Abstract

Entanglement is a fundamental concept in quantum mechanics. In this review, we study various aspects of coherence and entanglement, illustrated by several examples. We relate the concepts of loss of coherence and disentanglement, via a model of two two-level atoms in different types of reservoir, including cases of both independent and common baths. Finally, we relate decoherence and disentanglement, by focusing on the sudden death of the entanglement and the dependence of the death time with the distance of our initial condition from the decoherence-free subspace. In particular, we study the sudden death of the entanglement in a two-atom system with a common reservoir.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
    [CrossRef]
  2. A. Einstein, B. Podolosky, N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Phys. Rev. 47, 777–780 (1935).
    [CrossRef]
  3. M. Born, The Born–Einstein Letters 1916–1955 (Macmillan, 2005).
  4. J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics 1(3), 195–200 (1964).
  5. A. Aspect, J. Dalibard, G. Roger, “Experimental test of Bell’s inequalities using time-varying analyzers,” Phys. Rev. Lett. 49, 1804–1807 (1982).
    [CrossRef]
  6. C. H. Thompson, H. Holstein, “The chaotic ball model: local realism and Bell test “detection loophole,” arXiv.org, arXiv:quant-ph/0210150 (2002).
  7. E. Santos, “Optical tests of Bell’s inequalities not resting upon the absurd fair sampling assumption,” arXiv.org, arXiv:quant-ph/0401003 (2004).
  8. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
    [CrossRef] [PubMed]
  9. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991);
    [CrossRef] [PubMed]
  10. D. Deutsch, A. K. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
    [CrossRef] [PubMed]
  11. C. H. Bennett, S. J. Wiesner, “Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992).
    [CrossRef] [PubMed]
  12. J. H. Eberly, “Schmidt analysis of pure-state entanglement,” arXiv.org, arXiv:quant-ph/0508019 (2005).
  13. S. J. Freedman, J. F. Clauser, “Experimental test of local hidden-variable theories,” Phys. Rev. Lett. 28, 938–941 (1972).
    [CrossRef]
  14. J. H. Eberly, “Bell inequalities in quantum mechanics,” Am. J. Phys. 70, 276–279 (2002).
    [CrossRef]
  15. M. Orszag, Quantum Optics (Springer, 2000).
    [CrossRef]
  16. R. Grobe, K. Rzazewski, J. H. Eberly, “Measure of electron–electron correlation in atomic physics,” J. Phys. B 27, L503–L508 (1994).
    [CrossRef]
  17. A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVicenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
    [CrossRef] [PubMed]
  18. D. Bruss, “Characterizing entanglement,” J. Math. Phys. 43, 4237–4251 (2002).
    [CrossRef]
  19. R. Werner, “Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989).
    [CrossRef] [PubMed]
  20. A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett. 77, 1413–1415 (1996).
    [CrossRef] [PubMed]
  21. M. Horodecki, P. Horodecki, R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A 223, 1–8 (1996).
    [CrossRef]
  22. P. Horodecki, “Separability criterion and inseparable mixed states with positive partial transposition,” Phys. Lett. A 232, 333–339 (1997).
    [CrossRef]
  23. A. Sen, U. Sen, M. Lewestein, A. Sampera, “The separability versus entanglement problem,” arXiv.org, arXiv:quant-phys/0508032 (2005).
  24. C. Bennett, D. DiVicenzo, J. Smolin, W. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824–3851 (1996).
    [CrossRef] [PubMed]
  25. S. Hill, W. Wootters, “Entanglement of a pair of quantum bits,” Phys. Rev. Lett. 78, 5022–5025 (1997).
    [CrossRef]
  26. W. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998).
    [CrossRef]
  27. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
    [CrossRef] [PubMed]
  28. D. M. Mundarain, M. Orszag, “Entanglement distillation with local common reservoirs,” Phys. Rev. A 79, 022306 (2009).
    [CrossRef]
  29. D. M. Mundarain, M. Orszag, “Entanglement preservation by continuous distillation,” Phys. Rev. A 79, 052333 (2009).
    [CrossRef]
  30. W. H. Zurek, “Decoherence and the transition from quantum to classical,” Phys. Today 44(10), 36–44 (1991).
    [CrossRef]
  31. W. H. Zurek, “Pointer basis of quantum apparatus: into what mixture does the wave packet collapse?,” Phys. Rev. D 24, 1516–1525 (1981).
    [CrossRef]
  32. W. H. Zurek, “Environment-induced superselection rules,” Phys. Rev. D 26, 1862–1880 (1982).
    [CrossRef]
  33. N. Bohr, “The quantum postulate and the recent development of atomic theory,” Nature (London) 121, 580–590 (1928).
    [CrossRef]
  34. J. von Neumann, Mathematische Grundlagen der Quanten mechanik (Springer-Verlag, 1932).
  35. For generalized measurements and quantum state discrimination, see, for example, J. A. Bergou, U. Herzog, M. Hillery, in Quantum State Discrimination in Quantum State Estimation: M. Paris and J. Rehacek, eds., Vol. 649 of Lecture Notes in Physics (Springer-Verlag, 2004), pp. 417–456, and [36, 37].
    [CrossRef]
  36. J. A. Bergou, “Discrimination of quantum states,” to be published in J. Mod. Opt..
  37. J. Preskill, Quantum Information and Computation Vol. 229 of Lecture Notes in Physics (Springer-Verlag, 1998).
  38. E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, I.-O. Stamatescu, Decoherence and the Appearance of Classical World in Quantum Theory (Springer, 1997).
  39. E. Joos, H. D. Zeh, “The emergence of classical properties through interaction with the environment,” Z. Phys. B 59, 223–243 (1985).
    [CrossRef]
  40. A. Caldeira, A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587–616 (1983).
    [CrossRef]
  41. D. F. Walls, G. J. Milburn, “Effect of dissipation on quantum coherence,” Phys. Rev. A 31, 2403–2408 (1985).
    [CrossRef] [PubMed]
  42. D. F. Walls, G. J. Milburn, Quantum Optics (Springer, 1994).
    [CrossRef]
  43. V. Buzek, P. L. Knight, in Progress in Optics, E. Wolf (Elsevier, A1995), Vol. XXXIV, p. 1.
    [CrossRef]
  44. M. S. Kim, V. Buzek, “Schrödinger-cat states at finite temperature: influence of a finite-temperature heat bath on quantum interferences,” Phys. Rev. A 46, 4239–4251 (1992).
    [CrossRef] [PubMed]
  45. M. S. Kim, V. Buzek, “Decay of quantum coherences under the influence of a thermal heatbath: Schroedinger cat states at finite temperature,” J. Mod. Opt. 39, 1609–1614 (1992).
    [CrossRef]
  46. M. S. Kim, V. Buzek, “Photon statistics of superposition states in phase-sensitive reservoirs,” Phys. Rev. A 47, 610–619 (1993).
    [CrossRef] [PubMed]
  47. S. Schneider, G. J. Milburn, “Decoherence in ion traps due to laser intensity and phase fluctuations,” Phys. Rev. A 57, 3748–3752 (1998).
    [CrossRef]
  48. M. Murao, P. L. Knight, “Decoherence in nonclassical motional states of a trapped ion,” Phys. Rev. A 58, 663–669 (1998).
    [CrossRef]
  49. S. Schneider, G. J. Milburn, “Decoherence and fidelity in ion traps with fluctuating trap parameters,” Phys. Rev. A 59, 3766–3774 (1999).
    [CrossRef]
  50. D. A. Lidar, I. L. Chuang, K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. 81, 2594–2597 (1998).
    [CrossRef]
  51. D. A. Lidar, K. B. Whaley, “Decoherence-free subspaces and subsystems,” arXiv.org, arXiv:quant-phys/0301032 (2003).
  52. D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, D. J. Wineland, “A decoherence-free quantum memory using trapped ions,” Science 291, 1013–1015 (2001).
    [CrossRef] [PubMed]
  53. P. Zanardi, M. Rasetti, “Error avoiding quantum codes,” Mod. Phys. Lett. B 11, 1085–1093 (1997).
    [CrossRef]
  54. P. Zanardi, M. Rasetti, “Noiseless quantum codes,” Phys. Rev. Lett. 79, 3306–3309 (1997).
    [CrossRef]
  55. W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715 (2003).
    [CrossRef]
  56. T. Yu, J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93, 140404 (2004).
    [CrossRef] [PubMed]
  57. L. Diósi, “Progressive decoherence and total environmental disentanglement,” Lect. Notes Phys. 622, 157–163 (2003).
    [CrossRef]
  58. P. Marek, J. Lee, M. S. Kim, “Vacuum as a less hostile environment to entanglement,” Phys. Rev. A 77, 032302 (2008).
    [CrossRef]
  59. Y. X. Gong, Y.-S. Zhang, Y.-L. Dong, X.-L. Niu, Y.-F. Huang, G.-C. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A. 78, 042103 (2008).
    [CrossRef]
  60. D. Tolkunov, V. Privman, P. K. Aravind, “Decoherence of a measure of entanglement,” Phys. Rev. A. 71, 060308(R) (2005).
    [CrossRef]
  61. J. P. Paz, A. J. Roncaglia, “Dynamical phases for the evolution of the entanglement between two oscillators coupled to the same environment,” Phys. Rev. A. 79, 032102 (2009).
    [CrossRef]
  62. C. Cormick, J. P. Paz, “Decoherence of Bell states by local interactions with a dynamic spin environment,” Phys. Rev. A. 78, 012357 (2008).
    [CrossRef]
  63. T. Gorin, C. Pineda, T. H. Seligman, “Decoherence of an n-qubit quantum memory,” Phys. Rev. Lett. 99, 240405 (2007).
    [CrossRef]
  64. C. Pineda, T. Gorin, T. H. Seligman, “Decoherence of two-qubit systems: a random matrix description,” New J. Phys. 9, 106 (2007).
    [CrossRef]
  65. M. Ikram, F.-l. Li, M. Zubairy, “Disentanglement in a two-qubit system subjected to dissipation environments,” Phys. Rev. A. 75, 062336 (2007).
    [CrossRef]
  66. K.-L. Liu, H.-S. Goan, “Non-Markovian entanglement dynamics of quantum continuous variable systems in thermal environments,” Phys. Rev. A. 76, 022312 (2007).
    [CrossRef]
  67. A. Al-Qasimi, D. F. V. James, “Sudden death of entanglement at finite temperature,” Phys. Rev. A. 77, 012117 (2008).
    [CrossRef]
  68. M. Hernandez, M. Orszag, “Decoherence and disentanglement for two qubits in a common squeezed reservoir,” Phys. Rev. A. 78, 042114 (2008).
    [CrossRef]
  69. Z. Ficek, R. Tanás, “Dark periods and revivals of entanglement in a two-qubit system,” Phys. Rev. A. 74, 024304 (2006).
    [CrossRef]
  70. M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. Souto-Ribeiro, L. Davidovich, “Environment-induced death of entanglement,” Science 316, 579–582 (2007);
    [CrossRef] [PubMed]
  71. A. Salles, F. de Melo, M. P. Almeida, M. Hor-Meyll, S. P. Walborn, P. H. Souto-Ribeiro, L. Davidovich, “Experimental investigation of the dynamics of entanglement: sudden death, complementarity and continuous monitoring of the environment,” Phys. Rev. A. 78, 022322 (2008).
    [CrossRef]
  72. D. Braun, “Creation of entanglement by interaction with a common heat bath,” Phys. Rev. Lett. 89, 277901 (2002);
    [CrossRef]
  73. Z. Ficek, R. Tanas, “Entanglement induced by spontaneous emission in spatially extended two-atom systems,” J. Mod. Opt. 50, 2765–2779 (2003).
    [CrossRef]
  74. F. Benatti, R. Floreanini, M. Piani, “Environment induced entanglement in Markovian dissipative dynamics,” Phys. Rev. Lett. 91, 070402 (2003).
    [CrossRef] [PubMed]
  75. Z. Ficek, R. Tanás, “Delayed sudden birth of entanglement,” Phys. Rev. A. 77, 054301 (2008).
    [CrossRef]
  76. B. Bellomo, R. Lo Franco, G. Compagno, “Non-Markovian effects on the dynamics of entanglement,” Phys. Rev. Lett. 99, 160502 (2007).
    [CrossRef] [PubMed]
  77. S. Maniscalco, S. Olivares, M. G. A. Paris, “Entanglement oscillations in non-Markovian quantum channels,” Phys. Rev. A 75, 062119 (2007); see also the related work in [78, 79] and, for multipartite systems, [80].
    [CrossRef]
  78. L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, B. M. Garraway, “Sudden death and sudden birth of entanglement in common structured reservoirs,” Phys. Rev. A 79, 042302 (2009).
    [CrossRef]
  79. B. Bellomo, R. Lo Franco, S. Maniscalco, G. Compagno, “Entanglement trapping in structured environments,” Phys. Rev. A 78, 060302 (2008).
    [CrossRef]
  80. C. E. Lopez, G. Romero, F. Lastra, E. Solano, J. C. Retamal, “Sudden birth versus sudden death of entanglement in multipartite systems,” Phys. Rev. Lett. 101, 080503 (2008).
    [CrossRef] [PubMed]
  81. M. Franca-Santos, P. Milman, L. Davidovich, N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. A 73, 040305 (2006).
    [CrossRef]
  82. T. Yu, J. H. Eberly, “Evolution from entanglement to decoherence of standard bipartite mixed states,” arXiv.org,arXiv:quant-ph/0503089 (2006).
  83. F. Lastra, S. Wallentowitz, M. Orszag, M. Hernandez, “Quantum recoil effects in finite-time disentanglement of two distinguishable atoms,” J. Phys. B 42, 065504 (2009).
    [CrossRef]
  84. D. Mundarain, M. Orszag, “Decoherence-free subspace and entanglement by interaction with a common squeezed bath,” Phys. Rev. A 75, 040303(R) (2007).
    [CrossRef]
  85. D. Mundarain, M. Orszag, J. Stephany, “Total quantum Zeno effect and intelligent states for a two-level system in a squeezed bath,” Phys. Rev. A 74, 052107 (2006).
    [CrossRef]
  86. T. Yu, J. H. Eberly, “Negative entanglement measure and what it implies,” arXiv.org, arXiv:quant-ph/0703083 (2007).
  87. R. Tanas, Z. Ficek, “Stationary two-atom entanglement induced by non-classical two-photon correlations,” J. Opt. B 6, S610–S617 (2004).
    [CrossRef]
  88. A. Beige, S. Bose, D. Braun, S. F. Huelga, P. L. Knight, M. B. Plenio, V. Vedral, “Entangling atoms and ions in dissipative environments,” J. Mod. Opt 47, 2583––2598 (2000).
    [CrossRef]
  89. A. M. Basharov, “Entanglement of atomic states upon collective radiative decay,” JETP Lett. 75, 123–126 (2002).
    [CrossRef]
  90. L. Jakobczyk, “Entangling two qubits by dissipation,” J. Phys. A 35, 6383–6392 (2002).
    [CrossRef]
  91. M. Paternostro, S. M. Tame, G. M. Palma, M. S. Kim, “Entanglement generation and protection by detuning modulation,” Phys. Rev. A 74, 052317 (2006).
    [CrossRef]
  92. S. Natali, Z. Ficek, “Temporal and diffraction effects in entanglement creation in an optical cavity,” Phys. Rev. A 75, 042307 (2007).
    [CrossRef]
  93. L. Derkacz, L. Jakobczyk, “Vacuum-induced stationary entanglement in radiatively coupled three-level atoms,” arXiv.org, arXiv:0710.5048 (2007).
  94. R. H. Lehmberg, “Radiation from n-atom system. I. General formulation,” Phys. Rev. A 2, 883–888 (1970).
    [CrossRef]
  95. G. S. Agarwal, in Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches, edited by G. Hohler, Vol. 70 of Springer Tracts in Modern Physics (Springer-Verlag, 1974).

