Abstract

We present an experimental technique for a complete characterization of entanglement in a two-qutrit state generated using transverse spatial correlations of two parametric down-converted photons. We verify entanglement for a particular case via entanglement witness operators which are decomposed into a sum of local observables of single path and superposition projection operators. Experimentally, these operators are accomplished by using a spatial light modulator and a polarizing beam splitter which allow to modulate the amplitude of individually chosen path states. The quantification of entanglement is computed by the negativity obtained from the expectation values of the entanglement witnesses implemented.

© 2012 OSA

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    [CrossRef] [PubMed]
  2. M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A223, 1–8 (1996).
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  3. B. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A271, 319–326 (2000).
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  4. M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, “Optimization of entanglement witnesses,” Phys. Rev. A62, 052310 (2000).
    [CrossRef]
  5. O. Gühne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, “Experimental detection of entanglement via witness operators and local measurements,” J. Mod. Opt.50, 1079–1102 (2003).
    [CrossRef]
  6. M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. D’Ariano, and C. Macchiavello, “Detection of entanglement with polarized photons: experimental realization of an entanglement witness,” Phys. Rev. Lett.91, 227901 (2003).
    [CrossRef] [PubMed]
  7. M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bru, M. Lewenstein, and A. Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Phys. Rev. Lett.92, 87902 (2004).
    [CrossRef]
  8. N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett.95, 210502 (2005).
    [CrossRef] [PubMed]
  9. O. Gühne and G. Tóth, “Entanglement detection,” Physics Reports474, 1–75 (2009).
    [CrossRef]
  10. After submission of this work we became aware of an entanglement witness measurement in high dimensional orbital angular momentum states: M. Agnew, J. Leach, and R.W. Boyd, “Observation of entanglement witnesses for orbital angular momentum states,” Eur. Phys. J. D66, 1–4 (2012).
    [CrossRef]
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    [CrossRef]
  12. R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A34, 3962–3968 (1986).
    [CrossRef] [PubMed]
  13. Z. Y. Ou and L. Mandel, “Violation of bells inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett.61, 50–53 (1988).
    [CrossRef] [PubMed]
  14. Y. H. Shih and C. O. Alley, “New type of einstein-podolsky-rosen-bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett.61, 2921–2924 (1988).
    [CrossRef] [PubMed]
  15. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett.75, 4337–4341 (1995).
    [CrossRef] [PubMed]
  16. I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science328, 879–881 (2010).
    [CrossRef] [PubMed]
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  18. J. G. Rarity and P. R. Tapster, “Experimental violation of bells inequality based on phase and momentum,” Phys. Rev. Lett.64, 2495–2498 (1990).
    [CrossRef] [PubMed]
  19. A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett.102, 153902 (2009).
    [CrossRef] [PubMed]
  20. L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett.94, 100501 (2005).
    [CrossRef] [PubMed]
  21. M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe, and S. P. Kulik, “Anisotropically and high entanglement of biphoton states generated in spontaneous parametric down-conversion,” Phys. Rev. Lett.99, 063901 (2007).
    [CrossRef] [PubMed]
  22. J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett.82, 2594–2597 (1999).
    [CrossRef]
  23. J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett.62, 2205–2208 (1989).
    [CrossRef] [PubMed]
  24. A. Rossi, G. Vallone, F. De Martini, and P. Mataloni, “Generation of time-bin-entangled photons without temporal postselection,” Phys. Rev. A78, 012345 (2008).
    [CrossRef]
  25. L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A69, 042305 (2004).
    [CrossRef]
  26. J. Fuchs and C. Schweigert, Symmetries, Lie algebras and representations: A graduate course for physicists (Cambridge University Press, 1997).
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    [CrossRef] [PubMed]
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    [CrossRef]
  29. S. P. Walborn, C. H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Physics Reports, 495, 87–139 (2010).
    [CrossRef]
  30. E. J. S. Fonseca, J. C. Machado da Silva, C. H. Monken, and S. Pádua, “Controlling two-particle conditional interference” Phys. Rev. A61, 023801 (2000).
    [CrossRef]
  31. D. Greenberger, M. Horne, and A. Zeilinger, “Multiparticle interferometry and the superposition principle,” Phys. Today46, 22–22 (1993).
    [CrossRef]
  32. G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A65, 032314 (2002).
    [CrossRef]

2012 (1)

After submission of this work we became aware of an entanglement witness measurement in high dimensional orbital angular momentum states: M. Agnew, J. Leach, and R.W. Boyd, “Observation of entanglement witnesses for orbital angular momentum states,” Eur. Phys. J. D66, 1–4 (2012).
[CrossRef]

