luo, “Using measurement-induced disturbance to characterize correlations as classical or quantum,” Phys. Rev. A 77, 022301 (2008).
[Crossref]

N. Li and S. L. Luo, “Classical states versus separable states,” Phys. Rev. A 78, 024303 (2008).
[Crossref]

2007 (1)

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J. F. Roch, “Experimental Realization of Wheeler’s Delayed-Choice Gedanken Experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

2006 (1)

G. L. Long, “General quantum interference principle and duality computer,” Communications in Theoretical Physics 45, 825–844 (2006).
[Crossref]

2005 (1)

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

2004 (1)

S. Hamieh, R. Kobes, and H. Zaraket, “Positive-operator-valued measure optimization of classical correlations,” Phys. Rev. A 70, 052325 (2004).
[Crossref]

2003 (1)

V. Vedral, “Classical Correlations and Entanglement in Quantum Measurements,” Phys. Rev. Lett. 90, 050401 (2003).
[Crossref] [PubMed]

2001 (2)

H. Ollivier and W. H. Zurek, “Quantum Discord: A Measure of the Quantumness of Correlations,” Phys. Rev. Lett. 88, 017901 (2001).
[Crossref]

L. Henderson and V. Vedral, “Classical, quantum and total correlations,” J. Phys. A 34, 6899 (2001).
[Crossref]

2000 (1)

A. Chefles, “Quantum state discrimination,” Contemp. Phys. 41, 401 (2000).
[Crossref]

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 (1999).
[Crossref]

1998 (1)

S. Dürr, T. Nonn, and G. Rempe, “Fringe Visibility and Which-Way Information in an Atom Interferometer,” Phys. Rev. Lett. 81, 5705 (1998).
[Crossref]

1996 (1)

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

1995 (2)

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

E. P. Storey, S. M. Tan, M. J. Collett, and D. F. Walls, “Complementarity and uncertainty,” Nature (London) 375, 368 (1995).
[Crossref]

1992 (1)

P. Ghose and D. Home, “Wave-Particle Duality of Single-Photon States,” Foundations of Physics 22, 1435 (1992).
[Crossref]

1991 (1)

M. O. Scully, B. G. Englert, and H. Walther, “Quantum optical tests of complementarity,” Nature (London) 351, 111 (1991).
[Crossref]

1989 (1)

R. F. Werner, “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277 (1989).
[Crossref]

1988 (1)

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

1982 (1)

W. H. Zurek, “Environment-induced superselection rules,” Phys. Rev. D 26, 1862 (1982).
[Crossref]

1981 (1)

W. H. Zurek, “Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?” Phys. Rev. D 24, 1516 (1981).
[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 (1979).
[Crossref]

1928 (1)

N. Bohr, “The Quantum Postulate and the Recent Development of Atomic Theory,” Nature (London) 121, 580 (1928).
[Crossref]

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]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J. F. Roch, “Experimental Realization of Wheeler’s Delayed-Choice Gedanken Experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

Bekshaev, A. Y.

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

Bohr, N.

N. Bohr, “The Quantum Postulate and the Recent Development of Atomic Theory,” Nature (London) 121, 580 (1928).
[Crossref]

Brannan, T.

Z. Z. Qin, A. S. Prasad, T. Brannan, A. MacRae, A. Lezama, and A. Lvovsky, “Complete temporal characterization of a single photon,” Light: Science & Applications 4, e298 (2015).
[Crossref]

Brasil, C. A.

C. A. Brasil and L. A. D. de Castro, “Understanding the pointer states,” Eur. J. Phys. 36, 065024 (2015).
[Crossref]

Braverman, B.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

Brodutch, A.

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: Discord and related measures,” Rev. Mod. Phys. 84, 1655 (2012).
[Crossref]

Cable, H.

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: Discord and related measures,” Rev. Mod. Phys. 84, 1655 (2012).
[Crossref]

Chefles, A.

A. Chefles, “Quantum state discrimination,” Contemp. Phys. 41, 401 (2000).
[Crossref]

Collett, M. J.