2009 (5)

D. M. Mundarain, M. Orszag, “Entanglement distillation with local common reservoirs,” Phys. Rev. A 79, 022306 (2009).
[CrossRef]

D. M. Mundarain, M. Orszag, “Entanglement preservation by continuous distillation,” Phys. Rev. A 79, 052333 (2009).
[CrossRef]

J. P. Paz, A. J. Roncaglia, “Dynamical phases for the evolution of the entanglement between two oscillators coupled to the same environment,” Phys. Rev. A. 79, 032102 (2009).
[CrossRef]

L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, B. M. Garraway, “Sudden death and sudden birth of entanglement in common structured reservoirs,” Phys. Rev. A 79, 042302 (2009).
[CrossRef]

F. Lastra, S. Wallentowitz, M. Orszag, M. Hernandez, “Quantum recoil effects in finite-time disentanglement of two distinguishable atoms,” J. Phys. B 42, 065504 (2009).
[CrossRef]

2008 (9)

B. Bellomo, R. Lo Franco, S. Maniscalco, G. Compagno, “Entanglement trapping in structured environments,” Phys. Rev. A 78, 060302 (2008).
[CrossRef]

C. E. Lopez, G. Romero, F. Lastra, E. Solano, J. C. Retamal, “Sudden birth versus sudden death of entanglement in multipartite systems,” Phys. Rev. Lett. 101, 080503 (2008).
[CrossRef] [PubMed]

A. Salles, F. de Melo, M. P. Almeida, M. Hor-Meyll, S. P. Walborn, P. H. Souto-Ribeiro, L. Davidovich, “Experimental investigation of the dynamics of entanglement: sudden death, complementarity and continuous monitoring of the environment,” Phys. Rev. A. 78, 022322 (2008).
[CrossRef]

Z. Ficek, R. Tanás, “Delayed sudden birth of entanglement,” Phys. Rev. A. 77, 054301 (2008).
[CrossRef]

C. Cormick, J. P. Paz, “Decoherence of Bell states by local interactions with a dynamic spin environment,” Phys. Rev. A. 78, 012357 (2008).
[CrossRef]

P. Marek, J. Lee, M. S. Kim, “Vacuum as a less hostile environment to entanglement,” Phys. Rev. A 77, 032302 (2008).
[CrossRef]

Y. X. Gong, Y.-S. Zhang, Y.-L. Dong, X.-L. Niu, Y.-F. Huang, G.-C. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A. 78, 042103 (2008).
[CrossRef]

A. Al-Qasimi, D. F. V. James, “Sudden death of entanglement at finite temperature,” Phys. Rev. A. 77, 012117 (2008).
[CrossRef]

M. Hernandez, M. Orszag, “Decoherence and disentanglement for two qubits in a common squeezed reservoir,” Phys. Rev. A. 78, 042114 (2008).
[CrossRef]

2007 (9)

T. Gorin, C. Pineda, T. H. Seligman, “Decoherence of an n-qubit quantum memory,” Phys. Rev. Lett. 99, 240405 (2007).
[CrossRef]

C. Pineda, T. Gorin, T. H. Seligman, “Decoherence of two-qubit systems: a random matrix description,” New J. Phys. 9, 106 (2007).
[CrossRef]

M. Ikram, F.-l. Li, M. Zubairy, “Disentanglement in a two-qubit system subjected to dissipation environments,” Phys. Rev. A. 75, 062336 (2007).
[CrossRef]

K.-L. Liu, H.-S. Goan, “Non-Markovian entanglement dynamics of quantum continuous variable systems in thermal environments,” Phys. Rev. A. 76, 022312 (2007).
[CrossRef]

B. Bellomo, R. Lo Franco, G. Compagno, “Non-Markovian effects on the dynamics of entanglement,” Phys. Rev. Lett. 99, 160502 (2007).
[CrossRef] [PubMed]

S. Maniscalco, S. Olivares, M. G. A. Paris, “Entanglement oscillations in non-Markovian quantum channels,” Phys. Rev. A 75, 062119 (2007); see also the related work in [78, 79] and, for multipartite systems, [80].
[CrossRef]

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

D. Mundarain, M. Orszag, “Decoherence-free subspace and entanglement by interaction with a common squeezed bath,” Phys. Rev. A 75, 040303(R) (2007).
[CrossRef]

S. Natali, Z. Ficek, “Temporal and diffraction effects in entanglement creation in an optical cavity,” Phys. Rev. A 75, 042307 (2007).
[CrossRef]

2006 (4)

M. Paternostro, S. M. Tame, G. M. Palma, M. S. Kim, “Entanglement generation and protection by detuning modulation,” Phys. Rev. A 74, 052317 (2006).
[CrossRef]

D. Mundarain, M. Orszag, J. Stephany, “Total quantum Zeno effect and intelligent states for a two-level system in a squeezed bath,” Phys. Rev. A 74, 052107 (2006).
[CrossRef]

M. Franca-Santos, P. Milman, L. Davidovich, N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. A 73, 040305 (2006).
[CrossRef]

Z. Ficek, R. Tanás, “Dark periods and revivals of entanglement in a two-qubit system,” Phys. Rev. A. 74, 024304 (2006).
[CrossRef]

2005 (1)

D. Tolkunov, V. Privman, P. K. Aravind, “Decoherence of a measure of entanglement,” Phys. Rev. A. 71, 060308(R) (2005).
[CrossRef]

2004 (2)

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

R. Tanas, Z. Ficek, “Stationary two-atom entanglement induced by non-classical two-photon correlations,” J. Opt. B 6, S610–S617 (2004).
[CrossRef]

2003 (4)

W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715 (2003).
[CrossRef]

Z. Ficek, R. Tanas, “Entanglement induced by spontaneous emission in spatially extended two-atom systems,” J. Mod. Opt. 50, 2765–2779 (2003).
[CrossRef]

F. Benatti, R. Floreanini, M. Piani, “Environment induced entanglement in Markovian dissipative dynamics,” Phys. Rev. Lett. 91, 070402 (2003).
[CrossRef] [PubMed]

L. Diósi, “Progressive decoherence and total environmental disentanglement,” Lect. Notes Phys. 622, 157–163 (2003).
[CrossRef]

2002 (5)

D. Braun, “Creation of entanglement by interaction with a common heat bath,” Phys. Rev. Lett. 89, 277901 (2002);
[CrossRef]

J. H. Eberly, “Bell inequalities in quantum mechanics,” Am. J. Phys. 70, 276–279 (2002).
[CrossRef]

D. Bruss, “Characterizing entanglement,” J. Math. Phys. 43, 4237–4251 (2002).
[CrossRef]

A. M. Basharov, “Entanglement of atomic states upon collective radiative decay,” JETP Lett. 75, 123–126 (2002).
[CrossRef]

L. Jakobczyk, “Entangling two qubits by dissipation,” J. Phys. A 35, 6383–6392 (2002).
[CrossRef]

2001 (1)

D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, D. J. Wineland, “A decoherence-free quantum memory using trapped ions,” Science 291, 1013–1015 (2001).
[CrossRef] [PubMed]

2000 (1)

A. Beige, S. Bose, D. Braun, S. F. Huelga, P. L. Knight, M. B. Plenio, V. Vedral, “Entangling atoms and ions in dissipative environments,” J. Mod. Opt 47, 2583––2598 (2000).
[CrossRef]

1999 (1)

S. Schneider, G. J. Milburn, “Decoherence and fidelity in ion traps with fluctuating trap parameters,” Phys. Rev. A 59, 3766–3774 (1999).
[CrossRef]

1998 (4)

D. A. Lidar, I. L. Chuang, K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. 81, 2594–2597 (1998).
[CrossRef]

S. Schneider, G. J. Milburn, “Decoherence in ion traps due to laser intensity and phase fluctuations,” Phys. Rev. A 57, 3748–3752 (1998).
[CrossRef]

M. Murao, P. L. Knight, “Decoherence in nonclassical motional states of a trapped ion,” Phys. Rev. A 58, 663–669 (1998).
[CrossRef]

W. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998).
[CrossRef]

1997 (4)

S. Hill, W. Wootters, “Entanglement of a pair of quantum bits,” Phys. Rev. Lett. 78, 5022–5025 (1997).
[CrossRef]

P. Horodecki, “Separability criterion and inseparable mixed states with positive partial transposition,” Phys. Lett. A 232, 333–339 (1997).
[CrossRef]

P. Zanardi, M. Rasetti, “Error avoiding quantum codes,” Mod. Phys. Lett. B 11, 1085–1093 (1997).
[CrossRef]

P. Zanardi, M. Rasetti, “Noiseless quantum codes,” Phys. Rev. Lett. 79, 3306–3309 (1997).
[CrossRef]

1996 (5)

C. Bennett, D. DiVicenzo, J. Smolin, W. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824–3851 (1996).
[CrossRef] [PubMed]

A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett. 77, 1413–1415 (1996).
[CrossRef] [PubMed]

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

D. Deutsch, A. K. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
[CrossRef] [PubMed]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[CrossRef] [PubMed]

1995 (1)

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVicenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[CrossRef] [PubMed]

1994 (1)

R. Grobe, K. Rzazewski, J. H. Eberly, “Measure of electron–electron correlation in atomic physics,” J. Phys. B 27, L503–L508 (1994).
[CrossRef]

1993 (2)

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef] [PubMed]

M. S. Kim, V. Buzek, “Photon statistics of superposition states in phase-sensitive reservoirs,” Phys. Rev. A 47, 610–619 (1993).
[CrossRef] [PubMed]

1992 (3)

M. S. Kim, V. Buzek, “Schrödinger-cat states at finite temperature: influence of a finite-temperature heat bath on quantum interferences,” Phys. Rev. A 46, 4239–4251 (1992).
[CrossRef] [PubMed]

M. S. Kim, V. Buzek, “Decay of quantum coherences under the influence of a thermal heatbath: Schroedinger cat states at finite temperature,” J. Mod. Opt. 39, 1609–1614 (1992).
[CrossRef]

C. H. Bennett, S. J. Wiesner, “Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992).
[CrossRef] [PubMed]

1991 (2)

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991);
[CrossRef] [PubMed]

W. H. Zurek, “Decoherence and the transition from quantum to classical,” Phys. Today 44(10), 36–44 (1991).
[CrossRef]

1989 (1)

R. Werner, “Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989).
[CrossRef] [PubMed]

1985 (2)

D. F. Walls, G. J. Milburn, “Effect of dissipation on quantum coherence,” Phys. Rev. A 31, 2403–2408 (1985).
[CrossRef] [PubMed]

E. Joos, H. D. Zeh, “The emergence of classical properties through interaction with the environment,” Z. Phys. B 59, 223–243 (1985).
[CrossRef]

1983 (1)

A. Caldeira, A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587–616 (1983).
[CrossRef]

1982 (2)

W. H. Zurek, “Environment-induced superselection rules,” Phys. Rev. D 26, 1862–1880 (1982).
[CrossRef]

A. Aspect, J. Dalibard, G. Roger, “Experimental test of Bell’s inequalities using time-varying analyzers,” Phys. Rev. Lett. 49, 1804–1807 (1982).
[CrossRef]

1981 (1)

W. H. Zurek, “Pointer basis of quantum apparatus: into what mixture does the wave packet collapse?,” Phys. Rev. D 24, 1516–1525 (1981).
[CrossRef]

1972 (1)

S. J. Freedman, J. F. Clauser, “Experimental test of local hidden-variable theories,” Phys. Rev. Lett. 28, 938–941 (1972).
[CrossRef]

1970 (1)

R. H. Lehmberg, “Radiation from n-atom system. I. General formulation,” Phys. Rev. A 2, 883–888 (1970).
[CrossRef]

1964 (1)

J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics 1(3), 195–200 (1964).

1935 (2)

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
[CrossRef]

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

1928 (1)

N. Bohr, “The quantum postulate and the recent development of atomic theory,” Nature (London) 121, 580–590 (1928).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, in Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches, edited by G. Hohler, Vol. 70 of Springer Tracts in Modern Physics (Springer-Verlag, 1974).

Almeida, M. P.

A. Salles, F. de Melo, M. P. Almeida, M. Hor-Meyll, S. P. Walborn, P. H. Souto-Ribeiro, L. Davidovich, “Experimental investigation of the dynamics of entanglement: sudden death, complementarity and continuous monitoring of the environment,” Phys. Rev. A. 78, 022322 (2008).
[CrossRef]

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

Al-Qasimi, A.

A. Al-Qasimi, D. F. V. James, “Sudden death of entanglement at finite temperature,” Phys. Rev. A. 77, 012117 (2008).
[CrossRef]

Aravind, P. K.

D. Tolkunov, V. Privman, P. K. Aravind, “Decoherence of a measure of entanglement,” Phys. Rev. A. 71, 060308(R) (2005).
[CrossRef]

Aspect, A.