2010 (3)

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science328, 879–881 (2010).
[CrossRef] [PubMed]

W. M. Pimenta, B. Marques, M. A. Carvalho, M. R. Barros, J. G. Fonseca, J. Ferraz, M. Terra Cunha, and S. Pádua, “Minimal state tomography of spatial qubits using a spatial light modulator,” Opt. Express18, 24423–24433 (2010).
[CrossRef] [PubMed]

S. P. Walborn, C. H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Physics Reports, 495, 87–139 (2010).
[CrossRef]

2009 (2)

A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett.102, 153902 (2009).
[CrossRef] [PubMed]

O. Gühne and G. Tóth, “Entanglement detection,” Physics Reports474, 1–75 (2009).
[CrossRef]

2008 (1)

A. Rossi, G. Vallone, F. De Martini, and P. Mataloni, “Generation of time-bin-entangled photons without temporal postselection,” Phys. Rev. A78, 012345 (2008).
[CrossRef]

2007 (1)

M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe, and S. P. Kulik, “Anisotropically and high entanglement of biphoton states generated in spontaneous parametric down-conversion,” Phys. Rev. Lett.99, 063901 (2007).
[CrossRef] [PubMed]

2006 (1)

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys.8, 75 (2006).
[CrossRef]

2005 (2)

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett.95, 210502 (2005).
[CrossRef] [PubMed]

L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett.94, 100501 (2005).
[CrossRef] [PubMed]

2004 (2)

L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A69, 042305 (2004).
[CrossRef]

M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bru, M. Lewenstein, and A. Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Phys. Rev. Lett.92, 87902 (2004).
[CrossRef]

2003 (3)

O. Gühne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, “Experimental detection of entanglement via witness operators and local measurements,” J. Mod. Opt.50, 1079–1102 (2003).
[CrossRef]

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. D’Ariano, and C. Macchiavello, “Detection of entanglement with polarized photons: experimental realization of an entanglement witness,” Phys. Rev. Lett.91, 227901 (2003).
[CrossRef] [PubMed]

I. Moreno, P. Velásquez, C. Fernández-Pousa, M. Sánchez-López, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys.94, 3697–3702 (2003).
[CrossRef]

2002 (1)

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A65, 032314 (2002).
[CrossRef]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London)412, 313–316 (2001).
[CrossRef]

2000 (3)

B. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A271, 319–326 (2000).
[CrossRef]

M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, “Optimization of entanglement witnesses,” Phys. Rev. A62, 052310 (2000).
[CrossRef]

E. J. S. Fonseca, J. C. Machado da Silva, C. H. Monken, and S. Pádua, “Controlling two-particle conditional interference” Phys. Rev. A61, 023801 (2000).
[CrossRef]

1999 (1)

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett.82, 2594–2597 (1999).
[CrossRef]

1996 (2)

A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett.77, 1413–1415 (1996).
[CrossRef] [PubMed]

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A223, 1–8 (1996).
[CrossRef]

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett.75, 4337–4341 (1995).
[CrossRef] [PubMed]

1993 (1)

D. Greenberger, M. Horne, and A. Zeilinger, “Multiparticle interferometry and the superposition principle,” Phys. Today46, 22–22 (1993).
[CrossRef]

1990 (1)

J. G. Rarity and P. R. Tapster, “Experimental violation of bells inequality based on phase and momentum,” Phys. Rev. Lett.64, 2495–2498 (1990).
[CrossRef] [PubMed]

1989 (1)

J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett.62, 2205–2208 (1989).
[CrossRef] [PubMed]

1988 (2)

Z. Y. Ou and L. Mandel, “Violation of bells inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett.61, 50–53 (1988).
[CrossRef] [PubMed]

Y. H. Shih and C. O. Alley, “New type of einstein-podolsky-rosen-bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett.61, 2921–2924 (1988).
[CrossRef] [PubMed]

1986 (1)

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A34, 3962–3968 (1986).
[CrossRef] [PubMed]

Afek, I.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science328, 879–881 (2010).
[CrossRef] [PubMed]

Agnew, M.

After submission of this work we became aware of an entanglement witness measurement in high dimensional orbital angular momentum states: M. Agnew, J. Leach, and R.W. Boyd, “Observation of entanglement witnesses for orbital angular momentum states,” Eur. Phys. J. D66, 1–4 (2012).
[CrossRef]

Aguirre Gómez, J. G.

