Abstract

Most investigations of multipartite entanglement have been concerned with temporal modes of the electromagnetic field and have neglected its spatial structure. We present a simple model which allows us to generate tripartite entanglement between spatial modes by parametric down-conversion with two symmetrically tilted plane waves serving as a pump. The characteristics of this entanglement are investigated. We also discuss the generalization of our scheme to 2N+1 partite entanglement using 2N symmetrically tilted plane pump waves. Another interesting feature is the possibility of entanglement localization in just two spatial modes.

© 2010 Optical Society of America

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  1. V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2006).
    [CrossRef]
  2. G. Adesso, A. Serafini, and F. Illuminati, “Quantification and scaling of multipartite entanglement in continuous variable systems,” Phys. Rev. Lett. 93, 220504 (2004).
    [CrossRef] [PubMed]
  3. A. Serafini, G. Adesso, and F. Illuminati, “Unitarily localizable entanglement of Gaussian states,” Phys. Rev. A 71, 032349 (2005).
    [CrossRef]
  4. G. Adesso, A. Serafini, and F. Illuminati, “Multipartite entanglement in three-mode Gaussian states of continuous-variable systems: Quantification, sharing structure, and decoherence,” Phys. Rev. A 73, 032345 (2006).
    [CrossRef]
  5. G. Adesso and F. Illuminati, “Bipartite and multipartite entanglement of Gaussian states,” in Quantum Information with Continuous Variables of Atoms and Light, N.J.Cerf, G.Leuchs, and E.S.Polzik, eds. (Imperial College Press, 2007), pp. 1-21.
    [CrossRef]
  6. T. J. Osborne and F. Verstraete, “General monogamy inequality for bipartite qubit entanglement,” Phys. Rev. Lett. 96, 220503 (2006).
    [CrossRef] [PubMed]
  7. T. Hiroshima, G. Adesso, and F. Illuminati, “Monogamy inequality for distributed Gaussian entanglement,” Phys. Rev. Lett. 98, 050503 (2007).
    [CrossRef] [PubMed]
  8. Y. Lian, C. Xie, and K. Peng, “Continuous variable multipartite entanglement and optical implementations of quantum communication networks,” New J. Phys. 9, 314 (2007).
    [CrossRef]
  9. P. van Loock and S. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. 84, 3482 (2000).
    [CrossRef] [PubMed]
  10. O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
    [CrossRef]
  11. J. Guo, H. Zou, Z. Zhai, J. Zhang, and J. Gao, “Generation of continuous-variable tripartite entanglement using cascaded nonlinearities,” Phys. Rev. A 71, 034305 (2005).
    [CrossRef]
  12. Y. B. Yu, Z. D. Xie, X. Q. Yu, H. X. Li, P. Xu, H. M. Yao, and S. N. Zhu, “Generation of three-mode continuous-variable entanglement by cascaded nonlinear interactions in a quasiperiodic superlattice,” Phys. Rev. A 74, 042332 (2006).
    [CrossRef]
  13. A. Ferraro, M. G. A. Paris, M. Bondani, A. Allevi, E. Puddu, and A. Andreoni, “Three-mode entanglement by interlinked nonlinear interactions in optical χ(2) media,” J. Opt. Soc. Am. B 21, 1241-1249 (2004).
    [CrossRef]
  14. M. K. Olsen and A. S. Bradley, “Asymmetric polychromatic tripartite entanglement from interlinked χ(2) parametric interactions,” Phys. Rev. A 74, 063809 (2006).
    [CrossRef]
  15. A. V. Rodionov and A. S. Chirkin, “Entangled photon states in consecutive nonlinear optical interactions,” JETP Lett. 79, 253-256 (2004).
    [CrossRef]
  16. M. N. O'Sullivan-Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: Experimental realization of optically entangled d=3 and d=6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).
    [CrossRef] [PubMed]
  17. M. Stütz, S. Groblacher, T. Jennewein, and A. Zeilinger, “How to create and detect N-dimensional entangled photons with an active phase hologram,” Appl. Phys. Lett. 90, 261114 (2007).
    [CrossRef]
  18. P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping affects type-II spontaneous parametric down-conversion,” Phys. Rev. A 72, 033803 (2005).
    [CrossRef]
  19. J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
    [CrossRef] [PubMed]
  20. P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
    [CrossRef]
  21. M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539-1589 (1999).
    [CrossRef]
  22. A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett. 77, 1413 (1996).
    [CrossRef] [PubMed]
  23. M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A 223, 1-8 (1996).
    [CrossRef]
  24. R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726 (2000).
    [CrossRef] [PubMed]
  25. R. F. Werner and M. M. Wolf, “Bound entangled Gaussian states,” Phys. Rev. Lett. 86, 3658 (2001).
    [CrossRef] [PubMed]

