Abstract

We formulate an inseparability criterion based on the recently derived generalized Schrödinger–Robertson uncertainty relation (SRUR) [J. Phys. A 45, 195305 (2012)] together with the negativity of partial transpose (PT). This generalized SRUR systematically deals with two orthogonal quadrature amplitudes to higher orders, so it is relevant to characterize non-Gaussian quantum statistics. We first present a method that relies on the single-mode marginal distribution of two-mode fields under PT followed by beam-splitting operation. We then extend the SRUR to two-mode cases and develop a full two-mode version of the inseparability criterion. We find that our formulation can be useful to detect entanglement of non-Gaussian states even when, e.g., the entropic criterion that also involves higher-order moments fails.

© 2014 Optical Society of America

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  1. R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
    [CrossRef]
  2. L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
    [CrossRef]
  3. S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
    [CrossRef]
  4. M. G. Raymer, A. C. Funk, B. C. Sanders, and H. de Guise, “Separability criterion for separate quantum systems,” Phys. Rev. A 67, 052104 (2003).
    [CrossRef]
  5. G. S. Agarwal and A. Biswas, “Inseparability inequalities for higher order moments for bipartite systems,” New J. Phys. 7, 211 (2005).
    [CrossRef]
  6. E. Shchukin and W. Vogel, “Inseparability criteria for continuous bipartite quantum states,” Phys. Rev. Lett. 95, 230502 (2005).
    [CrossRef]
  7. M. Hillery and M. S. Zubairy, “Entanglement conditions for two-mode states,” Phys. Rev. Lett. 96, 050503 (2006).
    [CrossRef]
  8. H. Nha and J. Kim, “Entanglement criteria via the uncertainty relations in su(2) and su(1,1) algebras: detection of non-Gaussian entangled states,” Phys. Rev. A 74, 012317 (2006).
    [CrossRef]
  9. A. Miranowicz, M. Piani, P. Horodecki, and R. Horodecki, “Inseparability criteria based on matrices of moments,” Phys. Rev. A 80, 052303 (2009).
    [CrossRef]
  10. A. Miranowicz, M. Bartkowiak, X. Wang, Y.-X. Liu, and F. Nori, “Testing nonclassicality in multimode fields: a unified derivation of classical inequalities,” Phys. Rev. A 82, 013824 (2010).
    [CrossRef]
  11. S. P. Walborn, B. G. Taketani, A. Salles, F. Toscano, and R. L. de Matos Filho, “Entropic entanglement criteria for continuous variables,” Phys. Rev. Lett. 103, 160505 (2009).
    [CrossRef]
  12. H. Nha, S.-Y. Lee, S.-W. Ji, and M. S. Kim, “Efficient entanglement criteria beyond Gaussian limits using Gaussian measurements,” Phys. Rev. Lett. 108, 030503 (2012).
    [CrossRef]
  13. E. Shchukin, T. Richter, and W. Vogel, “Nonclassicality criteria in terms of moments,” Phys. Rev. A 71, 011802 (2005).
    [CrossRef]
  14. J. S. Ivan, N. Mukunda, and R. Simon, “Moments of non-Gaussian Wigner distributions and a generalized uncertainty principle: I. The single-mode case,” J. Phys. A 45, 195305 (2012).
    [CrossRef]
  15. J. S. Ivan, K. Kumar, N. Mukunda, and R. Simon, “Invariant theoretic approach to uncertainty relations for quantum systems,” arXiv:1205.5132 (2012).
  16. E. V. Shchukin and W. Vogel, “Nonclassical moments and their measurement,” Phys. Rev. A 72, 043808 (2005).
    [CrossRef]
  17. E. P. Menzel, F. Deppe, M. Mariantoni, M. A. Araque Caballero, A. Baust, T. Niemczyk, E. Hoffmann, A. Marx, E. Solano, and R. Gross, “Dual-path state reconstruction scheme for propagating quantum microwaves and detector noise tomography,” Phys. Rev. Lett. 105, 100401 (2010).
    [CrossRef]
  18. C. Eichler, D. Bozyigit, C. Lang, L. Steffen, J. Fink, and A. Wallraff, “Experimental state tomography of itinerant single microwave photons,” Phys. Rev. Lett. 106, 220503 U(2011).
    [CrossRef]
  19. A. Miranowicz and M. Piani, “Comment on “inseparability criteria for continuous bipartite quantum states,” Phys. Rev. Lett. 97, 058901 (2006).
    [CrossRef]
  20. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1985).
  21. A. Serafini and G. Adesso, “Standard forms and entanglement engineering of multimode Gaussian states under local operations,” J. Phys. A 40, 8041–8053 (2007).
    [CrossRef]
  22. R. Namiki, “Photonic families of non-Gaussian entangled states and entanglement criteria for continuous-variable systems,” Phys. Rev. A 85, 062307 (2012).
    [CrossRef]
  23. Q. Sun, H. Nha, and M. S. Zubairy, “Entanglement criteria and nonlocality for multimode continuous-variable systems,” Phys. Rev. A 80, 020101 (2009).
    [CrossRef]
  24. S.-Y. Lee, J. Park, H.-W. Lee, and H. Nha, “Generating arbitrary photon-number entangled states for continuous-variable quantum informatics,” Opt. Express 20, 14221–14233 (2012).
    [CrossRef]