L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett.94, 100501 (2005).
[CrossRef] [PubMed]

Alley, C. O.

Y. H. Shih and C. O. Alley, “New type of einstein-podolsky-rosen-bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett.61, 2921–2924 (1988).
[CrossRef] [PubMed]

Ambar, O.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science328, 879–881 (2010).
[CrossRef] [PubMed]

Barbieri, M.

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. D’Ariano, and C. Macchiavello, “Detection of entanglement with polarized photons: experimental realization of an entanglement witness,” Phys. Rev. Lett.91, 227901 (2003).
[CrossRef] [PubMed]

Barros, M. R.

Bourennane, M.

M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bru, M. Lewenstein, and A. Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Phys. Rev. Lett.92, 87902 (2004).
[CrossRef]

Boyd, R.W.

After submission of this work we became aware of an entanglement witness measurement in high dimensional orbital angular momentum states: M. Agnew, J. Leach, and R.W. Boyd, “Observation of entanglement witnesses for orbital angular momentum states,” Eur. Phys. J. D66, 1–4 (2012).
[CrossRef]

Brendel, J.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett.82, 2594–2597 (1999).
[CrossRef]

Bru, D.

M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bru, M. Lewenstein, and A. Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Phys. Rev. Lett.92, 87902 (2004).
[CrossRef]

Bruss, D.

O. Gühne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, “Experimental detection of entanglement via witness operators and local measurements,” J. Mod. Opt.50, 1079–1102 (2003).
[CrossRef]

Carvalho, M. A.

Chiuri, A.

A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett.102, 153902 (2009).
[CrossRef] [PubMed]

Cirac, J. I.

M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, “Optimization of entanglement witnesses,” Phys. Rev. A62, 052310 (2000).
[CrossRef]

D’Ariano, G.

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. D’Ariano, and C. Macchiavello, “Detection of entanglement with polarized photons: experimental realization of an entanglement witness,” Phys. Rev. Lett.91, 227901 (2003).
[CrossRef] [PubMed]

De Martini, F.

A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett.102, 153902 (2009).
[CrossRef] [PubMed]

A. Rossi, G. Vallone, F. De Martini, and P. Mataloni, “Generation of time-bin-entangled photons without temporal postselection,” Phys. Rev. A78, 012345 (2008).
[CrossRef]

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. D’Ariano, and C. Macchiavello, “Detection of entanglement with polarized photons: experimental realization of an entanglement witness,” Phys. Rev. Lett.91, 227901 (2003).
[CrossRef] [PubMed]

Di Nepi, G.

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. D’Ariano, and C. Macchiavello, “Detection of entanglement with polarized photons: experimental realization of an entanglement witness,” Phys. Rev. Lett.91, 227901 (2003).
[CrossRef] [PubMed]

Efremov, M. A.

M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe, and S. P. Kulik, “Anisotropically and high entanglement of biphoton states generated in spontaneous parametric down-conversion,” Phys. Rev. Lett.99, 063901 (2007).
[CrossRef] [PubMed]

Eibl, M.

M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bru, M. Lewenstein, and A. Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Phys. Rev. Lett.92, 87902 (2004).
[CrossRef]

Ekert, A.

O. Gühne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, “Experimental detection of entanglement via witness operators and local measurements,” J. Mod. Opt.50, 1079–1102 (2003).
[CrossRef]

Fedorov, M. V.

M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe, and S. P. Kulik, “Anisotropically and high entanglement of biphoton states generated in spontaneous parametric down-conversion,” Phys. Rev. Lett.99, 063901 (2007).
[CrossRef] [PubMed]

Fernández-Pousa, C.

I. Moreno, P. Velásquez, C. Fernández-Pousa, M. Sánchez-López, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys.94, 3697–3702 (2003).
[CrossRef]

Ferraz, J.

Fonseca, E. J. S.

E. J. S. Fonseca, J. C. Machado da Silva, C. H. Monken, and S. Pádua, “Controlling two-particle conditional interference” Phys. Rev. A61, 023801 (2000).
[CrossRef]

Fonseca, J. G.

Franson, J. D.

J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett.62, 2205–2208 (1989).
[CrossRef] [PubMed]

Fuchs, J.

J. Fuchs and C. Schweigert, Symmetries, Lie algebras and representations: A graduate course for physicists (Cambridge University Press, 1997).

Gaertner, S.

M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bru, M. Lewenstein, and A. Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Phys. Rev. Lett.92, 87902 (2004).
[CrossRef]

Ghosh, R.