2008 (1)

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[CrossRef] [PubMed]

2007 (3)

T. Hiroshima, G. Adesso, and F. Illuminati, “Monogamy inequality for distributed Gaussian entanglement,” Phys. Rev. Lett. 98, 050503 (2007).
[CrossRef] [PubMed]

Y. Lian, C. Xie, and K. Peng, “Continuous variable multipartite entanglement and optical implementations of quantum communication networks,” New J. Phys. 9, 314 (2007).
[CrossRef]

M. Stütz, S. Groblacher, T. Jennewein, and A. Zeilinger, “How to create and detect N-dimensional entangled photons with an active phase hologram,” Appl. Phys. Lett. 90, 261114 (2007).
[CrossRef]

2006 (5)

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2006).
[CrossRef]

G. Adesso, A. Serafini, and F. Illuminati, “Multipartite entanglement in three-mode Gaussian states of continuous-variable systems: Quantification, sharing structure, and decoherence,” Phys. Rev. A 73, 032345 (2006).
[CrossRef]

T. J. Osborne and F. Verstraete, “General monogamy inequality for bipartite qubit entanglement,” Phys. Rev. Lett. 96, 220503 (2006).
[CrossRef] [PubMed]

Y. B. Yu, Z. D. Xie, X. Q. Yu, H. X. Li, P. Xu, H. M. Yao, and S. N. Zhu, “Generation of three-mode continuous-variable entanglement by cascaded nonlinear interactions in a quasiperiodic superlattice,” Phys. Rev. A 74, 042332 (2006).
[CrossRef]

M. K. Olsen and A. S. Bradley, “Asymmetric polychromatic tripartite entanglement from interlinked χ(2) parametric interactions,” Phys. Rev. A 74, 063809 (2006).
[CrossRef]

2005 (4)

J. Guo, H. Zou, Z. Zhai, J. Zhang, and J. Gao, “Generation of continuous-variable tripartite entanglement using cascaded nonlinearities,” Phys. Rev. A 71, 034305 (2005).
[CrossRef]

M. N. O'Sullivan-Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: Experimental realization of optically entangled d=3 and d=6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef] [PubMed]

A. Serafini, G. Adesso, and F. Illuminati, “Unitarily localizable entanglement of Gaussian states,” Phys. Rev. A 71, 032349 (2005).
[CrossRef]

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping affects type-II spontaneous parametric down-conversion,” Phys. Rev. A 72, 033803 (2005).
[CrossRef]

2004 (4)

A. Ferraro, M. G. A. Paris, M. Bondani, A. Allevi, E. Puddu, and A. Andreoni, “Three-mode entanglement by interlinked nonlinear interactions in optical χ(2) media,” J. Opt. Soc. Am. B 21, 1241-1249 (2004).
[CrossRef]

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[CrossRef]

G. Adesso, A. Serafini, and F. Illuminati, “Quantification and scaling of multipartite entanglement in continuous variable systems,” Phys. Rev. Lett. 93, 220504 (2004).
[CrossRef] [PubMed]

A. V. Rodionov and A. S. Chirkin, “Entangled photon states in consecutive nonlinear optical interactions,” JETP Lett. 79, 253-256 (2004).
[CrossRef]

2003 (1)

P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
[CrossRef]

2001 (1)

R. F. Werner and M. M. Wolf, “Bound entangled Gaussian states,” Phys. Rev. Lett. 86, 3658 (2001).
[CrossRef] [PubMed]

2000 (2)

R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726 (2000).
[CrossRef] [PubMed]

P. van Loock and S. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. 84, 3482 (2000).
[CrossRef] [PubMed]

1999 (1)

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539-1589 (1999).
[CrossRef]

1996 (2)

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

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A 223, 1-8 (1996).
[CrossRef]

Adesso, G.