2012

H. Nha, S.-Y. Lee, S.-W. Ji, and M. S. Kim, “Efficient entanglement criteria beyond Gaussian limits using Gaussian measurements,” Phys. Rev. Lett. 108, 030503 (2012).
[CrossRef]

J. S. Ivan, N. Mukunda, and R. Simon, “Moments of non-Gaussian Wigner distributions and a generalized uncertainty principle: I. The single-mode case,” J. Phys. A 45, 195305 (2012).
[CrossRef]

R. Namiki, “Photonic families of non-Gaussian entangled states and entanglement criteria for continuous-variable systems,” Phys. Rev. A 85, 062307 (2012).
[CrossRef]

S.-Y. Lee, J. Park, H.-W. Lee, and H. Nha, “Generating arbitrary photon-number entangled states for continuous-variable quantum informatics,” Opt. Express 20, 14221–14233 (2012).
[CrossRef]

2011

C. Eichler, D. Bozyigit, C. Lang, L. Steffen, J. Fink, and A. Wallraff, “Experimental state tomography of itinerant single microwave photons,” Phys. Rev. Lett. 106, 220503 U(2011).
[CrossRef]

2010

E. P. Menzel, F. Deppe, M. Mariantoni, M. A. Araque Caballero, A. Baust, T. Niemczyk, E. Hoffmann, A. Marx, E. Solano, and R. Gross, “Dual-path state reconstruction scheme for propagating quantum microwaves and detector noise tomography,” Phys. Rev. Lett. 105, 100401 (2010).
[CrossRef]

A. Miranowicz, M. Bartkowiak, X. Wang, Y.-X. Liu, and F. Nori, “Testing nonclassicality in multimode fields: a unified derivation of classical inequalities,” Phys. Rev. A 82, 013824 (2010).
[CrossRef]

2009

S. P. Walborn, B. G. Taketani, A. Salles, F. Toscano, and R. L. de Matos Filho, “Entropic entanglement criteria for continuous variables,” Phys. Rev. Lett. 103, 160505 (2009).
[CrossRef]

A. Miranowicz, M. Piani, P. Horodecki, and R. Horodecki, “Inseparability criteria based on matrices of moments,” Phys. Rev. A 80, 052303 (2009).
[CrossRef]

Q. Sun, H. Nha, and M. S. Zubairy, “Entanglement criteria and nonlocality for multimode continuous-variable systems,” Phys. Rev. A 80, 020101 (2009).
[CrossRef]

2007

A. Serafini and G. Adesso, “Standard forms and entanglement engineering of multimode Gaussian states under local operations,” J. Phys. A 40, 8041–8053 (2007).
[CrossRef]

2006

A. Miranowicz and M. Piani, “Comment on “inseparability criteria for continuous bipartite quantum states,” Phys. Rev. Lett. 97, 058901 (2006).
[CrossRef]

M. Hillery and M. S. Zubairy, “Entanglement conditions for two-mode states,” Phys. Rev. Lett. 96, 050503 (2006).
[CrossRef]

