Abstract

Recently, it was shown that quantum entanglement is an indispensable part of the duality behavior of light. Here we report a surprisingly intimate connection between the stereographic projection and the duality–entanglement nature of a single photon. We show that the duality–entanglement relation [Optica 5, 942 (2018) [CrossRef]  ] naturally emerges from the stereographic projection geometry. We demonstrate that this geometry is complementarity sensitive, in the sense that it is sensitive to the particle nature, wave nature, and entanglement nature of a single photon.

© 2019 Optical Society of America

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References

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2019 (1)

2018 (3)

2017 (2)

G. Najarbashi and B. Seifi, Quantum Inf. Process. 16, 40 (2017).
[Crossref]

A. S. Rab, E. Polino, Z.-X. Man, N. B. An, Y.-J. Xia, N. Spagnolo, R. L. Franco, and F. Sciarrino, Nat. Commun. 8, 915 (2017).
[Crossref]

2016 (3)

E. Bagan, J. A. Bergou, S. S. Cottrell, and M. Hillery, Phys. Rev. Lett. 116, 160406 (2016).
[Crossref]

A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016).
[Crossref]

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, Phys. Rev. Lett. 117, 153901 (2016).
[Crossref]

2015 (2)

S. Cheng and M. J. Hall, Phys. Rev. A 92, 042101 (2015).
[Crossref]

M. Lostaglio, K. Korzekwa, D. Jennings, and T. Rudolph, Phys. Rev. X 5, 021001 (2015).
[Crossref]

2014 (1)

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

2013 (1)

Y.-C. Jeong, C. Di Franco, H.-T. Lim, M. Kim, and Y.-H. Kim, Nat. Commun. 4, 2471 (2013).
[Crossref]

2012 (3)

F. Kaiser, T. Coudreau, P. Milman, D. B. Ostrowsky, and S. Tanzilli, Science 338, 637 (2012).
[Crossref]

R. Menzel, D. Puhlmann, A. Heuer, and W. P. Schleich, Proc. Natl. Acad. Sci. 109, 9314 (2012).
[Crossref]

A. Peruzzo, P. Shadbolt, N. Brunner, S. Popescu, and J. L. O’Brien, Science 338, 634 (2012).
[Crossref]

2008 (2)

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, Phys. Rev. Lett. 100, 220402 (2008).
[Crossref]

D.-I. Chang, G. L. Khym, K. Kang, Y. Chung, H.-J. Lee, M. Seo, M. Heiblum, D. Mahalu, and V. Umansky, Nat. Phys. 4, 205 (2008).
[Crossref]

2007 (2)

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, Science 315, 966 (2007).
[Crossref]

G. Najarbashi, S. Ahadpour, M. Fasihi, and Y. Tavakoli, J. Phys. A 40, 6481 (2007).
[Crossref]

2002 (1)

J.-W. Lee, C. H. Kim, E. K. Lee, J. Kim, and S. Lee, Quantum Inf. Process. 1, 129 (2002).
[Crossref]

2001 (1)

R. Mosseri and R. Dandoloff, J. Phys. A 34, 10243 (2001).
[Crossref]

2000 (1)

Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. 84, 1 (2000).
[Crossref]

1999 (1)

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. Van der Zouw, and A. Zeilinger, Nature 401, 680 (1999).
[Crossref]

1998 (2)

S. Dürr, T. Nonn, and G. Rempe, Nature 395, 33 (1998).
[Crossref]

W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
[Crossref]

1996 (1)

B.-G. Englert, Phys. Rev. Lett. 77, 2154 (1996).
[Crossref]

1991 (1)

M. O. Scully, B.-G. Englert, and H. Walther, Nature 351, 111(1991).
[Crossref]

1979 (1)

W. K. Wootters and W. H. Zurek, Phys. Rev. D 19, 473 (1979).
[Crossref]

1923 (1)

L. De Broglie, Nature 112, 540 (1923).
[Crossref]

Agarwal, G.

X.-F. Qian and G. Agarwal, “Are quantum objects born with duality?” arXiv:1901.07595 (2019).

Ahadpour, S.

G. Najarbashi, S. Ahadpour, M. Fasihi, and Y. Tavakoli, J. Phys. A 40, 6481 (2007).
[Crossref]

An, N. B.