A. Aspect, J. Dalibard, G. Roger, “Experimental test of Bell’s inequalities using time-varying analyzers,” Phys. Rev. Lett. 49, 1804–1807 (1982).
[CrossRef]

Barenco, A.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVicenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[CrossRef] [PubMed]

Basharov, A. M.

A. M. Basharov, “Entanglement of atomic states upon collective radiative decay,” JETP Lett. 75, 123–126 (2002).
[CrossRef]

Beige, A.

A. Beige, S. Bose, D. Braun, S. F. Huelga, P. L. Knight, M. B. Plenio, V. Vedral, “Entangling atoms and ions in dissipative environments,” J. Mod. Opt 47, 2583––2598 (2000).
[CrossRef]

Bell, J. S.

J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics 1(3), 195–200 (1964).

Bellomo, B.

B. Bellomo, R. Lo Franco, S. Maniscalco, G. Compagno, “Entanglement trapping in structured environments,” Phys. Rev. A 78, 060302 (2008).
[CrossRef]

B. Bellomo, R. Lo Franco, G. Compagno, “Non-Markovian effects on the dynamics of entanglement,” Phys. Rev. Lett. 99, 160502 (2007).
[CrossRef] [PubMed]

Benatti, F.

F. Benatti, R. Floreanini, M. Piani, “Environment induced entanglement in Markovian dissipative dynamics,” Phys. Rev. Lett. 91, 070402 (2003).
[CrossRef] [PubMed]

Bennett, C.

C. Bennett, D. DiVicenzo, J. Smolin, W. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824–3851 (1996).
[CrossRef] [PubMed]

Bennett, C. H.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[CrossRef] [PubMed]

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVicenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[CrossRef] [PubMed]

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef] [PubMed]

C. H. Bennett, S. J. Wiesner, “Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992).
[CrossRef] [PubMed]

Bergou, J. A.

J. A. Bergou, “Discrimination of quantum states,” to be published in J. Mod. Opt..

For generalized measurements and quantum state discrimination, see, for example, J. A. Bergou, U. Herzog, M. Hillery, in Quantum State Discrimination in Quantum State Estimation: M. Paris and J. Rehacek, eds., Vol. 649 of Lecture Notes in Physics (Springer-Verlag, 2004), pp. 417–456, and [36, 37].
[CrossRef]

Bohr, N.

N. Bohr, “The quantum postulate and the recent development of atomic theory,” Nature (London) 121, 580–590 (1928).
[CrossRef]

Born, M.

M. Born, The Born–Einstein Letters 1916–1955 (Macmillan, 2005).

Bose, S.

A. Beige, S. Bose, D. Braun, S. F. Huelga, P. L. Knight, M. B. Plenio, V. Vedral, “Entangling atoms and ions in dissipative environments,” J. Mod. Opt 47, 2583––2598 (2000).
[CrossRef]

Brassard, G.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[CrossRef] [PubMed]

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef] [PubMed]

Braun, D.

D. Braun, “Creation of entanglement by interaction with a common heat bath,” Phys. Rev. Lett. 89, 277901 (2002);
[CrossRef]

A. Beige, S. Bose, D. Braun, S. F. Huelga, P. L. Knight, M. B. Plenio, V. Vedral, “Entangling atoms and ions in dissipative environments,” J. Mod. Opt 47, 2583––2598 (2000).
[CrossRef]

Bruss, D.

D. Bruss, “Characterizing entanglement,” J. Math. Phys. 43, 4237–4251 (2002).
[CrossRef]

Buzek, V.

M. S. Kim, V. Buzek, “Photon statistics of superposition states in phase-sensitive reservoirs,” Phys. Rev. A 47, 610–619 (1993).
[CrossRef] [PubMed]

M. S. Kim, V. Buzek, “Schrödinger-cat states at finite temperature: influence of a finite-temperature heat bath on quantum interferences,” Phys. Rev. A 46, 4239–4251 (1992).
[CrossRef] [PubMed]

M. S. Kim, V. Buzek, “Decay of quantum coherences under the influence of a thermal heatbath: Schroedinger cat states at finite temperature,” J. Mod. Opt. 39, 1609–1614 (1992).
[CrossRef]

V. Buzek, P. L. Knight, in Progress in Optics, E. Wolf (Elsevier, A1995), Vol. XXXIV, p. 1.
[CrossRef]

Caldeira, A.

A. Caldeira, A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587–616 (1983).
[CrossRef]

Chuang, I. L.

D. A. Lidar, I. L. Chuang, K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. 81, 2594–2597 (1998).
[CrossRef]

Clauser, J. F.

S. J. Freedman, J. F. Clauser, “Experimental test of local hidden-variable theories,” Phys. Rev. Lett. 28, 938–941 (1972).
[CrossRef]

Cleve, R.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVicenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[CrossRef] [PubMed]

Compagno, G.

B. Bellomo, R. Lo Franco, S. Maniscalco, G. Compagno, “Entanglement trapping in structured environments,” Phys. Rev. A 78, 060302 (2008).
[CrossRef]

B. Bellomo, R. Lo Franco, G. Compagno, “Non-Markovian effects on the dynamics of entanglement,” Phys. Rev. Lett. 99, 160502 (2007).
[CrossRef] [PubMed]

Cormick, C.

C. Cormick, J. P. Paz, “Decoherence of Bell states by local interactions with a dynamic spin environment,” Phys. Rev. A. 78, 012357 (2008).
[CrossRef]

Crepeau, C.

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef] [PubMed]

Dalibard, J.

A. Aspect, J. Dalibard, G. Roger, “Experimental test of Bell’s inequalities using time-varying analyzers,” Phys. Rev. Lett. 49, 1804–1807 (1982).
[CrossRef]

Davidovich, L.

A. Salles, F. de Melo, M. P. Almeida, M. Hor-Meyll, S. P. Walborn, P. H. Souto-Ribeiro, L. Davidovich, “Experimental investigation of the dynamics of entanglement: sudden death, complementarity and continuous monitoring of the environment,” Phys. Rev. A. 78, 022322 (2008).
[CrossRef]

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

M. Franca-Santos, P. Milman, L. Davidovich, N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. A 73, 040305 (2006).
[CrossRef]

de Melo, F.

A. Salles, F. de Melo, M. P. Almeida, M. Hor-Meyll, S. P. Walborn, P. H. Souto-Ribeiro, L. Davidovich, “Experimental investigation of the dynamics of entanglement: sudden death, complementarity and continuous monitoring of the environment,” Phys. Rev. A. 78, 022322 (2008).
[CrossRef]

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

Derkacz, L.

L. Derkacz, L. Jakobczyk, “Vacuum-induced stationary entanglement in radiatively coupled three-level atoms,” arXiv.org, arXiv:0710.5048 (2007).

Deutsch, D.

D. Deutsch, A. K. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
[CrossRef] [PubMed]

Diósi, L.

L. Diósi, “Progressive decoherence and total environmental disentanglement,” Lect. Notes Phys. 622, 157–163 (2003).
[CrossRef]

DiVicenzo, D.

C. Bennett, D. DiVicenzo, J. Smolin, W. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824–3851 (1996).
[CrossRef] [PubMed]

DiVicenzo, D. P.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVicenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[CrossRef] [PubMed]

Dong, Y.-L.

Y. X. Gong, Y.-S. Zhang, Y.-L. Dong, X.-L. Niu, Y.-F. Huang, G.-C. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A. 78, 042103 (2008).
[CrossRef]

Eberly, J. H.

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

J. H. Eberly, “Bell inequalities in quantum mechanics,” Am. J. Phys. 70, 276–279 (2002).
[CrossRef]

R. Grobe, K. Rzazewski, J. H. Eberly, “Measure of electron–electron correlation in atomic physics,” J. Phys. B 27, L503–L508 (1994).
[CrossRef]

J. H. Eberly, “Schmidt analysis of pure-state entanglement,” arXiv.org, arXiv:quant-ph/0508019 (2005).

T. Yu, J. H. Eberly, “Negative entanglement measure and what it implies,” arXiv.org, arXiv:quant-ph/0703083 (2007).

T. Yu, J. H. Eberly, “Evolution from entanglement to decoherence of standard bipartite mixed states,” arXiv.org,arXiv:quant-ph/0503089 (2006).

Einstein, A.

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

Ekert, A. K.

D. Deutsch, A. K. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
[CrossRef] [PubMed]

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991);
[CrossRef] [PubMed]

Ficek, Z.

Z. Ficek, R. Tanás, “Delayed sudden birth of entanglement,” Phys. Rev. A. 77, 054301 (2008).
[CrossRef]

S. Natali, Z. Ficek, “Temporal and diffraction effects in entanglement creation in an optical cavity,” Phys. Rev. A 75, 042307 (2007).
[CrossRef]

Z. Ficek, R. Tanás, “Dark periods and revivals of entanglement in a two-qubit system,” Phys. Rev. A. 74, 024304 (2006).
[CrossRef]

R. Tanas, Z. Ficek, “Stationary two-atom entanglement induced by non-classical two-photon correlations,” J. Opt. B 6, S610–S617 (2004).
[CrossRef]

Z. Ficek, R. Tanas, “Entanglement induced by spontaneous emission in spatially extended two-atom systems,” J. Mod. Opt. 50, 2765–2779 (2003).
[CrossRef]

Floreanini, R.

F. Benatti, R. Floreanini, M. Piani, “Environment induced entanglement in Markovian dissipative dynamics,” Phys. Rev. Lett. 91, 070402 (2003).
[CrossRef] [PubMed]

Franca-Santos, M.

M. Franca-Santos, P. Milman, L. Davidovich, N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. A 73, 040305 (2006).
[CrossRef]

Freedman, S. J.

S. J. Freedman, J. F. Clauser, “Experimental test of local hidden-variable theories,” Phys. Rev. Lett. 28, 938–941 (1972).
[CrossRef]

Garraway, B. M.

L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, B. M. Garraway, “Sudden death and sudden birth of entanglement in common structured reservoirs,” Phys. Rev. A 79, 042302 (2009).
[CrossRef]

Giulini, D.

E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, I.-O. Stamatescu, Decoherence and the Appearance of Classical World in Quantum Theory (Springer, 1997).

Goan, H.-S.

K.-L. Liu, H.-S. Goan, “Non-Markovian entanglement dynamics of quantum continuous variable systems in thermal environments,” Phys. Rev. A. 76, 022312 (2007).
[CrossRef]

Gong, Y. X.

Y. X. Gong, Y.-S. Zhang, Y.-L. Dong, X.-L. Niu, Y.-F. Huang, G.-C. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A. 78, 042103 (2008).
[CrossRef]

Gorin, T.

T. Gorin, C. Pineda, T. H. Seligman, “Decoherence of an n-qubit quantum memory,” Phys. Rev. Lett. 99, 240405 (2007).
[CrossRef]

C. Pineda, T. Gorin, T. H. Seligman, “Decoherence of two-qubit systems: a random matrix description,” New J. Phys. 9, 106 (2007).
[CrossRef]

Grobe, R.

R. Grobe, K. Rzazewski, J. H. Eberly, “Measure of electron–electron correlation in atomic physics,” J. Phys. B 27, L503–L508 (1994).
[CrossRef]

Guo, G.-C.

Y. X. Gong, Y.-S. Zhang, Y.-L. Dong, X.-L. Niu, Y.-F. Huang, G.-C. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A. 78, 042103 (2008).
[CrossRef]

Hernandez, M.

F. Lastra, S. Wallentowitz, M. Orszag, M. Hernandez, “Quantum recoil effects in finite-time disentanglement of two distinguishable atoms,” J. Phys. B 42, 065504 (2009).
[CrossRef]

M. Hernandez, M. Orszag, “Decoherence and disentanglement for two qubits in a common squeezed reservoir,” Phys. Rev. A. 78, 042114 (2008).
[CrossRef]

Herzog, U.

For generalized measurements and quantum state discrimination, see, for example, J. A. Bergou, U. Herzog, M. Hillery, in Quantum State Discrimination in Quantum State Estimation: M. Paris and J. Rehacek, eds., Vol. 649 of Lecture Notes in Physics (Springer-Verlag, 2004), pp. 417–456, and [36, 37].
[CrossRef]

Hill, S.

S. Hill, W. Wootters, “Entanglement of a pair of quantum bits,” Phys. Rev. Lett. 78, 5022–5025 (1997).
[CrossRef]

Hillery, M.

For generalized measurements and quantum state discrimination, see, for example, J. A. Bergou, U. Herzog, M. Hillery, in Quantum State Discrimination in Quantum State Estimation: M. Paris and J. Rehacek, eds., Vol. 649 of Lecture Notes in Physics (Springer-Verlag, 2004), pp. 417–456, and [36, 37].
[CrossRef]

Holstein, H.

C. H. Thompson, H. Holstein, “The chaotic ball model: local realism and Bell test “detection loophole,” arXiv.org, arXiv:quant-ph/0210150 (2002).

Hor-Meyll, M.

A. Salles, F. de Melo, M. P. Almeida, M. Hor-Meyll, S. P. Walborn, P. H. Souto-Ribeiro, L. Davidovich, “Experimental investigation of the dynamics of entanglement: sudden death, complementarity and continuous monitoring of the environment,” Phys. Rev. A. 78, 022322 (2008).
[CrossRef]

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

Horodecki, M.

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

Horodecki, P.

P. Horodecki, “Separability criterion and inseparable mixed states with positive partial transposition,” Phys. Lett. A 232, 333–339 (1997).
[CrossRef]

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

Horodecki, R.

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

Huang, Y.-F.

Y. X. Gong, Y.-S. Zhang, Y.-L. Dong, X.-L. Niu, Y.-F. Huang, G.-C. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A. 78, 042103 (2008).
[CrossRef]

Huelga, S. F.

A. Beige, S. Bose, D. Braun, S. F. Huelga, P. L. Knight, M. B. Plenio, V. Vedral, “Entangling atoms and ions in dissipative environments,” J. Mod. Opt 47, 2583––2598 (2000).
[CrossRef]

Ikram, M.

M. Ikram, F.-l. Li, M. Zubairy, “Disentanglement in a two-qubit system subjected to dissipation environments,” Phys. Rev. A. 75, 062336 (2007).
[CrossRef]

Itano, W. M.

D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, D. J. Wineland, “A decoherence-free quantum memory using trapped ions,” Science 291, 1013–1015 (2001).
[CrossRef] [PubMed]

Jakobczyk, L.

