Abstract

We experimentally simulate a quantum channel in a linear optical setup, which is modeled by a two-level system (i.e., qubit) interacting with a bosonic bath. Unlike the traditional works, we treat the system–bath interaction without applying the Born approximation, the Markov approximation, or the rotating-wave approximation (RWA). To the best of our knowledge, this is the first experimental simulation of a quantum channel without any of the approximations mentioned above by using linear optical devices. This non-RWA channel provides a more accurate picture of the quantum open-system dynamics. It not only reveals the effect of the counterrotating terms but also enables us to consider arbitrarily strong coupling regimes. With the proposed channel, we further experimentally investigate the dynamics of the quantum temporal steering (TS), i.e., a temporal analog of Einstein–Podolsky–Rosen steering. The experimental and theoretical results are in good agreement and show that the counterrotating terms significantly influence the TS dynamics. The TS in non-RWA and RWA channels presents different dynamics. However, we emphasize that the results without RWA are closer to realistic situations and thus more reliable. Due to the close relationship between TS and the security of the quantum cryptographic protocols, our findings are expected to have useful applications in secure quantum communications. This work also inspires future interest in studying other quantum coherence properties in the non-RWA channels.

© 2017 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  30. M. F. Pusey, “Negativity and steering: a stronger Peres conjecture,” Phys. Rev. A 88, 032313 (2013).
    [Crossref]
  31. P. Skrzypczyk, M. Navascués, and D. Cavalcanti, “Quantifying Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 180404 (2014).
    [Crossref]
  32. M. T. Quintino, T. Vértesi, and N. Brunner, “Joint measurability, Einstein–Podolsky–Rosen steering, and Bell nonlocality,” Phys. Rev. Lett. 113, 160402 (2014).
    [Crossref]
  33. R. Uola, T. Moroder, and O. Gühne, “Joint measurability of generalized measurements implies classicality,” Phys. Rev. Lett. 113, 160403 (2014).
    [Crossref]
  34. M. Piani and J. Watrous, “Necessary and sufficient quantum information characterization of Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 114, 060404 (2015).
    [Crossref]
  35. Y. N. Chen, C. M. Li, N. Lambert, S. L. Chen, Y. Ota, G. Y. Chen, and F. Nori, “Temporal steering inequality,” Phys. Rev. A 89, 032112 (2014).
    [Crossref]
  36. C. Emary, N. Lambert, and F. Nori, “Leggett–Garg inequalities,” Rep. Prog. Phys. 77, 016001 (2014).
    [Crossref]
  37. K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Temporal steering and security of quantum key distribution with mutually unbiased bases against individual attacks,” Phys. Rev. A 93, 062345 (2016).
    [Crossref]
  38. M. Tomamichel and R. Renner, “Uncertainty relation for smooth entropies,” Phys. Rev. Lett. 106, 110506 (2011).
    [Crossref]
  39. S. L. Chen, N. Lambert, C. M. Li, A. Miranowicz, Y. N. Chen, and F. Nori, “Quantifying non-Markovianity with temporal steering,” Phys. Rev. Lett. 116, 020503 (2016).
    [Crossref]
  40. C. M. Li, Y. N. Chen, N. Lambert, C. Y. Chiu, and F. Nori, “Certifying single-system steering for quantum-information processing,” Phys. Rev. A 92, 062310 (2015).
    [Crossref]
  41. H. S. Karthik, J. Prabhu Tej, A. R. Usha Devi, and A. K. Rajagopal, “Joint measurability and temporal steering,” J. Opt. Soc. Am. B 32, A34–A39 (2015).
    [Crossref]
  42. H. Y. Ku, S. L. Chen, H. B. Chen, N. Lambert, Y. N. Chen, and F. Nori, “Temporal steering in four dimensions with applications to coupled qubits and magnetoreception,” Phys. Rev. A 94, 062126 (2016).
    [Crossref]
  43. S. L. Chen, N. Lambert, C. M. Li, G. Y. Chen, Y. N. Chen, A. Miranowicz, and F. Nori, “Spatio-temporal steering for testing nonclassical correlations in quantum networks,” Sci. Rep. 7, 3728 (2017).
    [Crossref]
  44. K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Experimental temporal quantum steering,” Sci. Rep. 6, 38076 (2016).
    [Crossref]
  45. V. V. Albert, “Quantum Rabi model for n-state atoms,” Phys. Rev. Lett. 108, 180401 (2012).
    [Crossref]
  46. F. F. Fanchini, G. Karpat, B. Cakmak, L. K. Castelano, G. G. Aguilar, O. Jiménez Farías, S. P. Walborn, P. H. Souto Riberio, and M. C. de Oliveira, “Non-Markovianity through accessible information,” Phys. Rev. Lett. 112, 210402 (2014).
    [Crossref]
  47. O. J. Farí, G. H. Aguilar, A. Valdés-Hernández, P. H. Ribeiro, L. Davidovich, and S. P. Walborn, “Observation of the emergence of multipartite entanglement between a bipartite system and its environment,” Phys. Rev. Lett. 109, 150403 (2012).
    [Crossref]
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    [Crossref]
  49. T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93, 140404 (2004).
    [Crossref]

2017 (3)

I. de Vega and D. Alonso, “Dynamics of non-Markovian open quantum systems,” Rev. Mod. Phys. 89, 015001 (2017).
[Crossref]

L. Mancino, M. Sbroscia, I. Gianani, E. Roccia, and M. Barbieri, “Quantum simulation of single-qubit thermometry using linear optics,” Phys. Rev. Lett. 118, 130502 (2017).
[Crossref]