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A34, 3962–3968 (1986).
[CrossRef] [PubMed]

Gisin, N.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett.82, 2594–2597 (1999).
[CrossRef]

Greenberger, D.

D. Greenberger, M. Horne, and A. Zeilinger, “Multiparticle interferometry and the superposition principle,” Phys. Today46, 22–22 (1993).
[CrossRef]

Gröblacher, S.

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys.8, 75 (2006).
[CrossRef]

Gühne, O.

O. Gühne and G. Tóth, “Entanglement detection,” Physics Reports474, 1–75 (2009).
[CrossRef]

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett.95, 210502 (2005).
[CrossRef] [PubMed]

M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bru, M. Lewenstein, and A. Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Phys. Rev. Lett.92, 87902 (2004).
[CrossRef]

O. Gühne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, “Experimental detection of entanglement via witness operators and local measurements,” J. Mod. Opt.50, 1079–1102 (2003).
[CrossRef]

Hong, C. K.

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A34, 3962–3968 (1986).
[CrossRef] [PubMed]

Horne, M.

D. Greenberger, M. Horne, and A. Zeilinger, “Multiparticle interferometry and the superposition principle,” Phys. Today46, 22–22 (1993).
[CrossRef]

Horodecki, M.

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A223, 1–8 (1996).
[CrossRef]

Horodecki, P.

M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, “Optimization of entanglement witnesses,” Phys. Rev. A62, 052310 (2000).
[CrossRef]

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A223, 1–8 (1996).
[CrossRef]

Horodecki, R.

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A223, 1–8 (1996).
[CrossRef]

Hyllus, P.

M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bru, M. Lewenstein, and A. Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Phys. Rev. Lett.92, 87902 (2004).
[CrossRef]

O. Gühne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, “Experimental detection of entanglement via witness operators and local measurements,” J. Mod. Opt.50, 1079–1102 (2003).
[CrossRef]

Jennewein, T.

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys.8, 75 (2006).
[CrossRef]

Kiesel, N.

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett.95, 210502 (2005).
[CrossRef] [PubMed]

Kraus, B.

M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, “Optimization of entanglement witnesses,” Phys. Rev. A62, 052310 (2000).
[CrossRef]

Kulik, S. P.

M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe, and S. P. Kulik, “Anisotropically and high entanglement of biphoton states generated in spontaneous parametric down-conversion,” Phys. Rev. Lett.99, 063901 (2007).
[CrossRef] [PubMed]

Kurtsiefer, C.

M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bru, M. Lewenstein, and A. Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Phys. Rev. Lett.92, 87902 (2004).
[CrossRef]

Kwiat, P. G.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett.75, 4337–4341 (1995).
[CrossRef] [PubMed]

Leach, J.

After submission of this work we became aware of an entanglement witness measurement in high dimensional orbital angular momentum states: M. Agnew, J. Leach, and R.W. Boyd, “Observation of entanglement witnesses for orbital angular momentum states,” Eur. Phys. J. D66, 1–4 (2012).
[CrossRef]

Lewenstein, M.

M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bru, M. Lewenstein, and A. Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Phys. Rev. Lett.92, 87902 (2004).
[CrossRef]

O. Gühne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, “Experimental detection of entanglement via witness operators and local measurements,” J. Mod. Opt.50, 1079–1102 (2003).
[CrossRef]

M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, “Optimization of entanglement witnesses,” Phys. Rev. A62, 052310 (2000).
[CrossRef]

Lima, G.

L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett.94, 100501 (2005).
[CrossRef] [PubMed]

Macchiavello, C.

O. Gühne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, “Experimental detection of entanglement via witness operators and local measurements,” J. Mod. Opt.50, 1079–1102 (2003).
[CrossRef]

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. D’Ariano, and C. Macchiavello, “Detection of entanglement with polarized photons: experimental realization of an entanglement witness,” Phys. Rev. Lett.91, 227901 (2003).
[CrossRef] [PubMed]

Machado da Silva, J. C.

E. J. S. Fonseca, J. C. Machado da Silva, C. H. Monken, and S. Pádua, “Controlling two-particle conditional interference” Phys. Rev. A61, 023801 (2000).
[CrossRef]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London)412, 313–316 (2001).
[CrossRef]

Mandel, L.

Z. Y. Ou and L. Mandel, “Violation of bells inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett.61, 50–53 (1988).
[CrossRef] [PubMed]

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A34, 3962–3968 (1986).
[CrossRef] [PubMed]

Marques, B.

Mataloni, P.