L. Jakobczyk, “Entangling two qubits by dissipation,” J. Phys. A 35, 6383–6392 (2002).
[CrossRef]

L. Derkacz, L. Jakobczyk, “Vacuum-induced stationary entanglement in radiatively coupled three-level atoms,” arXiv.org, arXiv:0710.5048 (2007).

James, D. F. V.

A. Al-Qasimi, D. F. V. James, “Sudden death of entanglement at finite temperature,” Phys. Rev. A. 77, 012117 (2008).
[CrossRef]

Joos, E.

E. Joos, H. D. Zeh, “The emergence of classical properties through interaction with the environment,” Z. Phys. B 59, 223–243 (1985).
[CrossRef]

E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, I.-O. Stamatescu, Decoherence and the Appearance of Classical World in Quantum Theory (Springer, 1997).

Jozsa, R.

D. Deutsch, A. K. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
[CrossRef] [PubMed]

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef] [PubMed]

Kiefer, C.

E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, I.-O. Stamatescu, Decoherence and the Appearance of Classical World in Quantum Theory (Springer, 1997).

Kielpinski, D.

D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, D. J. Wineland, “A decoherence-free quantum memory using trapped ions,” Science 291, 1013–1015 (2001).
[CrossRef] [PubMed]

Kim, M. S.

P. Marek, J. Lee, M. S. Kim, “Vacuum as a less hostile environment to entanglement,” Phys. Rev. A 77, 032302 (2008).
[CrossRef]

M. Paternostro, S. M. Tame, G. M. Palma, M. S. Kim, “Entanglement generation and protection by detuning modulation,” Phys. Rev. A 74, 052317 (2006).
[CrossRef]

M. S. Kim, V. Buzek, “Photon statistics of superposition states in phase-sensitive reservoirs,” Phys. Rev. A 47, 610–619 (1993).
[CrossRef] [PubMed]

M. S. Kim, V. Buzek, “Decay of quantum coherences under the influence of a thermal heatbath: Schroedinger cat states at finite temperature,” J. Mod. Opt. 39, 1609–1614 (1992).
[CrossRef]

M. S. Kim, V. Buzek, “Schrödinger-cat states at finite temperature: influence of a finite-temperature heat bath on quantum interferences,” Phys. Rev. A 46, 4239–4251 (1992).
[CrossRef] [PubMed]

Knight, P. L.

A. Beige, S. Bose, D. Braun, S. F. Huelga, P. L. Knight, M. B. Plenio, V. Vedral, “Entangling atoms and ions in dissipative environments,” J. Mod. Opt 47, 2583––2598 (2000).
[CrossRef]

M. Murao, P. L. Knight, “Decoherence in nonclassical motional states of a trapped ion,” Phys. Rev. A 58, 663–669 (1998).
[CrossRef]

V. Buzek, P. L. Knight, in Progress in Optics, E. Wolf (Elsevier, A1995), Vol. XXXIV, p. 1.
[CrossRef]

Kupsch, J.

E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, I.-O. Stamatescu, Decoherence and the Appearance of Classical World in Quantum Theory (Springer, 1997).

Lastra, F.

F. Lastra, S. Wallentowitz, M. Orszag, M. Hernandez, “Quantum recoil effects in finite-time disentanglement of two distinguishable atoms,” J. Phys. B 42, 065504 (2009).
[CrossRef]

C. E. Lopez, G. Romero, F. Lastra, E. Solano, J. C. Retamal, “Sudden birth versus sudden death of entanglement in multipartite systems,” Phys. Rev. Lett. 101, 080503 (2008).
[CrossRef] [PubMed]

Lee, J.

P. Marek, J. Lee, M. S. Kim, “Vacuum as a less hostile environment to entanglement,” Phys. Rev. A 77, 032302 (2008).
[CrossRef]

Leggett, A. J.

A. Caldeira, A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587–616 (1983).
[CrossRef]

Lehmberg, R. H.

R. H. Lehmberg, “Radiation from n-atom system. I. General formulation,” Phys. Rev. A 2, 883–888 (1970).
[CrossRef]

Lewestein, M.

A. Sen, U. Sen, M. Lewestein, A. Sampera, “The separability versus entanglement problem,” arXiv.org, arXiv:quant-phys/0508032 (2005).

Li, F.-l.

M. Ikram, F.-l. Li, M. Zubairy, “Disentanglement in a two-qubit system subjected to dissipation environments,” Phys. Rev. A. 75, 062336 (2007).
[CrossRef]

Lidar, D. A.

D. A. Lidar, I. L. Chuang, K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. 81, 2594–2597 (1998).
[CrossRef]

D. A. Lidar, K. B. Whaley, “Decoherence-free subspaces and subsystems,” arXiv.org, arXiv:quant-phys/0301032 (2003).

Liu, K.-L.

K.-L. Liu, H.-S. Goan, “Non-Markovian entanglement dynamics of quantum continuous variable systems in thermal environments,” Phys. Rev. A. 76, 022312 (2007).
[CrossRef]

Lo Franco, R.

B. Bellomo, R. Lo Franco, S. Maniscalco, G. Compagno, “Entanglement trapping in structured environments,” Phys. Rev. A 78, 060302 (2008).
[CrossRef]

B. Bellomo, R. Lo Franco, G. Compagno, “Non-Markovian effects on the dynamics of entanglement,” Phys. Rev. Lett. 99, 160502 (2007).
[CrossRef] [PubMed]

Lopez, C. E.

C. E. Lopez, G. Romero, F. Lastra, E. Solano, J. C. Retamal, “Sudden birth versus sudden death of entanglement in multipartite systems,” Phys. Rev. Lett. 101, 080503 (2008).
[CrossRef] [PubMed]

Macchiavello, C.

D. Deutsch, A. K. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
[CrossRef] [PubMed]

Maniscalco, S.

L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, B. M. Garraway, “Sudden death and sudden birth of entanglement in common structured reservoirs,” Phys. Rev. A 79, 042302 (2009).
[CrossRef]

B. Bellomo, R. Lo Franco, S. Maniscalco, G. Compagno, “Entanglement trapping in structured environments,” Phys. Rev. A 78, 060302 (2008).
[CrossRef]

S. Maniscalco, S. Olivares, M. G. A. Paris, “Entanglement oscillations in non-Markovian quantum channels,” Phys. Rev. A 75, 062119 (2007); see also the related work in [78, 79] and, for multipartite systems, [80].
[CrossRef]

Marek, P.

P. Marek, J. Lee, M. S. Kim, “Vacuum as a less hostile environment to entanglement,” Phys. Rev. A 77, 032302 (2008).
[CrossRef]

Margolus, N.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVicenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[CrossRef] [PubMed]

Mazzola, L.

L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, B. M. Garraway, “Sudden death and sudden birth of entanglement in common structured reservoirs,” Phys. Rev. A 79, 042302 (2009).
[CrossRef]

Meyer, V.

D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, D. J. Wineland, “A decoherence-free quantum memory using trapped ions,” Science 291, 1013–1015 (2001).
[CrossRef] [PubMed]

Milburn, G. J.

S. Schneider, G. J. Milburn, “Decoherence and fidelity in ion traps with fluctuating trap parameters,” Phys. Rev. A 59, 3766–3774 (1999).
[CrossRef]

S. Schneider, G. J. Milburn, “Decoherence in ion traps due to laser intensity and phase fluctuations,” Phys. Rev. A 57, 3748–3752 (1998).
[CrossRef]

D. F. Walls, G. J. Milburn, “Effect of dissipation on quantum coherence,” Phys. Rev. A 31, 2403–2408 (1985).
[CrossRef] [PubMed]

D. F. Walls, G. J. Milburn, Quantum Optics (Springer, 1994).
[CrossRef]

Milman, P.

M. Franca-Santos, P. Milman, L. Davidovich, N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. A 73, 040305 (2006).
[CrossRef]

Monroe, C.

D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, D. J. Wineland, “A decoherence-free quantum memory using trapped ions,” Science 291, 1013–1015 (2001).
[CrossRef] [PubMed]

Mundarain, D.

D. Mundarain, M. Orszag, “Decoherence-free subspace and entanglement by interaction with a common squeezed bath,” Phys. Rev. A 75, 040303(R) (2007).
[CrossRef]

D. Mundarain, M. Orszag, J. Stephany, “Total quantum Zeno effect and intelligent states for a two-level system in a squeezed bath,” Phys. Rev. A 74, 052107 (2006).
[CrossRef]

Mundarain, D. M.

D. M. Mundarain, M. Orszag, “Entanglement distillation with local common reservoirs,” Phys. Rev. A 79, 022306 (2009).
[CrossRef]

D. M. Mundarain, M. Orszag, “Entanglement preservation by continuous distillation,” Phys. Rev. A 79, 052333 (2009).
[CrossRef]

Murao, M.

M. Murao, P. L. Knight, “Decoherence in nonclassical motional states of a trapped ion,” Phys. Rev. A 58, 663–669 (1998).
[CrossRef]

Natali, S.

S. Natali, Z. Ficek, “Temporal and diffraction effects in entanglement creation in an optical cavity,” Phys. Rev. A 75, 042307 (2007).
[CrossRef]

Niu, X.-L.

Y. X. Gong, Y.-S. Zhang, Y.-L. Dong, X.-L. Niu, Y.-F. Huang, G.-C. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A. 78, 042103 (2008).
[CrossRef]

Olivares, S.

S. Maniscalco, S. Olivares, M. G. A. Paris, “Entanglement oscillations in non-Markovian quantum channels,” Phys. Rev. A 75, 062119 (2007); see also the related work in [78, 79] and, for multipartite systems, [80].
[CrossRef]

Orszag, M.

D. M. Mundarain, M. Orszag, “Entanglement preservation by continuous distillation,” Phys. Rev. A 79, 052333 (2009).
[CrossRef]

D. M. Mundarain, M. Orszag, “Entanglement distillation with local common reservoirs,” Phys. Rev. A 79, 022306 (2009).
[CrossRef]

F. Lastra, S. Wallentowitz, M. Orszag, M. Hernandez, “Quantum recoil effects in finite-time disentanglement of two distinguishable atoms,” J. Phys. B 42, 065504 (2009).
[CrossRef]

M. Hernandez, M. Orszag, “Decoherence and disentanglement for two qubits in a common squeezed reservoir,” Phys. Rev. A. 78, 042114 (2008).
[CrossRef]

D. Mundarain, M. Orszag, “Decoherence-free subspace and entanglement by interaction with a common squeezed bath,” Phys. Rev. A 75, 040303(R) (2007).
[CrossRef]

D. Mundarain, M. Orszag, J. Stephany, “Total quantum Zeno effect and intelligent states for a two-level system in a squeezed bath,” Phys. Rev. A 74, 052107 (2006).
[CrossRef]

M. Orszag, Quantum Optics (Springer, 2000).
[CrossRef]

Palma, G. M.

M. Paternostro, S. M. Tame, G. M. Palma, M. S. Kim, “Entanglement generation and protection by detuning modulation,” Phys. Rev. A 74, 052317 (2006).
[CrossRef]

Paris, M. G. A.

S. Maniscalco, S. Olivares, M. G. A. Paris, “Entanglement oscillations in non-Markovian quantum channels,” Phys. Rev. A 75, 062119 (2007); see also the related work in [78, 79] and, for multipartite systems, [80].
[CrossRef]

Paternostro, M.

M. Paternostro, S. M. Tame, G. M. Palma, M. S. Kim, “Entanglement generation and protection by detuning modulation,” Phys. Rev. A 74, 052317 (2006).
[CrossRef]

Paz, J. P.

J. P. Paz, A. J. Roncaglia, “Dynamical phases for the evolution of the entanglement between two oscillators coupled to the same environment,” Phys. Rev. A. 79, 032102 (2009).
[CrossRef]

C. Cormick, J. P. Paz, “Decoherence of Bell states by local interactions with a dynamic spin environment,” Phys. Rev. A. 78, 012357 (2008).
[CrossRef]

Peres, A.

A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett. 77, 1413–1415 (1996).
[CrossRef] [PubMed]

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef] [PubMed]

Piani, M.

F. Benatti, R. Floreanini, M. Piani, “Environment induced entanglement in Markovian dissipative dynamics,” Phys. Rev. Lett. 91, 070402 (2003).
[CrossRef] [PubMed]

Piilo, J.

L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, B. M. Garraway, “Sudden death and sudden birth of entanglement in common structured reservoirs,” Phys. Rev. A 79, 042302 (2009).
[CrossRef]

Pineda, C.

C. Pineda, T. Gorin, T. H. Seligman, “Decoherence of two-qubit systems: a random matrix description,” New J. Phys. 9, 106 (2007).
[CrossRef]

T. Gorin, C. Pineda, T. H. Seligman, “Decoherence of an n-qubit quantum memory,” Phys. Rev. Lett. 99, 240405 (2007).
[CrossRef]

Plenio, M. B.

A. Beige, S. Bose, D. Braun, S. F. Huelga, P. L. Knight, M. B. Plenio, V. Vedral, “Entangling atoms and ions in dissipative environments,” J. Mod. Opt 47, 2583––2598 (2000).
[CrossRef]

Podolosky, B.

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

Popescu, S.

D. Deutsch, A. K. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
[CrossRef] [PubMed]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[CrossRef] [PubMed]

Preskill, J.

J. Preskill, Quantum Information and Computation Vol. 229 of Lecture Notes in Physics (Springer-Verlag, 1998).

Privman, V.

D. Tolkunov, V. Privman, P. K. Aravind, “Decoherence of a measure of entanglement,” Phys. Rev. A. 71, 060308(R) (2005).
[CrossRef]

Rasetti, M.

P. Zanardi, M. Rasetti, “Error avoiding quantum codes,” Mod. Phys. Lett. B 11, 1085–1093 (1997).
[CrossRef]

P. Zanardi, M. Rasetti, “Noiseless quantum codes,” Phys. Rev. Lett. 79, 3306–3309 (1997).
[CrossRef]

Retamal, J. C.

C. E. Lopez, G. Romero, F. Lastra, E. Solano, J. C. Retamal, “Sudden birth versus sudden death of entanglement in multipartite systems,” Phys. Rev. Lett. 101, 080503 (2008).
[CrossRef] [PubMed]

Roger, G.