A. S. Rab, E. Polino, Z.-X. Man, N. B. An, Y.-J. Xia, N. Spagnolo, R. L. Franco, and F. Sciarrino, Nat. Commun. 8, 915 (2017).
[Crossref]

Arndt, M.

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. Van der Zouw, and A. Zeilinger, Nature 401, 680 (1999).
[Crossref]

Aspect, A.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, Phys. Rev. Lett. 100, 220402 (2008).
[Crossref]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, Science 315, 966 (2007).
[Crossref]

Bagan, E.

E. Bagan, J. A. Bergou, S. S. Cottrell, and M. Hillery, Phys. Rev. Lett. 116, 160406 (2016).
[Crossref]

Baumgratz, T.

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

Bergou, J. A.

E. Bagan, J. A. Bergou, S. S. Cottrell, and M. Hillery, Phys. Rev. Lett. 116, 160406 (2016).
[Crossref]

Bohr, N.

N. Bohr, The Quantum Postulate and the Recent Development of Atomic Theory (1928).

Brunner, N.

A. Peruzzo, P. Shadbolt, N. Brunner, S. Popescu, and J. L. O’Brien, Science 338, 634 (2012).
[Crossref]

Chang, D.-I.

D.-I. Chang, G. L. Khym, K. Kang, Y. Chung, H.-J. Lee, M. Seo, M. Heiblum, D. Mahalu, and V. Umansky, Nat. Phys. 4, 205 (2008).
[Crossref]

Cheng, S.

S. Cheng and M. J. Hall, Phys. Rev. A 92, 042101 (2015).
[Crossref]

Chung, Y.

D.-I. Chang, G. L. Khym, K. Kang, Y. Chung, H.-J. Lee, M. Seo, M. Heiblum, D. Mahalu, and V. Umansky, Nat. Phys. 4, 205 (2008).
[Crossref]

Cottrell, S. S.

E. Bagan, J. A. Bergou, S. S. Cottrell, and M. Hillery, Phys. Rev. Lett. 116, 160406 (2016).
[Crossref]

Coudreau, T.

F. Kaiser, T. Coudreau, P. Milman, D. B. Ostrowsky, and S. Tanzilli, Science 338, 637 (2012).
[Crossref]

Cramer, M.

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

Dandoloff, R.

R. Mosseri and R. Dandoloff, J. Phys. A 34, 10243 (2001).
[Crossref]

De Broglie, L.

L. De Broglie, Nature 112, 540 (1923).
[Crossref]

Di Franco, C.

Y.-C. Jeong, C. Di Franco, H.-T. Lim, M. Kim, and Y.-H. Kim, Nat. Commun. 4, 2471 (2013).
[Crossref]

Dürr, S.

S. Dürr, T. Nonn, and G. Rempe, Nature 395, 33 (1998).
[Crossref]

Eberly, J.

Eberly, J. H.

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, Phys. Rev. Lett. 117, 153901 (2016).
[Crossref]

Englert, B.-G.

B.-G. Englert, Phys. Rev. Lett. 77, 2154 (1996).
[Crossref]

M. O. Scully, B.-G. Englert, and H. Walther, Nature 351, 111(1991).
[Crossref]

Fasihi, M.

G. Najarbashi, S. Ahadpour, M. Fasihi, and Y. Tavakoli, J. Phys. A 40, 6481 (2007).
[Crossref]

Feynman, R. P.

R. P. Feynman, R. B. Leighton, and M. Sands, Lectures on Physics, Vol. iii (1965).

Franco, R. L.

A. S. Rab, E. Polino, Z.-X. Man, N. B. An, Y.-J. Xia, N. Spagnolo, R. L. Franco, and F. Sciarrino, Nat. Commun. 8, 915 (2017).
[Crossref]

Grangier, P.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, Phys. Rev. Lett. 100, 220402 (2008).
[Crossref]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, Science 315, 966 (2007).
[Crossref]

Grosshans, F.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, Phys. Rev. Lett. 100, 220402 (2008).
[Crossref]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, Science 315, 966 (2007).
[Crossref]

Hall, M. J.

S. Cheng and M. J. Hall, Phys. Rev. A 92, 042101 (2015).
[Crossref]

Heiblum, M.