H. Nha and J. Kim, “Entanglement criteria via the uncertainty relations in su(2) and su(1,1) algebras: detection of non-Gaussian entangled states,” Phys. Rev. A 74, 012317 (2006).
[CrossRef]

2005

G. S. Agarwal and A. Biswas, “Inseparability inequalities for higher order moments for bipartite systems,” New J. Phys. 7, 211 (2005).
[CrossRef]

E. Shchukin and W. Vogel, “Inseparability criteria for continuous bipartite quantum states,” Phys. Rev. Lett. 95, 230502 (2005).
[CrossRef]

E. V. Shchukin and W. Vogel, “Nonclassical moments and their measurement,” Phys. Rev. A 72, 043808 (2005).
[CrossRef]

E. Shchukin, T. Richter, and W. Vogel, “Nonclassicality criteria in terms of moments,” Phys. Rev. A 71, 011802 (2005).
[CrossRef]

2003

M. G. Raymer, A. C. Funk, B. C. Sanders, and H. de Guise, “Separability criterion for separate quantum systems,” Phys. Rev. A 67, 052104 (2003).
[CrossRef]

2002

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[CrossRef]

2000

R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
[CrossRef]

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef]

Adesso, G.

A. Serafini and G. Adesso, “Standard forms and entanglement engineering of multimode Gaussian states under local operations,” J. Phys. A 40, 8041–8053 (2007).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and A. Biswas, “Inseparability inequalities for higher order moments for bipartite systems,” New J. Phys. 7, 211 (2005).
[CrossRef]

Araque Caballero, M. A.

E. P. Menzel, F. Deppe, M. Mariantoni, M. A. Araque Caballero, A. Baust, T. Niemczyk, E. Hoffmann, A. Marx, E. Solano, and R. Gross, “Dual-path state reconstruction scheme for propagating quantum microwaves and detector noise tomography,” Phys. Rev. Lett. 105, 100401 (2010).
[CrossRef]

Bartkowiak, M.

A. Miranowicz, M. Bartkowiak, X. Wang, Y.-X. Liu, and F. Nori, “Testing nonclassicality in multimode fields: a unified derivation of classical inequalities,” Phys. Rev. A 82, 013824 (2010).
[CrossRef]

Baust, A.

E. P. Menzel, F. Deppe, M. Mariantoni, M. A. Araque Caballero, A. Baust, T. Niemczyk, E. Hoffmann, A. Marx, E. Solano, and R. Gross, “Dual-path state reconstruction scheme for propagating quantum microwaves and detector noise tomography,” Phys. Rev. Lett. 105, 100401 (2010).
[CrossRef]

Biswas, A.

G. S. Agarwal and A. Biswas, “Inseparability inequalities for higher order moments for bipartite systems,” New J. Phys. 7, 211 (2005).
[CrossRef]

Bozyigit, D.

C. Eichler, D. Bozyigit, C. Lang, L. Steffen, J. Fink, and A. Wallraff, “Experimental state tomography of itinerant single microwave photons,” Phys. Rev. Lett. 106, 220503 U(2011).
[CrossRef]

Cirac, J. I.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef]

de Guise, H.

M. G. Raymer, A. C. Funk, B. C. Sanders, and H. de Guise, “Separability criterion for separate quantum systems,” Phys. Rev. A 67, 052104 (2003).
[CrossRef]

de Matos Filho, R. L.

S. P. Walborn, B. G. Taketani, A. Salles, F. Toscano, and R. L. de Matos Filho, “Entropic entanglement criteria for continuous variables,” Phys. Rev. Lett. 103, 160505 (2009).
[CrossRef]

Deppe, F.

E. P. Menzel, F. Deppe, M. Mariantoni, M. A. Araque Caballero, A. Baust, T. Niemczyk, E. Hoffmann, A. Marx, E. Solano, and R. Gross, “Dual-path state reconstruction scheme for propagating quantum microwaves and detector noise tomography,” Phys. Rev. Lett. 105, 100401 (2010).
[CrossRef]

Duan, L.-M.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef]

Eichler, C.