A. Aspect, J. Dalibard, G. Roger, “Experimental test of Bell’s inequalities using time-varying analyzers,” Phys. Rev. Lett. 49, 1804–1807 (1982).
[CrossRef]

Romero, G.

C. E. Lopez, G. Romero, F. Lastra, E. Solano, J. C. Retamal, “Sudden birth versus sudden death of entanglement in multipartite systems,” Phys. Rev. Lett. 101, 080503 (2008).
[CrossRef] [PubMed]

Roncaglia, A. J.

J. P. Paz, A. J. Roncaglia, “Dynamical phases for the evolution of the entanglement between two oscillators coupled to the same environment,” Phys. Rev. A. 79, 032102 (2009).
[CrossRef]

Rosen, N.

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

Rowe, M. A.

D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, D. J. Wineland, “A decoherence-free quantum memory using trapped ions,” Science 291, 1013–1015 (2001).
[CrossRef] [PubMed]

Rzazewski, K.

R. Grobe, K. Rzazewski, J. H. Eberly, “Measure of electron–electron correlation in atomic physics,” J. Phys. B 27, L503–L508 (1994).
[CrossRef]

Sackett, C. A.

D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, D. J. Wineland, “A decoherence-free quantum memory using trapped ions,” Science 291, 1013–1015 (2001).
[CrossRef] [PubMed]

Salles, A.

A. Salles, F. de Melo, M. P. Almeida, M. Hor-Meyll, S. P. Walborn, P. H. Souto-Ribeiro, L. Davidovich, “Experimental investigation of the dynamics of entanglement: sudden death, complementarity and continuous monitoring of the environment,” Phys. Rev. A. 78, 022322 (2008).
[CrossRef]

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

Sampera, A.

A. Sen, U. Sen, M. Lewestein, A. Sampera, “The separability versus entanglement problem,” arXiv.org, arXiv:quant-phys/0508032 (2005).

Sanpera, A.

D. Deutsch, A. K. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
[CrossRef] [PubMed]

Santos, E.

E. Santos, “Optical tests of Bell’s inequalities not resting upon the absurd fair sampling assumption,” arXiv.org, arXiv:quant-ph/0401003 (2004).

Schneider, S.

S. Schneider, G. J. Milburn, “Decoherence and fidelity in ion traps with fluctuating trap parameters,” Phys. Rev. A 59, 3766–3774 (1999).
[CrossRef]

S. Schneider, G. J. Milburn, “Decoherence in ion traps due to laser intensity and phase fluctuations,” Phys. Rev. A 57, 3748–3752 (1998).
[CrossRef]

Schrödinger, E.

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
[CrossRef]

Schumacher, B.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[CrossRef] [PubMed]

Seligman, T. H.

T. Gorin, C. Pineda, T. H. Seligman, “Decoherence of an n-qubit quantum memory,” Phys. Rev. Lett. 99, 240405 (2007).
[CrossRef]

C. Pineda, T. Gorin, T. H. Seligman, “Decoherence of two-qubit systems: a random matrix description,” New J. Phys. 9, 106 (2007).
[CrossRef]

Sen, A.

A. Sen, U. Sen, M. Lewestein, A. Sampera, “The separability versus entanglement problem,” arXiv.org, arXiv:quant-phys/0508032 (2005).

Sen, U.

A. Sen, U. Sen, M. Lewestein, A. Sampera, “The separability versus entanglement problem,” arXiv.org, arXiv:quant-phys/0508032 (2005).

Shor, P.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVicenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[CrossRef] [PubMed]

Sleator, T.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVicenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[CrossRef] [PubMed]

Smolin, J.

C. Bennett, D. DiVicenzo, J. Smolin, W. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824–3851 (1996).
[CrossRef] [PubMed]

Smolin, J. A.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[CrossRef] [PubMed]

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVicenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[CrossRef] [PubMed]

Solano, E.

C. E. Lopez, G. Romero, F. Lastra, E. Solano, J. C. Retamal, “Sudden birth versus sudden death of entanglement in multipartite systems,” Phys. Rev. Lett. 101, 080503 (2008).
[CrossRef] [PubMed]

Souto-Ribeiro, P. H.

A. Salles, F. de Melo, M. P. Almeida, M. Hor-Meyll, S. P. Walborn, P. H. Souto-Ribeiro, L. Davidovich, “Experimental investigation of the dynamics of entanglement: sudden death, complementarity and continuous monitoring of the environment,” Phys. Rev. A. 78, 022322 (2008).
[CrossRef]

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

Stamatescu, I.-O.

E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, I.-O. Stamatescu, Decoherence and the Appearance of Classical World in Quantum Theory (Springer, 1997).

Stephany, J.

D. Mundarain, M. Orszag, J. Stephany, “Total quantum Zeno effect and intelligent states for a two-level system in a squeezed bath,” Phys. Rev. A 74, 052107 (2006).
[CrossRef]

Suominen, K.-A.

L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, B. M. Garraway, “Sudden death and sudden birth of entanglement in common structured reservoirs,” Phys. Rev. A 79, 042302 (2009).
[CrossRef]

Tame, S. M.

M. Paternostro, S. M. Tame, G. M. Palma, M. S. Kim, “Entanglement generation and protection by detuning modulation,” Phys. Rev. A 74, 052317 (2006).
[CrossRef]

Tanas, R.

R. Tanas, Z. Ficek, “Stationary two-atom entanglement induced by non-classical two-photon correlations,” J. Opt. B 6, S610–S617 (2004).
[CrossRef]

Z. Ficek, R. Tanas, “Entanglement induced by spontaneous emission in spatially extended two-atom systems,” J. Mod. Opt. 50, 2765–2779 (2003).
[CrossRef]

Tanás, R.

Z. Ficek, R. Tanás, “Delayed sudden birth of entanglement,” Phys. Rev. A. 77, 054301 (2008).
[CrossRef]

Z. Ficek, R. Tanás, “Dark periods and revivals of entanglement in a two-qubit system,” Phys. Rev. A. 74, 024304 (2006).
[CrossRef]

Thompson, C. H.

C. H. Thompson, H. Holstein, “The chaotic ball model: local realism and Bell test “detection loophole,” arXiv.org, arXiv:quant-ph/0210150 (2002).

Tolkunov, D.

D. Tolkunov, V. Privman, P. K. Aravind, “Decoherence of a measure of entanglement,” Phys. Rev. A. 71, 060308(R) (2005).
[CrossRef]

Vedral, V.

A. Beige, S. Bose, D. Braun, S. F. Huelga, P. L. Knight, M. B. Plenio, V. Vedral, “Entangling atoms and ions in dissipative environments,” J. Mod. Opt 47, 2583––2598 (2000).
[CrossRef]

von Neumann, J.

J. von Neumann, Mathematische Grundlagen der Quanten mechanik (Springer-Verlag, 1932).

Walborn, S. P.

A. Salles, F. de Melo, M. P. Almeida, M. Hor-Meyll, S. P. Walborn, P. H. Souto-Ribeiro, L. Davidovich, “Experimental investigation of the dynamics of entanglement: sudden death, complementarity and continuous monitoring of the environment,” Phys. Rev. A. 78, 022322 (2008).
[CrossRef]

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

Wallentowitz, S.

F. Lastra, S. Wallentowitz, M. Orszag, M. Hernandez, “Quantum recoil effects in finite-time disentanglement of two distinguishable atoms,” J. Phys. B 42, 065504 (2009).
[CrossRef]

Walls, D. F.

D. F. Walls, G. J. Milburn, “Effect of dissipation on quantum coherence,” Phys. Rev. A 31, 2403–2408 (1985).
[CrossRef] [PubMed]

D. F. Walls, G. J. Milburn, Quantum Optics (Springer, 1994).
[CrossRef]

Weinfurter, H.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVicenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[CrossRef] [PubMed]

Werner, R.

R. Werner, “Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989).
[CrossRef] [PubMed]

Whaley, K. B.

D. A. Lidar, I. L. Chuang, K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. 81, 2594–2597 (1998).
[CrossRef]

D. A. Lidar, K. B. Whaley, “Decoherence-free subspaces and subsystems,” arXiv.org, arXiv:quant-phys/0301032 (2003).

Wiesner, S. J.

C. H. Bennett, S. J. Wiesner, “Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992).
[CrossRef] [PubMed]

Wineland, D. J.

D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, D. J. Wineland, “A decoherence-free quantum memory using trapped ions,” Science 291, 1013–1015 (2001).
[CrossRef] [PubMed]

Wootters, W.

W. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998).
[CrossRef]

S. Hill, W. Wootters, “Entanglement of a pair of quantum bits,” Phys. Rev. Lett. 78, 5022–5025 (1997).
[CrossRef]

C. Bennett, D. DiVicenzo, J. Smolin, W. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824–3851 (1996).
[CrossRef] [PubMed]

Wootters, W. K.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[CrossRef] [PubMed]

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef] [PubMed]

Yu, T.

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

T. Yu, J. H. Eberly, “Negative entanglement measure and what it implies,” arXiv.org, arXiv:quant-ph/0703083 (2007).

T. Yu, J. H. Eberly, “Evolution from entanglement to decoherence of standard bipartite mixed states,” arXiv.org,arXiv:quant-ph/0503089 (2006).

Zagury, N.

M. Franca-Santos, P. Milman, L. Davidovich, N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. A 73, 040305 (2006).
[CrossRef]

Zanardi, P.

P. Zanardi, M. Rasetti, “Error avoiding quantum codes,” Mod. Phys. Lett. B 11, 1085–1093 (1997).
[CrossRef]

P. Zanardi, M. Rasetti, “Noiseless quantum codes,” Phys. Rev. Lett. 79, 3306–3309 (1997).
[CrossRef]

Zeh, H. D.

E. Joos, H. D. Zeh, “The emergence of classical properties through interaction with the environment,” Z. Phys. B 59, 223–243 (1985).
[CrossRef]

E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, I.-O. Stamatescu, Decoherence and the Appearance of Classical World in Quantum Theory (Springer, 1997).

Zhang, Y.-S.

Y. X. Gong, Y.-S. Zhang, Y.-L. Dong, X.-L. Niu, Y.-F. Huang, G.-C. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A. 78, 042103 (2008).
[CrossRef]

Zubairy, M.

M. Ikram, F.-l. Li, M. Zubairy, “Disentanglement in a two-qubit system subjected to dissipation environments,” Phys. Rev. A. 75, 062336 (2007).
[CrossRef]

Zurek, W. H.

W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715 (2003).
[CrossRef]

W. H. Zurek, “Decoherence and the transition from quantum to classical,” Phys. Today 44(10), 36–44 (1991).
[CrossRef]

W. H. Zurek, “Environment-induced superselection rules,” Phys. Rev. D 26, 1862–1880 (1982).
[CrossRef]

W. H. Zurek, “Pointer basis of quantum apparatus: into what mixture does the wave packet collapse?,” Phys. Rev. D 24, 1516–1525 (1981).
[CrossRef]

Am. J. Phys. (1)

J. H. Eberly, “Bell inequalities in quantum mechanics,” Am. J. Phys. 70, 276–279 (2002).
[CrossRef]

J. Phys. B (1)

F. Lastra, S. Wallentowitz, M. Orszag, M. Hernandez, “Quantum recoil effects in finite-time disentanglement of two distinguishable atoms,” J. Phys. B 42, 065504 (2009).
[CrossRef]

J. Math. Phys. (1)

D. Bruss, “Characterizing entanglement,” J. Math. Phys. 43, 4237–4251 (2002).
[CrossRef]

J. Mod. Opt. (1)

Z. Ficek, R. Tanas, “Entanglement induced by spontaneous emission in spatially extended two-atom systems,” J. Mod. Opt. 50, 2765–2779 (2003).
[CrossRef]

J. Mod. Opt (1)

A. Beige, S. Bose, D. Braun, S. F. Huelga, P. L. Knight, M. B. Plenio, V. Vedral, “Entangling atoms and ions in dissipative environments,” J. Mod. Opt 47, 2583––2598 (2000).
[CrossRef]

J. Mod. Opt. (1)

M. S. Kim, V. Buzek, “Decay of quantum coherences under the influence of a thermal heatbath: Schroedinger cat states at finite temperature,” J. Mod. Opt. 39, 1609–1614 (1992).
[CrossRef]

J. Opt. B (1)

R. Tanas, Z. Ficek, “Stationary two-atom entanglement induced by non-classical two-photon correlations,” J. Opt. B 6, S610–S617 (2004).
[CrossRef]

J. Phys. B (1)

R. Grobe, K. Rzazewski, J. H. Eberly, “Measure of electron–electron correlation in atomic physics,” J. Phys. B 27, L503–L508 (1994).
[CrossRef]

J. Phys. A (1)

L. Jakobczyk, “Entangling two qubits by dissipation,” J. Phys. A 35, 6383–6392 (2002).
[CrossRef]

JETP Lett. (1)

A. M. Basharov, “Entanglement of atomic states upon collective radiative decay,” JETP Lett. 75, 123–126 (2002).
[CrossRef]

Lect. Notes Phys. (1)

L. Diósi, “Progressive decoherence and total environmental disentanglement,” Lect. Notes Phys. 622, 157–163 (2003).
[CrossRef]

Mod. Phys. Lett. B (1)

P. Zanardi, M. Rasetti, “Error avoiding quantum codes,” Mod. Phys. Lett. B 11, 1085–1093 (1997).
[CrossRef]

Nature (London) (1)

N. Bohr, “The quantum postulate and the recent development of atomic theory,” Nature (London) 121, 580–590 (1928).
[CrossRef]

New J. Phys. (1)

C. Pineda, T. Gorin, T. H. Seligman, “Decoherence of two-qubit systems: a random matrix description,” New J. Phys. 9, 106 (2007).
[CrossRef]

Phys. Lett. A (1)

P. Horodecki, “Separability criterion and inseparable mixed states with positive partial transposition,” Phys. Lett. A 232, 333–339 (1997).
[CrossRef]

Phys. Rev. (1)

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

Phys. Rev. A (3)

S. Natali, Z. Ficek, “Temporal and diffraction effects in entanglement creation in an optical cavity,” Phys. Rev. A 75, 042307 (2007).
[CrossRef]

D. Mundarain, M. Orszag, “Decoherence-free subspace and entanglement by interaction with a common squeezed bath,” Phys. Rev. A 75, 040303(R) (2007).
[CrossRef]

M. Franca-Santos, P. Milman, L. Davidovich, N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. A 73, 040305 (2006).
[CrossRef]