A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett.102, 153902 (2009).
[CrossRef] [PubMed]

A. Rossi, G. Vallone, F. De Martini, and P. Mataloni, “Generation of time-bin-entangled photons without temporal postselection,” Phys. Rev. A78, 012345 (2008).
[CrossRef]

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. D’Ariano, and C. Macchiavello, “Detection of entanglement with polarized photons: experimental realization of an entanglement witness,” Phys. Rev. Lett.91, 227901 (2003).
[CrossRef] [PubMed]

Mateos, F.

I. Moreno, P. Velásquez, C. Fernández-Pousa, M. Sánchez-López, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys.94, 3697–3702 (2003).
[CrossRef]

Mattle, K.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett.75, 4337–4341 (1995).
[CrossRef] [PubMed]

Monken, C. H.

S. P. Walborn, C. H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Physics Reports, 495, 87–139 (2010).
[CrossRef]

L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett.94, 100501 (2005).
[CrossRef] [PubMed]

E. J. S. Fonseca, J. C. Machado da Silva, C. H. Monken, and S. Pádua, “Controlling two-particle conditional interference” Phys. Rev. A61, 023801 (2000).
[CrossRef]

Moreno, I.

I. Moreno, P. Velásquez, C. Fernández-Pousa, M. Sánchez-López, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys.94, 3697–3702 (2003).
[CrossRef]

Moreva, E. V.

M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe, and S. P. Kulik, “Anisotropically and high entanglement of biphoton states generated in spontaneous parametric down-conversion,” Phys. Rev. Lett.99, 063901 (2007).
[CrossRef] [PubMed]

Neves, L.

L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett.94, 100501 (2005).
[CrossRef] [PubMed]

L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A69, 042305 (2004).
[CrossRef]

Ou, Z. Y.

Z. Y. Ou and L. Mandel, “Violation of bells inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett.61, 50–53 (1988).
[CrossRef] [PubMed]

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A34, 3962–3968 (1986).
[CrossRef] [PubMed]

Pádua, S.

W. M. Pimenta, B. Marques, M. A. Carvalho, M. R. Barros, J. G. Fonseca, J. Ferraz, M. Terra Cunha, and S. Pádua, “Minimal state tomography of spatial qubits using a spatial light modulator,” Opt. Express18, 24423–24433 (2010).
[CrossRef] [PubMed]

S. P. Walborn, C. H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Physics Reports, 495, 87–139 (2010).
[CrossRef]

L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett.94, 100501 (2005).
[CrossRef] [PubMed]

L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A69, 042305 (2004).
[CrossRef]

E. J. S. Fonseca, J. C. Machado da Silva, C. H. Monken, and S. Pádua, “Controlling two-particle conditional interference” Phys. Rev. A61, 023801 (2000).
[CrossRef]

Peres, A.

A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett.77, 1413–1415 (1996).
[CrossRef] [PubMed]

Pimenta, W. M.

Rarity, J. G.

J. G. Rarity and P. R. Tapster, “Experimental violation of bells inequality based on phase and momentum,” Phys. Rev. Lett.64, 2495–2498 (1990).
[CrossRef] [PubMed]

Rossi, A.

A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett.102, 153902 (2009).
[CrossRef] [PubMed]

A. Rossi, G. Vallone, F. De Martini, and P. Mataloni, “Generation of time-bin-entangled photons without temporal postselection,” Phys. Rev. A78, 012345 (2008).
[CrossRef]

Saavedra, C.

L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett.94, 100501 (2005).
[CrossRef] [PubMed]

L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A69, 042305 (2004).
[CrossRef]

Sánchez-López, M.

I. Moreno, P. Velásquez, C. Fernández-Pousa, M. Sánchez-López, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys.94, 3697–3702 (2003).
[CrossRef]

Sanpera, A.

M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bru, M. Lewenstein, and A. Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Phys. Rev. Lett.92, 87902 (2004).
[CrossRef]

O. Gühne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, “Experimental detection of entanglement via witness operators and local measurements,” J. Mod. Opt.50, 1079–1102 (2003).
[CrossRef]

Schmid, C.

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett.95, 210502 (2005).
[CrossRef] [PubMed]

Schweigert, C.

J. Fuchs and C. Schweigert, Symmetries, Lie algebras and representations: A graduate course for physicists (Cambridge University Press, 1997).

Sergienko, A. V.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett.75, 4337–4341 (1995).
[CrossRef] [PubMed]

Shih, Y.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett.75, 4337–4341 (1995).
[CrossRef] [PubMed]

Shih, Y. H.