E. P. Storey, S. M. Tan, M. J. Collett, and D. F. Walls, “Complementarity and uncertainty,” Nature (London) 375, 368 (1995).
[Crossref]

De Abajo, F. J. G.

I. Silveiro, J. M. P. Ortega, and F. J. G. De Abajo, “Quantum nonlocal effects in individual and interacting graphene nanoribbons,” Light: Science & Applications,  4, e241 (2015).
[Crossref]

de Castro, L. A. D.

C. A. Brasil and L. A. D. de Castro, “Understanding the pointer states,” Eur. J. Phys. 36, 065024 (2015).
[Crossref]

Deng, Z. T.

H. Yan, K. Y. Liao, Z. T. Deng, J. Y. He, Z. Y. Xue, Z. M. Zhang, and S. L. Zhu, “Experimental observation of simultaneous wave and particle behavior in a narrowband single-photon wave packet,” Phys. Rev. A 91, 042132 (2015).
[Crossref]

Du, J. F.

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

Dürr, S.

S. Dürr, T. Nonn, and G. Rempe, “Fringe Visibility and Which-Way Information in an Atom Interferometer,” Phys. Rev. Lett. 81, 5705 (1998).
[Crossref]

Englert, B. 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 (1999).
[Crossref]

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

M. O. Scully, B. G. Englert, and H. Walther, “Quantum optical tests of complementarity,” Nature (London) 351, 111 (1991).
[Crossref]

Fan, D. H.

W. J. Guo, D. H. Fan, and L. F. Wei, “Experimentally testing Bell’s theorem based on Hardy’s nonlocal ladder proofs,” Science China Physics, Mechanics & Astronomy,  58, 1–5 (2015).
[Crossref]

Gao, K. L.

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

Ghose, P.

P. Ghose and D. Home, “Wave-Particle Duality of Single-Photon States,” Foundations of Physics 22, 1435 (1992).
[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]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J. F. Roch, “Experimental Realization of Wheeler’s Delayed-Choice Gedanken Experiment,” Science 315, 966 (2007).
[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 (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]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J. F. Roch, “Experimental Realization of Wheeler’s Delayed-Choice Gedanken Experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

Guo, W. J.

W. J. Guo, D. H. Fan, and L. F. Wei, “Experimentally testing Bell’s theorem based on Hardy’s nonlocal ladder proofs,” Science China Physics, Mechanics & Astronomy,  58, 1–5 (2015).
[Crossref]

Hamieh, S.

S. Hamieh, R. Kobes, and H. Zaraket, “Positive-operator-valued measure optimization of classical correlations,” Phys. Rev. A 70, 052325 (2004).
[Crossref]

He, J. Y.

H. Yan, K. Y. Liao, Z. T. Deng, J. Y. He, Z. Y. Xue, Z. M. Zhang, and S. L. Zhu, “Experimental observation of simultaneous wave and particle behavior in a narrowband single-photon wave packet,” Phys. Rev. A 91, 042132 (2015).
[Crossref]

Helstrom, C. W.

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

Henderson, L.

L. Henderson and V. Vedral, “Classical, quantum and total correlations,” J. Phys. A 34, 6899 (2001).
[Crossref]

Home, D.

P. Ghose and D. Home, “Wave-Particle Duality of Single-Photon States,” Foundations of Physics 22, 1435 (1992).
[Crossref]

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]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J. F. Roch, “Experimental Realization of Wheeler’s Delayed-Choice Gedanken Experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

Jaeger, G.

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

Kobes, R.

S. Hamieh, R. Kobes, and H. Zaraket, “Positive-operator-valued measure optimization of classical correlations,” Phys. Rev. A 70, 052325 (2004).
[Crossref]

Kocsis, S.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

Kofman, A. G.

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[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 (1999).
[Crossref]

Lezama, A.

Z. Z. Qin, A. S. Prasad, T. Brannan, A. MacRae, A. Lezama, and A. Lvovsky, “Complete temporal characterization of a single photon,” Light: Science & Applications 4, e298 (2015).
[Crossref]

Li, H.