Phys. Rev. Lett. (2)

A. Aspect, J. Dalibard, G. Roger, “Experimental test of Bell’s inequalities using time-varying analyzers,” Phys. Rev. Lett. 49, 1804–1807 (1982).
[CrossRef]

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722–725 (1996).
[CrossRef] [PubMed]

Phys. Lett. A (1)

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

Phys. Rev. A (2)

S. Schneider, G. J. Milburn, “Decoherence and fidelity in ion traps with fluctuating trap parameters,” Phys. Rev. A 59, 3766–3774 (1999).
[CrossRef]

S. Schneider, G. J. Milburn, “Decoherence in ion traps due to laser intensity and phase fluctuations,” Phys. Rev. A 57, 3748–3752 (1998).
[CrossRef]

Phys. Rev. A. (2)

M. Ikram, F.-l. Li, M. Zubairy, “Disentanglement in a two-qubit system subjected to dissipation environments,” Phys. Rev. A. 75, 062336 (2007).
[CrossRef]

M. Hernandez, M. Orszag, “Decoherence and disentanglement for two qubits in a common squeezed reservoir,” Phys. Rev. A. 78, 042114 (2008).
[CrossRef]

Phys. Rev. Lett. (2)

F. Benatti, R. Floreanini, M. Piani, “Environment induced entanglement in Markovian dissipative dynamics,” Phys. Rev. Lett. 91, 070402 (2003).
[CrossRef] [PubMed]

C. E. Lopez, G. Romero, F. Lastra, E. Solano, J. C. Retamal, “Sudden birth versus sudden death of entanglement in multipartite systems,” Phys. Rev. Lett. 101, 080503 (2008).
[CrossRef] [PubMed]

Phys. Rev. A (16)

D. Mundarain, M. Orszag, J. Stephany, “Total quantum Zeno effect and intelligent states for a two-level system in a squeezed bath,” Phys. Rev. A 74, 052107 (2006).
[CrossRef]

R. H. Lehmberg, “Radiation from n-atom system. I. General formulation,” Phys. Rev. A 2, 883–888 (1970).
[CrossRef]

M. Paternostro, S. M. Tame, G. M. Palma, M. S. Kim, “Entanglement generation and protection by detuning modulation,” Phys. Rev. A 74, 052317 (2006).
[CrossRef]

S. Maniscalco, S. Olivares, M. G. A. Paris, “Entanglement oscillations in non-Markovian quantum channels,” Phys. Rev. A 75, 062119 (2007); see also the related work in [78, 79] and, for multipartite systems, [80].
[CrossRef]

L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, B. M. Garraway, “Sudden death and sudden birth of entanglement in common structured reservoirs,” Phys. Rev. A 79, 042302 (2009).
[CrossRef]

B. Bellomo, R. Lo Franco, S. Maniscalco, G. Compagno, “Entanglement trapping in structured environments,” Phys. Rev. A 78, 060302 (2008).
[CrossRef]

M. Murao, P. L. Knight, “Decoherence in nonclassical motional states of a trapped ion,” Phys. Rev. A 58, 663–669 (1998).
[CrossRef]

M. S. Kim, V. Buzek, “Photon statistics of superposition states in phase-sensitive reservoirs,” Phys. Rev. A 47, 610–619 (1993).
[CrossRef] [PubMed]

D. F. Walls, G. J. Milburn, “Effect of dissipation on quantum coherence,” Phys. Rev. A 31, 2403–2408 (1985).
[CrossRef] [PubMed]

M. S. Kim, V. Buzek, “Schrödinger-cat states at finite temperature: influence of a finite-temperature heat bath on quantum interferences,” Phys. Rev. A 46, 4239–4251 (1992).
[CrossRef] [PubMed]

P. Marek, J. Lee, M. S. Kim, “Vacuum as a less hostile environment to entanglement,” Phys. Rev. A 77, 032302 (2008).
[CrossRef]

D. M. Mundarain, M. Orszag, “Entanglement distillation with local common reservoirs,” Phys. Rev. A 79, 022306 (2009).
[CrossRef]

D. M. Mundarain, M. Orszag, “Entanglement preservation by continuous distillation,” Phys. Rev. A 79, 052333 (2009).
[CrossRef]

C. Bennett, D. DiVicenzo, J. Smolin, W. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824–3851 (1996).
[CrossRef] [PubMed]

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVicenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457–3467 (1995).
[CrossRef] [PubMed]

R. Werner, “Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989).
[CrossRef] [PubMed]

Phys. Rev. A. (9)

Y. X. Gong, Y.-S. Zhang, Y.-L. Dong, X.-L. Niu, Y.-F. Huang, G.-C. Guo, “Dependence of the decoherence of polarization states in phase-damping channels on the frequency spectrum envelope of photons,” Phys. Rev. A. 78, 042103 (2008).
[CrossRef]

D. Tolkunov, V. Privman, P. K. Aravind, “Decoherence of a measure of entanglement,” Phys. Rev. A. 71, 060308(R) (2005).
[CrossRef]

J. P. Paz, A. J. Roncaglia, “Dynamical phases for the evolution of the entanglement between two oscillators coupled to the same environment,” Phys. Rev. A. 79, 032102 (2009).
[CrossRef]

C. Cormick, J. P. Paz, “Decoherence of Bell states by local interactions with a dynamic spin environment,” Phys. Rev. A. 78, 012357 (2008).
[CrossRef]

A. Salles, F. de Melo, M. P. Almeida, M. Hor-Meyll, S. P. Walborn, P. H. Souto-Ribeiro, L. Davidovich, “Experimental investigation of the dynamics of entanglement: sudden death, complementarity and continuous monitoring of the environment,” Phys. Rev. A. 78, 022322 (2008).
[CrossRef]

Z. Ficek, R. Tanás, “Delayed sudden birth of entanglement,” Phys. Rev. A. 77, 054301 (2008).
[CrossRef]

Z. Ficek, R. Tanás, “Dark periods and revivals of entanglement in a two-qubit system,” Phys. Rev. A. 74, 024304 (2006).
[CrossRef]

K.-L. Liu, H.-S. Goan, “Non-Markovian entanglement dynamics of quantum continuous variable systems in thermal environments,” Phys. Rev. A. 76, 022312 (2007).
[CrossRef]

A. Al-Qasimi, D. F. V. James, “Sudden death of entanglement at finite temperature,” Phys. Rev. A. 77, 012117 (2008).
[CrossRef]

Phys. Rev. D (2)

W. H. Zurek, “Pointer basis of quantum apparatus: into what mixture does the wave packet collapse?,” Phys. Rev. D 24, 1516–1525 (1981).
[CrossRef]

W. H. Zurek, “Environment-induced superselection rules,” Phys. Rev. D 26, 1862–1880 (1982).
[CrossRef]

Phys. Rev. Lett. (14)

S. Hill, W. Wootters, “Entanglement of a pair of quantum bits,” Phys. Rev. Lett. 78, 5022–5025 (1997).
[CrossRef]

W. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998).
[CrossRef]

S. J. Freedman, J. F. Clauser, “Experimental test of local hidden-variable theories,” Phys. Rev. Lett. 28, 938–941 (1972).
[CrossRef]

A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett. 77, 1413–1415 (1996).
[CrossRef] [PubMed]

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[CrossRef] [PubMed]

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991);
[CrossRef] [PubMed]

D. Deutsch, A. K. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77, 2818–2821 (1996).
[CrossRef] [PubMed]

C. H. Bennett, S. J. Wiesner, “Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992).
[CrossRef] [PubMed]

P. Zanardi, M. Rasetti, “Noiseless quantum codes,” Phys. Rev. Lett. 79, 3306–3309 (1997).
[CrossRef]

B. Bellomo, R. Lo Franco, G. Compagno, “Non-Markovian effects on the dynamics of entanglement,” Phys. Rev. Lett. 99, 160502 (2007).
[CrossRef] [PubMed]

D. Braun, “Creation of entanglement by interaction with a common heat bath,” Phys. Rev. Lett. 89, 277901 (2002);
[CrossRef]

T. Gorin, C. Pineda, T. H. Seligman, “Decoherence of an n-qubit quantum memory,” Phys. Rev. Lett. 99, 240405 (2007).
[CrossRef]

D. A. Lidar, I. L. Chuang, K. B. Whaley, “Decoherence-free subspaces for quantum computation,” Phys. Rev. Lett. 81, 2594–2597 (1998).
[CrossRef]

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

Phys. Today (1)

W. H. Zurek, “Decoherence and the transition from quantum to classical,” Phys. Today 44(10), 36–44 (1991).
[CrossRef]

Physica A (1)

A. Caldeira, A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587–616 (1983).
[CrossRef]

Physics (1)

J. S. Bell, “On the Einstein–Podolsky–Rosen paradox,” Physics 1(3), 195–200 (1964).

Proc. Cambridge Philos. Soc. (1)

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
[CrossRef]

Rev. Mod. Phys. (1)

W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys. 75, 715 (2003).
[CrossRef]

Science (2)

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

D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, D. J. Wineland, “A decoherence-free quantum memory using trapped ions,” Science 291, 1013–1015 (2001).
[CrossRef] [PubMed]

Z. Phys. B (1)

E. Joos, H. D. Zeh, “The emergence of classical properties through interaction with the environment,” Z. Phys. B 59, 223–243 (1985).
[CrossRef]

Other (18)

D. F. Walls, G. J. Milburn, Quantum Optics (Springer, 1994).
[CrossRef]

V. Buzek, P. L. Knight, in Progress in Optics, E. Wolf (Elsevier, A1995), Vol. XXXIV, p. 1.
[CrossRef]

D. A. Lidar, K. B. Whaley, “Decoherence-free subspaces and subsystems,” arXiv.org, arXiv:quant-phys/0301032 (2003).

M. Born, The Born–Einstein Letters 1916–1955 (Macmillan, 2005).

J. H. Eberly, “Schmidt analysis of pure-state entanglement,” arXiv.org, arXiv:quant-ph/0508019 (2005).

C. H. Thompson, H. Holstein, “The chaotic ball model: local realism and Bell test “detection loophole,” arXiv.org, arXiv:quant-ph/0210150 (2002).

E. Santos, “Optical tests of Bell’s inequalities not resting upon the absurd fair sampling assumption,” arXiv.org, arXiv:quant-ph/0401003 (2004).

M. Orszag, Quantum Optics (Springer, 2000).
[CrossRef]

A. Sen, U. Sen, M. Lewestein, A. Sampera, “The separability versus entanglement problem,” arXiv.org, arXiv:quant-phys/0508032 (2005).

J. von Neumann, Mathematische Grundlagen der Quanten mechanik (Springer-Verlag, 1932).

For generalized measurements and quantum state discrimination, see, for example, J. A. Bergou, U. Herzog, M. Hillery, in Quantum State Discrimination in Quantum State Estimation: M. Paris and J. Rehacek, eds., Vol. 649 of Lecture Notes in Physics (Springer-Verlag, 2004), pp. 417–456, and [36, 37].
[CrossRef]

J. A. Bergou, “Discrimination of quantum states,” to be published in J. Mod. Opt..

J. Preskill, Quantum Information and Computation Vol. 229 of Lecture Notes in Physics (Springer-Verlag, 1998).

E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, I.-O. Stamatescu, Decoherence and the Appearance of Classical World in Quantum Theory (Springer, 1997).

T. Yu, J. H. Eberly, “Evolution from entanglement to decoherence of standard bipartite mixed states,” arXiv.org,arXiv:quant-ph/0503089 (2006).

T. Yu, J. H. Eberly, “Negative entanglement measure and what it implies,” arXiv.org, arXiv:quant-ph/0703083 (2007).

G. S. Agarwal, in Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches, edited by G. Hohler, Vol. 70 of Springer Tracts in Modern Physics (Springer-Verlag, 1974).

L. Derkacz, L. Jakobczyk, “Vacuum-induced stationary entanglement in radiatively coupled three-level atoms,” arXiv.org, arXiv:0710.5048 (2007).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (23)

Fig. 1
Fig. 1

Fidelity after an iteration of the purification algorithm. Note that f > f for f > 1 2 .

Fig. 2
Fig. 2

Fidelity after one iteration of Bennett’s algorithm. Note that G 1 > G for G > 1 2 .

Fig. 3
Fig. 3

Schematic of an assembly of two qubits A and B, located in (a) two independent and spatially separated reservoirs and (b) a common reservoir.

Fig. 4
Fig. 4

(a) Time evolution of the concurrence in a vacuum reservoir when the atoms are initially in the entangled mixed state [Eq. (103)]. (b) Two typical behaviors, sudden death and asymptotic decay in time, for qubits interacting with two independent vacuum reservoirs.

Fig. 5
Fig. 5

Entanglement evolution for the initial state α | 00 ± β | 11 with (left) n = 0.01 and (right) n = 1 . In both cases, ESD occurs for all ranges of 0 < β < 1 .

Fig. 6
Fig. 6

Concurrence for Werner states, Eq. (108), with (left) n = 0.01 and (right) n = 1 .

Fig. 7
Fig. 7

Death time for (a) state α | 00 ± β | 11 and (b) a Werner state. We plotted the disentanglement time for different values of the parameter n. In the case of n = 0 , there exists a range in the initial conditions for which the ESD is not permitted, decaying asymptotically. For larger values of n, the disentanglement time decreases; i.e., the sudden death occurs faster.

Fig. 8
Fig. 8

Negative eigenvalue ( λ ) of ρ P T , for the separability criterion, for | ϕ 3 as the initial state. This eigenvalue is always negative, indicating entanglement at all times.

Fig. 9
Fig. 9

Time evolution of the concurrence for initial | Ψ 1 ( 0 ) with ϵ = 0.28 (solid curve), ϵ = 0.345 (dashed curve), ϵ = 0.9 (dotted curve),

Fig. 10
Fig. 10

(a) Death time. (b) Revival time of the entanglement as a function of ϵ, with initial | Ψ a .

Fig. 11
Fig. 11

(a) Time evolution of the concurrence with initial | Ψ b for ϵ = 0.3 (solid curve), ϵ = 0.5 (dotted), ϵ = 0.707 (dashed), ϵ = 0.9 (dashed–dotted). (b) Death–revival time, as given by Eq. (130), versus ϵ.

Fig. 12
Fig. 12

Time evolution of concurrence for initial | ϕ 3 , with N = 0.1 (dashed curve), N = 0.5 (solid), N = 1 (dotted).