T. Hiroshima, G. Adesso, and F. Illuminati, “Monogamy inequality for distributed Gaussian entanglement,” Phys. Rev. Lett. 98, 050503 (2007).
[CrossRef] [PubMed]

G. Adesso, A. Serafini, and F. Illuminati, “Multipartite entanglement in three-mode Gaussian states of continuous-variable systems: Quantification, sharing structure, and decoherence,” Phys. Rev. A 73, 032345 (2006).
[CrossRef]

A. Serafini, G. Adesso, and F. Illuminati, “Unitarily localizable entanglement of Gaussian states,” Phys. Rev. A 71, 032349 (2005).
[CrossRef]

G. Adesso, A. Serafini, and F. Illuminati, “Quantification and scaling of multipartite entanglement in continuous variable systems,” Phys. Rev. Lett. 93, 220504 (2004).
[CrossRef] [PubMed]

G. Adesso and F. Illuminati, “Bipartite and multipartite entanglement of Gaussian states,” in Quantum Information with Continuous Variables of Atoms and Light, N.J.Cerf, G.Leuchs, and E.S.Polzik, eds. (Imperial College Press, 2007), pp. 1-21.
[CrossRef]

Aiello, A.

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[CrossRef] [PubMed]

Allevi, A.

Andreoni, A.

Bondani, M.

Boyd, R. W.

M. N. O'Sullivan-Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: Experimental realization of optically entangled d=3 and d=6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef] [PubMed]

Bradley, A. S.

M. K. Olsen and A. S. Bradley, “Asymmetric polychromatic tripartite entanglement from interlinked χ(2) parametric interactions,” Phys. Rev. A 74, 063809 (2006).
[CrossRef]

Braunstein, S.

P. van Loock and S. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. 84, 3482 (2000).
[CrossRef] [PubMed]

Chirkin, A. S.

A. V. Rodionov and A. S. Chirkin, “Entangled photon states in consecutive nonlinear optical interactions,” JETP Lett. 79, 253-256 (2004).
[CrossRef]

Coffman, V.

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2006).
[CrossRef]

Eliel, E. R.

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[CrossRef] [PubMed]

Feng, S.

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[CrossRef]

Ferraro, A.

Furusawa, A.

P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
[CrossRef]

Gao, J.

J. Guo, H. Zou, Z. Zhai, J. Zhang, and J. Gao, “Generation of continuous-variable tripartite entanglement using cascaded nonlinearities,” Phys. Rev. A 71, 034305 (2005).
[CrossRef]

Groblacher, S.

M. Stütz, S. Groblacher, T. Jennewein, and A. Zeilinger, “How to create and detect N-dimensional entangled photons with an active phase hologram,” Appl. Phys. Lett. 90, 261114 (2007).
[CrossRef]

Guo, J.

J. Guo, H. Zou, Z. Zhai, J. Zhang, and J. Gao, “Generation of continuous-variable tripartite entanglement using cascaded nonlinearities,” Phys. Rev. A 71, 034305 (2005).
[CrossRef]

Hiroshima, T.

T. Hiroshima, G. Adesso, and F. Illuminati, “Monogamy inequality for distributed Gaussian entanglement,” Phys. Rev. Lett. 98, 050503 (2007).
[CrossRef] [PubMed]

Horodecki, M.

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A 223, 1-8 (1996).
[CrossRef]

Horodecki, P.

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A 223, 1-8 (1996).
[CrossRef]

Horodecki, R.

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A 223, 1-8 (1996).
[CrossRef]

Howell, J. C.

M. N. O'Sullivan-Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: Experimental realization of optically entangled d=3 and d=6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef] [PubMed]

Illuminati, F.

T. Hiroshima, G. Adesso, and F. Illuminati, “Monogamy inequality for distributed Gaussian entanglement,” Phys. Rev. Lett. 98, 050503 (2007).
[CrossRef] [PubMed]

G. Adesso, A. Serafini, and F. Illuminati, “Multipartite entanglement in three-mode Gaussian states of continuous-variable systems: Quantification, sharing structure, and decoherence,” Phys. Rev. A 73, 032345 (2006).
[CrossRef]

A. Serafini, G. Adesso, and F. Illuminati, “Unitarily localizable entanglement of Gaussian states,” Phys. Rev. A 71, 032349 (2005).
[CrossRef]

G. Adesso, A. Serafini, and F. Illuminati, “Quantification and scaling of multipartite entanglement in continuous variable systems,” Phys. Rev. Lett. 93, 220504 (2004).
[CrossRef] [PubMed]

G. Adesso and F. Illuminati, “Bipartite and multipartite entanglement of Gaussian states,” in Quantum Information with Continuous Variables of Atoms and Light, N.J.Cerf, G.Leuchs, and E.S.Polzik, eds. (Imperial College Press, 2007), pp. 1-21.
[CrossRef]

Jennewein, T.