S. L. Chen, N. Lambert, C. M. Li, G. Y. Chen, Y. N. Chen, A. Miranowicz, and F. Nori, “Spatio-temporal steering for testing nonclassical correlations in quantum networks,” Sci. Rep. 7, 3728 (2017).
[Crossref]

2016 (7)

K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Experimental temporal quantum steering,” Sci. Rep. 6, 38076 (2016).
[Crossref]

S. L. Chen, N. Lambert, C. M. Li, A. Miranowicz, Y. N. Chen, and F. Nori, “Quantifying non-Markovianity with temporal steering,” Phys. Rev. Lett. 116, 020503 (2016).
[Crossref]

H. Y. Ku, S. L. Chen, H. B. Chen, N. Lambert, Y. N. Chen, and F. Nori, “Temporal steering in four dimensions with applications to coupled qubits and magnetoreception,” Phys. Rev. A 94, 062126 (2016).
[Crossref]

Z. Sun, L. W. Zhou, G. Y. Xiao, D. Poletti, and J. B. Gong, “Finite-time Landau–Zener processes and counterdiabatic driving in open systems: beyond Born, Markov, and rotating-wave approximations,” Phys. Rev. A 93, 012121 (2016).
[Crossref]

H. Zhu, M. Hayashi, and L. Chen, “Universal steering inequalities,” Phys. Rev. Lett. 116, 070403 (2016).
[Crossref]

K. Sun, X. J. Ye, J. S. Xu, X. Y. Xu, J. S. Tang, Y. C. Wu, J. L. Chen, F. C. Li, and C. G. Guo, “Experimental quantification of asymmetric Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
[Crossref]

K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Temporal steering and security of quantum key distribution with mutually unbiased bases against individual attacks,” Phys. Rev. A 93, 062345 (2016).
[Crossref]

2015 (7)

M. Piani and J. Watrous, “Necessary and sufficient quantum information characterization of Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 114, 060404 (2015).
[Crossref]

M. Marciniak, A. Rutkowski, Z. Yin, M. Horodecki, and R. Horodecki, “Unbounded violation of quantum steering inequalities,” Phys. Rev. Lett. 115, 170401 (2015).
[Crossref]

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, L. Rosales-Zarate, G. Adesso, and M. D. Reid, “Secure continuous variable teleportation and Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 115, 180502 (2015).
[Crossref]

C. M. Li, Y. N. Chen, N. Lambert, C. Y. Chiu, and F. Nori, “Certifying single-system steering for quantum-information processing,” Phys. Rev. A 92, 062310 (2015).
[Crossref]

H. S. Karthik, J. Prabhu Tej, A. R. Usha Devi, and A. K. Rajagopal, “Joint measurability and temporal steering,” J. Opt. Soc. Am. B 32, A34–A39 (2015).
[Crossref]

Z. Sun, J. Liu, J. Ma, and X. Wang, “Quantum speed limits in open systems: non-Markovian dynamics without rotating-wave approximation,” Sci. Rep. 5, 8444 (2015).
[Crossref]

2014 (8)

F. F. Fanchini, G. Karpat, B. Cakmak, L. K. Castelano, G. G. Aguilar, O. Jiménez Farías, S. P. Walborn, P. H. Souto Riberio, and M. C. de Oliveira, “Non-Markovianity through accessible information,” Phys. Rev. Lett. 112, 210402 (2014).
[Crossref]

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

P. Skrzypczyk, M. Navascués, and D. Cavalcanti, “Quantifying Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 180404 (2014).
[Crossref]

M. T. Quintino, T. Vértesi, and N. Brunner, “Joint measurability, Einstein–Podolsky–Rosen steering, and Bell nonlocality,” Phys. Rev. Lett. 113, 160402 (2014).
[Crossref]

R. Uola, T. Moroder, and O. Gühne, “Joint measurability of generalized measurements implies classicality,” Phys. Rev. Lett. 113, 160403 (2014).
[Crossref]

Y. N. Chen, C. M. Li, N. Lambert, S. L. Chen, Y. Ota, G. Y. Chen, and F. Nori, “Temporal steering inequality,” Phys. Rev. A 89, 032112 (2014).
[Crossref]

C. Emary, N. Lambert, and F. Nori, “Leggett–Garg inequalities,” Rep. Prog. Phys. 77, 016001 (2014).
[Crossref]

J. S. Xu, M. H. Yung, X. Y. Xu, S. Boixo, Z. W. Zhou, C. F. Li, A. Aspuru-Guzik, and G. C. Guo, “Demon-like algorithmic quantum cooling and its realization with quantum optics,” Nat. Photonics 8, 113–118 (2014).
[Crossref]

2013 (3)

D. Kast and J. Ankerhold, “Persistence of coherent quantum dynamics at strong dissipation,” Phys. Rev. Lett. 110, 010402 (2013).
[Crossref]

M. F. Pusey, “Negativity and steering: a stronger Peres conjecture,” Phys. Rev. A 88, 032313 (2013).
[Crossref]

J. S. Xu, K. Sun, C. F. Li, X. Y. Xu, G. C. Guo, E. Andersson, R. Lo Franco, and G. Compagno, “Experimental recovery of quantum correlations in absence of system-environment back-action,” Nat. Commun. 4, 2851 (2013).
[Crossref]

2012 (7)

V. Handchen, 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, 598–601 (2012).
[Crossref]

W. M. Zhang, P. Y. Lo, H. N. Xiong, M. W.-Y. Tu, and F. Nori, “General non-Markovian dynamics of open quantum systems,” Phys. Rev. Lett. 109, 170402 (2012).
[Crossref]