Fig. 13
Fig. 13

(a) Death time. (b) Revival time versus N for the initial state | ϕ 3 .

Fig. 14
Fig. 14

(a) Death time. (b) Revival time versus N for the initial state | ϕ 4 .

Fig. 15
Fig. 15

Time evolution of concurrence for | Ψ a ( t ) as initial state and N = 0.1 : ϵ = 0.1 (long dashed curve), ϵ = 0.2 (dashed–dotted), ϵ = 0.29 (dashed), ϵ = 0.5 (dotted), ϵ = 0.9 (solid).

Fig. 16
Fig. 16

(a) Death time. (b) Revival time, with initial state | Ψ a and N = 0 (solid curve), N = 0.1 (dotted), N = 0.2 (dashed).

Fig. 17
Fig. 17

Time evolution of the concurrence for | Ψ 2 ( t ) as initial state and N = 0.1 : ϵ = 0 (solid curve), ϵ = 0.4 (closely spaced dotted), ϵ = 0.54 (dashed-dotted line), ϵ = 0.6 (long dashed), ϵ c = 0.707 (dashed), ϵ = 0.9 (widely spaced dotted).

Fig. 18
Fig. 18

k 1 versus ϵ for N between 0 and 2.

Fig. 19
Fig. 19

(a) Time evolution of concurrence for initial | 0 ( α | 1 + β | 0 ) with different values of α. (b) Time evolution of concurrence for initial ( α | 0 + β | 1 ) | 1 with different values of α. The reservoir is at T = 0 .

Fig. 20
Fig. 20

Concurrence versus time for different initial conditions α 1 = α 2 α . The reservoir is at T = 0 .

Fig. 21
Fig. 21

Evolution of the concurrence for the initial conditions (a) | 11 , and (b) | | 00 , and different values of n.

Fig. 22
Fig. 22

Concurrence versus time for the initial conditions | Ψ ( 0 ) = | 10 and | Ψ ( 0 ) = | 01 . The parameter of each curve is the reservoir temperature (average thermal photon number).

Fig. 23
Fig. 23

Trajectories between the PPT (separable) and the NPT (entangled) areas: 1, Initial and final states are both entangled; 2, Initial entanglement going asymptotically to zero; 3, Initial entanglement with sudden death; 4, Initial entanglement with sudden death and revival; 5, Periodic death and revival; 6, Entanglement generation starting from a separable state; 7, Time delayed generation of entanglement starting from a separable state.

Tables (1)

Tables Icon

Table 1 Effect of Bilateral C-NOT Gate on Input Source and Target, Yielding Final States

Equations (208)

Equations on this page are rendered with MathJax. Learn more.