T. Xin, H. Li, B. X. Wang, and G. L. Long, “Realization of an entanglement-assisted quantum delayed-choice experiment,” Phys. Rev. A 92, 022126 (2015).
[Crossref]

Li, J. L.

G. L. Long, W. Qin, Z. Yang, and J. L. Li, “Realistic Interpretation of Quantum Mechanics and Encounter-Delayed-Choice Experiment,” https://arxiv.org/abs/1410.4129 (2014).

Li, N.

N. Li and S. L. Luo, “Classical states versus separable states,” Phys. Rev. A 78, 024303 (2008).
[Crossref]

Li, T. C.

T. C. Li and Z. Q. Yin, “Quantum superposition, entanglement, and state teleportation of a microorganism on an electromechanical oscillator,” Science Bulletin 61, 163–171 (2016).
[Crossref]

Liao, K. Y.

H. Yan, K. Y. Liao, Z. T. Deng, J. Y. He, Z. Y. Xue, Z. M. Zhang, and S. L. Zhu, “Experimental observation of simultaneous wave and particle behavior in a narrowband single-photon wave packet,” Phys. Rev. A 91, 042132 (2015).
[Crossref]

Liu, M. L.

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

Long, G. L.

T. Xin, H. Li, B. X. Wang, and G. L. Long, “Realization of an entanglement-assisted quantum delayed-choice experiment,” Phys. Rev. A 92, 022126 (2015).
[Crossref]

G. L. Long, “General quantum interference principle and duality computer,” Communications in Theoretical Physics 45, 825–844 (2006).
[Crossref]

G. L. Long, W. Qin, Z. Yang, and J. L. Li, “Realistic Interpretation of Quantum Mechanics and Encounter-Delayed-Choice Experiment,” https://arxiv.org/abs/1410.4129 (2014).

Luo, S. L.

N. Li and S. L. Luo, “Classical states versus separable states,” Phys. Rev. A 78, 024303 (2008).
[Crossref]

S. L. Luo, “Quantum discord for two-qubit systems,” Phys. Rev. A 77, 042303 (2008).
[Crossref]

S. L. Luo, “Using measurement-induced disturbance to characterize correlations as classical or quantum,” Phys. Rev. A 77, 022301 (2008).
[Crossref]

Lvovsky, A.

Z. Z. Qin, A. S. Prasad, T. Brannan, A. MacRae, A. Lezama, and A. Lvovsky, “Complete temporal characterization of a single photon,” Light: Science & Applications 4, e298 (2015).
[Crossref]

MacRae, A.

Z. Z. Qin, A. S. Prasad, T. Brannan, A. MacRae, A. Lezama, and A. Lvovsky, “Complete temporal characterization of a single photon,” Light: Science & Applications 4, e298 (2015).
[Crossref]

Mirin, R. P.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

Modi, K.

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: Discord and related measures,” Rev. Mod. Phys. 84, 1655 (2012).
[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 (1998).
[Crossref]

Nori, F.

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

Ollivier, H.

H. Ollivier and W. H. Zurek, “Quantum Discord: A Measure of the Quantumness of Correlations,” Phys. Rev. Lett. 88, 017901 (2001).
[Crossref]

Ortega, J. M. P.

I. Silveiro, J. M. P. Ortega, and F. J. G. De Abajo, “Quantum nonlocal effects in individual and interacting graphene nanoribbons,” Light: Science & Applications,  4, e241 (2015).
[Crossref]

Paterek, T.

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: Discord and related measures,” Rev. Mod. Phys. 84, 1655 (2012).
[Crossref]

Peng, X. H.

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

Prasad, A. S.

Z. Z. Qin, A. S. Prasad, T. Brannan, A. MacRae, A. Lezama, and A. Lvovsky, “Complete temporal characterization of a single photon,” Light: Science & Applications 4, e298 (2015).
[Crossref]

Qin, W.

G. L. Long, W. Qin, Z. Yang, and J. L. Li, “Realistic Interpretation of Quantum Mechanics and Encounter-Delayed-Choice Experiment,” https://arxiv.org/abs/1410.4129 (2014).

Qin, Z. Z.