M. Stütz, S. Groblacher, T. Jennewein, and A. Zeilinger, “How to create and detect N-dimensional entangled photons with an active phase hologram,” Appl. Phys. Lett. 90, 261114 (2007).
[CrossRef]

Jennings, G.

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[CrossRef]

Khan, I. A.

M. N. O'Sullivan-Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: Experimental realization of optically entangled d=3 and d=6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef] [PubMed]

Kolobov, M. I.

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539-1589 (1999).
[CrossRef]

Kundu, J.

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2006).
[CrossRef]

Lee, P. S. K.

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping affects type-II spontaneous parametric down-conversion,” Phys. Rev. A 72, 033803 (2005).
[CrossRef]

Li, H. X.

Y. B. Yu, Z. D. Xie, X. Q. Yu, H. X. Li, P. Xu, H. M. Yao, and S. N. Zhu, “Generation of three-mode continuous-variable entanglement by cascaded nonlinear interactions in a quasiperiodic superlattice,” Phys. Rev. A 74, 042332 (2006).
[CrossRef]

Lian, Y.

Y. Lian, C. Xie, and K. Peng, “Continuous variable multipartite entanglement and optical implementations of quantum communication networks,” New J. Phys. 9, 314 (2007).
[CrossRef]

Oemrawsingh, S. S. R.

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[CrossRef] [PubMed]

Olsen, M. K.

M. K. Olsen and A. S. Bradley, “Asymmetric polychromatic tripartite entanglement from interlinked χ(2) parametric interactions,” Phys. Rev. A 74, 063809 (2006).
[CrossRef]

Osborne, T. J.

T. J. Osborne and F. Verstraete, “General monogamy inequality for bipartite qubit entanglement,” Phys. Rev. Lett. 96, 220503 (2006).
[CrossRef] [PubMed]

O'Sullivan-Hale, M. N.

M. N. O'Sullivan-Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: Experimental realization of optically entangled d=3 and d=6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).
[CrossRef] [PubMed]

Paris, M. G. A.

Peng, K.

Y. Lian, C. Xie, and K. Peng, “Continuous variable multipartite entanglement and optical implementations of quantum communication networks,” New J. Phys. 9, 314 (2007).
[CrossRef]

Peres, A.

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

Pfister, O.

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[CrossRef]

Pooser, R.

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[CrossRef]

Pors, J. B.

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[CrossRef] [PubMed]

Puddu, E.

Rodionov, A. V.

A. V. Rodionov and A. S. Chirkin, “Entangled photon states in consecutive nonlinear optical interactions,” JETP Lett. 79, 253-256 (2004).
[CrossRef]

Serafini, A.

G. Adesso, A. Serafini, and F. Illuminati, “Multipartite entanglement in three-mode Gaussian states of continuous-variable systems: Quantification, sharing structure, and decoherence,” Phys. Rev. A 73, 032345 (2006).
[CrossRef]

A. Serafini, G. Adesso, and F. Illuminati, “Unitarily localizable entanglement of Gaussian states,” Phys. Rev. A 71, 032349 (2005).
[CrossRef]

G. Adesso, A. Serafini, and F. Illuminati, “Quantification and scaling of multipartite entanglement in continuous variable systems,” Phys. Rev. Lett. 93, 220504 (2004).
[CrossRef] [PubMed]

Simon, R.

R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726 (2000).
[CrossRef] [PubMed]

Stütz, M.

M. Stütz, S. Groblacher, T. Jennewein, and A. Zeilinger, “How to create and detect N-dimensional entangled photons with an active phase hologram,” Appl. Phys. Lett. 90, 261114 (2007).
[CrossRef]

't Hooft, G. W.

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[CrossRef] [PubMed]

van Exter, M. P.

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[CrossRef] [PubMed]

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping affects type-II spontaneous parametric down-conversion,” Phys. Rev. A 72, 033803 (2005).
[CrossRef]

van Loock, P.