J. Ma, Z. Sun, X. Wang, and F. Nori, “Entanglement dynamics of two qubits in a common bath,” Phys. Rev. A 85, 062323 (2012).
[Crossref]

J. Larson, “Absence of vacuum induced berry phases without the rotating wave approximation in cavity QED,” Phys. Rev. Lett. 108, 033601 (2012).
[Crossref]

Z. Sun, J. Ma, X. Wang, and F. Nori, “Photon-assisted Landau–Zener transition: role of coherent superposition states,” Phys. Rev. A 86, 012107 (2012).
[Crossref]

O. J. Farí, G. H. Aguilar, A. Valdés-Hernández, P. H. Ribeiro, L. Davidovich, and S. P. Walborn, “Observation of the emergence of multipartite entanglement between a bipartite system and its environment,” Phys. Rev. Lett. 109, 150403 (2012).
[Crossref]

V. V. Albert, “Quantum Rabi model for n-state atoms,” Phys. Rev. Lett. 108, 180401 (2012).
[Crossref]

2011 (2)

B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
[Crossref]

M. Tomamichel and R. Renner, “Uncertainty relation for smooth entropies,” Phys. Rev. Lett. 106, 110506 (2011).
[Crossref]

2010 (3)

J. S. Xu, X. Y. Xu, C. F. Li, C. J. Zhang, X. B. Zou, and G. C. Guo, “Experimental investigation of classical and quantum correlations under decoherence,” Nat. Commun. 1, 7 (2010).
[Crossref]

M. Tanaka and Y. Tanimura, “Multistate electron transfer dynamics in the condensed phase: exact calculations from the reduced hierarchy equations of motion approach,” J. Chem. Phys. 132, 214502 (2010).
[Crossref]

A. G. Dijkstra and Y. Tanimura, “Non-Markovian entanglement dynamics in the presence of system–bath coherence,” Phys. Rev. Lett. 104, 250401 (2010).
[Crossref]

2009 (1)

J. S. Xu, C. F. Li, X. Y. Xu, C. H. Shi, X. B. Zou, and G. C. Guo, “Experimental characterization of entanglement dynamics in noisy channels,” Phys. Rev. Lett. 103, 240502 (2009).
[Crossref]

2008 (1)

J. S. Jin, X. Zheng, and Y. J. Yan, “Exact dynamics of dissipative electronic systems and quantum transport: hierarchical equations of motion approach,” J. Chem. Phys. 128, 234703 (2008).
[Crossref]

2007 (5)

A. Ishizaki and Y. Tanimura, “Dynamics of a multimode system coupled to multiple heat baths probed by two-dimensional infrared spectroscopy,” J. Phys. Chem. A 111, 9269–9276 (2007).
[Crossref]

M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316, 579–582 (2007).
[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]

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]

A. Acin, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref]

2004 (1)

T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93, 140404 (2004).
[Crossref]

1990 (1)

Y. Tanimura, “Nonperturbative expansion method for a quantum system coupled to a harmonic-oscillator bath,” Phys. Rev. A 41, 6676–6687 (1990).
[Crossref]

Acin, A.

A. Acin, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref]

Adesso, G.

Q. Y. He, L. Rosales-Zarate, G. Adesso, and M. D. Reid, “Secure continuous variable teleportation and Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 115, 180502 (2015).
[Crossref]

Aguilar, G. G.

F. F. Fanchini, G. Karpat, B. Cakmak, L. K. Castelano, G. G. Aguilar, O. Jiménez Farías, S. P. Walborn, P. H. Souto Riberio, and M. C. de Oliveira, “Non-Markovianity through accessible information,” Phys. Rev. Lett. 112, 210402 (2014).
[Crossref]

Aguilar, G. H.

O. J. Farí, G. H. Aguilar, A. Valdés-Hernández, P. H. Ribeiro, L. Davidovich, and S. P. Walborn, “Observation of the emergence of multipartite entanglement between a bipartite system and its environment,” Phys. Rev. Lett. 109, 150403 (2012).
[Crossref]

Albert, V. V.

V. V. Albert, “Quantum Rabi model for n-state atoms,” Phys. Rev. Lett. 108, 180401 (2012).
[Crossref]

Almeida, M. P.

M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316, 579–582 (2007).
[Crossref]

Alonso, D.

I. de Vega and D. Alonso, “Dynamics of non-Markovian open quantum systems,” Rev. Mod. Phys. 89, 015001 (2017).
[Crossref]

Andersson, E.

J. S. Xu, K. Sun, C. F. Li, X. Y. Xu, G. C. Guo, E. Andersson, R. Lo Franco, and G. Compagno, “Experimental recovery of quantum correlations in absence of system-environment back-action,” Nat. Commun. 4, 2851 (2013).
[Crossref]

Ankerhold, J.

D. Kast and J. Ankerhold, “Persistence of coherent quantum dynamics at strong dissipation,” Phys. Rev. Lett. 110, 010402 (2013).
[Crossref]

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).
[Crossref]

Aspuru-Guzik, A.

J. S. Xu, M. H. Yung, X. Y. Xu, S. Boixo, Z. W. Zhou, C. F. Li, A. Aspuru-Guzik, and G. C. Guo, “Demon-like algorithmic quantum cooling and its realization with quantum optics,” Nat. Photonics 8, 113–118 (2014).
[Crossref]

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).
[Crossref]

Barbieri, M.