D.-I. Chang, G. L. Khym, K. Kang, Y. Chung, H.-J. Lee, M. Seo, M. Heiblum, D. Mahalu, and V. Umansky, Nat. Phys. 4, 205 (2008).
[Crossref]

Heuer, A.

R. Menzel, D. Puhlmann, A. Heuer, and W. P. Schleich, Proc. Natl. Acad. Sci. 109, 9314 (2012).
[Crossref]

Hillery, M.

E. Bagan, J. A. Bergou, S. S. Cottrell, and M. Hillery, Phys. Rev. Lett. 116, 160406 (2016).
[Crossref]

Jacques, V.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, Phys. Rev. Lett. 100, 220402 (2008).
[Crossref]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, Science 315, 966 (2007).
[Crossref]

Jennings, D.

M. Lostaglio, K. Korzekwa, D. Jennings, and T. Rudolph, Phys. Rev. X 5, 021001 (2015).
[Crossref]

Jeong, Y.-C.

Y.-C. Jeong, C. Di Franco, H.-T. Lim, M. Kim, and Y.-H. Kim, Nat. Commun. 4, 2471 (2013).
[Crossref]

Kaiser, F.

F. Kaiser, T. Coudreau, P. Milman, D. B. Ostrowsky, and S. Tanzilli, Science 338, 637 (2012).
[Crossref]

Kang, K.

D.-I. Chang, G. L. Khym, K. Kang, Y. Chung, H.-J. Lee, M. Seo, M. Heiblum, D. Mahalu, and V. Umansky, Nat. Phys. 4, 205 (2008).
[Crossref]

Keller, C.

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. Van der Zouw, and A. Zeilinger, Nature 401, 680 (1999).
[Crossref]

Khym, G. L.

D.-I. Chang, G. L. Khym, K. Kang, Y. Chung, H.-J. Lee, M. Seo, M. Heiblum, D. Mahalu, and V. Umansky, Nat. Phys. 4, 205 (2008).
[Crossref]

Kim, C. H.

J.-W. Lee, C. H. Kim, E. K. Lee, J. Kim, and S. Lee, Quantum Inf. Process. 1, 129 (2002).
[Crossref]

Kim, J.

J.-W. Lee, C. H. Kim, E. K. Lee, J. Kim, and S. Lee, Quantum Inf. Process. 1, 129 (2002).
[Crossref]

Kim, M.

Y.-C. Jeong, C. Di Franco, H.-T. Lim, M. Kim, and Y.-H. Kim, Nat. Commun. 4, 2471 (2013).
[Crossref]

Kim, Y.-H.

Y.-C. Jeong, C. Di Franco, H.-T. Lim, M. Kim, and Y.-H. Kim, Nat. Commun. 4, 2471 (2013).
[Crossref]

Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. 84, 1 (2000).
[Crossref]

Korzekwa, K.

M. Lostaglio, K. Korzekwa, D. Jennings, and T. Rudolph, Phys. Rev. X 5, 021001 (2015).
[Crossref]

Kulik, S. P.

Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. 84, 1 (2000).
[Crossref]

Lee, E. K.

J.-W. Lee, C. H. Kim, E. K. Lee, J. Kim, and S. Lee, Quantum Inf. Process. 1, 129 (2002).
[Crossref]

Lee, H.-J.

D.-I. Chang, G. L. Khym, K. Kang, Y. Chung, H.-J. Lee, M. Seo, M. Heiblum, D. Mahalu, and V. Umansky, Nat. Phys. 4, 205 (2008).
[Crossref]

Lee, J.-W.

J.-W. Lee, C. H. Kim, E. K. Lee, J. Kim, and S. Lee, Quantum Inf. Process. 1, 129 (2002).
[Crossref]

Lee, S.

J.-W. Lee, C. H. Kim, E. K. Lee, J. Kim, and S. Lee, Quantum Inf. Process. 1, 129 (2002).
[Crossref]

Leighton, R. B.

R. P. Feynman, R. B. Leighton, and M. Sands, Lectures on Physics, Vol. iii (1965).

Lim, H.-T.

Y.-C. Jeong, C. Di Franco, H.-T. Lim, M. Kim, and Y.-H. Kim, Nat. Commun. 4, 2471 (2013).
[Crossref]

Lostaglio, M.

M. Lostaglio, K. Korzekwa, D. Jennings, and T. Rudolph, Phys. Rev. X 5, 021001 (2015).
[Crossref]

Mahalu, D.

