Abstract

Quantum steering is a stronger form of quantum entanglement and is considered to rigorously address the nonlocal correlation in the original Einstein–Podolsky–Rosen paradox. We study the quantum steerability of two classes of continuous-variable (CV) entangled states that are most often employed for CV quantum informatics-entangled coherent states (ECSs) and two-mode-squeezed vacuum (TMSV). We test the two steering criteria by utilizing Heisenberg and entropic uncertainty relations, respectively, and find that the latter can reveal the steerability of ECSs of any size and parity, whereas the former scarcely can. The steering behaviors of the two states are investigated and compared when they experience a noisy environment, i.e., an amplitude damping channel. When the noise is added asymmetrically, e.g., when only one party of the bipartite system is subject to noise, quantum steering can be possible only in one direction, e.g., Alice on Bob’s state but not Bob on Alice’s state. We show the emergence of this one-directional steering for both ECSs and the TMSV in a certain parameter range under decoherence.

© 2013 Optical Society of America

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  1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).
  2. E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935) [“Die gegenwrtige situation in der quantenmechanik,” Naturwissenschaften 23, 807–812 (1935)].
    [CrossRef]
  3. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
    [CrossRef]
  4. M. D. Reid, “Demonstration of the Einstein–Podolsky–Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40, 913–923 (1989).
    [CrossRef]
  5. Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
    [CrossRef]
  6. J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein–Podolsky–Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
    [CrossRef]
  7. D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845 (2010).
    [CrossRef]
  8. S. P. Walborn, A. Salles, R. M. Gomes, F. Toscano, and P. H. Souto Ribeiro, “Revealing hidden Einstein–Podolsky–Rosen nonlocality,” Phys. Rev. Lett. 106, 130402 (2011).
    [CrossRef]
  9. A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).
    [CrossRef]
  10. H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
    [CrossRef]
  11. E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
    [CrossRef]
  12. S. L. W. Midgley, A. J. Ferris, and M. K. Olsen, “Asymmetric Gaussian steering: when Alice and Bob disagree,” Phys. Rev. A 81, 022101 (2010).
    [CrossRef]
  13. V. Händchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 596–599 (2012).
    [CrossRef]
  14. D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).
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    [CrossRef]
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  17. Q. Y. He, P. D. Drummond, and M. D. Reid, “Entanglement, EPR steering, and Bell-nonlocality criteria for multipartite higher-spin systems,” Phys. Rev. A 83, 032120 (2011).
    [CrossRef]
  18. I. Białynicki-Birula and J. Mycielski, “Uncertainty relations for information entropy in wave mechanics,” Commun. Math. Phys. 44, 129–132 (1975).
    [CrossRef]
  19. M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
    [CrossRef]
  20. N.-Q. Jiang, H.-Y. Fan, L.-S. Xi, L.-Y. Tang, and X.-Z. Yuan, “Evolution of a two-mode squeezed vacuum in the amplitude dissipative channel,” Chin. Phys. B 20, 120302 (2011).
    [CrossRef]

2012

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).
[CrossRef]

V. Händchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 596–599 (2012).
[CrossRef]

See references in B. C. Sanders, “Review of entangled coherent states,” J. Phys. A 45, 244002 (2012).
[CrossRef]

2011

Q. Y. He, P. D. Drummond, and M. D. Reid, “Entanglement, EPR steering, and Bell-nonlocality criteria for multipartite higher-spin systems,” Phys. Rev. A 83, 032120 (2011).
[CrossRef]

S. P. Walborn, A. Salles, R. M. Gomes, F. Toscano, and P. H. Souto Ribeiro, “Revealing hidden Einstein–Podolsky–Rosen nonlocality,” Phys. Rev. Lett. 106, 130402 (2011).
[CrossRef]

N.-Q. Jiang, H.-Y. Fan, L.-S. Xi, L.-Y. Tang, and X.-Z. Yuan, “Evolution of a two-mode squeezed vacuum in the amplitude dissipative channel,” Chin. Phys. B 20, 120302 (2011).
[CrossRef]

2010

S. L. W. Midgley, A. J. Ferris, and M. K. Olsen, “Asymmetric Gaussian steering: when Alice and Bob disagree,” Phys. Rev. A 81, 022101 (2010).
[CrossRef]

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845 (2010).
[CrossRef]

2009

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[CrossRef]

2007

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[CrossRef]

2006

M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
[CrossRef]

2004

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein–Podolsky–Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[CrossRef]

1992

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef]

1989

M. D. Reid, “Demonstration of the Einstein–Podolsky–Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40, 913–923 (1989).
[CrossRef]

1975

I. Białynicki-Birula and J. Mycielski, “Uncertainty relations for information entropy in wave mechanics,” Commun. Math. Phys. 44, 129–132 (1975).
[CrossRef]

1935

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935) [“Die gegenwrtige situation in der quantenmechanik,” Naturwissenschaften 23, 807–812 (1935)].
[CrossRef]

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Bennet, A. J.

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).
[CrossRef]

Bennink, R. S.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein–Podolsky–Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[CrossRef]

Bentley, S. J.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein–Podolsky–Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[CrossRef]

Bialynicki-Birula, I.

I. Białynicki-Birula and J. Mycielski, “Uncertainty relations for information entropy in wave mechanics,” Commun. Math. Phys. 44, 129–132 (1975).
[CrossRef]

Boyd, R. W.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein–Podolsky–Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[CrossRef]

Branciard, C.

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).
[CrossRef]

Cavalcanti, E. G.

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).
[CrossRef]

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[CrossRef]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Cirac, J. I.

M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
[CrossRef]

Doherty, A. C.

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[CrossRef]

Drummond, P. D.

Q. Y. He, P. D. Drummond, and M. D. Reid, “Entanglement, EPR steering, and Bell-nonlocality criteria for multipartite higher-spin systems,” Phys. Rev. A 83, 032120 (2011).
[CrossRef]

Eberle, T.

V. Händchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 596–599 (2012).
[CrossRef]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Evans, D. A.

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).
[CrossRef]

Fan, H.-Y.

N.-Q. Jiang, H.-Y. Fan, L.-S. Xi, L.-Y. Tang, and X.-Z. Yuan, “Evolution of a two-mode squeezed vacuum in the amplitude dissipative channel,” Chin. Phys. B 20, 120302 (2011).
[CrossRef]

Ferris, A. J.

S. L. W. Midgley, A. J. Ferris, and M. K. Olsen, “Asymmetric Gaussian steering: when Alice and Bob disagree,” Phys. Rev. A 81, 022101 (2010).
[CrossRef]

Franz, T.

V. Händchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 596–599 (2012).
[CrossRef]

Giedke, G.

M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
[CrossRef]

Gomes, R. M.

S. P. Walborn, A. Salles, R. M. Gomes, F. Toscano, and P. H. Souto Ribeiro, “Revealing hidden Einstein–Podolsky–Rosen nonlocality,” Phys. Rev. Lett. 106, 130402 (2011).
[CrossRef]

Händchen, V.

V. Händchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 596–599 (2012).
[CrossRef]

He, Q. Y.

Q. Y. He, P. D. Drummond, and M. D. Reid, “Entanglement, EPR steering, and Bell-nonlocality criteria for multipartite higher-spin systems,” Phys. Rev. A 83, 032120 (2011).
[CrossRef]

Howell, J. C.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein–Podolsky–Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[CrossRef]

Jiang, N.-Q.

N.-Q. Jiang, H.-Y. Fan, L.-S. Xi, L.-Y. Tang, and X.-Z. Yuan, “Evolution of a two-mode squeezed vacuum in the amplitude dissipative channel,” Chin. Phys. B 20, 120302 (2011).
[CrossRef]

Jones, S. J.

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845 (2010).
[CrossRef]

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[CrossRef]

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[CrossRef]

Kimble, H. J.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef]

Midgley, S. L. W.

S. L. W. Midgley, A. J. Ferris, and M. K. Olsen, “Asymmetric Gaussian steering: when Alice and Bob disagree,” Phys. Rev. A 81, 022101 (2010).
[CrossRef]

Milburn, G. J.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

Mycielski, J.

I. Białynicki-Birula and J. Mycielski, “Uncertainty relations for information entropy in wave mechanics,” Commun. Math. Phys. 44, 129–132 (1975).
[CrossRef]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Olsen, M. K.

S. L. W. Midgley, A. J. Ferris, and M. K. Olsen, “Asymmetric Gaussian steering: when Alice and Bob disagree,” Phys. Rev. A 81, 022101 (2010).
[CrossRef]

Ou, Z. Y.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef]

Peng, K. C.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef]

Pereira, S. F.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[CrossRef]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Pryde, G. J.

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).
[CrossRef]

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845 (2010).
[CrossRef]

Reid, M. D.

Q. Y. He, P. D. Drummond, and M. D. Reid, “Entanglement, EPR steering, and Bell-nonlocality criteria for multipartite higher-spin systems,” Phys. Rev. A 83, 032120 (2011).
[CrossRef]

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[CrossRef]

M. D. Reid, “Demonstration of the Einstein–Podolsky–Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40, 913–923 (1989).
[CrossRef]

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Salles, A.

S. P. Walborn, A. Salles, R. M. Gomes, F. Toscano, and P. H. Souto Ribeiro, “Revealing hidden Einstein–Podolsky–Rosen nonlocality,” Phys. Rev. Lett. 106, 130402 (2011).
[CrossRef]

Samblowski, A.

V. Händchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 596–599 (2012).
[CrossRef]

Sanders, B. C.

See references in B. C. Sanders, “Review of entangled coherent states,” J. Phys. A 45, 244002 (2012).
[CrossRef]

Saunders, D. J.

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).
[CrossRef]

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845 (2010).
[CrossRef]

Schnabel, R.

V. Händchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 596–599 (2012).
[CrossRef]

Schrödinger, E.

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935) [“Die gegenwrtige situation in der quantenmechanik,” Naturwissenschaften 23, 807–812 (1935)].
[CrossRef]

Souto Ribeiro, P. H.

S. P. Walborn, A. Salles, R. M. Gomes, F. Toscano, and P. H. Souto Ribeiro, “Revealing hidden Einstein–Podolsky–Rosen nonlocality,” Phys. Rev. Lett. 106, 130402 (2011).
[CrossRef]

Steinlechner, S.

V. Händchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 596–599 (2012).
[CrossRef]

Tang, L.-Y.

N.-Q. Jiang, H.-Y. Fan, L.-S. Xi, L.-Y. Tang, and X.-Z. Yuan, “Evolution of a two-mode squeezed vacuum in the amplitude dissipative channel,” Chin. Phys. B 20, 120302 (2011).
[CrossRef]

Toscano, F.

S. P. Walborn, A. Salles, R. M. Gomes, F. Toscano, and P. H. Souto Ribeiro, “Revealing hidden Einstein–Podolsky–Rosen nonlocality,” Phys. Rev. Lett. 106, 130402 (2011).
[CrossRef]

Walborn, S. P.

S. P. Walborn, A. Salles, R. M. Gomes, F. Toscano, and P. H. Souto Ribeiro, “Revealing hidden Einstein–Podolsky–Rosen nonlocality,” Phys. Rev. Lett. 106, 130402 (2011).
[CrossRef]

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994).

Werner, R. F.

V. Händchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 596–599 (2012).
[CrossRef]

Wiseman, H. M.

A. J. Bennet, D. A. Evans, D. J. Saunders, C. Branciard, E. G. Cavalcanti, H. M. Wiseman, and G. J. Pryde, “Arbitrarily loss-tolerant Einstein–Podolsky–Rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole,” Phys. Rev. X 2, 031003 (2012).
[CrossRef]

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845 (2010).
[CrossRef]

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[CrossRef]

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[CrossRef]

Wolf, M. M.

M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
[CrossRef]

Xi, L.-S.

N.-Q. Jiang, H.-Y. Fan, L.-S. Xi, L.-Y. Tang, and X.-Z. Yuan, “Evolution of a two-mode squeezed vacuum in the amplitude dissipative channel,” Chin. Phys. B 20, 120302 (2011).
[CrossRef]

Yuan, X.-Z.

N.-Q. Jiang, H.-Y. Fan, L.-S. Xi, L.-Y. Tang, and X.-Z. Yuan, “Evolution of a two-mode squeezed vacuum in the amplitude dissipative channel,” Chin. Phys. B 20, 120302 (2011).
[CrossRef]

Chin. Phys. B

N.-Q. Jiang, H.-Y. Fan, L.-S. Xi, L.-Y. Tang, and X.-Z. Yuan, “Evolution of a two-mode squeezed vacuum in the amplitude dissipative channel,” Chin. Phys. B 20, 120302 (2011).
[CrossRef]

Commun. Math. Phys.

I. Białynicki-Birula and J. Mycielski, “Uncertainty relations for information entropy in wave mechanics,” Commun. Math. Phys. 44, 129–132 (1975).
[CrossRef]

J. Phys. A

See references in B. C. Sanders, “Review of entangled coherent states,” J. Phys. A 45, 244002 (2012).
[CrossRef]

Nat. Photonics

V. Händchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein–Podolsky–Rosen steering,” Nat. Photonics 6, 596–599 (2012).
[CrossRef]

Nat. Phys.

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845 (2010).
[CrossRef]

Phys. Rev.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Phys. Rev. A

M. D. Reid, “Demonstration of the Einstein–Podolsky–Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40, 913–923 (1989).
[CrossRef]

E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[CrossRef]

S. L. W. Midgley, A. J. Ferris, and M. K. Olsen, “Asymmetric Gaussian steering: when Alice and Bob disagree,” Phys. Rev. A 81, 022101 (2010).
[CrossRef]

Q. Y. He, P. D. Drummond, and M. D. Reid, “Entanglement, EPR steering, and Bell-nonlocality criteria for multipartite higher-spin systems,” Phys. Rev. A 83, 032120 (2011).
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Figures (3)

Fig. 1.
Fig. 1.

(a) Product of inferred uncertainties and (b) the sum of inferred Shannon entropies of ECSs |Φ±(α,α) plotted against the amplitude α. In each panel, the two curves represent the cases of |Φ+(α,α) (blue solid) and |Φ(α,α) (red dashed), and horizontal (gray solid) lines indicate the corresponding uncertainty bounds 1/2 and ln(eπ).

Fig. 2.
Fig. 2.

Sum of inferred entropies of xPDF and pPDF for (a) ECSs and (b) TMSV plotted against the normalized time r for n^(0)=4 (blue solid) and n^(0)=10 (red dashed). The horizontal (gray solid) line indicates the entropic uncertainty bound ln(eπ). The inset in (a) shows a magnified region where the violation—equivalently, steerability—starts to disappear. See the main text for the detailed violation time of ESI.

Fig. 3.
Fig. 3.

Sum of inferred entropies for xPDF and pPDF of (a) ECSs and (b) TMSV when n^(0)=4. The solid curves denote the case when Alice steers Bob’s state (i=B, j=A), and the dashed ones the other way around (i=A, j=B). The horizontal solid lines indicate the entropic uncertainty bound as before. The term two-way means that both Alice and Bob can steer each other’s state; the term one-way in the colored region, that Alice can steer but Bob cannot; finally, no-way, that no one can steer.

Equations (59)

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|TMSV=S^AB(s)|00AB=sechsea^b^tanhs|00AB,
S^AB(s)=exp[s(a^b^a^b^)]
|Φ±(α,β)=N±(α,β)(|αA|βB±|αA|βB),
P(a,b)=λP(λ)P(a|λ)PQ(b|λ),
P(b)=aP(a,b)=λP(λ)PQ(b|λ),
P(b|a)=P(a,b)/P(a)=λP(λ)P(a|λ)P(a)PQ(b|λ).
P(a,b)=λP(λ)P(a|λ)P(b|λ),
P(a,b)=λP(λ)PQ(a|λ)PQ(b|λ),
nonlocalitysteerabilityentanglement
x|α=1π4e12x2+2αxα2,p|α=1π4e12p2i2αp.
P(xA,xB)=xA,xB|ρ|xA,xB,
P(xA)=dxBP(xA,xB),
P(xB|xA)=P(xA,xB)/P(xA),
P(xA,xB)=dpAdpBW(xA,pA,xB,pB),
P(pA,pB)=dxAdxBW(xA,pA,xB,pB).
Δinf2(XB|XA)dxAP(xA)Δinf2(XB|XA=xA)=dxAP(xA)dxBP(xB|xA)(xBmest)2.
mest=XBAdxBP(xB|xA)xB.
Δinf(XB|XA)Δinf(PB|PA)12.
Δinf(XB|XA)=Δinf(PB|PA)=12coshs.
Δinf(XB|XA)Δinf(PB|PA)=12coshs12,
xA,xB|Φ±(α,β)=N±(α,β)π[e12(xA2α)212(xB2β)2±e12(xA+2α)212(xB+2β)2],
pA,pB|Φ±(α,β)=N±(α,β)πeα2β2[e12(pA+i2α)212(pB+i2β)2±e12(pAi2α)212(pBi2β)2],
PΦ±(xA,xB)=|xA,xB|Φ±(α,β)|2,PΦ±(pA,pB)=|pA,pB|Φ±(α,β)|2.
Δinf(XB|PA)Δinf(PB|XA)12,
H(X)+H(P)ln(eπ),
H(X)H[P(x)]=dxP(x)lnP(x)
H(XB|XA)dxAP(xA)H(XB|xA),
H(XB|xA)H[P(xB|xA)]=dxBP(xB|xA)lnP(xB|xA),
P(xB|xA)=λP(λ|xA)P(xB|λ)
H(XB|xA)=H[λP(λ|xA)P(xB|λ)]λP(λ|xA)H[P(XB|λ)],
H(XB|XA)λP(λ)H[P(XB|λ)].
H(XB|XA)+H(PB|PA)ln(eπ).
H(XB|PA)+H(PB|XA)ln(eπ).
dρdτ=L(a^)[ρ]+L(b^)[ρ],
L(o^)[ρ]=o^ρo^12ρo^o^12o^o^ρ.
ρ(τ)=n=0K^n(a^)K^n(b^)ρ(0)K^n(a^)K^n(b^)
K^n(o^)=rnn!to^o^o^n(t=eτ/2,r=1t2).
| α β | β | α ADC | t α t β | t β | t α .
ρ ECS ± ( 0 ) = | Φ ± ( α , β ) Φ ± ( α , β ) | ADC ρ ECS ± ( τ ) = F ± | Φ ± ( t α , t β ) Φ ± ( t α , t β ) | + ( 1 F ± ) | Φ ( t α , t β ) Φ ( t α , t β ) | ,
F ± = [ N ± ( α , β ) N ± ( t α , t β ) ] 2 1 + e 2 r 2 ( α 2 + β 2 ) 2 .
P ECS ± ( x A , x B ) = F ± P Φ ± ( x A , x B ) + ( 1 F ± ) P Φ ( x A , x B ) ,
ρTMSV(τ)=(A2B2)eBa^b^:eA(a^a^+b^b^):eBa^b^,
A=1r2tanh2s1r4tanh2s,B=t2tanhs1r4tanh2s.
WTMSV(xA,pA,xB,pB)=1π2A2B2(2A)2B2×eA(2A)+B2(2A)2B2(xA2+pA2+xB2+pB2)+2B(2A)2B2(xAxBpApB).
PTMSV(xA,xB)=1πA2B2(2A)2B2eA(2A)+B2(2A)2B2(xA2+xB2)+2B(2A)2B2xAxB.
Δinf2(XB|XA)=Δinf2(PB|PA)=t4+r4+2t2r2cosh2s2(t2cosh2s+r2)
Δinf(XB|XA)Δinf(PB|PA)<12r2<12.
H(XB|XA)=ln[2πeΔinf(XA|XA)],H(PB|PA)=ln[2πeΔinf(PA|PA)].
dρdτ=L(b^)[ρ].
ρ(τ)=n=0K^n(b^)ρ(0)K^n(b^)
H(XA|XB)+H(PA|PB)ln(eπ).
ρECS±(τ)=F±|Φ±(α,tα)Φ±(α,tα)|+(1F±)|Φ(α,tα)Φ(α,tα)|
F±=[N±(α,α)N±(α,tα)]21+e2r2α22.
ρTMSV(τ)=(AB2)eBa^b^:eAa^a^b^b^:eBa^b^
WTMSV(xA,pA,xB,pB)=1π2AB22AB2×e2A+B22AB2(xA2+pA2)A+B22AB2(xB2+pB2)+2B2AB2(xAxBpApB).
PTMSV(xA,xB)=1πAB22AB2×e2A+B22AB2xA2A+B22AB2xB2+2B2AB2xAxB,
PTMSV(pA,pB)=(xApA,xBpB)
Δinf2(XB|XA)=Δinf2(PB|PA)=t2sech2s+r22,Δinf(XB|XA)Δinf(PB|PA)<12r2<1,
Δinf2(XA|XB)=Δinf2(PA|PB)=t2+r2cosh2s2(t2cosh2s+r2),Δinf(XA|XB)Δinf(PA|PB)<12r2<12.

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