Y. H. Shih and C. O. Alley, “New type of einstein-podolsky-rosen-bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett.61, 2921–2924 (1988).
[CrossRef] [PubMed]

Silberberg, Y.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science328, 879–881 (2010).
[CrossRef] [PubMed]

Souto Ribeiro, P. H.

S. P. Walborn, C. H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Physics Reports, 495, 87–139 (2010).
[CrossRef]

Straupe, S. S.

M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe, and S. P. Kulik, “Anisotropically and high entanglement of biphoton states generated in spontaneous parametric down-conversion,” Phys. Rev. Lett.99, 063901 (2007).
[CrossRef] [PubMed]

Tapster, P. R.

J. G. Rarity and P. R. Tapster, “Experimental violation of bells inequality based on phase and momentum,” Phys. Rev. Lett.64, 2495–2498 (1990).
[CrossRef] [PubMed]

Terhal, B.

B. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A271, 319–326 (2000).
[CrossRef]

Terra Cunha, M.

Tittel, W.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett.82, 2594–2597 (1999).
[CrossRef]

Tóth, G.

O. Gühne and G. Tóth, “Entanglement detection,” Physics Reports474, 1–75 (2009).
[CrossRef]

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett.95, 210502 (2005).
[CrossRef] [PubMed]

Ursin, R.

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett.95, 210502 (2005).
[CrossRef] [PubMed]

Vallone, G.

A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett.102, 153902 (2009).
[CrossRef] [PubMed]

A. Rossi, G. Vallone, F. De Martini, and P. Mataloni, “Generation of time-bin-entangled photons without temporal postselection,” Phys. Rev. A78, 012345 (2008).
[CrossRef]

Vaziri, A.

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys.8, 75 (2006).
[CrossRef]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London)412, 313–316 (2001).
[CrossRef]

Velásquez, P.

I. Moreno, P. Velásquez, C. Fernández-Pousa, M. Sánchez-López, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys.94, 3697–3702 (2003).
[CrossRef]

Vidal, G.

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A65, 032314 (2002).
[CrossRef]

Volkov, P. A.

M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe, and S. P. Kulik, “Anisotropically and high entanglement of biphoton states generated in spontaneous parametric down-conversion,” Phys. Rev. Lett.99, 063901 (2007).
[CrossRef] [PubMed]

Walborn, S. P.

S. P. Walborn, C. H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Physics Reports, 495, 87–139 (2010).
[CrossRef]

Weber, U.

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett.95, 210502 (2005).
[CrossRef] [PubMed]

Weihs, G.

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys.8, 75 (2006).
[CrossRef]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London)412, 313–316 (2001).
[CrossRef]

Weinfurter, H.

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett.95, 210502 (2005).
[CrossRef] [PubMed]

M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bru, M. Lewenstein, and A. Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Phys. Rev. Lett.92, 87902 (2004).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett.75, 4337–4341 (1995).
[CrossRef] [PubMed]

Werner, R. F.

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A65, 032314 (2002).
[CrossRef]

Zbinden, H.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett.82, 2594–2597 (1999).
[CrossRef]

Zeilinger, A.

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys.8, 75 (2006).
[CrossRef]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London)412, 313–316 (2001).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett.75, 4337–4341 (1995).
[CrossRef] [PubMed]

D. Greenberger, M. Horne, and A. Zeilinger, “Multiparticle interferometry and the superposition principle,” Phys. Today46, 22–22 (1993).
[CrossRef]

Eur. Phys. J. D (1)

After submission of this work we became aware of an entanglement witness measurement in high dimensional orbital angular momentum states: M. Agnew, J. Leach, and R.W. Boyd, “Observation of entanglement witnesses for orbital angular momentum states,” Eur. Phys. J. D66, 1–4 (2012).
[CrossRef]

J. Appl. Phys. (1)

I. Moreno, P. Velásquez, C. Fernández-Pousa, M. Sánchez-López, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys.94, 3697–3702 (2003).
[CrossRef]

J. Mod. Opt. (1)

O. Gühne, P. Hyllus, D. Bruss, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera, “Experimental detection of entanglement via witness operators and local measurements,” J. Mod. Opt.50, 1079–1102 (2003).
[CrossRef]

Nature (London) (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London)412, 313–316 (2001).
[CrossRef]

New J. Phys. (1)

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys.8, 75 (2006).
[CrossRef]

Opt. Express (1)

Phys. Lett. A (2)

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A223, 1–8 (1996).
[CrossRef]

B. Terhal, “Bell inequalities and the separability criterion,” Phys. Lett. A271, 319–326 (2000).
[CrossRef]

Phys. Rev. A (6)