Z. Z. Qin, A. S. Prasad, T. Brannan, A. MacRae, A. Lezama, and A. Lvovsky, “Complete temporal characterization of a single photon,” Light: Science & Applications 4, e298 (2015).
[Crossref]

Ravets, S.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

Rempe, G.

S. Dürr, T. Nonn, and G. Rempe, “Fringe Visibility and Which-Way Information in an Atom Interferometer,” Phys. Rev. Lett. 81, 5705 (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]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J. F. Roch, “Experimental Realization of Wheeler’s Delayed-Choice Gedanken Experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

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 (1999).
[Crossref]

Scully, M. O.

M. O. Scully, B. G. Englert, and H. Walther, “Quantum optical tests of complementarity,” Nature (London) 351, 111 (1991).
[Crossref]

Shalm, L. K.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

Shimony, A.

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

Silveiro, I.

I. Silveiro, J. M. P. Ortega, and F. J. G. De Abajo, “Quantum nonlocal effects in individual and interacting graphene nanoribbons,” Light: Science & Applications,  4, e241 (2015).
[Crossref]

Steinberg, A. M.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

Stevens, M. J.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

Storey, E. P.

E. P. Storey, S. M. Tan, M. J. Collett, and D. F. Walls, “Complementarity and uncertainty,” Nature (London) 375, 368 (1995).
[Crossref]

Suter, D.

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

Tan, S. M.

E. P. Storey, S. M. Tan, M. J. Collett, and D. F. Walls, “Complementarity and uncertainty,” Nature (London) 375, 368 (1995).
[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]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J. F. Roch, “Experimental Realization of Wheeler’s Delayed-Choice Gedanken Experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

Vaidman, L.

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

Vedral, V.

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: Discord and related measures,” Rev. Mod. Phys. 84, 1655 (2012).
[Crossref]

V. Vedral, “Classical Correlations and Entanglement in Quantum Measurements,” Phys. Rev. Lett. 90, 050401 (2003).
[Crossref] [PubMed]

L. Henderson and V. Vedral, “Classical, quantum and total correlations,” J. Phys. A 34, 6899 (2001).
[Crossref]

Walls, D. F.

E. P. Storey, S. M. Tan, M. J. Collett, and D. F. Walls, “Complementarity and uncertainty,” Nature (London) 375, 368 (1995).
[Crossref]

Walther, H.

M. O. Scully, B. G. Englert, and H. Walther, “Quantum optical tests of complementarity,” Nature (London) 351, 111 (1991).
[Crossref]

Wang, B. X.

T. Xin, H. Li, B. X. Wang, and G. L. Long, “Realization of an entanglement-assisted quantum delayed-choice experiment,” Phys. Rev. A 92, 022126 (2015).
[Crossref]

Wei, L. F.

W. J. Guo, D. H. Fan, and L. F. Wei, “Experimentally testing Bell’s theorem based on Hardy’s nonlocal ladder proofs,” Science China Physics, Mechanics & Astronomy,  58, 1–5 (2015).
[Crossref]

Werner, R. F.

R. F. Werner, “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277 (1989).
[Crossref]

Wootters, W. K.

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 (1979).
[Crossref]

Wu, E.

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]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J. F. Roch, “Experimental Realization of Wheeler’s Delayed-Choice Gedanken Experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

Xin, T.

T. Xin, H. Li, B. X. Wang, and G. L. Long, “Realization of an entanglement-assisted quantum delayed-choice experiment,” Phys. Rev. A 92, 022126 (2015).
[Crossref]

Xue, Z. Y.

H. Yan, K. Y. Liao, Z. T. Deng, J. Y. He, Z. Y. Xue, Z. M. Zhang, and S. L. Zhu, “Experimental observation of simultaneous wave and particle behavior in a narrowband single-photon wave packet,” Phys. Rev. A 91, 042132 (2015).
[Crossref]

Yan, H.

H. Yan, K. Y. Liao, Z. T. Deng, J. Y. He, Z. Y. Xue, Z. M. Zhang, and S. L. Zhu, “Experimental observation of simultaneous wave and particle behavior in a narrowband single-photon wave packet,” Phys. Rev. A 91, 042132 (2015).
[Crossref]

Yang, Z.