P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
[CrossRef]

P. van Loock and S. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. 84, 3482 (2000).
[CrossRef] [PubMed]

Verstraete, F.

T. J. Osborne and F. Verstraete, “General monogamy inequality for bipartite qubit entanglement,” Phys. Rev. Lett. 96, 220503 (2006).
[CrossRef] [PubMed]

Werner, R. F.

R. F. Werner and M. M. Wolf, “Bound entangled Gaussian states,” Phys. Rev. Lett. 86, 3658 (2001).
[CrossRef] [PubMed]

Woerdman, J. P.

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[CrossRef] [PubMed]

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping affects type-II spontaneous parametric down-conversion,” Phys. Rev. A 72, 033803 (2005).
[CrossRef]

Wolf, M. M.

R. F. Werner and M. M. Wolf, “Bound entangled Gaussian states,” Phys. Rev. Lett. 86, 3658 (2001).
[CrossRef] [PubMed]

Wootters, W. K.

V. Coffman, J. Kundu, and W. K. Wootters, “Distributed entanglement,” Phys. Rev. A 61, 052306 (2006).
[CrossRef]

Xie, C.

Y. Lian, C. Xie, and K. Peng, “Continuous variable multipartite entanglement and optical implementations of quantum communication networks,” New J. Phys. 9, 314 (2007).
[CrossRef]

Xie, D.

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[CrossRef]

Xie, Z. D.

Y. B. Yu, Z. D. Xie, X. Q. Yu, H. X. Li, P. Xu, H. M. Yao, and S. N. Zhu, “Generation of three-mode continuous-variable entanglement by cascaded nonlinear interactions in a quasiperiodic superlattice,” Phys. Rev. A 74, 042332 (2006).
[CrossRef]

Xu, P.

Y. B. Yu, Z. D. Xie, X. Q. Yu, H. X. Li, P. Xu, H. M. Yao, and S. N. Zhu, “Generation of three-mode continuous-variable entanglement by cascaded nonlinear interactions in a quasiperiodic superlattice,” Phys. Rev. A 74, 042332 (2006).
[CrossRef]

Yao, H. M.

Y. B. Yu, Z. D. Xie, X. Q. Yu, H. X. Li, P. Xu, H. M. Yao, and S. N. Zhu, “Generation of three-mode continuous-variable entanglement by cascaded nonlinear interactions in a quasiperiodic superlattice,” Phys. Rev. A 74, 042332 (2006).
[CrossRef]

Yu, X. Q.

Y. B. Yu, Z. D. Xie, X. Q. Yu, H. X. Li, P. Xu, H. M. Yao, and S. N. Zhu, “Generation of three-mode continuous-variable entanglement by cascaded nonlinear interactions in a quasiperiodic superlattice,” Phys. Rev. A 74, 042332 (2006).
[CrossRef]

Yu, Y. B.

Y. B. Yu, Z. D. Xie, X. Q. Yu, H. X. Li, P. Xu, H. M. Yao, and S. N. Zhu, “Generation of three-mode continuous-variable entanglement by cascaded nonlinear interactions in a quasiperiodic superlattice,” Phys. Rev. A 74, 042332 (2006).
[CrossRef]

Zeilinger, A.

M. Stütz, S. Groblacher, T. Jennewein, and A. Zeilinger, “How to create and detect N-dimensional entangled photons with an active phase hologram,” Appl. Phys. Lett. 90, 261114 (2007).
[CrossRef]

Zhai, Z.

J. Guo, H. Zou, Z. Zhai, J. Zhang, and J. Gao, “Generation of continuous-variable tripartite entanglement using cascaded nonlinearities,” Phys. Rev. A 71, 034305 (2005).
[CrossRef]

Zhang, J.

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Figures (2)

Fig. 1
Fig. 1

Scheme for generating tripartite entanglement between spatial modes. The tilted pumps have wave vectors k p ( ± q 0 ) . The transverse (vertical) components are ± q 0 .

Fig. 2
Fig. 2

Generalization of the scheme for 2 N symmetrically tilted pump waves with N = 2 and 4. The little circles represent the projections of the pump wave vectors in the x y plane of the crystal entering face.