C. Eichler, D. Bozyigit, C. Lang, L. Steffen, J. Fink, and A. Wallraff, “Experimental state tomography of itinerant single microwave photons,” Phys. Rev. Lett. 106, 220503 U(2011).
[CrossRef]

Fink, J.

C. Eichler, D. Bozyigit, C. Lang, L. Steffen, J. Fink, and A. Wallraff, “Experimental state tomography of itinerant single microwave photons,” Phys. Rev. Lett. 106, 220503 U(2011).
[CrossRef]

Funk, A. C.

M. G. Raymer, A. C. Funk, B. C. Sanders, and H. de Guise, “Separability criterion for separate quantum systems,” Phys. Rev. A 67, 052104 (2003).
[CrossRef]

Giedke, G.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef]

Giovannetti, V.

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[CrossRef]

Gross, R.

E. P. Menzel, F. Deppe, M. Mariantoni, M. A. Araque Caballero, A. Baust, T. Niemczyk, E. Hoffmann, A. Marx, E. Solano, and R. Gross, “Dual-path state reconstruction scheme for propagating quantum microwaves and detector noise tomography,” Phys. Rev. Lett. 105, 100401 (2010).
[CrossRef]

Hillery, M.

M. Hillery and M. S. Zubairy, “Entanglement conditions for two-mode states,” Phys. Rev. Lett. 96, 050503 (2006).
[CrossRef]

Hoffmann, E.

E. P. Menzel, F. Deppe, M. Mariantoni, M. A. Araque Caballero, A. Baust, T. Niemczyk, E. Hoffmann, A. Marx, E. Solano, and R. Gross, “Dual-path state reconstruction scheme for propagating quantum microwaves and detector noise tomography,” Phys. Rev. Lett. 105, 100401 (2010).
[CrossRef]

Horn, R. A.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1985).

Horodecki, P.

A. Miranowicz, M. Piani, P. Horodecki, and R. Horodecki, “Inseparability criteria based on matrices of moments,” Phys. Rev. A 80, 052303 (2009).
[CrossRef]

Horodecki, R.

A. Miranowicz, M. Piani, P. Horodecki, and R. Horodecki, “Inseparability criteria based on matrices of moments,” Phys. Rev. A 80, 052303 (2009).
[CrossRef]

Ivan, J. S.

J. S. Ivan, N. Mukunda, and R. Simon, “Moments of non-Gaussian Wigner distributions and a generalized uncertainty principle: I. The single-mode case,” J. Phys. A 45, 195305 (2012).
[CrossRef]

J. S. Ivan, K. Kumar, N. Mukunda, and R. Simon, “Invariant theoretic approach to uncertainty relations for quantum systems,” arXiv:1205.5132 (2012).

Ji, S.-W.

H. Nha, S.-Y. Lee, S.-W. Ji, and M. S. Kim, “Efficient entanglement criteria beyond Gaussian limits using Gaussian measurements,” Phys. Rev. Lett. 108, 030503 (2012).
[CrossRef]

Johnson, C. R.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1985).

Kim, J.

H. Nha and J. Kim, “Entanglement criteria via the uncertainty relations in su(2) and su(1,1) algebras: detection of non-Gaussian entangled states,” Phys. Rev. A 74, 012317 (2006).
[CrossRef]

Kim, M. S.

H. Nha, S.-Y. Lee, S.-W. Ji, and M. S. Kim, “Efficient entanglement criteria beyond Gaussian limits using Gaussian measurements,” Phys. Rev. Lett. 108, 030503 (2012).
[CrossRef]

Kumar, K.

J. S. Ivan, K. Kumar, N. Mukunda, and R. Simon, “Invariant theoretic approach to uncertainty relations for quantum systems,” arXiv:1205.5132 (2012).

Lang, C.

C. Eichler, D. Bozyigit, C. Lang, L. Steffen, J. Fink, and A. Wallraff, “Experimental state tomography of itinerant single microwave photons,” Phys. Rev. Lett. 106, 220503 U(2011).
[CrossRef]

Lee, H.-W.

Lee, S.-Y.