L. Mancino, M. Sbroscia, I. Gianani, E. Roccia, and M. Barbieri, “Quantum simulation of single-qubit thermometry using linear optics,” Phys. Rev. Lett. 118, 130502 (2017).
[Crossref]

Bartkiewicz, K.

K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Experimental temporal quantum steering,” Sci. Rep. 6, 38076 (2016).
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K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Temporal steering and security of quantum key distribution with mutually unbiased bases against individual attacks,” Phys. Rev. A 93, 062345 (2016).
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K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Temporal steering and security of quantum key distribution with mutually unbiased bases against individual attacks,” Phys. Rev. A 93, 062345 (2016).
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S. L. Chen, N. Lambert, C. M. Li, G. Y. Chen, Y. N. Chen, A. Miranowicz, and F. Nori, “Spatio-temporal steering for testing nonclassical correlations in quantum networks,” Sci. Rep. 7, 3728 (2017).
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H. Y. Ku, S. L. Chen, H. B. Chen, N. Lambert, Y. N. Chen, and F. Nori, “Temporal steering in four dimensions with applications to coupled qubits and magnetoreception,” Phys. Rev. A 94, 062126 (2016).
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S. L. Chen, N. Lambert, C. M. Li, A. Miranowicz, Y. N. Chen, and F. Nori, “Quantifying non-Markovianity with temporal steering,” Phys. Rev. Lett. 116, 020503 (2016).
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S. L. Chen, N. Lambert, C. M. Li, G. Y. Chen, Y. N. Chen, A. Miranowicz, and F. Nori, “Spatio-temporal steering for testing nonclassical correlations in quantum networks,” Sci. Rep. 7, 3728 (2017).
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H. Y. Ku, S. L. Chen, H. B. Chen, N. Lambert, Y. N. Chen, and F. Nori, “Temporal steering in four dimensions with applications to coupled qubits and magnetoreception,” Phys. Rev. A 94, 062126 (2016).
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S. L. Chen, N. Lambert, C. M. Li, A. Miranowicz, Y. N. Chen, and F. Nori, “Quantifying non-Markovianity with temporal steering,” Phys. Rev. Lett. 116, 020503 (2016).
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Y. N. Chen, C. M. Li, N. Lambert, S. L. Chen, Y. Ota, G. Y. Chen, and F. Nori, “Temporal steering inequality,” Phys. Rev. A 89, 032112 (2014).
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J. S. Xu, M. H. Yung, X. Y. Xu, S. Boixo, Z. W. Zhou, C. F. Li, A. Aspuru-Guzik, and G. C. Guo, “Demon-like algorithmic quantum cooling and its realization with quantum optics,” Nat. Photonics 8, 113–118 (2014).
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J. S. Xu, K. Sun, C. F. Li, X. Y. Xu, G. C. Guo, E. Andersson, R. Lo Franco, and G. Compagno, “Experimental recovery of quantum correlations in absence of system-environment back-action,” Nat. Commun. 4, 2851 (2013).
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B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
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J. S. Xu, X. Y. Xu, C. F. Li, C. J. Zhang, X. B. Zou, and G. C. Guo, “Experimental investigation of classical and quantum correlations under decoherence,” Nat. Commun. 1, 7 (2010).
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J. S. Xu, C. F. Li, X. Y. Xu, C. H. Shi, X. B. Zou, and G. C. Guo, “Experimental characterization of entanglement dynamics in noisy channels,” Phys. Rev. Lett. 103, 240502 (2009).
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V. Handchen, 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, 598–601 (2012).
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H. Zhu, M. Hayashi, and L. Chen, “Universal steering inequalities,” Phys. Rev. Lett. 116, 070403 (2016).
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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. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316, 579–582 (2007).
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M. Marciniak, A. Rutkowski, Z. Yin, M. Horodecki, and R. Horodecki, “Unbounded violation of quantum steering inequalities,” Phys. Rev. Lett. 115, 170401 (2015).
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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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F. F. Fanchini, G. Karpat, B. Cakmak, L. K. Castelano, G. G. Aguilar, O. Jiménez Farías, S. P. Walborn, P. H. Souto Riberio, and M. C. de Oliveira, “Non-Markovianity through accessible information,” Phys. Rev. Lett. 112, 210402 (2014).
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B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
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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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S. L. Chen, N. Lambert, C. M. Li, G. Y. Chen, Y. N. Chen, A. Miranowicz, and F. Nori, “Spatio-temporal steering for testing nonclassical correlations in quantum networks,” Sci. Rep. 7, 3728 (2017).
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H. Y. Ku, S. L. Chen, H. B. Chen, N. Lambert, Y. N. Chen, and F. Nori, “Temporal steering in four dimensions with applications to coupled qubits and magnetoreception,” Phys. Rev. A 94, 062126 (2016).
[Crossref]

S. L. Chen, N. Lambert, C. M. Li, A. Miranowicz, Y. N. Chen, and F. Nori, “Quantifying non-Markovianity with temporal steering,” Phys. Rev. Lett. 116, 020503 (2016).
[Crossref]

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[Crossref]

Y. N. Chen, C. M. Li, N. Lambert, S. L. Chen, Y. Ota, G. Y. Chen, and F. Nori, “Temporal steering inequality,” Phys. Rev. A 89, 032112 (2014).
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K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Temporal steering and security of quantum key distribution with mutually unbiased bases against individual attacks,” Phys. Rev. A 93, 062345 (2016).
[Crossref]

K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Experimental temporal quantum steering,” Sci. Rep. 6, 38076 (2016).
[Crossref]