Equations (29)

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E p ( x ) = α ( e i q 0 x + e i q 0 x ) ,
a ̂ ( z , q ) a ̂ 0 ( z ) ,
a ̂ ( z , ± q 0 + q ) a ̂ ± ( z ) .
d d z a ̂ 0 = α λ ( a ̂ + + a ̂ ) e i Δ z ,
d d z a ̂ + = α λ a ̂ 0 e i Δ z ,
d d z a ̂ = α λ a ̂ 0 e i Δ z ,
a ̂ + ( l ) = 1 2 [ U ( l ) + 1 ] a ̂ + ( 0 ) + 1 2 [ U ( l ) 1 ] a ̂ ( 0 ) + 1 2 V ( l ) a ̂ 0 ( 0 ) ,
a ̂ ( l ) = 1 2 [ U ( l ) 1 ] a ̂ + ( 0 ) + 1 2 [ U ( l ) + 1 ] a ̂ ( 0 ) + 1 2 V ( l ) a ̂ 0 ( 0 ) ,
a ̂ 0 ( l ) = U ( l ) a ̂ 0 ( 0 ) + 1 2 V ( l ) [ a ̂ + ( 0 ) + a ̂ ( 0 ) ] ,
U ( z ) = e i Δ z 2 ( cosh ( Γ z ) i Δ 2 Γ sinh ( Γ z ) ) ,
V ( z ) = e i Δ z 2 2 α λ Γ sinh ( Γ z ) ,
x ̂ 0 ( l ) 2 Re a ̂ 0 ( l ) ,
p ̂ 0 ( l ) 2 Im a ̂ 0 ( l ) ,
x ̂ ± ( l ) 2 Re a ̂ ± ( l ) ,
p ̂ ± ( l ) 2 Im a ̂ ± ( l ) .
σ i j = Tr [ ρ ( Δ ξ ̂ i Δ ξ ̂ j + Δ ξ ̂ j Δ ξ ̂ i ) 2 ] ,
σ = ( cosh ( 2 r ) 0 sinh ( 2 r ) 2 0 sinh ( 2 r ) 2 0 0 cosh ( 2 r ) 0 sinh ( 2 r ) 2 0 sinh ( 2 r ) 2 sinh ( 2 r ) 2 0 cosh 2 r 0 sinh 2 r 0 0 sinh ( 2 r ) 2 0 cosh 2 r 0 sinh 2 r sinh ( 2 r ) 2 0 sinh 2 r 0 cosh 2 r 0 0 sinh ( 2 r ) 2 0 sinh 2 r 0 cosh 2 r ) .
C { x ̂ 0 ( l ) x ̂ + ( l ) + x ̂ ( l ) 2 } 2 ρ + { p ̂ 0 ( l ) + p ̂ ( l ) + p ̂ ( l ) 2 } 2 ρ 1 2 .
C = 4 { cosh ( 2 r ) sinh ( 2 r ) } = 4 e 2 r ,
α λ l > 3 ln 2 2 2 0.735 .
x ̂ 0 ( l ) x ̂ 0 ( l ) , p ̂ 0 ( l ) p ̂ 0 ( l ) ,
x ̂ 1 ( l ) x ̂ + ( l ) + x ̂ ( l ) 2 , p ̂ 1 ( l ) p ̂ + ( l ) + p ̂ ( l ) 2 ,
x ̂ 2 ( l ) x ̂ + ( l ) x ̂ ( l ) 2 , p ̂ 2 ( l ) p ̂ + ( l ) p ̂ ( l ) 2 ,
σ = S T σ S ,
S = I 1 ( 1 2 0 1 2 0 0 1 2 0 1 2 1 2 0 1 2 0 0 1 2 0 1 2 ) ,
σ = ( cosh ( 2 r ) 0 sinh ( 2 r ) 0 0 cosh ( 2 r ) 0 sinh ( 2 r ) sinh ( 2 r ) 0 cosh ( 2 r ) 0 0 sinh ( 2 r ) 0 cosh ( 2 r ) ) I 1 .
σ ̃ = ( cosh ( 2 r ) 0 sinh ( 2 r ) 0 0 cosh ( 2 r ) 0 sinh ( 2 r ) sinh ( 2 r ) 0 cosh ( 2 r ) 0 0 sinh ( 2 r ) 0 cosh ( 2 r ) ) I 1 .
E N ( ρ ) = ln ρ ̃ 1 = max ( 0 , ln ν ̃ ) .
E N ( ρ ) = 2 r .

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