| ψ = | ψ A | ϕ B ,
1 2 ( | 0 A | 1 B | 1 A | 0 B )
| Φ ± 1 2 ( | 00 ± | 11 ) ,
| Ψ ± 1 2 ( | 01 ± | 10 ) .
ρ C = 1 2 ( | H A V B H A V B | ) + 1 2 ( | V A H B V A H B | ) ,
ρ QM = | ψ + ψ + | ,
| ψ + = 1 2 ( | H A V B + | V A H B ) .
Tr A { ρ | H A H A | } Tr { ρ | H A H A | } = | V B V B | ,
ρ C = 1 2 ( 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 ) , ρ QM = 1 2 ( 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 ) .
( ρ C ) A = ( ρ QM ) A = 1 2 I ,
| ψ A B = i = 1 d A μ = 1 d B M ( i , μ ) | i | μ ,
M = i = 1 d A μ = 1 d B M ( i , μ ) | i μ | .
M ( i , μ ) = k = 1 min ( d A , d B ) u i k d k k v k μ ,
| ψ A B = i = 1 d A μ = 1 d B k = 1 min ( d A , d B ) u i k d k k v k μ | i | μ
= k = 1 min ( d A , d B ) d k k ( i = 1 d A u i k | i ) ( μ = 1 d B v k μ | μ ) .
| a k = ( i = 1 d A u i k | i ) ,
| b k = ( μ = 1 d B v k μ | μ ) ,
| ψ A B = k = 1 min ( d A , d B ) d k k | a k | b k .
ρ = i , k λ i λ k | a i a k | | b i b k | ,
ρ A = k λ k | a k a k | ,
ρ B = k λ k | b k b k | ,
0 λ k 1 and k λ k = 1 .
K = 1 Tr A ρ A 2 = 1 Tr B ρ B 2 = 1 n λ n 2 .
1 K D .
S [ p ρ 1 + ( 1 p ) ρ 2 ] p S ( ρ 1 ) + ( 1 p ) S ( ρ 2 ) ,
E ( ψ ) = S ( ρ A ) = S ( ρ B ) = k = 1 d λ k log 2 ( λ k ) .
E ( ψ ) k p k E ( ψ k ) .
ρ = i p i | a i a i | | b i b i | ;
ρ m μ , n ν T A = ρ n μ , m ν ,
ρ sep T A = i p i ( | a i a i | ) T | b i b i | .
E ( ρ ) = i p i E ( | ψ i ) .
E ( ρ ) = min i p i E ( ψ i ) ,
E ( ψ ) = Tr ( ρ A log ρ A ) = Tr ( ρ B log ρ B ) .
ρ ̃ A B = ( σ y σ y ) ρ A B * ( σ y σ y ) ,
C ( ρ ) = max { 0 , λ 1 λ 2 λ 3 λ 4 } ,
E ( ρ A B ) = E ( C ( ρ A B ) ) ,
E ( C ) = H [ 1 2 + 1 2 1 C 2 ] ,
H ( x ) = x log 2 x ( 1 x ) log 2 ( 1 x ) .
ρ ( 0 ) = ( ρ 11 0 0 ρ 14 0 ρ 22 ρ 23 0 0 ρ 32 ρ 33 0 ρ 41 0 0 ρ 44 ) ,
C = 2 max { 0 , ρ 23 ρ 32 ρ 11 ρ 44 , ρ 14 ρ 41 ρ 22 ρ 33 } .
ρ = f | 0 0 | + ( 1 f ) | 1 1 | ,
ρ = ( f | 0 0 | + ( 1 f ) | 1 1 | ) S ( f | 0 0 | + ( 1 f ) | 1 1 | ) T ,
C-NOT ( | 0 0 | ) C ( | 0 0 | ) T ( | 0 0 | ) C ( | 0 0 | ) T ,
C-NOT ( | 1 1 | ) C ( | 0 0 | ) T ( | 1 1 | ) C ( | 1 1 | ) T .
ρ = ( f 2 | 0 0 | + ( 1 f ) 2 | 1 1 | ) S | 0 0 | T + f ( 1 f ) ( | 0 0 | + | 1 1 | ) S | 1 1 | T .
ρ = f 2 f 2 + ( 1 f ) 2 | 0 0 | + ( 1 f ) 2 f 2 + ( 1 f ) 2 | 1 1 | = f | 0 0 | + ( 1 f ) | 1 1 | ,
f n + 1 = f n 2 f n 2 + ( 1 f n ) 2 .
ρ G = G | ψ ψ | + ( 1 G ) 3 [ | ψ + ψ + | + | ϕ ϕ | + | ϕ + ϕ + | ] ,
ρ G = G | ϕ + ϕ + | + ( 1 G ) 3 [ | ϕ ϕ | + | ψ + ψ + | + | ψ ψ | ] .
| 0 A 1 | 0 A 2 | 0 A 1 | 0 A 2 ,
| 0 A 1 | 1 A 2 | 0 A 1 | 1 A 2 ,
| 1 A 1 | 0 A 2 | 1 A 1 | 1 A 2 ,
| 1 A 1 | 1 A 2 | 1 A 1 | 0 A 2 ,
ρ T = F 1 | ϕ + T ϕ + | + F 2 | ϕ T ϕ | + F 3 | ψ + T ψ + | + F 4 | ψ T ψ | ,
F 1 = G 2 | ϕ + C ϕ + | + G ( 1 G ) 3 | ϕ C ϕ | + ( 1 G ) 2 9 | ψ + C ψ + | + ( 1 G ) 2 9 | ψ C ψ | ,
F 2 = ( 1 G ) 2 9 | ϕ + C ϕ + | + G ( 1 G ) 3 | ϕ C ϕ | + ( 1 G ) 2 9 | ψ + C ψ + | + ( 1 G ) 2 9 | ψ C ψ | ,
F 3 = ( 1 G ) 2 9 | ϕ + C ϕ + | + G ( 1 G ) 3 | ϕ C ϕ | + ( 1 G ) 2 9 | ψ + C ψ + | + ( 1 G ) 2 9 | ψ C ψ | ,
F 4 = ( 1 G ) 2 9 | ϕ + C ϕ + | + G ( 1 G ) 3 | ϕ C ϕ | + ( 1 G ) 2 9 | ψ + C ψ + | + ( 1 G ) 2 9 | ψ C ψ | .
ρ C = G 1 | ϕ + C ϕ + | + G 2 | ϕ C ϕ | + G 3 | ψ + C ψ + | + G 4 | ψ C ψ | ,
G 1 = G 2 + ( 1 G ) 2 9 G 2 + 2 G ( 1 G ) 3 + 5 ( 1 G ) 2 9 .
ρ C final = G 1 | ψ C ψ | + G 2 | ψ + C ψ + | + G 3 | ϕ C ϕ | + G 4 | ϕ + C ϕ + | .
| s 1 | d 2 | s 1 | d 1 ,
| s 2 | d 2 | s 2 | d 2 .
| ψ initial = α | s 1 + β | s 2 ,
| ψ initial = ( α | s 1 + β | s 2 ) | d 2 after meas α | s 1 | d 1 + β | s 2 | d 2 | Ψ c ,
ρ c = | Ψ c Ψ c | = | α | 2 | s 1 s 1 | | d 1 d 1 | + | β | 2 | s 2 s 2 | | d 2 d 2 | + α * β | s 2 s 1 | | d 2 d 1 | + α β * | s 1 s 2 | | d 1 d 2 |
Non unitary ρ r = | α | 2 | s 1 s 1 | | d 1 d 1 | + | β | 2 | s 2 s 2 | | d 2 d 2 | .
| s 3 = 1 2 ( | s 1 + | s 2 ) ,
| s 4 = 1 2 ( | s 1 | s 2 ) ,
| Ψ c = 1 2 ( | s 1 | d 1 | s 2 | d 2 ) = 1 2 [ 1 2 ( | s 3 + | s 4 ) | d 1 1 2 ( | s 3 | s 4 ) | d 2 ] = 1 2 [ | s 3 | d 3 + | s 4 | d 4 ] ,
| d 3 = 1 2 ( | d 1 | d 2 ) ,
| d 4 = 1 2 ( | d 1 + | d 2 ) .
( ρ c ) diag = 1 2 | s 1 s 1 | | d 1 d 1 | + 1 2 | s 2 s 2 | | d 2 d 2 | ,
( ρ c ) diag = 1 2 | s 3 s 3 | | d 3 d 3 | + 1 2 | s 4 s 4 | | d 4 d 4 | .
| Ψ c | ϵ 0 = [ α | s 1 | d 1 + β | s 2 | d 2 ] | ϵ 0 ( α | s 1 | d 1 | ϵ 1 + β | s 2 | d 2 | ϵ 2 ) = | ψ ,
ρ SD = Tr ϵ | ψ ψ | = i ϵ i | ψ ψ | ϵ i = ρ r ,
H int = n | m m | D m ,
| m | ϕ 0 t exp ( i H int t ) | m | ϕ 0 = | m exp ( i D m t ) | ϕ 0 = | m | ϕ m ( t ) .
m F m | m | ϕ 0 m F m | m | ϕ m ( t ) ;
ρ S = Tr envir m , p F m F p * | m p | | ϕ m ( t ) ϕ p ( t ) | = m , p F m F p * | m p | ϕ m ( t ) | ϕ p ( t ) ,
ϕ m ( t ) | ϕ p ( t ) = δ p m
ρ S m | F m | 2 | m m | .
m F m | m | ϕ m | ϵ 0 m F m | m | ϕ m | ϵ m ;
ρ system-apparatus = m | F m | 2 | m m | | ϕ m ϕ m | .
t c = γ 1 2 | α | 2 .
| ψ ( 0 ) = N ( | α 1 + | α 2 ) ,
d ρ d t = γ 2 ( 2 a ρ a a a ρ ρ a a ) .
X N ( η , t ) = Tr ( ρ ( t ) exp ( η a ) exp ( η * a ) ) .
X N ( η , t ) t = Tr [ d ρ d t exp ( η a ) exp ( η * a ) ] = γ 2 Tr [ ( 2 a ρ a a a ρ ρ a a ) exp ( η a ) exp ( η * a ) ] = γ 2 Tr [ 2 ρ a exp ( η a ) exp ( η * a ) a ρ exp ( η a ) exp ( η * a ) a a ρ a a exp ( η a ) exp ( η * a ) ] = γ 2 { η Tr [ ρ a exp ( η a ) exp ( η * a ) ] η * Tr [ ρ exp ( η a ) exp ( η * a ) a ] } X N ( η , t ) t = γ 2 [ η X N ( η , t ) η + η * X N ( η , t ) η * ] .
[ a , f ( a , a ) ] = f ( a , a ) a ,
[ a , f ( a , a ) ] = f ( a , a ) a ,
[ a a , exp ( η a ) exp ( η * a ) ] = η a exp ( η a ) exp ( η * a ) + η * exp ( η a ) exp ( η * a ) a .
X N ( η , t ) = X N ( η exp ( γ t 2 ) , 0 ) = X N ( η ( t ) , 0 ) .
X N ( η , t ) t = X N ( η , t ) η ( t ) η ( t ) t + X N ( η , t ) η * ( t ) η * ( t ) t = γ 2 [ η X N ( η , t ) η + η * X N ( η , t ) η * ] .
X N ( η , 0 ) = Tr ( ρ ( 0 ) ) exp ( η a ) exp ( η * a ) = N 2 Tr i , j [ | α i α j | exp ( η a ) exp ( η * a ) ] = N 2 i , j [ α j | exp ( η a ) exp ( η * a ) | α i ] = N 2 i , j [ α j | α i exp ( η α j * η * α i ) ] .
X N ( η , t ) = N 2 i , j [ α j | α i exp ( η α j * η * α i ) exp ( γ t 2 ) ] .
ρ = N 2 i , j = 1 2 α i | α j ( 1 exp ( γ t ) ) | α j exp ( γ t 2 ) α i exp ( γ t 2 ) | .
α | α = exp ( 2 | α | 2 ) ,
ρ = N 2 { | α exp ( γ t 2 ) α exp ( γ t 2 ) | + | α exp ( γ t 2 ) α exp ( γ t 2 ) | } + N 2 exp [ 2 | α | 2 ( 1 exp ( γ t ) ) ] { | α exp ( γ t 2 ) α exp ( γ t 2 ) | + | α exp ( γ t 2 ) α exp ( γ t 2 ) | } .
exp ( 2 | α | 2 γ t ) exp ( t t c ) ,
t c = γ 1 2 | α | 2 .
t c ( sq ) = γ 1 2 [ N + 2 α 2 ( N M + 1 2 ) ] ,
t c ( sq ) = γ 1 2 [ N ] ,
| ψ = a | 0 + b | 1 ,
| 0 j | 0 j ,
| 1 j exp [ i ϕ ] | 1 j ,
R z ( ϕ ) = [ 1 0 0 exp ( i ϕ ) ]
ρ j = R z ( ϕ ) | ψ j ψ | R z ( ϕ ) T p ( ϕ ) d ϕ ,
p ( ϕ ) = 1 4 π γ exp ( ϕ 2 4 γ ) ,
ρ j = [ | a | 2 a b * exp ( γ ) a * b exp ( γ ) | b | 2 ] ,
| 0 1 | 0 2 = | 00 | 00 ,
| 0 1 | 1 2 = | 01 exp ( i ϕ ) | 0 1 | 1 2 = exp ( i ϕ ) | 01 ,
| 1 1 | 0 2 = | 10 exp ( i ϕ ) | 1 1 | 0 2 = exp ( i ϕ ) | 10 ,
| 1 1 | 1 2 = | 11 exp ( 2 i ϕ ) | 1 1 | 1 2 = exp ( 2 i ϕ ) | 11 .
| χ = α | 10 + β | 01 exp ( i ϕ ) ( α | 10 + β | 01 ) = exp ( i ϕ ) | χ .
DFS 2 = { | 10 , | 01 } .
{ | 001 , | 010 , | 100 } = DFS 3 ( 1 ) , { | 110 , | 101 , | 011 } = DFS 3 ( 2 ) ,
H = H S I B + I S H B + H I ,
ρ S B ( t ) = U ( t ) ρ S B ( 0 ) U ( t ) ,
ρ S B ( t ) = U ( t ) ρ S ( 0 ) ρ B ( 0 ) U ( t ) .
H I = α S α B ,
ρ ( t ) = T r B [ U ( t ) ρ ( 0 ) ρ B ( 0 ) U ( t ) ] .
ρ = μ | U ( t ) ( ρ ( 0 ) ν λ ν | ν ν | ) U ( t ) | μ = a A a ρ ( 0 ) A a ,
A a = λ ν μ | U ( t ) | ν , a = μ , ν .
a A a A a = I S .
S α | k ̃ = c α | k ̃ , α , | k ̃ .
d ρ d t = i [ H s , ρ ] + L D ( ρ ) ,
L D ( ρ ( t ) ) = 1 2 α , β = 1 M d α , β ( [ F α , ρ F β ] + [ F α ρ , F β ] ) ,
H ̃ ( DFS ) H ( total system Hilbert space ) .
ρ = k , j = 1 N ρ k j | k ̃ j ̃ | .
F α | k ̃ = j = 1 N C k j α | j ̃ .
L D ( ρ ) = 1 2 α , β = 1 M d α , β k , j , m , n = 1 N ρ k j ( 2 C j m β * C k n α | n ̃ m ̃ | C m n β * C k n α | m ̃ j ̃ | C j m β * C n m α | k ̃ n ̃ | ) = 0 .
C k n α = C n α δ k , n ,
k , j = 1 N ρ k j | k ̃ j ̃ | ( 2 C j β * C k α C k β * C k α C j β * C j α ) = 0 .
( 2 C j β * C k α C k β * C k α C j β * C j α ) = 0 ,
2 = Z * + Z 1 ,
F α | k ̃ = C α | k ̃ for α , k .
[ F α , F β ] | k ̃ = 0 .
[ F α , F β ] = γ = 1 M f α , β γ F γ ,
γ = 1 M f α , β γ C γ = 0 ,
ρ ̂ t = Γ 2 i , j = 1 2 [ ( N + 1 ) ( 2 σ i ρ ̂ σ j σ i σ j ρ ̂ ρ ̂ σ i σ j ) + N ( 2 σ i ρ ̂ σ j σ i σ j ρ ̂ ρ ̂ σ i σ j ) M ( 2 σ i ρ ̂ σ j σ i σ j ρ ̂ ρ ̂ σ i σ j ) M * ( 2 σ i ρ ̂ σ j σ i σ j ρ ̂ ρ ̂ σ i σ j ) ] ,
C 1 ( t ) = 2 e Γ t [ | ρ 23 | ρ 11 ( ρ 44 + ( ρ 22 + ρ 33 ) ω 2 + ρ 11 ω 4 ) ] ,
C 2 ( t ) = 2 e Γ t [ | ρ 14 | ρ 22 ρ 33 + ρ 11 ( ρ 22 + ρ 33 ) ω 2 + ρ 11 2 ω 4 ] ,
t d 1 = 1 Γ ln ( ρ 11 ( 2 ρ 11 + ρ 22 + ρ 33 + ( ρ 22 + ρ 33 ) 2 4 ( ρ 11 ρ 44 | ρ 23 | 2 ) ) 2 ( ρ 11 | ρ 23 | 2 ) ) .
t d 2 = 1 Γ ln ( ρ 11 ( 2 ρ 11 + ρ 22 + ρ 33 + ( ρ 22 ρ 33 ) 2 + ( 4 | ρ 23 | 2 ) ) 2 ( ρ 22 ρ 33 + ρ 11 ( ρ 11 + ρ 22 + ρ 33 ) | ρ 14 | 2 ) ) ,
ρ 11 | ρ 23 | 2 , the entanglement decays asymptotically ;
ρ 11 > | ρ 23 | 2 , the entanglement decays in a finite time ,
ρ 22 ρ 33 + ρ 11 ( 1 ρ 44 ) | ρ 14 | 2 , the entanglement decays asymptotically ;
ρ 22 ρ 33 + ρ 11 ( 1 ρ 44 ) > | ρ 14 | 2 , the entanglement decays in a finite time ,
C = max { 0 , 2 | β | e Γ t ( | α | | β | ( e Γ t 1 ) ) } ,
ρ ( 0 ) = 1 3 ( a 0 0 0 0 1 1 0 0 1 1 0 0 0 0 1 a ) .
C = max { 0 , 2 3 e Γ t ( 1 a ( 3 2 ( 1 + a ) e Γ t + a e 2 Γ t ) ) } ,
t d = 1 Γ ln ( a + 1 2 a + a 2 a ) .
C 1 ( t ) = 2 | ρ 23 | e ( 2 n + 1 ) t 2 ( 2 n + 1 ) 2 { [ ( ( ρ 11 ρ 22 ρ 33 + ρ 44 ) n 2 + ( 2 ρ 11 ρ 22 ρ 33 ) n + ρ 11 ) e 2 ( 2 n + 1 ) t + 2 n ( ( ρ 11 ρ 44 ) n + 1 2 ( 2 ρ 11 + ρ 22 + ρ 33 ) ) e ( 2 n + 1 ) t + n 2 ] [ ( ( ρ 11 ρ 22 ρ 33 + ρ 44 ) n 2 + ( 2 ρ 11 ρ 22 ρ 33 ) n + ρ 11 ) e 2 ( 2 n + 1 ) t 2 ( n + 1 ) ( ( ρ 11 ρ 44 ) n + 1 2 ( 2 ρ 11 + ρ 22 + ρ 33 ) ) e ( 2 n + 1 ) t + ( n + 1 ) 2 ] } 1 2 ,
C 2 ( t ) = 2 | ρ 14 | e ( 2 n + 1 ) t 2 ( 2 n + 1 ) 2 { [ ( ( ρ 11 ρ 22 ρ 33 + ρ 44 ) n 2 + ( 2 ρ 11 ρ 22 ρ 33 ) n + ρ 11 ) e 2 ( 2 n + 1 ) t + ( 2 ( ρ 22 ρ 33 ) n 2 + ( ρ 11 + 2 ρ 22 2 ρ 33 ρ 44 ) n + ρ 11 + ρ 22 ) e ( 2 n + 1 ) t + n ( n + 1 ) ] [ ( ( ρ 11 ρ 22 ρ 33 + ρ 44 ) n 2 + ( 2 ρ 11 ρ 22 ρ 33 ) n + a 0 ) e 2 ( 2 n + 1 ) t + ( 2 ( ρ 33 ρ 22 ) n 2 + ( ρ 11 2 ρ 22 + 2 ρ 33 ρ 44 ) n + ρ 11 + ρ 33 ) e ( 2 n + 1 ) t + n ( n + 1 ) ] } 1 2 ,
ρ ( 0 ) = 1 4 ( 1 a 0 0 0 0 1 + a 2 a 0 0 2 a 1 + a 0 0 0 0 1 a ) .
ρ t = 1 2 Γ ( 2 S ρ S S S ρ ρ S S ) ,
S = N + 1 ( σ 1 + σ 2 ) N e i Ψ ( σ 1 + σ 2 ) = cosh ( r ) ( σ 1 + σ 2 ) sinh ( r ) e i Ψ ( σ 1 + σ 2 ) ,
| ϕ 1 = 1 N 2 + M 2 ( N | + + + M e i Ψ | ) ,
| ϕ 2 = 1 2 ( | + | + ) .
| ϕ 3 = 1 2 ( | + + | + ) ,
| ϕ 4 = 1 N 2 + M 2 ( M | + + N e i Ψ | ) .
C ( ρ 1 ( t ) ) = 2 N ( N + 1 ) 2 N + 1 ,
C ( ρ 2 ( t ) ) = 1 .
ρ 3 ( t ) = ( ( e 2 t 1 ) e 2 t 0 0 0 0 0 0 0 0 0 e 2 t 0 0 0 0 0 ) .
ρ 4 ( t ) = ( ( 1 2 t + e 2 t ) e 2 t 0 0 0 0 0 0 0 0 0 2 t e 2 t 0 0 0 0 e 2 t ) ,
| Ψ a = ϵ | ϕ 1 + 1 ϵ 2 | ϕ 4 ,
| Ψ b = ϵ | ϕ 2 + 1 ϵ 2 | ϕ 3 ,
| Ψ a ( 0 ) = ϵ | + 1 ϵ 2 | + + ,
ρ a ( t ) = ( ( 2 t 1 + 2 t ϵ 2 + ϵ 2 + e 2 t ) e 2 t 0 0 ϵ 1 ϵ 2 e t 0 0 0 0 0 0 2 t ( 1 ϵ 2 ) e 2 t 0 ϵ 1 ϵ 2 e t 0 0 ( 1 ϵ 2 ) e 2 t ) ,
C ( ρ a ) = max { 0 , 2 ( ( ϵ 1 ϵ 2 ) e t t e 2 t ( 1 ϵ 2 ) ) } ,
t e t = ϵ 1 ϵ 2 .
| Ψ b ( 0 ) = 1 2 [ ( ϵ + 1 ϵ 2 ) | + ( ϵ 1 ϵ 2 ) | + ] ,
ρ b ( t ) = ( ( e 2 t ϵ 2 e 2 t 1 + ϵ 2 ) e 2 t 0 0 0 0 ϵ 2 ϵ 1 ϵ 2 e t 0 0 ϵ 1 ϵ 2 e t ( 1 ϵ 2 ) e 2 t 0 0 0 0 0 ) ,
C ( ρ b ( t ) ) = max { 0 , e 2 t | ϵ 2 e 2 t 1 + ϵ 2 | } ,
t = 1 2 ln ( 1 ϵ 2 ϵ 2 ) .
C 1 ( ρ ( t ) ) = | ρ 33 ( t ) ρ 22 ( t ) | 2 ( N ( ρ 11 ( t ) + ρ 44 ( t ) ) + ρ 44 ( t ) + 2 ρ 14 ( t ) N ( N + 1 ) ) 2 N + 1 ( ( N ( ρ 11 ( t ) + ρ 44 ( t ) ) + ρ 11 ( t ) 2 ρ 14 ( t ) N ( N + 1 ) ) 2 N + 1 ) ,
C 2 ( ρ ( t ) ) = 2 2 N + 1 | N ( N + 1 ) ( ρ 11 ( t ) ρ 44 ( t ) ) + ρ 14 ( t ) | ( ρ 22 ( t ) 2 ρ 23 ( t ) + ρ 33 ( t ) ) ( ρ 22 ( t ) + 2 ρ 23 e ( t ) + ρ 33 ( t ) ) ,
| Ψ a ( 0 ) = ϵ | ϕ 1 + 1 ϵ 2 | ϕ 4
C ( ρ a ( 0 ) ) = | 2 ϵ 1 ϵ 2 + 4 N ( N + 1 ) ( ϵ 2 1 2 ) | 2 N + 1 .
| Ψ b ( 0 ) = ϵ | ϕ 2 + 1 ϵ 2 | ϕ 3 .
| ϕ 1 1 2 ( | + + + | ) ,
| Ψ a = k 1 | + + + k 2 | ,
k 1 = ϵ N + M 1 ϵ 2 N 2 + M 2 , k 2 = ϵ M N 1 ϵ 2 N 2 + M 2 .
ρ 11 ( t ) = ρ 11 e 2 Γ t ,
ρ 22 ( t ) = 1 4 ( ρ 22 + ρ 33 ρ 23 ρ 32 + 2 ( ρ 22 ρ 33 ) e Γ t + ( 4 ρ 11 t + ρ 22 + ρ 33 + ρ 23 + ρ 32 ) e 2 Γ t ) ,
ρ 33 ( t ) = 1 4 ( ρ 22 + ρ 33 ρ 23 ρ 32 + 2 ( ρ 33 ρ 22 ) e Γ t + ( 4 ρ 11 t + ρ 22 + ρ 33 + ρ 23 + ρ 32 ) e 2 Γ t ) ,
ρ 44 ( t ) = 1 2