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J. S. Xu, M. H. Yung, X. Y. Xu, S. Boixo, Z. W. Zhou, C. F. Li, A. Aspuru-Guzik, and G. C. Guo, “Demon-like algorithmic quantum cooling and its realization with quantum optics,” Nat. Photonics 8, 113–118 (2014).
[Crossref]

J. S. Xu, K. Sun, C. F. Li, X. Y. Xu, G. C. Guo, E. Andersson, R. Lo Franco, and G. Compagno, “Experimental recovery of quantum correlations in absence of system-environment back-action,” Nat. Commun. 4, 2851 (2013).
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B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
[Crossref]

J. S. Xu, X. Y. Xu, C. F. Li, C. J. Zhang, X. B. Zou, and G. C. Guo, “Experimental investigation of classical and quantum correlations under decoherence,” Nat. Commun. 1, 7 (2010).
[Crossref]

J. S. Xu, C. F. Li, X. Y. Xu, C. H. Shi, X. B. Zou, and G. C. Guo, “Experimental characterization of entanglement dynamics in noisy channels,” Phys. Rev. Lett. 103, 240502 (2009).
[Crossref]

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S. L. Chen, N. Lambert, C. M. Li, G. Y. Chen, Y. N. Chen, A. Miranowicz, and F. Nori, “Spatio-temporal steering for testing nonclassical correlations in quantum networks,” Sci. Rep. 7, 3728 (2017).
[Crossref]

S. L. Chen, N. Lambert, C. M. Li, A. Miranowicz, Y. N. Chen, and F. Nori, “Quantifying non-Markovianity with temporal steering,” Phys. Rev. Lett. 116, 020503 (2016).
[Crossref]

C. M. Li, Y. N. Chen, N. Lambert, C. Y. Chiu, and F. Nori, “Certifying single-system steering for quantum-information processing,” Phys. Rev. A 92, 062310 (2015).
[Crossref]

Y. N. Chen, C. M. Li, N. Lambert, S. L. Chen, Y. Ota, G. Y. Chen, and F. Nori, “Temporal steering inequality,” Phys. Rev. A 89, 032112 (2014).
[Crossref]

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[Crossref]

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M. Marciniak, A. Rutkowski, Z. Yin, M. Horodecki, and R. Horodecki, “Unbounded violation of quantum steering inequalities,” Phys. Rev. Lett. 115, 170401 (2015).
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A. Acin, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
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S. L. Chen, N. Lambert, C. M. Li, G. Y. Chen, Y. N. Chen, A. Miranowicz, and F. Nori, “Spatio-temporal steering for testing nonclassical correlations in quantum networks,” Sci. Rep. 7, 3728 (2017).
[Crossref]

K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Experimental temporal quantum steering,” Sci. Rep. 6, 38076 (2016).
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K. Bartkiewicz, A. Černoch, K. Lemr, A. Miranowicz, and F. Nori, “Temporal steering and security of quantum key distribution with mutually unbiased bases against individual attacks,” Phys. Rev. A 93, 062345 (2016).
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R. Uola, T. Moroder, and O. Gühne, “Joint measurability of generalized measurements implies classicality,” Phys. Rev. Lett. 113, 160403 (2014).
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P. Skrzypczyk, M. Navascués, and D. Cavalcanti, “Quantifying Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 180404 (2014).
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S. L. Chen, N. Lambert, C. M. Li, A. Miranowicz, Y. N. Chen, and F. Nori, “Quantifying non-Markovianity with temporal steering,” Phys. Rev. Lett. 116, 020503 (2016).
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J. Ma, Z. Sun, X. Wang, and F. Nori, “Entanglement dynamics of two qubits in a common bath,” Phys. Rev. A 85, 062323 (2012).
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W. M. Zhang, P. Y. Lo, H. N. Xiong, M. W.-Y. Tu, and F. Nori, “General non-Markovian dynamics of open quantum systems,” Phys. Rev. Lett. 109, 170402 (2012).
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J. S. Xu, C. F. Li, X. Y. Xu, C. H. Shi, X. B. Zou, and G. C. Guo, “Experimental characterization of entanglement dynamics in noisy channels,” Phys. Rev. Lett. 103, 240502 (2009).
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J. S. Xu, K. Sun, C. F. Li, X. Y. Xu, G. C. Guo, E. Andersson, R. Lo Franco, and G. Compagno, “Experimental recovery of quantum correlations in absence of system-environment back-action,” Nat. Commun. 4, 2851 (2013).
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Z. Sun, J. Liu, J. Ma, and X. Wang, “Quantum speed limits in open systems: non-Markovian dynamics without rotating-wave approximation,” Sci. Rep. 5, 8444 (2015).
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J. Ma, Z. Sun, X. Wang, and F. Nori, “Entanglement dynamics of two qubits in a common bath,” Phys. Rev. A 85, 062323 (2012).
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Z. Sun, J. Ma, X. Wang, and F. Nori, “Photon-assisted Landau–Zener transition: role of coherent superposition states,” Phys. Rev. A 86, 012107 (2012).
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M. Tanaka and Y. Tanimura, “Multistate electron transfer dynamics in the condensed phase: exact calculations from the reduced hierarchy equations of motion approach,” J. Chem. Phys. 132, 214502 (2010).
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K. Sun, X. J. Ye, J. S. Xu, X. Y. Xu, J. S. Tang, Y. C. Wu, J. L. Chen, F. C. Li, and C. G. Guo, “Experimental quantification of asymmetric Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
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M. Tanaka and Y. Tanimura, “Multistate electron transfer dynamics in the condensed phase: exact calculations from the reduced hierarchy equations of motion approach,” J. Chem. Phys. 132, 214502 (2010).
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A. Ishizaki and Y. Tanimura, “Dynamics of a multimode system coupled to multiple heat baths probed by two-dimensional infrared spectroscopy,” J. Phys. Chem. A 111, 9269–9276 (2007).
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Y. Tanimura, “Nonperturbative expansion method for a quantum system coupled to a harmonic-oscillator bath,” Phys. Rev. A 41, 6676–6687 (1990).
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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. Tomamichel and R. Renner, “Uncertainty relation for smooth entropies,” Phys. Rev. Lett. 106, 110506 (2011).
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W. M. Zhang, P. Y. Lo, H. N. Xiong, M. W.-Y. Tu, and F. Nori, “General non-Markovian dynamics of open quantum systems,” Phys. Rev. Lett. 109, 170402 (2012).
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R. Uola, T. Moroder, and O. Gühne, “Joint measurability of generalized measurements implies classicality,” Phys. Rev. Lett. 113, 160403 (2014).
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Valdés-Hernández, A.