S.-Y. Lee, J. Park, H.-W. Lee, and H. Nha, “Generating arbitrary photon-number entangled states for continuous-variable quantum informatics,” Opt. Express 20, 14221–14233 (2012).
[CrossRef]

H. Nha, S.-Y. Lee, S.-W. Ji, and M. S. Kim, “Efficient entanglement criteria beyond Gaussian limits using Gaussian measurements,” Phys. Rev. Lett. 108, 030503 (2012).
[CrossRef]

Liu, Y.-X.

A. Miranowicz, M. Bartkowiak, X. Wang, Y.-X. Liu, and F. Nori, “Testing nonclassicality in multimode fields: a unified derivation of classical inequalities,” Phys. Rev. A 82, 013824 (2010).
[CrossRef]

Mancini, S.

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[CrossRef]

Mariantoni, M.

E. P. Menzel, F. Deppe, M. Mariantoni, M. A. Araque Caballero, A. Baust, T. Niemczyk, E. Hoffmann, A. Marx, E. Solano, and R. Gross, “Dual-path state reconstruction scheme for propagating quantum microwaves and detector noise tomography,” Phys. Rev. Lett. 105, 100401 (2010).
[CrossRef]

Marx, A.

E. P. Menzel, F. Deppe, M. Mariantoni, M. A. Araque Caballero, A. Baust, T. Niemczyk, E. Hoffmann, A. Marx, E. Solano, and R. Gross, “Dual-path state reconstruction scheme for propagating quantum microwaves and detector noise tomography,” Phys. Rev. Lett. 105, 100401 (2010).
[CrossRef]

Menzel, E. P.

E. P. Menzel, F. Deppe, M. Mariantoni, M. A. Araque Caballero, A. Baust, T. Niemczyk, E. Hoffmann, A. Marx, E. Solano, and R. Gross, “Dual-path state reconstruction scheme for propagating quantum microwaves and detector noise tomography,” Phys. Rev. Lett. 105, 100401 (2010).
[CrossRef]

Miranowicz, A.

A. Miranowicz, M. Bartkowiak, X. Wang, Y.-X. Liu, and F. Nori, “Testing nonclassicality in multimode fields: a unified derivation of classical inequalities,” Phys. Rev. A 82, 013824 (2010).
[CrossRef]

A. Miranowicz, M. Piani, P. Horodecki, and R. Horodecki, “Inseparability criteria based on matrices of moments,” Phys. Rev. A 80, 052303 (2009).
[CrossRef]

A. Miranowicz and M. Piani, “Comment on “inseparability criteria for continuous bipartite quantum states,” Phys. Rev. Lett. 97, 058901 (2006).
[CrossRef]

Mukunda, N.

J. S. Ivan, N. Mukunda, and R. Simon, “Moments of non-Gaussian Wigner distributions and a generalized uncertainty principle: I. The single-mode case,” J. Phys. A 45, 195305 (2012).
[CrossRef]

J. S. Ivan, K. Kumar, N. Mukunda, and R. Simon, “Invariant theoretic approach to uncertainty relations for quantum systems,” arXiv:1205.5132 (2012).

Namiki, R.

R. Namiki, “Photonic families of non-Gaussian entangled states and entanglement criteria for continuous-variable systems,” Phys. Rev. A 85, 062307 (2012).
[CrossRef]

Nha, H.

S.-Y. Lee, J. Park, H.-W. Lee, and H. Nha, “Generating arbitrary photon-number entangled states for continuous-variable quantum informatics,” Opt. Express 20, 14221–14233 (2012).
[CrossRef]

H. Nha, S.-Y. Lee, S.-W. Ji, and M. S. Kim, “Efficient entanglement criteria beyond Gaussian limits using Gaussian measurements,” Phys. Rev. Lett. 108, 030503 (2012).
[CrossRef]

Q. Sun, H. Nha, and M. S. Zubairy, “Entanglement criteria and nonlocality for multimode continuous-variable systems,” Phys. Rev. A 80, 020101 (2009).
[CrossRef]

H. Nha and J. Kim, “Entanglement criteria via the uncertainty relations in su(2) and su(1,1) algebras: detection of non-Gaussian entangled states,” Phys. Rev. A 74, 012317 (2006).
[CrossRef]

Niemczyk, T.