D.-I. Chang, G. L. Khym, K. Kang, Y. Chung, H.-J. Lee, M. Seo, M. Heiblum, D. Mahalu, and V. Umansky, Nat. Phys. 4, 205 (2008).
[Crossref]

Maleki, A.

Maleki, Y.

Malhotra, T.

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, Phys. Rev. Lett. 117, 153901 (2016).
[Crossref]

Man, Z.-X.

A. S. Rab, E. Polino, Z.-X. Man, N. B. An, Y.-J. Xia, N. Spagnolo, R. L. Franco, and F. Sciarrino, Nat. Commun. 8, 915 (2017).
[Crossref]

Menzel, R.

R. Menzel, D. Puhlmann, A. Heuer, and W. P. Schleich, Proc. Natl. Acad. Sci. 109, 9314 (2012).
[Crossref]

Milman, P.

F. Kaiser, T. Coudreau, P. Milman, D. B. Ostrowsky, and S. Tanzilli, Science 338, 637 (2012).
[Crossref]

Mosseri, R.

R. Mosseri and R. Dandoloff, J. Phys. A 34, 10243 (2001).
[Crossref]

Nairz, O.

M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. Van der Zouw, and A. Zeilinger, Nature 401, 680 (1999).
[Crossref]

Najarbashi, G.

G. Najarbashi and B. Seifi, Quantum Inf. Process. 16, 40 (2017).
[Crossref]

G. Najarbashi, S. Ahadpour, M. Fasihi, and Y. Tavakoli, J. Phys. A 40, 6481 (2007).
[Crossref]

Nonn, T.

S. Dürr, T. Nonn, and G. Rempe, Nature 395, 33 (1998).
[Crossref]

O’Brien, J. L.

A. Peruzzo, P. Shadbolt, N. Brunner, S. Popescu, and J. L. O’Brien, Science 338, 634 (2012).
[Crossref]

Ostrowsky, D. B.

F. Kaiser, T. Coudreau, P. Milman, D. B. Ostrowsky, and S. Tanzilli, Science 338, 637 (2012).
[Crossref]

Peruzzo, A.

A. Peruzzo, P. Shadbolt, N. Brunner, S. Popescu, and J. L. O’Brien, Science 338, 634 (2012).
[Crossref]

Plenio, M. B.

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

Polino, E.

A. S. Rab, E. Polino, Z.-X. Man, N. B. An, Y.-J. Xia, N. Spagnolo, R. L. Franco, and F. Sciarrino, Nat. Commun. 8, 915 (2017).
[Crossref]

Popescu, S.

A. Peruzzo, P. Shadbolt, N. Brunner, S. Popescu, and J. L. O’Brien, Science 338, 634 (2012).
[Crossref]

Puhlmann, D.

R. Menzel, D. Puhlmann, A. Heuer, and W. P. Schleich, Proc. Natl. Acad. Sci. 109, 9314 (2012).
[Crossref]

Qian, X.-F.

X.-F. Qian, A. Vamivakas, and J. Eberly, Optica 5, 942 (2018).
[Crossref]

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, Phys. Rev. Lett. 117, 153901 (2016).
[Crossref]

X.-F. Qian and G. Agarwal, “Are quantum objects born with duality?” arXiv:1901.07595 (2019).

Rab, A. S.

A. S. Rab, E. Polino, Z.-X. Man, N. B. An, Y.-J. Xia, N. Spagnolo, R. L. Franco, and F. Sciarrino, Nat. Commun. 8, 915 (2017).
[Crossref]

Rempe, G.

S. Dürr, T. Nonn, and G. Rempe, Nature 395, 33 (1998).
[Crossref]

Roch, J.-F.

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, Phys. Rev. Lett. 100, 220402 (2008).
[Crossref]

V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J.-F. Roch, Science 315, 966 (2007).
[Crossref]

Rudolph, T.

M. Lostaglio, K. Korzekwa, D. Jennings, and T. Rudolph, Phys. Rev. X 5, 021001 (2015).
[Crossref]

Sands, M.

R. P. Feynman, R. B. Leighton, and M. Sands, Lectures on Physics, Vol. iii (1965).

Schleich, W. P.

R. Menzel, D. Puhlmann, A. Heuer, and W. P. Schleich, Proc. Natl. Acad. Sci. 109, 9314 (2012).
[Crossref]

Sciarrino, F.