O. J. Farí, G. H. Aguilar, A. Valdés-Hernández, P. H. Ribeiro, L. Davidovich, and S. P. Walborn, “Observation of the emergence of multipartite entanglement between a bipartite system and its environment,” Phys. Rev. Lett. 109, 150403 (2012).
[Crossref]

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M. T. Quintino, T. Vértesi, and N. Brunner, “Joint measurability, Einstein–Podolsky–Rosen steering, and Bell nonlocality,” Phys. Rev. Lett. 113, 160402 (2014).
[Crossref]

J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 112, 200402 (2014).
[Crossref]

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F. F. Fanchini, G. Karpat, B. Cakmak, L. K. Castelano, G. G. Aguilar, O. Jiménez Farías, S. P. Walborn, P. H. Souto Riberio, and M. C. de Oliveira, “Non-Markovianity through accessible information,” Phys. Rev. Lett. 112, 210402 (2014).
[Crossref]

O. J. Farí, G. H. Aguilar, A. Valdés-Hernández, P. H. Ribeiro, L. Davidovich, and S. P. Walborn, “Observation of the emergence of multipartite entanglement between a bipartite system and its environment,” Phys. Rev. Lett. 109, 150403 (2012).
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M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316, 579–582 (2007).
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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).
[Crossref]

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Z. Sun, J. Liu, J. Ma, and X. Wang, “Quantum speed limits in open systems: non-Markovian dynamics without rotating-wave approximation,” Sci. Rep. 5, 8444 (2015).
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Z. Sun, J. Ma, X. Wang, and F. Nori, “Photon-assisted Landau–Zener transition: role of coherent superposition states,” Phys. Rev. A 86, 012107 (2012).
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J. Ma, Z. Sun, X. Wang, and F. Nori, “Entanglement dynamics of two qubits in a common bath,” Phys. Rev. A 85, 062323 (2012).
[Crossref]

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M. Piani and J. Watrous, “Necessary and sufficient quantum information characterization of Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 114, 060404 (2015).
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V. Handchen, 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, 598–601 (2012).
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Z. Sun, L. W. Zhou, G. Y. Xiao, D. Poletti, and J. B. Gong, “Finite-time Landau–Zener processes and counterdiabatic driving in open systems: beyond Born, Markov, and rotating-wave approximations,” Phys. Rev. A 93, 012121 (2016).
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W. M. Zhang, P. Y. Lo, H. N. Xiong, M. W.-Y. Tu, and F. Nori, “General non-Markovian dynamics of open quantum systems,” Phys. Rev. Lett. 109, 170402 (2012).
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K. Sun, X. J. Ye, J. S. Xu, X. Y. Xu, J. S. Tang, Y. C. Wu, J. L. Chen, F. C. Li, and C. G. Guo, “Experimental quantification of asymmetric Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
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J. S. Xu, K. Sun, C. F. Li, X. Y. Xu, G. C. Guo, E. Andersson, R. Lo Franco, and G. Compagno, “Experimental recovery of quantum correlations in absence of system-environment back-action,” Nat. Commun. 4, 2851 (2013).
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J. S. Xu, X. Y. Xu, C. F. Li, C. J. Zhang, X. B. Zou, and G. C. Guo, “Experimental investigation of classical and quantum correlations under decoherence,” Nat. Commun. 1, 7 (2010).
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J. S. Xu, C. F. Li, X. Y. Xu, C. H. Shi, X. B. Zou, and G. C. Guo, “Experimental characterization of entanglement dynamics in noisy channels,” Phys. Rev. Lett. 103, 240502 (2009).
[Crossref]

Xu, X. Y.

K. Sun, X. J. Ye, J. S. Xu, X. Y. Xu, J. S. Tang, Y. C. Wu, J. L. Chen, F. C. Li, and C. G. Guo, “Experimental quantification of asymmetric Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
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J. S. Xu, M. H. Yung, X. Y. Xu, S. Boixo, Z. W. Zhou, C. F. Li, A. Aspuru-Guzik, and G. C. Guo, “Demon-like algorithmic quantum cooling and its realization with quantum optics,” Nat. Photonics 8, 113–118 (2014).
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J. S. Xu, K. Sun, C. F. Li, X. Y. Xu, G. C. Guo, E. Andersson, R. Lo Franco, and G. Compagno, “Experimental recovery of quantum correlations in absence of system-environment back-action,” Nat. Commun. 4, 2851 (2013).
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J. S. Xu, X. Y. Xu, C. F. Li, C. J. Zhang, X. B. Zou, and G. C. Guo, “Experimental investigation of classical and quantum correlations under decoherence,” Nat. Commun. 1, 7 (2010).
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J. S. Xu, C. F. Li, X. Y. Xu, C. H. Shi, X. B. Zou, and G. C. Guo, “Experimental characterization of entanglement dynamics in noisy channels,” Phys. Rev. Lett. 103, 240502 (2009).
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Yan, Y. J.

J. S. Jin, X. Zheng, and Y. J. Yan, “Exact dynamics of dissipative electronic systems and quantum transport: hierarchical equations of motion approach,” J. Chem. Phys. 128, 234703 (2008).
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Ye, X. J.

K. Sun, X. J. Ye, J. S. Xu, X. Y. Xu, J. S. Tang, Y. C. Wu, J. L. Chen, F. C. Li, and C. G. Guo, “Experimental quantification of asymmetric Einstein–Podolsky–Rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
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Yin, Z.

M. Marciniak, A. Rutkowski, Z. Yin, M. Horodecki, and R. Horodecki, “Unbounded violation of quantum steering inequalities,” Phys. Rev. Lett. 115, 170401 (2015).
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T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93, 140404 (2004).
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J. S. Xu, M. H. Yung, X. Y. Xu, S. Boixo, Z. W. Zhou, C. F. Li, A. Aspuru-Guzik, and G. C. Guo, “Demon-like algorithmic quantum cooling and its realization with quantum optics,” Nat. Photonics 8, 113–118 (2014).
[Crossref]

Zhang, C. J.

J. S. Xu, X. Y. Xu, C. F. Li, C. J. Zhang, X. B. Zou, and G. C. Guo, “Experimental investigation of classical and quantum correlations under decoherence,” Nat. Commun. 1, 7 (2010).
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W. M. Zhang, P. Y. Lo, H. N. Xiong, M. W.-Y. Tu, and F. Nori, “General non-Markovian dynamics of open quantum systems,” Phys. Rev. Lett. 109, 170402 (2012).
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Zheng, X.

J. S. Jin, X. Zheng, and Y. J. Yan, “Exact dynamics of dissipative electronic systems and quantum transport: hierarchical equations of motion approach,” J. Chem. Phys. 128, 234703 (2008).
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Z. Sun, L. W. Zhou, G. Y. Xiao, D. Poletti, and J. B. Gong, “Finite-time Landau–Zener processes and counterdiabatic driving in open systems: beyond Born, Markov, and rotating-wave approximations,” Phys. Rev. A 93, 012121 (2016).
[Crossref]

Zhou, Z. W.

J. S. Xu, M. H. Yung, X. Y. Xu, S. Boixo, Z. W. Zhou, C. F. Li, A. Aspuru-Guzik, and G. C. Guo, “Demon-like algorithmic quantum cooling and its realization with quantum optics,” Nat. Photonics 8, 113–118 (2014).
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J. S. Xu, X. Y. Xu, C. F. Li, C. J. Zhang, X. B. Zou, and G. C. Guo, “Experimental investigation of classical and quantum correlations under decoherence,” Nat. Commun. 1, 7 (2010).
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J. S. Xu, C. F. Li, X. Y. Xu, C. H. Shi, X. B. Zou, and G. C. Guo, “Experimental characterization of entanglement dynamics in noisy channels,” Phys. Rev. Lett. 103, 240502 (2009).
[Crossref]

J. Chem. Phys. (2)

J. S. Jin, X. Zheng, and Y. J. Yan, “Exact dynamics of dissipative electronic systems and quantum transport: hierarchical equations of motion approach,” J. Chem. Phys. 128, 234703 (2008).
[Crossref]

M. Tanaka and Y. Tanimura, “Multistate electron transfer dynamics in the condensed phase: exact calculations from the reduced hierarchy equations of motion approach,” J. Chem. Phys. 132, 214502 (2010).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. Chem. A (1)

A. Ishizaki and Y. Tanimura, “Dynamics of a multimode system coupled to multiple heat baths probed by two-dimensional infrared spectroscopy,” J. Phys. Chem. A 111, 9269–9276 (2007).
[Crossref]

Nat. Commun. (2)

J. S. Xu, X. Y. Xu, C. F. Li, C. J. Zhang, X. B. Zou, and G. C. Guo, “Experimental investigation of classical and quantum correlations under decoherence,” Nat. Commun. 1, 7 (2010).
[Crossref]

J. S. Xu, K. Sun, C. F. Li, X. Y. Xu, G. C. Guo, E. Andersson, R. Lo Franco, and G. Compagno, “Experimental recovery of quantum correlations in absence of system-environment back-action,” Nat. Commun. 4, 2851 (2013).
[Crossref]

Nat. Photonics (2)

J. S. Xu, M. H. Yung, X. Y. Xu, S. Boixo, Z. W. Zhou, C. F. Li, A. Aspuru-Guzik, and G. C. Guo, “Demon-like algorithmic quantum cooling and its realization with quantum optics,” Nat. Photonics 8, 113–118 (2014).
[Crossref]

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[Crossref]

Nat. Phys. (2)

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]

B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
[Crossref]

Phys. Rev. A (10)

J. Ma, Z. Sun, X. Wang, and F. Nori, “Entanglement dynamics of two qubits in a common bath,” Phys. Rev. A 85, 062323 (2012).
[Crossref]

Z. Sun, L. W. Zhou, G. Y. Xiao, D. Poletti, and J. B. Gong, “Finite-time Landau–Zener processes and counterdiabatic driving in open systems: beyond Born, Markov, and rotating-wave approximations,” Phys. Rev. A 93, 012121 (2016).
[Crossref]

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Supplementary Material (1)

NameDescription
» Supplement 1       Brief introduction of the hierarchy equation method

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Figures (3)

Fig. 1.
Fig. 1.

Experimental setup and the stages of the experiment. (a) Photon pairs with an 810 nm wavelength are produced via spontaneous parametric downconversion. One of the two photons is used as the trigger for the coincident counts. The other photon is led to the preparation unit, consisting of a polarized beam splitter (PBS1), a half-wave plate (HWP), and a quarter-wave plate (QWP). In the TS problem, this photon is prepared into one of the six eigenstates of the Pauli operators σx,y,z. (b) Simulation of the quantum channel without RWA in Eq. (5). The angles of HWP1,2,3,4 are adjusted in [0,π/4], while the angles of HWP5,6,7,8 are set at π/4. Two Soleil–Babinet compensators (SBC1 and SBC2) add relative phases to the passing components H and V, respectively. The birefringent calcite beam displacers (BD1,2,3,4) couple the polarization states |H and |V with the spacial modes |ip (i=0,1,2,3). (c) Quantum state tomography is implemented by two QWPs, four HWPs, and two PBSs. Finally, two single-photon detectors equipped with two 10 nm interference filters (IFs) are used for the photon counting.

Fig. 2.
Fig. 2.

TS parameter S2 versus scaled time ω0t in the non-RWA and RWA channels. Parts (a) and (b) correspond to the measuring bases |+(|) and |0(|1), which are the eigenstates of σx and σz, respectively. Parts (c) and (d) correspond to the measuring bases |+(|) and |R(|L), which are the eigenstates of σx and σy, respectively. The channel parameters, i.e., the system–bath coupling parameters γ=2.5ω0 and the broadening width of the bath mode λ=0.05ω0, which result in an effective strength of the system–bath coupling, i.e., geff=0.25ω0. Horizontal red dashed lines indicate the steering limit. Vertical dashed lines point out the steerable durations corresponding to S2>1. Inset: enlarged drawing with more data for the peaks of S2 close to or beyond the steering limit.

Fig. 3.
Fig. 3.

TS weight versus scaled time ω0t in the non-RWA and RWA channels. The measuring bases in (a) and (b) are |+(|) and |0(|1), which are the eigenstates of σx and σz, respectively. Parts (c) and (d) correspond to the measuring bases |+(|) and |R(|L), which are the eigenstates of σx and σy, respectively. The values of parameters γ and λ are chosen the same as those in Fig. 2.

Equations (17)

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H=HS+HB+HInt,
HInt=kσx(gkbk+gk*bk)
HIntRWA=k(gkσ+bk+gk*σbk),
J(ω)=12πγλ2(ωω0)2+λ2,
|gS|vacBp|gS|evenB,g+1p|eS|oddB,g,|eS|vacBq|eS|evenB,e+1q|gS|oddB,e,
ρS(t)=[ρ11(t)ρ12(t)ρ12*(t)ρ22(t)],
|H|0pcos2θ1eiϕ1|H|0p+sin2θ1eiϕ2|V|Ψ1,3,|V|0pcos2θ2eiϕ3|V|Ψ2,0sin2θ2eiϕ4|H|3p,
|H|0peiϕ1|H|0p,|V|0pcos(2θ2)eiϕ3|V|0p+sin(2θ2)eiϕ4|H|3p,
|H|0p|H|0p,|V|0psin(2θ4)|V|0pcos(2θ4)|V|2p,
SNi=1NE(Bi,tBAi,tA2)1,
E(Bi,tBAi,tA2)a=±1P(a|Ai,tA)Bi,tBAi,tA=a2,
Bi,tBAi,tA=ab=±1bP(Bi,tB=b|Ai,tA=a),
WTS1maxTrλϱλ,
ς˜a|AiλDλ(a|Ai)ϱλ0,a,Ai,
ϱλ0,
|HPBS2,HWP1cos2θ1|H|2+sin2θ1|V|1SBC1,2cos2θ1eiϕ1|H|2+sin2θ1eiϕ2|V|1BD1,2cos2θ1eiϕ1|H|0+sin2θ1eiϕ2|V|1HWP5,3cos2θ1eiϕ1|V|0+sin2θ1eiϕ2(cos2θ3|Vsin2θ3|H)|1BD3,4cos2θ1eiϕ1|V|0+sin2θ1eiϕ2(cos2θ3|V|1sin2θ3|H|3)HWP7,8cos2θ1eiϕ1|H|0+sin2θ1eiϕ2(cos2θ3|V|1sin2θ3|V|3),
|VPBS2,HWP2cos2θ2|V|2sin2θ2|H|1SBC1,2cos2θ2eiϕ3|V|2sin2θ2eiϕ4|H|1BD1,2cos2θ2eiϕ3|V|2sin2θ2eiϕ4|H|3HWP6,4cos2θ2eiϕ3(cos2θ4|Vsin2θ4|H)|2sin2θ2eiϕ4|V|3BD3,4cos2θ2eiϕ3(cos2θ4|V|2sin2θ4|H|0)sin2θ2eiϕ4|V|3HWP7,8sin2θ2eiϕ4|H|3+cos2θ2eiϕ3(cos2θ4|V|2sin2θ4|V|0),