E. P. Menzel, F. Deppe, M. Mariantoni, M. A. Araque Caballero, A. Baust, T. Niemczyk, E. Hoffmann, A. Marx, E. Solano, and R. Gross, “Dual-path state reconstruction scheme for propagating quantum microwaves and detector noise tomography,” Phys. Rev. Lett. 105, 100401 (2010).
[CrossRef]

Nori, F.

A. Miranowicz, M. Bartkowiak, X. Wang, Y.-X. Liu, and F. Nori, “Testing nonclassicality in multimode fields: a unified derivation of classical inequalities,” Phys. Rev. A 82, 013824 (2010).
[CrossRef]

Park, J.

Piani, M.

A. Miranowicz, M. Piani, P. Horodecki, and R. Horodecki, “Inseparability criteria based on matrices of moments,” Phys. Rev. A 80, 052303 (2009).
[CrossRef]

A. Miranowicz and M. Piani, “Comment on “inseparability criteria for continuous bipartite quantum states,” Phys. Rev. Lett. 97, 058901 (2006).
[CrossRef]

Raymer, M. G.

M. G. Raymer, A. C. Funk, B. C. Sanders, and H. de Guise, “Separability criterion for separate quantum systems,” Phys. Rev. A 67, 052104 (2003).
[CrossRef]

Richter, T.

E. Shchukin, T. Richter, and W. Vogel, “Nonclassicality criteria in terms of moments,” Phys. Rev. A 71, 011802 (2005).
[CrossRef]

Salles, A.

S. P. Walborn, B. G. Taketani, A. Salles, F. Toscano, and R. L. de Matos Filho, “Entropic entanglement criteria for continuous variables,” Phys. Rev. Lett. 103, 160505 (2009).
[CrossRef]

Sanders, B. C.

M. G. Raymer, A. C. Funk, B. C. Sanders, and H. de Guise, “Separability criterion for separate quantum systems,” Phys. Rev. A 67, 052104 (2003).
[CrossRef]

Serafini, A.

A. Serafini and G. Adesso, “Standard forms and entanglement engineering of multimode Gaussian states under local operations,” J. Phys. A 40, 8041–8053 (2007).
[CrossRef]

Shchukin, E.

E. Shchukin, T. Richter, and W. Vogel, “Nonclassicality criteria in terms of moments,” Phys. Rev. A 71, 011802 (2005).
[CrossRef]

E. Shchukin and W. Vogel, “Inseparability criteria for continuous bipartite quantum states,” Phys. Rev. Lett. 95, 230502 (2005).
[CrossRef]

Shchukin, E. V.

E. V. Shchukin and W. Vogel, “Nonclassical moments and their measurement,” Phys. Rev. A 72, 043808 (2005).
[CrossRef]

Simon, R.

J. S. Ivan, N. Mukunda, and R. Simon, “Moments of non-Gaussian Wigner distributions and a generalized uncertainty principle: I. The single-mode case,” J. Phys. A 45, 195305 (2012).
[CrossRef]

R. Simon, “Peres-Horodecki separability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2726–2729 (2000).
[CrossRef]

J. S. Ivan, K. Kumar, N. Mukunda, and R. Simon, “Invariant theoretic approach to uncertainty relations for quantum systems,” arXiv:1205.5132 (2012).

Solano, E.

E. P. Menzel, F. Deppe, M. Mariantoni, M. A. Araque Caballero, A. Baust, T. Niemczyk, E. Hoffmann, A. Marx, E. Solano, and R. Gross, “Dual-path state reconstruction scheme for propagating quantum microwaves and detector noise tomography,” Phys. Rev. Lett. 105, 100401 (2010).
[CrossRef]

Steffen, L.

C. Eichler, D. Bozyigit, C. Lang, L. Steffen, J. Fink, and A. Wallraff, “Experimental state tomography of itinerant single microwave photons,” Phys. Rev. Lett. 106, 220503 U(2011).
[CrossRef]

Sun, Q.

Q. Sun, H. Nha, and M. S. Zubairy, “Entanglement criteria and nonlocality for multimode continuous-variable systems,” Phys. Rev. A 80, 020101 (2009).
[CrossRef]

Taketani, B. G.

