Distillation of continuous-variable entanglement with optical means

Annals of Physics

Volumn 311, Issue 2, 2004, Pages 431-458

  Eisert, J ^a,b   Browne, D E ^b   Scheel, S ^b   Plenio, M B ^b  

^a UNIVERSITY OF POTSDAM   (Germany)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Continuous variable entanglement; Entanglement distillation; Gaussian operations; Gaussian states; Non Gaussian operations; Preservation of coherence; Quantum information theory; Quantum optics

Indexed keywords

EID: 2442668731     PISSN: 00034916     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.aop.2003.12.008     Document Type: Article

References (56)

1. C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, I.A. Smolin, W.K. Wootters, Phys. Rev. Lett. 76 (1996) 722.
2. C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wootters, Phys. Rev. A 54 (1996) 3824.
3. H.J. Briegel, W. Dür, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 81 (1998) 5932.
4. H. Aschauer, H.J. Briegel, Phys. Rev. Lett. 88 (2002) 047902.
5. E.M. Rains, Phys. Rev. A 60 (1999) 173.
6. E.M. Rains, Phys. Rev. A 60 (1999) 179.
7. J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, A. Zeilinger, Nature 423 (2003) 417.
8. Z. Zhao, T. Yang, Y.A. Chen, A.-N. Zhang, J.-W. Pan, Phys. Rev. Lett. 90 (2003) 207901.
9. Z. Zhao, J.-W. Pan, M.S. Zhan, Phys. Rev. A 64 (2001) 014301.
10. T. Yamamoto, M. Koashi, N. Imoto, Phys. Rev. A 64 (2001) 012304.
11. J.-W. Pan, C. Simon, C. Brukner, A. Zeilinger, Nature 410 (2001) 1067.
12. X.B. Wang, Phys. Rev. A 68 (2003) 042304.
13. D. Browne, J. Eisert, S. Scheel, M.B. Plenio, Phys. Rev. A 67 (2003) 062320.
14. S. Scheel, L. Knöll, T. Opatrný, D.-G. Welsch, Phys. Rev. A 62 (2000) 043803.
15. M. Belsey, D.T. Smithey, M.G. Raymer, Phys. Rev. A 46 (1992) 414.
16. M.S. Kim, W. Son, V. Bužek, P.L. Knight, Phys. Rev. A 65 (2002) 032323.
17. M.M. Wolf, J. Eisert, M.B. Plenio, Phys. Rev. Lett. 90 (2003) 047904.
18. R.E. Slusher, L.W. Hollberg, B. Yurke, J.C. Mertz, J.F. Valley, Phys. Rev. Lett. 55 (1985) 2409.
19. M.D. Reid, D.F. Walls, Phys. Rev. A 34 (1986) 1260.
20. L.A. Wu, H.J. Kimble, J.L. Hall, H. Wu, Phys. Rev. Lett. 57 (1986) 2520.
21. A. Furusawa, J. Sørensen, S.L. Braunstein, C. Fuchs, H.J. Kimble, E.S. Polzik, Science 282 (1998) 706.
22. Ch. Silberhorn, P.K. Lam, O. Weiss, F. König, N. Korolkova, G. Leuchs, Phys. Rev. Lett. 86 (2001) 4267.
23. A. Kuzmich, I.A. Walmsley, L. Mandel, Phys. Rev. A 64 (2001) 063804.
24. A. Kuzmich, E.S. Polzik, Phys. Rev. Lett. 85 (2000) 5639.
25. C. Schori, J.L. Sørensen, E.S. Polzik, Phys. Rev. A 66 (2002) 033802.
26. J. Eisert, S. Scheel, M.B. Plenio, Phys. Rev. Lett. 89 (2002) 137903.
27. J. Fiurášek, Phys. Rev. Lett. 89 (2002) 137904.
28. G. Giedke, J.I. Cirac, Phys. Rev. A 66 (2002) 032316.
29. G. Giedke, J. Eisert, J.I. Cirac, M.B. Plenio, Quant. Inf. Comp. 3 (2003) 211.
30. G. Giedke, L.-M. Duan, J.I. Cirac, P. Zoller, Quant. Inf. Comp. 1 (2001) 79.
31. D. Gottesman, J. Preskill, Phys. Rev. A 63 (2001) 22309.
32. T.C. Ralph, P.K. Lam, Phys. Rev. Lett. 81 (1998) 5668.
33. M. Hillery, Phys. Rev. A 61 (2000) 022309.
34. S. Lorenz, Ch. Silberhorn, N. Korolkova, R.S. Windeler, G. Leuchs, Appl. Phys. B 73 (2001) 855.
35. F. Grosshans, P. Grangier, Phys. Rev. Lett. 88 (2002) 057902.
36. Ch. Silberhorn, T.C. Ralph, N. Lütkenhaus, G. Leuchs, Phys. Rev. