Abstract

Multipartite quantum entanglement is a key resource for ensuring security in quantum network. We show that by using a unified parameter in terms of reduced noise variances one can determine different types of tripartite entanglement of a given state generated in a hybrid optomechanical system, where an atomic ensemble is located inside a single-mode cavity with a movable mirror, with different thresholds for each type. In particular, the special quantum states which allow both entanglement and steering genuinely shared among atom-light-mirror modes can be observed, even though there is no direct interaction between the mirror and the atomic ensemble. We further show the robustness against mechanical thermal noise and damping, the relaxation time of atomic ensemble, as well as the effect of gain factors involved in the criteria. Our analysis provides an experimentally achievable method to determine the type of tripartite quantum correlation in a way.

© 2015 Optical Society of America

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References

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  58. For two-mode Gaussian states, the condition EA|B(g) < 1 is necessary and sufficient to confirm steering of A by B [16, 17, 24], and the minimum value of EA|B(g) can be used to quantify the Gaussian EPR steering [22].
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2015 (9)

S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. - A. Bachor, M D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167–172 (2015).
[Crossref]

Q. Y. He, Q. H. Gong, and M. D. Reid, “Classifying directional Gaussian entanglement, Einstein-Podolsky-Rosen steering, and discord,” Phys. Rev. Lett. 114, 060402 (2015).
[Crossref] [PubMed]

I. Kogias, A. R. Lee, S. Ragy, and G. Adesso, “Quantification of Gaussian quantum steering,” Phys. Rev. Lett. 114, 060403 (2015).
[Crossref] [PubMed]

C. M. Li, K. Chen, Y. N. Chen, Q. Zhang, Y. A. Chen, and J. W. Pan, “Genuine high-order Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 115, 010402 (2015).
[Crossref] [PubMed]

D. Cavalcanti, P. Skrzypczyk, G. H. Aguilar, R. V. Nery, P. H. Souto Ribeiro, and S. P. Walborn, “Detection of entanglement in asymmetric quantum networks and multipartite quantum steering,” Nat. Commun. 6, 7941 (2015).
[Crossref] [PubMed]

M. Wang, Q. H. Gong, Z. Ficek, and Q. Y. He, “Efficient scheme for perfect collective Einstein-Podolsky-Rosen steering,” Sci. Rep. 5, 12346 (2015).
[Crossref] [PubMed]

Y. Yan, G. X. Li, and Q. L. Wu, “Entanglement and Einstein-Podolsky-Rosen steering between a nanomechanical resonator and a cavity coupled with two quantum dots,” Opt. Express 23, 21306–21322 (2015).
[Crossref] [PubMed]

Y. D. Wang, S. Chesi, and A. A. Clerk, “Bipartite and tripartite output entanglement in three-mode optomechanical systems,” Phys. Rev. A 91, 013807 (2015).
[Crossref]

Q. Lin and B. He, “Optomechanical entanglement under pulse drive,” Opt. Express 23, 24497–24507 (2015).
[Crossref] [PubMed]

2014 (6)

Q. Y. He and Z. Ficek, “Einstein-Podolsky-Rosen paradox and quantum steering in a three-mode optomechanical system,” Phys. Rev. A 89, 022332 (2014).
[Crossref]

M. Wang, Q. H. Gong, Z. Ficek, and Q.Y. He, “Role of thermal noise in tripartite quantum steering,” Phys. Rev. A 90, 023801 (2014).
[Crossref]

M. Wang, Q. H. Gong, and Q.Y. He, “Collective multipartite Einstein-Podolsky-Rosen steering: more secure optical networks,” Opt. Lett. 39, 6703–6706 (2014).
[Crossref] [PubMed]

R. Y. Teh and M. D. Reid, “Criteria for genuine N-partite continuous-variable entanglement and Einstein-Podolsky-Rosen steering,” Phys. Rev. A 90, 062337 (2014).
[Crossref]

A. Carmele, B. Vogell, K. Stannigel, and P. Zoller, “Opto-nanomechanics strongly coupled to a Rydberg super-atom: coherent versus incoherent dynamics,” New J. Phys. 16, 063042 (2014).
[Crossref]

K. Sun, J. S. Xu, X. J. Ye, Y. C. Wu, J. L. Chen, C. F. Li, and G. C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014).
[Crossref] [PubMed]

2013 (8)

S. Steinlechner, J. Bauchrowitz, T. Eberle, and R. Schnabel, “Strong Einstein-Podolsky-Rosen steering with unconditional entangled states,” Phys. Rev. A 87, 022104 (2013).
[Crossref]

L. K. Shalm, D. R. Hamel, Z. Yan, C. Simon, K. J. Resch, and T. Jennewein, “Three-photon energy–time entanglement,” Nat. Phys. 9, 19–22 (2013).
[Crossref]

S. M. Giampaolo and B. C. Hiesmayr, “Genuine multipartite entanglement in the XY model,” Phys. Rev. A 88, 052305 (2013).
[Crossref]

J. D. Bancal, J. Barrett, N. Gisin, and S. Pironio, “Definitions of multipartite nonlocality,” Phys. Rev. A 88, 014102 (2013).
[Crossref]

F. Levi and F. Mintert, “Hierarchies of multipartite entanglement,” Phys. Rev. Lett. 110, 150402 (2013).
[Crossref] [PubMed]

L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013).
[Crossref] [PubMed]

Q. Y. He and M. D. Reid, “Genuine multipartite Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 111, 250403 (2013).
[Crossref]

T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
[Crossref] [PubMed]

2012 (8)

L. H. Sun, G. X. Li, and Z. Ficek, “First-order coherence versus entanglement in a nanomechanical cavity,” Phys. Rev. A 85, 022327 (2012).
[Crossref]

S. Singh, H. Jing, E. M. Wright, and P. Meystre, “Quantum-state transfer between a Bose-Einstein condensate and an optomechanical mirror,” Phys. Rev. A 86, 021801(R) (2012).
[Crossref]

R. Gallego, L. E. Würflinger, A. Acín, and M. Navascués, “Operational framework for nonlocality,” Phys. Rev. Lett. 109, 070401 (2012).
[Crossref] [PubMed]

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]

C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301(R) (2012).
[Crossref]

D. H. Smith, G. Gillett, M. P. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nat. Commun. 3, 625 (2012).
[Crossref] [PubMed]

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).

B. Wittmann, S. Ramelow, F. Steinlechner, N. K. Langford, N. Brunner, H. M. Wiseman, R. Ursin, and A. Zeilinger, “Loophole-free Einstein–Podolsky–Rosen experiment via quantum steering,” New J. Phys. 14, 053030 (2012).
[Crossref]

2011 (5)

J. D. Bancal, N. Gisin, Y. C. Liang, and S. Pironio, “Device-independent witnesses of genuine multipartite entanglement,” Phys. Rev. Lett. 106, 250404 (2011).
[Crossref] [PubMed]

M. Huber, H. Schimpf, A. Gabriel, C. Spengler, D. Brup, and B. C. Hiesmayr, “Experimentally implementable criteria revealing substructures of genuine multipartite entanglement,” Phys. Rev. A 83, 022328 (2011).
[Crossref]

S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84, 052327 (2011).
[Crossref]

J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref] [PubMed]

J. D. Teufel, T. Donner, D. L Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

2010 (6)

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

T. P. Purdy, D. W. C. Brooks, T. Botter, N. Brahms, Z.-Y. Ma, and D. M. Stamper-Kurn, “Tunable cavity optomechanics with ultracold atoms,” Phys. Rev. Lett. 105, 133602 (2010).
[Crossref]

R. Kanamoto and P. Meystre, “Optomechanics of a quantum-degenerate Fermi gas,” Phys. Rev. Lett. 104, 063601 (2010).
[Crossref]

K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041 (2010).
[Crossref]

B. Hage, A. Samblowski, and R. Schnabel, “Towards Einstein-Podolsky-Rosen quantum channel multiplexing,” Phys. Rev. A 81, 062301 (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, 845–849 (2010).
[Crossref]

2009 (2)

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]

D. L. Deng, Z. S. Zhou, and J. L. Chen, “Svetlichny’s approach to detecting genuine multipartite entanglement in arbitrarily-high-dimensional systems by a Bell-type inequality,” Phys. Rev. A 80, 022109 (2009).
[Crossref]

2008 (2)

C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307(R) (2008).
[Crossref]

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[Crossref] [PubMed]

2007 (2)

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] [PubMed]

S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007).
[Crossref]

2004 (1)

H. Yonezawa, T. Aoki, and A. Furusawa, “Demonstration of a quantum teleportation network for continuous variables,” Nature 431, 430–433 (2004).
[Crossref] [PubMed]

2003 (1)

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

2000 (1)

P. van Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: A quantum teleportation network,” Phys. Rev. Lett. 84, 3482–3485 (2000).
[Crossref] [PubMed]

1999 (2)

M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
[Crossref]

R. Cleve, D. Gottesman, and H.-K. Lo, “How to share a quantum secret,” Phys. Rev. Lett. 83, 648–651 (1999).
[Crossref]

1989 (2)

M. D. Reid, “Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40, 913–923 (1989).
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1964 (1)

J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics 1, 195–200 (1964).

1935 (2)

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Aguilar, G. H.

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Ansmann, M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
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Aoki, T.

H. Yonezawa, T. Aoki, and A. Furusawa, “Demonstration of a quantum teleportation network for continuous variables,” Nature 431, 430–433 (2004).
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Armstrong, S.

S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. - A. Bachor, M D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167–172 (2015).
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Aspelmeyer, M.

J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
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S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84, 052327 (2011).
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Bachor, H. - A.

S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. - A. Bachor, M D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167–172 (2015).
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J. D. Bancal, J. Barrett, N. Gisin, and S. Pironio, “Definitions of multipartite nonlocality,” Phys. Rev. A 88, 014102 (2013).
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J. D. Bancal, N. Gisin, Y. C. Liang, and S. Pironio, “Device-independent witnesses of genuine multipartite entanglement,” Phys. Rev. Lett. 106, 250404 (2011).
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J. D. Bancal, J. Barrett, N. Gisin, and S. Pironio, “Definitions of multipartite nonlocality,” Phys. Rev. A 88, 014102 (2013).
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S. Steinlechner, J. Bauchrowitz, T. Eberle, and R. Schnabel, “Strong Einstein-Podolsky-Rosen steering with unconditional entangled states,” Phys. Rev. A 87, 022104 (2013).
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J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics 1, 195–200 (1964).

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M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
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A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
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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).

D. H. Smith, G. Gillett, M. P. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nat. Commun. 3, 625 (2012).
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C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301(R) (2012).
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T. P. Purdy, D. W. C. Brooks, T. Botter, N. Brahms, Z.-Y. Ma, and D. M. Stamper-Kurn, “Tunable cavity optomechanics with ultracold atoms,” Phys. Rev. Lett. 105, 133602 (2010).
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Brup, D.

M. Huber, H. Schimpf, A. Gabriel, C. Spengler, D. Brup, and B. C. Hiesmayr, “Experimentally implementable criteria revealing substructures of genuine multipartite entanglement,” Phys. Rev. A 83, 022328 (2011).
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M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999).
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D. H. Smith, G. Gillett, M. P. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nat. Commun. 3, 625 (2012).
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A. Carmele, B. Vogell, K. Stannigel, and P. Zoller, “Opto-nanomechanics strongly coupled to a Rydberg super-atom: coherent versus incoherent dynamics,” New J. Phys. 16, 063042 (2014).
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D. Cavalcanti, P. Skrzypczyk, G. H. Aguilar, R. V. Nery, P. H. Souto Ribeiro, and S. P. Walborn, “Detection of entanglement in asymmetric quantum networks and multipartite quantum steering,” Nat. Commun. 6, 7941 (2015).
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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).

C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301(R) (2012).
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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).
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J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
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Chen, J. L.

K. Sun, J. S. Xu, X. J. Ye, Y. C. Wu, J. L. Chen, C. F. Li, and G. C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014).
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D. L. Deng, Z. S. Zhou, and J. L. Chen, “Svetlichny’s approach to detecting genuine multipartite entanglement in arbitrarily-high-dimensional systems by a Bell-type inequality,” Phys. Rev. A 80, 022109 (2009).
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Chen, K.

C. M. Li, K. Chen, Y. N. Chen, Q. Zhang, Y. A. Chen, and J. W. Pan, “Genuine high-order Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 115, 010402 (2015).
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Chen, Y. A.

C. M. Li, K. Chen, Y. N. Chen, Q. Zhang, Y. A. Chen, and J. W. Pan, “Genuine high-order Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 115, 010402 (2015).
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Chen, Y. N.

C. M. Li, K. Chen, Y. N. Chen, Q. Zhang, Y. A. Chen, and J. W. Pan, “Genuine high-order Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 115, 010402 (2015).
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Y. D. Wang, S. Chesi, and A. A. Clerk, “Bipartite and tripartite output entanglement in three-mode optomechanical systems,” Phys. Rev. A 91, 013807 (2015).
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Cicak, K.

J. D. Teufel, T. Donner, D. L Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
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Cleland, A. N.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
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Clerk, A. A.

Y. D. Wang, S. Chesi, and A. A. Clerk, “Bipartite and tripartite output entanglement in three-mode optomechanical systems,” Phys. Rev. A 91, 013807 (2015).
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Cleve, R.

R. Cleve, D. Gottesman, and H.-K. Lo, “How to share a quantum secret,” Phys. Rev. Lett. 83, 648–651 (1999).
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de Almeida, M. P.

D. H. Smith, G. Gillett, M. P. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nat. Commun. 3, 625 (2012).
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Deng, D. L.

D. L. Deng, Z. S. Zhou, and J. L. Chen, “Svetlichny’s approach to detecting genuine multipartite entanglement in arbitrarily-high-dimensional systems by a Bell-type inequality,” Phys. Rev. A 80, 022109 (2009).
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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).
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S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007).
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Donner, T.

J. D. Teufel, T. Donner, D. L Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
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F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
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Eberle, T.

S. Steinlechner, J. Bauchrowitz, T. Eberle, and R. Schnabel, “Strong Einstein-Podolsky-Rosen steering with unconditional entangled states,” Phys. Rev. A 87, 022104 (2013).
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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).
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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).
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Esslinger, T.

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
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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).

Fedrizzi, A.

D. H. Smith, G. Gillett, M. P. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nat. Commun. 3, 625 (2012).
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Ficek, Z.

M. Wang, Q. H. Gong, Z. Ficek, and Q. Y. He, “Efficient scheme for perfect collective Einstein-Podolsky-Rosen steering,” Sci. Rep. 5, 12346 (2015).
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Q. Y. He and Z. Ficek, “Einstein-Podolsky-Rosen paradox and quantum steering in a three-mode optomechanical system,” Phys. Rev. A 89, 022332 (2014).
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M. Wang, Q. H. Gong, Z. Ficek, and Q.Y. He, “Role of thermal noise in tripartite quantum steering,” Phys. Rev. A 90, 023801 (2014).
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L. H. Sun, G. X. Li, and Z. Ficek, “First-order coherence versus entanglement in a nanomechanical cavity,” Phys. Rev. A 85, 022327 (2012).
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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).
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Furusawa, A.

H. Yonezawa, T. Aoki, and A. Furusawa, “Demonstration of a quantum teleportation network for continuous variables,” Nature 431, 430–433 (2004).
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P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
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Gabriel, A.

M. Huber, H. Schimpf, A. Gabriel, C. Spengler, D. Brup, and B. C. Hiesmayr, “Experimentally implementable criteria revealing substructures of genuine multipartite entanglement,” Phys. Rev. A 83, 022328 (2011).
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Gallego, R.

R. Gallego, L. E. Würflinger, A. Acín, and M. Navascués, “Operational framework for nonlocality,” Phys. Rev. Lett. 109, 070401 (2012).
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Gerrits, T.

D. H. Smith, G. Gillett, M. P. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nat. Commun. 3, 625 (2012).
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Giampaolo, S. M.

S. M. Giampaolo and B. C. Hiesmayr, “Genuine multipartite entanglement in the XY model,” Phys. Rev. A 88, 052305 (2013).
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Gillett, G.

D. H. Smith, G. Gillett, M. P. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nat. Commun. 3, 625 (2012).
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Gisin, N.

J. D. Bancal, J. Barrett, N. Gisin, and S. Pironio, “Definitions of multipartite nonlocality,” Phys. Rev. A 88, 014102 (2013).
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J. D. Bancal, N. Gisin, Y. C. Liang, and S. Pironio, “Device-independent witnesses of genuine multipartite entanglement,” Phys. Rev. Lett. 106, 250404 (2011).
[Crossref] [PubMed]

Gong, Q. H.

Q. Y. He, Q. H. Gong, and M. D. Reid, “Classifying directional Gaussian entanglement, Einstein-Podolsky-Rosen steering, and discord,” Phys. Rev. Lett. 114, 060402 (2015).
[Crossref] [PubMed]

S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. - A. Bachor, M D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167–172 (2015).
[Crossref]

M. Wang, Q. H. Gong, Z. Ficek, and Q. Y. He, “Efficient scheme for perfect collective Einstein-Podolsky-Rosen steering,” Sci. Rep. 5, 12346 (2015).
[Crossref] [PubMed]

M. Wang, Q. H. Gong, Z. Ficek, and Q.Y. He, “Role of thermal noise in tripartite quantum steering,” Phys. Rev. A 90, 023801 (2014).
[Crossref]

M. Wang, Q. H. Gong, and Q.Y. He, “Collective multipartite Einstein-Podolsky-Rosen steering: more secure optical networks,” Opt. Lett. 39, 6703–6706 (2014).
[Crossref] [PubMed]

Gottesman, D.

R. Cleve, D. Gottesman, and H.-K. Lo, “How to share a quantum secret,” Phys. Rev. Lett. 83, 648–651 (1999).
[Crossref]

Gröblacher, S.

J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref] [PubMed]

Guo, G. C.

K. Sun, J. S. Xu, X. J. Ye, Y. C. Wu, J. L. Chen, C. F. Li, and G. C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014).
[Crossref] [PubMed]

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B. Hage, A. Samblowski, and R. Schnabel, “Towards Einstein-Podolsky-Rosen quantum channel multiplexing,” Phys. Rev. A 81, 062301 (2010).
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L. K. Shalm, D. R. Hamel, Z. Yan, C. Simon, K. J. Resch, and T. Jennewein, “Three-photon energy–time entanglement,” Nat. Phys. 9, 19–22 (2013).
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Hammerer, K.

S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84, 052327 (2011).
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K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041 (2010).
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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]

Harlow, J. W.

J. D. Teufel, T. Donner, D. L Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

He, B.

He, Q. Y.

M. Wang, Q. H. Gong, Z. Ficek, and Q. Y. He, “Efficient scheme for perfect collective Einstein-Podolsky-Rosen steering,” Sci. Rep. 5, 12346 (2015).
[Crossref] [PubMed]

S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. - A. Bachor, M D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167–172 (2015).
[Crossref]

Q. Y. He, Q. H. Gong, and M. D. Reid, “Classifying directional Gaussian entanglement, Einstein-Podolsky-Rosen steering, and discord,” Phys. Rev. Lett. 114, 060402 (2015).
[Crossref] [PubMed]

Q. Y. He and Z. Ficek, “Einstein-Podolsky-Rosen paradox and quantum steering in a three-mode optomechanical system,” Phys. Rev. A 89, 022332 (2014).
[Crossref]

Q. Y. He and M. D. Reid, “Genuine multipartite Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 111, 250403 (2013).
[Crossref]

He, Q.Y.

M. Wang, Q. H. Gong, and Q.Y. He, “Collective multipartite Einstein-Podolsky-Rosen steering: more secure optical networks,” Opt. Lett. 39, 6703–6706 (2014).
[Crossref] [PubMed]

M. Wang, Q. H. Gong, Z. Ficek, and Q.Y. He, “Role of thermal noise in tripartite quantum steering,” Phys. Rev. A 90, 023801 (2014).
[Crossref]

Hiesmayr, B. C.

S. M. Giampaolo and B. C. Hiesmayr, “Genuine multipartite entanglement in the XY model,” Phys. Rev. A 88, 052305 (2013).
[Crossref]

M. Huber, H. Schimpf, A. Gabriel, C. Spengler, D. Brup, and B. C. Hiesmayr, “Experimentally implementable criteria revealing substructures of genuine multipartite entanglement,” Phys. Rev. A 83, 022328 (2011).
[Crossref]

Hill, J. T.

J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref] [PubMed]

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C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307(R) (2008).
[Crossref]

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A. Carmele, B. Vogell, K. Stannigel, and P. Zoller, “Opto-nanomechanics strongly coupled to a Rydberg super-atom: coherent versus incoherent dynamics,” New J. Phys. 16, 063042 (2014).
[Crossref]

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D. Cavalcanti, P. Skrzypczyk, G. H. Aguilar, R. V. Nery, P. H. Souto Ribeiro, and S. P. Walborn, “Detection of entanglement in asymmetric quantum networks and multipartite quantum steering,” Nat. Commun. 6, 7941 (2015).
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C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301(R) (2012).
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A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Wang, M.

M. Wang, Q. H. Gong, Z. Ficek, and Q. Y. He, “Efficient scheme for perfect collective Einstein-Podolsky-Rosen steering,” Sci. Rep. 5, 12346 (2015).
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S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. - A. Bachor, M D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167–172 (2015).
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M. Wang, Q. H. Gong, Z. Ficek, and Q.Y. He, “Role of thermal noise in tripartite quantum steering,” Phys. Rev. A 90, 023801 (2014).
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M. Wang, Q. H. Gong, and Q.Y. He, “Collective multipartite Einstein-Podolsky-Rosen steering: more secure optical networks,” Opt. Lett. 39, 6703–6706 (2014).
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A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
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D. H. Smith, G. Gillett, M. P. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nat. Commun. 3, 625 (2012).
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A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
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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).
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R. F. Werner, “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989).
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D. H. Smith, G. Gillett, M. P. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nat. Commun. 3, 625 (2012).
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J. D. Teufel, T. Donner, D. L Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
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S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84, 052327 (2011).
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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).

D. H. Smith, G. Gillett, M. P. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nat. Commun. 3, 625 (2012).
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C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301(R) (2012).
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B. Wittmann, S. Ramelow, F. Steinlechner, N. K. Langford, N. Brunner, H. M. Wiseman, R. Ursin, and A. Zeilinger, “Loophole-free Einstein–Podolsky–Rosen experiment via quantum steering,” New J. Phys. 14, 053030 (2012).
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D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845–849 (2010).
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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).
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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).
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S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007).
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Wittmann, B.

B. Wittmann, S. Ramelow, F. Steinlechner, N. K. Langford, N. Brunner, H. M. Wiseman, R. Ursin, and A. Zeilinger, “Loophole-free Einstein–Podolsky–Rosen experiment via quantum steering,” New J. Phys. 14, 053030 (2012).
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S. Singh, H. Jing, E. M. Wright, and P. Meystre, “Quantum-state transfer between a Bose-Einstein condensate and an optomechanical mirror,” Phys. Rev. A 86, 021801(R) (2012).
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Wu, Y. C.

K. Sun, J. S. Xu, X. J. Ye, Y. C. Wu, J. L. Chen, C. F. Li, and G. C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014).
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R. Gallego, L. E. Würflinger, A. Acín, and M. Navascués, “Operational framework for nonlocality,” Phys. Rev. Lett. 109, 070401 (2012).
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K. Sun, J. S. Xu, X. J. Ye, Y. C. Wu, J. L. Chen, C. F. Li, and G. C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014).
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Yan, Z.

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Ye, X. J.

K. Sun, J. S. Xu, X. J. Ye, Y. C. Wu, J. L. Chen, C. F. Li, and G. C. Guo, “Experimental demonstration of the Einstein-Podolsky-Rosen steering game based on the all-versus-nothing proof,” Phys. Rev. Lett. 113, 140402 (2014).
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B. Wittmann, S. Ramelow, F. Steinlechner, N. K. Langford, N. Brunner, H. M. Wiseman, R. Ursin, and A. Zeilinger, “Loophole-free Einstein–Podolsky–Rosen experiment via quantum steering,” New J. Phys. 14, 053030 (2012).
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Zhang, Q.

C. M. Li, K. Chen, Y. N. Chen, Q. Zhang, Y. A. Chen, and J. W. Pan, “Genuine high-order Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 115, 010402 (2015).
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D. L. Deng, Z. S. Zhou, and J. L. Chen, “Svetlichny’s approach to detecting genuine multipartite entanglement in arbitrarily-high-dimensional systems by a Bell-type inequality,” Phys. Rev. A 80, 022109 (2009).
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A. Carmele, B. Vogell, K. Stannigel, and P. Zoller, “Opto-nanomechanics strongly coupled to a Rydberg super-atom: coherent versus incoherent dynamics,” New J. Phys. 16, 063042 (2014).
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Nat. Commun. (2)

D. H. Smith, G. Gillett, M. P. de Almeida, C. Branciard, A. Fedrizzi, T. J. Weinhold, A. Lita, B. Calkins, T. Gerrits, H. M. Wiseman, S. W. Nam, and A. G. White, “Conclusive quantum steering with superconducting transition-edge sensors,” Nat. Commun. 3, 625 (2012).
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D. Cavalcanti, P. Skrzypczyk, G. H. Aguilar, R. V. Nery, P. H. Souto Ribeiro, and S. P. Walborn, “Detection of entanglement in asymmetric quantum networks and multipartite quantum steering,” Nat. Commun. 6, 7941 (2015).
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Nat. Photonics (1)

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. (3)

D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using Bell-local states,” Nat. Phys. 6, 845–849 (2010).
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L. K. Shalm, D. R. Hamel, Z. Yan, C. Simon, K. J. Resch, and T. Jennewein, “Three-photon energy–time entanglement,” Nat. Phys. 9, 19–22 (2013).
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S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. - A. Bachor, M D. Reid, and P. K. Lam, “Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11, 167–172 (2015).
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Nature (4)

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New J. Phys. (2)

A. Carmele, B. Vogell, K. Stannigel, and P. Zoller, “Opto-nanomechanics strongly coupled to a Rydberg super-atom: coherent versus incoherent dynamics,” New J. Phys. 16, 063042 (2014).
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B. Wittmann, S. Ramelow, F. Steinlechner, N. K. Langford, N. Brunner, H. M. Wiseman, R. Ursin, and A. Zeilinger, “Loophole-free Einstein–Podolsky–Rosen experiment via quantum steering,” New J. Phys. 14, 053030 (2012).
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Opt. Express (2)

Opt. Lett. (1)

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C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301(R) (2012).
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S. J. Jones, H. M. Wiseman, and A. C. Doherty, “Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering,” Phys. Rev. A 76, 052116 (2007).
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Y. D. Wang, S. Chesi, and A. A. Clerk, “Bipartite and tripartite output entanglement in three-mode optomechanical systems,” Phys. Rev. A 91, 013807 (2015).
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Q. Y. He and Z. Ficek, “Einstein-Podolsky-Rosen paradox and quantum steering in a three-mode optomechanical system,” Phys. Rev. A 89, 022332 (2014).
[Crossref]

M. Wang, Q. H. Gong, Z. Ficek, and Q.Y. He, “Role of thermal noise in tripartite quantum steering,” Phys. Rev. A 90, 023801 (2014).
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C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77, 050307(R) (2008).
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S. G. Hofer, W. Wieczorek, M. Aspelmeyer, and K. Hammerer, “Quantum entanglement and teleportation in pulsed cavity optomechanics,” Phys. Rev. A 84, 052327 (2011).
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L. H. Sun, G. X. Li, and Z. Ficek, “First-order coherence versus entanglement in a nanomechanical cavity,” Phys. Rev. A 85, 022327 (2012).
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S. Singh, H. Jing, E. M. Wright, and P. Meystre, “Quantum-state transfer between a Bose-Einstein condensate and an optomechanical mirror,” Phys. Rev. A 86, 021801(R) (2012).
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R. Gallego, L. E. Würflinger, A. Acín, and M. Navascués, “Operational framework for nonlocality,” Phys. Rev. Lett. 109, 070401 (2012).
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J. D. Bancal, N. Gisin, Y. C. Liang, and S. Pironio, “Device-independent witnesses of genuine multipartite entanglement,” Phys. Rev. Lett. 106, 250404 (2011).
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C. M. Li, K. Chen, Y. N. Chen, Q. Zhang, Y. A. Chen, and J. W. Pan, “Genuine high-order Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 115, 010402 (2015).
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Phys. Rev. X (1)

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).

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Sci. Rep. (1)

M. Wang, Q. H. Gong, Z. Ficek, and Q. Y. He, “Efficient scheme for perfect collective Einstein-Podolsky-Rosen steering,” Sci. Rep. 5, 12346 (2015).
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Other (1)

For two-mode Gaussian states, the condition EA|B(g) < 1 is necessary and sufficient to confirm steering of A by B [16, 17, 24], and the minimum value of EA|B(g) can be used to quantify the Gaussian EPR steering [22].

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of a hybrid atomic optomechanical system. The cavity mode has a frequency ωc and a decay rate κ, driven by a laser pulse of frequency ωL. The movable mirror oscillates with frequency ωm and dissipation rate γm. Each of the N identical two-level atoms includes a ground state |gj〉 and an excited state |ej〉, with transition frequency ωa and relaxation time γa. The collective dipole lowering, raising, and population difference operators can be defined as S = j | g j e j |, S + = j | e j g j |, and S z = j ( | e j e j | | g j g j | ), respectively. We assume that all the atoms are initially prepared in their ground state, so that Sz ≃ 〈Sz〉 ≃ −N. At the same time, the initial assumption κga assures that the photons inside the cavity may escape from the cavity quickly before making an effective Rabi cycle between the two levels. Thus, most of atoms stay in the ground state such that the condition of low atomic excitation limit still stands. In this case, we may represent the collective dipole lowering and raising operators in terms of annihilation and creation operators c a = S / | S z | and c a = S + / | S z |, respectively.

Fig. 2
Fig. 2

The effect of initial mechanical thermal noise nm0 on the EPR-type tripartite entanglement parameter Em|ac. Left: the 3D plot of Em|ac given in Eq. (16) as function of nm0 and squeezing parameter r = (GGA)τ for fixed coupling strength α = 2; Right: the contour plot of Em|ac. The value of Em|ac smaller than the thresholds 2, 1, and 0.5, reveals the tripartite inseparability, genuine tripartite entanglement and the steering of the mirror by the cavity field and atomic modes, and genuine tripartite steering, respectively. Here, the cavity field and the atomic mode are assumed initially in a vacuum state na0 = nc0 = 0.

Fig. 3
Fig. 3

The effect of the relaxation of the atom γa when γm = 0 (a) and the mechanical damping γm when γa = 0 (b) on the EPR-type tripartite entanglement parameter Em|ac. Here, we set the coupling strength α = 2, the initial thermal noise nm0,a0,c0 = 0 and the reservoir bath m,a,c = 0.

Fig. 4
Fig. 4

Same to Fig. 2 but the mirror in contact with a nonzero temperature T reservoir with mean number of the thermal phonons m = nm0 = n, showing the decoherence effects due to mechanical coupling to a heat bath. Here, we set α = 2, γm = 6 × 10−5κ, and γa = 10−4κ. The cavity field and the atomic modes are in the ordinary zero temperature environment a,c = na0,c0 = 0.

Fig. 5
Fig. 5

(a) The effect of gains in inequalities (1114) on the tripartite entanglement parameter Em|ac(g) (black solid) comparing with the parameter Em|ac shown in Fig. 3 when n = 0 (blue dashed). The inset shows the optimal gains that can minimize the reduced noise variance of the mirror Em|ac(g), in terms of the measurements performed on the cavity field and atomic modes. (b) the tripartite entanglement parameter Em|ac(g) (black solid) versus four different thresholds determined by the corresponding gains for each type of tripartite entanglement. Here, we set α = 2, γm = 6 × 10−5κ, and γa = 10−4κ, a,m,c = na0,m0,c0 = 0.

Equations (18)

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H = Δ c a c a c + ω m b m b m + Δ a c a c a + g 0 a c a c ( b m + b m ) + g a ( c a a c + a c c a ) + i [ E ( t ) a c E * ( t ) a c ] .
δ a ˙ c = κ δ a c i g a δ c a i g ( δ b m + δ b m ) 2 κ a in , δ b ˙ m = γ m δ b m i g ( δ a c + δ a c ) 2 γ m b in , δ c ˙ a = γ a δ c a i g a δ a c 2 γ a c in ,
a c ( t ) i g a κ c a ( t ) i g κ b m ( t ) 2 κ a in ( t ) .
c ˙ a = ( G a + γ a ) c a GG a b m + i 2 G a a in 2 γ a c in , b ˙ m = ( G γ m ) b m + GG a c a i 2 G a in 2 γ m b in ,
Y ˙ ( t ) = M Y ( t ) + η ( t ) ,
M = ( G a γ a GG a GG a G γ m ) ,
c a ( t ) = e λ u t v 2 + e λ v t u 2 v 2 + u 2 c a ( 0 ) + e λ u t e λ v t v 2 + u 2 b m ( 0 ) + i v 2 2 G a 2 G v 2 + u 2 e λ u t 0 t d t a in ( t ) e λ u t + i u 2 2 G a + 2 G v 2 + u 2 e λ v t 0 t d t a in ( t ) e λ v t 2 γ a v 2 + u 2 [ u 2 e λ v t 0 t d t c in ( t ) e λ v t + v 2 e λ u t 0 t d t c in ( t ) e λ u t ] + 2 γ m v 2 + u 2 [ e λ v t 0 t d t b in ( t ) e λ v t e λ u t 0 t d t b in ( t ) e λ u t ] , b m ( t ) = ( e λ u t e λ v t ) u 2 v 2 v 2 + u 2 c a ( 0 ) + e λ v t v 2 + e λ u t u 2 v 2 + u 2 b m ( 0 ) + i u 2 v 2 2 G a 2 G v 2 + u 2 e λ u t 0 t d t a in ( t ) e λ u t i v 2 u 2 2 G a + 2 G v 2 + u 2 e λ v t 0 t d t a in ( t ) e λ v t + u 2 v 2 2 γ a v 2 + u 2 [ e λ v t 0 t d t c in ( t ) e λ v t e λ u t 0 t d t c in ( t ) e λ u t ] 2 γ m v 2 + u 2 [ u 2 e λ u t 0 t d t b in ( t ) e λ u t + v 2 e λ v t 0 t d t b in ( t ) e λ v t ] .
A in = 2 ( G G a ) 1 e 2 ( G G a ) τ 0 τ d t a in ( t ) e ( G G a ) t , A out = 2 ( G G a ) e 2 ( G G a ) τ 1 0 τ d t a out ( t ) e ( G G a ) t ,
A out = e r A in i e 2 r 1 ( α B in + β C in ) , B out = ( α 2 e r β 2 ) B in + α β ( e r 1 ) C in + i α e 2 r 1 A in , C out = ( α 2 β 2 e r ) C in α β ( e r 1 ) B in + i β e 2 r 1 A in ,
E m | ac ( g ) = Δ inf , ac 2 X m + Δ inf , ac 2 P m = Δ 2 ( X m + g a P a + g c X c ) + Δ 2 ( P m + h a X a + h c P c ) ,
E m | ac ( g ) min { 1 + | g a h a + g c h c | , | g a h a | + | 1 + g c h c | , | g c h c | + | 1 + g a h a | } .
E m | ac ( g ) min { 1 , | g a h a | , | g c h c | } .
E m | ac ( g ) 1 + | g a h a | + | g c h c | ,
E m | ac ( g ) 1 .
E m | ac = Δ 2 ( X m + P a + X c 2 ) + Δ 2 ( P m + X a P c 2 ) .
E m | ac = [ γ e 2 r 1 e r 2 ] 2 ( 2 n a 0 + 1 ) + [ α γ ( e r 1 ) α 2 e 2 r 1 + 1 ] 2 ( 2 n m 0 + 1 ) + [ β γ ( e r 1 ) β 2 e 2 r 1 + 1 2 ] 2 ( 2 n c 0 + 1 ) ,
A in 1 = Γ 1 0 τ d t a in ( t ) e λ u t , A in 2 = Γ 2 0 τ d t a in ( t ) e λ v t , A in 3 = Γ 3 0 τ d t a in ( t ) e λ u t , B m 1 = Γ 1 0 τ d t b in ( t ) e λ u t , B m 2 = Γ 2 0 τ d t b in ( t ) e λ v t , B m 3 = Γ 3 0 τ d t b in ( t ) e λ u t , C 1 = Γ 1 0 τ d t c in ( t ) e λ u t , C 2 = Γ 2 0 τ d t c in ( t ) e λ v t , C 3 = Γ 3 0 τ d t c in ( t ) e λ u t ,
C out = e λ u τ v 2 + e λ v τ u 2 v 2 + u 2 C in + e λ u τ e λ v τ v 2 + u 2 B in + i v 2 2 G a 2 G Γ 3 ( v 2 + u 2 ) A in 1 + i e λ v τ ( u 2 2 G a + 2 G ) Γ 2 ( v 2 + u 2 ) A in 2 1 Γ 3 ( v 2 + u 2 ) ( v 2 2 γ a C 1 + 2 γ m B m 1 ) + e λ v τ Γ 2 ( v 2 + u 2 ) ( 2 γ n B m 2 u 2 2 γ a C 2 ) , B out = ( e λ u τ e λ v τ ) u 2 v 2 v 2 + u 2 C in + e λ v τ v 2 + e λ v τ u 2 v 2 + u 2 B in i u 2 v 2 2 G a u 2 2 G Γ 3 ( v 2 + u 2 ) A in 1 + i e λ v τ ( v 2 u 2 2 G a + v 2 2 G ) Γ 2 ( v 2 + u 2 ) A in 2 u 2 Γ 3 ( v 2 + u 2 ) ( v 2 2 γ a C 1 + 2 γ m B m 1 ) + v 2 e λ v τ Γ 2 ( v 2 + u 2 ) ( u 2 2 γ a C 2 2 γ m B m 2 ) , A out = 1 u 2 + v 2 ( i 2 f 1 v 2 Γ 3 + i 2 f 2 u 2 Γ 3 Γ 5 ) C in + 1 u 2 + v 2 ( i 2 f 1 Γ 3 i 2 f 2 Γ 3 Γ 5 ) B in + e λ u τ λ u ( u 2 + v 2 ) ( h 1 A in 1 i f 1 v 2 γ a C 1 i f 1 γ m B m 1 ) + 1 λ u ( u 2 + v 2 ) ( h 1 A in 3 + i f 1 v 2 γ a C 3 + i f 1 γ m B m 3 ) + 2 e λ u τ Γ 4 ( λ u + λ v ) ( u 2 + v 2 ) ( h 2 A in 2 i f 2 u 2 γ a C 2 + i f 2 γ m B m 2 ) + 2 ( λ u + λ v ) ( u 2 + v 2 ) ( h 2 A in 3 i f 2 u 2 γ a C 3 i f 2 γ m B m 3 ) A in 3 ,

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