A. S. Rab, E. Polino, Z.-X. Man, N. B. An, Y.-J. Xia, N. Spagnolo, R. L. Franco, and F. Sciarrino, Nat. Commun. 8, 915 (2017).
[Crossref]

Scully, M. O.

Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. 84, 1 (2000).
[Crossref]

M. O. Scully, B.-G. Englert, and H. Walther, Nature 351, 111(1991).
[Crossref]

Seifi, B.

G. Najarbashi and B. Seifi, Quantum Inf. Process. 16, 40 (2017).
[Crossref]

Seo, M.

D.-I. Chang, G. L. Khym, K. Kang, Y. Chung, H.-J. Lee, M. Seo, M. Heiblum, D. Mahalu, and V. Umansky, Nat. Phys. 4, 205 (2008).
[Crossref]

Shadbolt, P.

A. Peruzzo, P. Shadbolt, N. Brunner, S. Popescu, and J. L. O’Brien, Science 338, 634 (2012).
[Crossref]

Shih, Y.

Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. 84, 1 (2000).
[Crossref]

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Figures (1)

Fig. 1.
Fig. 1. Geometry of the unit ball B 3 , sliced as shells of the radius 1 C 2 . The boundary of the ball gives the S 2 sphere of the pure states, shown as the red (largest) shell. The inner shells represent entangled photon states, and the smaller shells correspond to the larger entanglement. The center of the ball gives the maximally entangled states. The north and the south poles are associated with the pure particle nature of the photon with D = 1 . The points on the x 0 axis (shown in blue in the left panel) are the states with no wave nature. The great disc on x 0 = 0 corresponds to the states with no particle natures, where the boundary of the disc gives the great circle of the geometry corresponding to states having only the wave nature.

Equations (22)

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| ψ = μ | 0 | χ 1 + ν | 1 | χ 2 ,
| ψ = α 0 | 0 | e + α 1 | 0 | f + α 2 | 1 | e + α 3 | 1 | f .
V = p D max p D min p D max + p D min .
V = 2 | α ¯ 2 α 0 + α ¯ 3 α 1 | .
D = | p 0 p 1 | p 0 + p 1 ,
D = | ( | α 0 | 2 + | α 1 | 2 ) ( | α 2 | 2 + | α 3 | 2 ) | .
ρ ph = ( | α 0 | 2 + | α 1 | 2 ( α ¯ 2 α 0 + α ¯ 3 α 1 ) ( α 2 α ¯ 0 + α 3 α ¯ 1 ) | α 2 | 2 + | α 3 | 2 ) .
F ( | ψ ) := | ψ q = q 1 | 0 q + q 2 | 1 q = ( q 1 q 2 ) ,
q 1 = α 0 + α 1 e ^ 2 , q 2 = α 2 + α 3 e ^ 2 ,
P : ( q 1 , q 2 ) Q = q 1 q 2 1 = 1 | q 2 | 2 ( π 1 + π 2 e ^ 2 ) ,
Q : Q { { x i } , i = 0 , 1 , 2 , 3 , 4 } with i = 0 4 x i 2 = 1 ,
x 0 = ψ | ( σ z I ) | ψ = | q 1 | 2 | q 2 | 2 ,
x 1 = ψ | ( σ x I ) | ψ = 2 Re ( π 1 ) ,
x 2 = ψ | ( σ y I ) | ψ = 2 Im ( π 1 ) ,
x 3 = Re [ ψ | J ( σ y σ y ) | ψ ] = 2 Re ( π 2 ) ,
x 4 = Im [ ψ | J ( σ y σ y ) | ψ ] = 2 Im ( π 2 ) .
D 2 = x 0 2 ,
V 2 = x 1 2 + x 2 2 ,
C 2 = x 3 2 + x 4 2 .
Q = 1 | q 2 | 2 ( Re ( π 2 ) e ^ 2 + Im ( π 2 ) e ^ 3 ) = 1 2 | q 2 | 2 ( x 3 e ^ 2 + x 4 e ^ 3 ) .
| ψ = cos ( θ / 2 ) | 0 + e i φ sin ( θ / 2 ) | 1 .
V 2 + D 2 = 2 Tr ( ρ ph 2 ) 1 ,

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