S. P. Walborn, B. G. Taketani, A. Salles, F. Toscano, and R. L. de Matos Filho, “Entropic entanglement criteria for continuous variables,” Phys. Rev. Lett. 103, 160505 (2009).
[CrossRef]

Tombesi, P.

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[CrossRef]

Toscano, F.

S. P. Walborn, B. G. Taketani, A. Salles, F. Toscano, and R. L. de Matos Filho, “Entropic entanglement criteria for continuous variables,” Phys. Rev. Lett. 103, 160505 (2009).
[CrossRef]

Vitali, D.

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[CrossRef]

Vogel, W.

E. V. Shchukin and W. Vogel, “Nonclassical moments and their measurement,” Phys. Rev. A 72, 043808 (2005).
[CrossRef]

E. Shchukin and W. Vogel, “Inseparability criteria for continuous bipartite quantum states,” Phys. Rev. Lett. 95, 230502 (2005).
[CrossRef]

E. Shchukin, T. Richter, and W. Vogel, “Nonclassicality criteria in terms of moments,” Phys. Rev. A 71, 011802 (2005).
[CrossRef]

Walborn, S. P.

S. P. Walborn, B. G. Taketani, A. Salles, F. Toscano, and R. L. de Matos Filho, “Entropic entanglement criteria for continuous variables,” Phys. Rev. Lett. 103, 160505 (2009).
[CrossRef]

Wallraff, A.

C. Eichler, D. Bozyigit, C. Lang, L. Steffen, J. Fink, and A. Wallraff, “Experimental state tomography of itinerant single microwave photons,” Phys. Rev. Lett. 106, 220503 U(2011).
[CrossRef]

Wang, X.

A. Miranowicz, M. Bartkowiak, X. Wang, Y.-X. Liu, and F. Nori, “Testing nonclassicality in multimode fields: a unified derivation of classical inequalities,” Phys. Rev. A 82, 013824 (2010).
[CrossRef]

Zoller, P.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef]

Zubairy, M. S.

Q. Sun, H. Nha, and M. S. Zubairy, “Entanglement criteria and nonlocality for multimode continuous-variable systems,” Phys. Rev. A 80, 020101 (2009).
[CrossRef]

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Figures (2)

Fig. 1.
Fig. 1.

(a) Sum of entropies subtracted by the entropic bound, i.e., H[P(x)]+H[P(p+)]ln(πe), as a function of the amplitude α and the coherence parameter p of the dephased cat ρcat. (b) Determinant of the fourth-order MM of marginal PT dephased cat, i.e., det[M1|(1/2)(ρcatMΓ)], using the same observables x and p+ as in (a). In each panel, the contours are 0.2, 0.1, 0.05, 0.01, 0.001, and 0 from above [note that 0 is missing in (b)], and a negative value implies that the respective criterion detects entanglement. Notice that the entropic criterion cannot detect entanglement for small α, while the fourth-order MM criterion does for all α and p.

Fig. 2.
Fig. 2.

(a) Negative determinant of the submatrix of the fourth-order MM of PT beam-split number state B^|n,m. From below, the lines denote the case of n=1, 2, 3, 4, 5. (b) Determinant of the submatrix of fourth-order MM of PT number-correlated state c0|0,0+c1|1,1+c2|2,2. The contours are 10, 20, 30, 40, and 50; the darker the region is, the larger its absolute value is. For both cases, a negative value indicates that the entanglement of the corresponding state is detected.

Equations (49)