M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, “Optimization of entanglement witnesses,” Phys. Rev. A62, 052310 (2000).
[CrossRef]

A. Rossi, G. Vallone, F. De Martini, and P. Mataloni, “Generation of time-bin-entangled photons without temporal postselection,” Phys. Rev. A78, 012345 (2008).
[CrossRef]

L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A69, 042305 (2004).
[CrossRef]

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A34, 3962–3968 (1986).
[CrossRef] [PubMed]

E. J. S. Fonseca, J. C. Machado da Silva, C. H. Monken, and S. Pádua, “Controlling two-particle conditional interference” Phys. Rev. A61, 023801 (2000).
[CrossRef]

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A65, 032314 (2002).
[CrossRef]

Phys. Rev. Lett. (13)

A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett.77, 1413–1415 (1996).
[CrossRef] [PubMed]

Z. Y. Ou and L. Mandel, “Violation of bells inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett.61, 50–53 (1988).
[CrossRef] [PubMed]

Y. H. Shih and C. O. Alley, “New type of einstein-podolsky-rosen-bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett.61, 2921–2924 (1988).
[CrossRef] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett.75, 4337–4341 (1995).
[CrossRef] [PubMed]

M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. D’Ariano, and C. Macchiavello, “Detection of entanglement with polarized photons: experimental realization of an entanglement witness,” Phys. Rev. Lett.91, 227901 (2003).
[CrossRef] [PubMed]

M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bru, M. Lewenstein, and A. Sanpera, “Experimental detection of multipartite entanglement using witness operators,” Phys. Rev. Lett.92, 87902 (2004).
[CrossRef]

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental analysis of a four-qubit photon cluster state,” Phys. Rev. Lett.95, 210502 (2005).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, “Experimental violation of bells inequality based on phase and momentum,” Phys. Rev. Lett.64, 2495–2498 (1990).
[CrossRef] [PubMed]

A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett.102, 153902 (2009).
[CrossRef] [PubMed]

L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett.94, 100501 (2005).
[CrossRef] [PubMed]

M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe, and S. P. Kulik, “Anisotropically and high entanglement of biphoton states generated in spontaneous parametric down-conversion,” Phys. Rev. Lett.99, 063901 (2007).
[CrossRef] [PubMed]

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett.82, 2594–2597 (1999).
[CrossRef]

J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett.62, 2205–2208 (1989).
[CrossRef] [PubMed]

Phys. Today (1)

D. Greenberger, M. Horne, and A. Zeilinger, “Multiparticle interferometry and the superposition principle,” Phys. Today46, 22–22 (1993).
[CrossRef]

Physics Reports (2)

S. P. Walborn, C. H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Physics Reports, 495, 87–139 (2010).
[CrossRef]

O. Gühne and G. Tóth, “Entanglement detection,” Physics Reports474, 1–75 (2009).
[CrossRef]

Science (1)

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science328, 879–881 (2010).
[CrossRef] [PubMed]

Other (1)

J. Fuchs and C. Schweigert, Symmetries, Lie algebras and representations: A graduate course for physicists (Cambridge University Press, 1997).

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Figures (3)

Fig. 1
Fig. 1

Experimental setup for the generation and entanglement detection of a two-qutrit state. An entangled photon source is made from type I SPDC by pumping a Bismuth Borate (BiBO) crystal using a CW blue laser beam. The two-qutrit state is produced by placing a 3-slit aperture into the photon paths. A spatial light modulator (SLM) together with a polarizing beam splitter (PBS) are used for the characterization and entanglement witness implementation of the state. The measurement operators are implemented by using convergent lenses Ls and Li when a three-slit image or an interference pattern are created at the detector planes Ds and Di. C is a photon counter which records single and coincidence counts.

Fig. 2
Fig. 2

Experimental setups to demonstrate the (a) characterization with a minimum modulation (gray level=240), and the selection of path states (b) {|0〉, |2〉}s ⊗ {|0〉, |2〉}i, (c) {|1〉, |2〉}s ⊗ {|0〉, |1〉}i, and (d) {|0〉, |1〉}s ⊗ {|1〉, |2〉}i, by setting the gray levels 240 and 0 for minimum and maximum attenuations, respectively, just behind the slits at the LCD.