G. L. Long, W. Qin, Z. Yang, and J. L. Li, “Realistic Interpretation of Quantum Mechanics and Encounter-Delayed-Choice Experiment,” https://arxiv.org/abs/1410.4129 (2014).

Yasin, A.

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

Yin, Z. Q.

T. C. Li and Z. Q. Yin, “Quantum superposition, entanglement, and state teleportation of a microorganism on an electromechanical oscillator,” Science Bulletin 61, 163–171 (2016).
[Crossref]

Zaraket, H.

S. Hamieh, R. Kobes, and H. Zaraket, “Positive-operator-valued measure optimization of classical correlations,” Phys. Rev. A 70, 052325 (2004).
[Crossref]

Zhang, Z. M.

H. Yan, K. Y. Liao, Z. T. Deng, J. Y. He, Z. Y. Xue, Z. M. Zhang, and S. L. Zhu, “Experimental observation of simultaneous wave and particle behavior in a narrowband single-photon wave packet,” Phys. Rev. A 91, 042132 (2015).
[Crossref]

Zhu, S. L.

H. Yan, K. Y. Liao, Z. T. Deng, J. Y. He, Z. Y. Xue, Z. M. Zhang, and S. L. Zhu, “Experimental observation of simultaneous wave and particle behavior in a narrowband single-photon wave packet,” Phys. Rev. A 91, 042132 (2015).
[Crossref]

Zhu, X. W.

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

Zurek, W. H.

H. Ollivier and W. H. Zurek, “Quantum Discord: A Measure of the Quantumness of Correlations,” Phys. Rev. Lett. 88, 017901 (2001).
[Crossref]

W. H. Zurek, “Environment-induced superselection rules,” Phys. Rev. D 26, 1862 (1982).
[Crossref]

W. H. Zurek, “Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?” Phys. Rev. D 24, 1516 (1981).
[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 (1979).
[Crossref]

Communications in Theoretical Physics (1)

G. L. Long, “General quantum interference principle and duality computer,” Communications in Theoretical Physics 45, 825–844 (2006).
[Crossref]

Contemp. Phys. (1)

A. Chefles, “Quantum state discrimination,” Contemp. Phys. 41, 401 (2000).
[Crossref]

Eur. J. Phys. (1)

C. A. Brasil and L. A. D. de Castro, “Understanding the pointer states,” Eur. J. Phys. 36, 065024 (2015).
[Crossref]

Foundations of Physics (1)

P. Ghose and D. Home, “Wave-Particle Duality of Single-Photon States,” Foundations of Physics 22, 1435 (1992).
[Crossref]

J. Phys. A (1)

L. Henderson and V. Vedral, “Classical, quantum and total correlations,” J. Phys. A 34, 6899 (2001).
[Crossref]

Light: Science & Applications (2)

Z. Z. Qin, A. S. Prasad, T. Brannan, A. MacRae, A. Lezama, and A. Lvovsky, “Complete temporal characterization of a single photon,” Light: Science & Applications 4, e298 (2015).
[Crossref]

I. Silveiro, J. M. P. Ortega, and F. J. G. De Abajo, “Quantum nonlocal effects in individual and interacting graphene nanoribbons,” Light: Science & Applications,  4, e241 (2015).
[Crossref]

Nature (London) (3)

M. O. Scully, B. G. Englert, and H. Walther, “Quantum optical tests of complementarity,” Nature (London) 351, 111 (1991).
[Crossref]

E. P. Storey, S. M. Tan, M. J. Collett, and D. F. Walls, “Complementarity and uncertainty,” Nature (London) 375, 368 (1995).
[Crossref]

N. Bohr, “The Quantum Postulate and the Recent Development of Atomic Theory,” Nature (London) 121, 580 (1928).
[Crossref]

New J. Phys. (1)

K. Y. Bliokh, A. Y. Bekshaev, A. G. Kofman, and F. Nori, “Photon trajectories, anomalous velocities and weak measurements: a classical interpretation,” New J. Phys. 15, 073022 (2013).
[Crossref]

Phys. Lett. A (1)

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

Phys. Rev. A (10)

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

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

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

T. Xin, H. Li, B. X. Wang, and G. L. Long, “Realization of an entanglement-assisted quantum delayed-choice experiment,” Phys. Rev. A 92, 022126 (2015).
[Crossref]

H. Yan, K. Y. Liao, Z. T. Deng, J. Y. He, Z. Y. Xue, Z. M. Zhang, and S. L. Zhu, “Experimental observation of simultaneous wave and particle behavior in a narrowband single-photon wave packet,” Phys. Rev. A 91, 042132 (2015).
[Crossref]

S. L. Luo, “Quantum discord for two-qubit systems,” Phys. Rev. A 77, 042303 (2008).
[Crossref]

S. L. Luo, “Using measurement-induced disturbance to characterize correlations as classical or quantum,” Phys. Rev. A 77, 022301 (2008).
[Crossref]

N. Li and S. L. Luo, “Classical states versus separable states,” Phys. Rev. A 78, 024303 (2008).
[Crossref]

R. F. Werner, “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277 (1989).
[Crossref]

S. Hamieh, R. Kobes, and H. Zaraket, “Positive-operator-valued measure optimization of classical correlations,” Phys. Rev. A 70, 052325 (2004).
[Crossref]

Phys. Rev. D (3)

W. H. Zurek, “Environment-induced superselection rules,” Phys. Rev. D 26, 1862 (1982).
[Crossref]

W. H. Zurek, “Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?” Phys. Rev. D 24, 1516 (1981).
[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 (1979).
[Crossref]

Phys. Rev. Lett. (5)

B. G. Englert, “Fringe Visibility and Which-Way Information: An Inequality,” Phys. Rev. Lett. 77, 2154 (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 (1998).
[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]

V. Vedral, “Classical Correlations and Entanglement in Quantum Measurements,” Phys. Rev. Lett. 90, 050401 (2003).
[Crossref] [PubMed]

H. Ollivier and W. H. Zurek, “Quantum Discord: A Measure of the Quantumness of Correlations,” Phys. Rev. Lett. 88, 017901 (2001).
[Crossref]

Rev. Mod. Phys. (1)

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: Discord and related measures,” Rev. Mod. Phys. 84, 1655 (2012).
[Crossref]

Science (2)

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, “Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer,” Science 332, 1170 (2011).
[Crossref] [PubMed]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J. F. Roch, “Experimental Realization of Wheeler’s Delayed-Choice Gedanken Experiment,” Science 315, 966 (2007).
[Crossref] [PubMed]

Science Bulletin (1)

T. C. Li and Z. Q. Yin, “Quantum superposition, entanglement, and state teleportation of a microorganism on an electromechanical oscillator,” Science Bulletin 61, 163–171 (2016).
[Crossref]

Science China Physics, Mechanics & Astronomy (1)

W. J. Guo, D. H. Fan, and L. F. Wei, “Experimentally testing Bell’s theorem based on Hardy’s nonlocal ladder proofs,” Science China Physics, Mechanics & Astronomy,  58, 1–5 (2015).
[Crossref]

Other (2)

G. L. Long, W. Qin, Z. Yang, and J. L. Li, “Realistic Interpretation of Quantum Mechanics and Encounter-Delayed-Choice Experiment,” https://arxiv.org/abs/1410.4129 (2014).

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

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Figures (3)

Fig. 1
Fig. 1 Schematic of a symmetric Mach-Zehnder interferometer. Here, PS refers to phase shifter, BS to beam splitters, and HR to high reflector.
Fig. 2
Fig. 2 The relationship between the classical correlations and the fringe visibility V. ρ′ refers to the entangles state in Eq. (13), and ρ″ to the correlated state in Eq. (15).
Fig. 3
Fig. 3 The relationship between the quantum correlations and the fringe visibility V. ρ′ refers to the entangles state in Eq. (13), and ρ″ to the correlated state in Eq. (15).