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F^F^=Tr(ρF^F^)=cMf^(ρ)c0,
Mf^(ρ)=[f^0f^0f^0f^1f^0f^nf^1f^0f^1f^1f^1f^nf^nf^0f^nf^1f^nf^n],
Mf^(ρ)0.
|ΔA^ΔA^ΔA^ΔB^ΔB^ΔA^ΔB^ΔB^|0.
ΔA^ΔA^ΔB^ΔB^|ΔA^ΔB^|2
ΔA^ΔA^ΔB^ΔB^Im2ΔA^ΔB^
X^=[x^p^]X^=S[x^p^],
[X^i,X^j]=iΩij,Ω=[0110].
U^(S)[x^p^]U^(S)=S[x^p^].
ρ=U^(S)ρU^(S)Wρ(X)=Wρ(S1X),
(T^12m)=[x^p^],(T^1m)=[x^212(x^p^+p^x^)p^2],(T^32m)=[x^313(x^2p^+x^p^x^+p^x^2)13(x^p^2+p^x^p^+p^2x^)p^3],.
τ^jmτ^jm=j=|jj|j+j(i2)j+jjCmmm+mjjjτ^j,m+m×(j+j+j+1)!(2j+1)(j+jj)!(j+jj)!(j+jj)!,
τ^jm=T^jm/(j+m)!(jm)!,
T^12mT^1m=T^32m+m+i2(2mm)T^12m+mT^1mT^1m=T^2m+m+i(mm)T^1m+m+(1)m4(2δm,0)δm+m,0T^00.
T^jmρ=dxdpWρ(x,p)xj+mpjm.
f^=(ΔT^1212,ΔT^1212,ΔT^11,ΔT^10,ΔT^11,)
Mf^=(ΔT^jmΔT^jm)
Mjm,jm=Vjm,jm+i2Ωjm,jm,
Vjm,jm=12{T^jm,T^jm}T^jmT^jm,
Ωjm,jm=1i[T^jm,T^jm].
M12(ρ)[(Δx^)2Δx^Δp^Δp^Δx^(Δp^)2].
f^=(ΔT^12m,ΔT^1m)(ΔT^1212,ΔT^1212,ΔT^1,1,ΔT^10,ΔT^11)
M1(ρ)[M12(ρ)M12,1(ρ)M1,12(ρ)M1,1(ρ)],
M12,1(ρ)=M1,12(ρ)=[Δx^ΔT^11Δx^ΔT^10Δx^ΔT^11Δp^ΔT^11Δp^ΔT^10Δp^ΔT^11].
MJ+12=[MJM12:J,J+12MJ+12,12:JMJ+12,J+12].
MJ(ρ)=VJ(ρ)+i2ΩJ(ρ)0,
MJ+12|JMJ+12,J+12MJ+12,12:JMJ1MJ+12,12:J0.
U^(S)T^jmU^(S)=m=jjKmm(j)(S)T^jm.
[xp]=S[xp][x2xpp2]=K(1)(S)[x2xpp2],
S=[abcd]K(1)(S)=[a22abb2acad+bcbdc22cdd2],
ρU^(S)ρU^(S)MJKJ(S)MJKJT(S),
KJ(S)=K(12)(S)K(1)(S)K(J)(S).
M12SM12ST,M1K(1)M1K(1)T,M1,12K(1)M1,12ST.
x±=x1±x22,p±=p1±p22,
W(x+,p+,x,p)W(x+,p,x,p+).
W(x±,p)=dxdp±W(x+,p,x,p+)
MJ(ρMΓ)=VJ(ρMΓ)+i2ΩJ(ρMΓ)0.
ρcat=N[|α,αα,α|+|α,αα,α|p(|α,αα,α|+|α,αα,α|)],
H[P(x±)]+H[P(p)]ln(πe),
H[P(q)]=dqP(q)lnP(q)
M1|12=M1,1M1,12M121M12,10.
M1|12=M1,1=12[1i1i1+2di(1+4d)1i(1+4d)1+8d(14Nα2)],
T^1212A,T^1212A,T^1212B,T^1212B,T^11A,T^10A,T^11A,T^1212AT^1212B,T^1212AT^1212B,T^1212AT^1212B,T^1212AT^1212B,T^11B,T^10B,T^11B,
KJ(S)=KA(12)(S)KB(12)(S)KA(1)(S)[KA(12)(S)KB(12)(S)]KB(1)(S)KB(J)(S).
(ΔX^)2+(ΔX^)2|c1d2c2d1|+|c3d4c4d3|
Mf^(ρΓ)0
(ΔX^)2+(ΔX^)2c2+c2,
Δa^Δa^Δb^Δb^Re2Δa^Δb^,
a^2,a^a^,a^2,a^b^,a^b^,b^2,a^b^,a^b^,b^b^,b^2.

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