Fig. 3
Fig. 3

At the image plane, (a) a double slit image is recorded by idler detector when the photon path state |1〉 is attenuated by using a gray level=0 of the SLM. As a result, (b) a two-slit interference pattern is obtained at the Fourier plane with Di scanning in the x-direction at the idler arm, while Ds is kept fixed at xs = 0 mm. The path phase differences of ±π and ±π/2 with respect to the central maximum give the states related to the measurement operators �� 02 ± and �� 02 ±. Continuous curves represent the theoretical fits of the experimental data.

Tables (3)

Tables Icon

Table 1 Nonpositive Eigensystem of ρTB

Tables Icon

Table 2 Probabilities for all basis states of �� = 3.

Tables Icon

Table 3 Expectation values of path superposition operators

Equations (14)

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| Ψ = a | 02 + b | 11 + c | 20 | a | 2 + | b | 2 + | c | 2 .
𝒲 j = 1 2 ( | l l l l | + | m m m m | e 2 i ( θ j ) | m l l m | e 2 i ( θ j ) | l m m l | ) ,
𝒳 l m ± = | x l m ± x l m ± | ; | x l m ± = 1 2 ( | l ± | m ) , 𝓎 l m ± = | y l m ± y l m ± | ; | y l m ± = 1 2 ( | l ± i | m ) .
λ 1 = ( 0 1 0 1 0 0 0 0 0 ) , λ 2 = ( 0 i 0 i 0 0 0 0 0 ) , λ 3 = ( 1 0 0 0 1 0 0 0 0 ) , λ 4 = ( 0 0 1 0 0 0 1 0 0 ) , λ 5 = ( 0 0 i 0 0 0 i 0 0 ) , λ 6 = ( 0 0 0 0 0 1 0 1 0 ) , λ 7 = ( 0 0 0 0 0 i 0 i 0 ) , λ 8 = 1 3 ( 1 0 0 0 1 0 0 0 2 ) .
λ 1 = 𝒳 01 + 𝒳 01 , λ 2 = 𝓎 01 + 𝓎 01 , λ 3 = | 0 0 | | 1 1 | , λ 4 = 𝒳 02 + 𝒳 02 , λ 5 = 𝓎 02 + 𝓎 02 , λ 6 = 𝒳 12 + 𝒳 12 , λ 7 = 𝓎 12 + 𝓎 12 , λ 8 = 1 3 ( | 0 0 | + | 1 1 | 2 | 2 2 | ) .
𝒪 Q 3 ( d ) = 1 2 k = 1 9 λ k Tr ( λ k 𝒪 Q 3 ) ,
𝒪 n Q 3 ( d ) = 1 2 n k 1 = 1 9 k n = 1 9 λ k 1 λ k n Tr ( λ k 1 λ k n 𝒪 n Q 3 ) .
𝒲 j ( d ) = 1 4 k 1 = 1 9 k 2 = 1 9 λ k 1 λ k 2 Tr ( λ k 1 λ k 2 𝒲 j ) .
𝒲 j ( d ) 1 = 1 2 ( | l l | | l l | + | m m | | m m | ) ,
𝒲 j ( d ) 2 = 1 4 ( 𝒳 l m 𝒳 l m + 𝒳 l m 𝒳 l m + + 𝒳 l m + 𝒳 l m 𝒳 l m + 𝒳 l m + 𝓎 l m 𝓎 l m + 𝓎 l m + 𝓎 l m + + 𝓎 l m + 𝓎 l m 𝓎 l m + 𝓎 l m + ) ,
𝒲 j ( d ) 3 = 1 4 ( 𝒳 l m 𝓎 l m + 𝒳 l m 𝓎 l m + + 𝒳 l m + 𝓎 l m 𝒳 l m + 𝓎 l m + + 𝓎 l m 𝒳 l m 𝓎 l m 𝒳 l m + 𝓎 l m + 𝒳 l m + 𝓎 l m + 𝒳 l m + ) .
𝒲 setup ( j ) = 1 4 ( 2 | l l | | l l | + 2 | m m | | m m | 𝒳 l m + 𝒳 l m + + 𝒳 l m + 𝒳 l m + 𝒳 l m 𝒳 l m + 𝒳 l m 𝒳 l m 𝓎 l m + 𝓎 l m + + 𝓎 l m + 𝓎 l m + 𝓎 l m 𝓎 l m + 𝓎 l m 𝓎 l m ) .
| Ψ ( 0.57 ± 0.02 ) | 02 + ( 0.59 ± 0.02 ) | 11 + ( 0.56 ± 0.02 ) | 20 ,
Ψ = 1 3 2 3 ( 𝒲 setup ( 1 ) + 𝒲 setup ( 2 ) + 𝒲 setup ( 3 ) ) ,

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