Equations (22)

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

| a 1 2 ( | a + | b ) , | b 1 2 ( | a | b )
ρ in Q = 1 2 ( 1 + S x σ x + S y σ y + S z σ z )
ρ f = 1 4 ( 1 S x ) ( 1 + σ x ) ρ in D + 1 4 ( 1 + S x ) ( 1 σ x ) U ρ in D U 1 4 e i ϕ ( S z i S y ) ( σ z i σ y ) ρ in D U 1 4 e i ϕ ( S z + i S y ) ( σ z + i σ y ) U ρ in D ,
ρ f Q = 1 4 ( 1 S x ) ( 1 + σ x ) + 1 4 ( 1 + S x ) ( 1 σ x ) 1 4 e i ϕ ( S z i S y ) ( σ z i σ y ) Tr D ( ρ in D U ) 1 4 e i ϕ ( S z + i S y ) ( σ z + i σ y ) Tr D ( U ρ in D )
P a = Tr Q [ 1 2 ( 1 + σ z ) ρ f Q ] = 1 2 1 2 S y 2 + S z 2 | Tr D ( U ρ in D ) | cos ( α + β + ϕ )
V P max a P min a P max a + P min a S y 2 + S z 2 | Tr D ( U ρ in D ) |
ρ f D = 1 S x 2 ρ in D + 1 + S x 2 U ρ in D U ,
D = Tr D | 1 S x 2 ρ in D 1 + S x 2 U ρ in D U |
D 2 + 1 P 2 V 0 2 V 2 1 ,
| M a = sin γ 1 V 2 | d + e i φ ( cos γ sin γ V 1 V 2 ) U | d , | M b = e i φ cos γ 1 V 2 | d ( sin γ + cos γ V 1 V 2 ) U | d
| M a = 1 m 1 + m 2 | d e i φ 1 m 1 m 2 U | d , | M b = e i φ 1 m 1 m 2 | d 1 m 1 + m 2 U | d
J ( ρ ) = max [ S ( ρ Q ) S ( ρ Q | { k } ) ] ,
| Ψ = 1 2 ( | a | d + | b U | d ) .
J ( ρ ) = 1 + V 2 log ( 1 + V 2 ) 1 V 2 log ( 1 V 2 )
ρ = 1 2 ( | a a | | d d | + | b b | U | d d | U ) ,
S ( ρ Q ) S ( ρ Q | { k } ) = 1 + 1 2 [ cos 2 γ ] log [ cos 2 γ ] + 1 2 [ sin 2 γ ] log [ sin 2 γ ] + 1 2 [ ( 1 V 2 cos γ V sin γ ) 2 ] log [ ( 1 V 2 cos γ V sin γ ) 2 ] + 1 2 [ ( 1 V 2 sin γ + V cos γ ) 2 ] log [ ( 1 V 2 sin γ + V cos γ ) 2 ] 1 2 [ ( 1 V 2 sin γ + V cos γ ) 2 + cos 2 γ ] log [ ( 1 V 2 sin γ + V cos γ ) 2 + cos 2 γ ] 1 2 [ ( 1 V 2 cos γ V sin γ ) 2 + sin 2 γ ] log [ ( 1 V 2 cos γ V sin γ ) 2 + sin 2 γ ]
γ = arcsin { 1 2 [ 1 + ( 1 V 2 ) 1 2 ] } 1 2 ,
J ( ρ ) = 1 + 1 V 2 2 log ( 1 + 1 V 2 ) + 1 1 V 2 2 log ( 1 1 V 2 ) ,
D ( ρ ) = ( ρ ) J ( ρ ) .
( ρ ) = S ( ρ Q ) + S ( ρ D ) S ( ρ ) ,
D ( ρ ) = 1 + V 2 log ( 1 + V 2 ) 1 V 2 log ( 1 V 2 ) .
D ( ρ ) = 1 + V 2 log ( 1 + V 2 ) 1 V 2 log ( 1 V 2 ) 1 + 1 V 2 2 log ( 1 + 1 V 2 ) 1 1 V 2 2 log ( 1 1 V 2 ) .

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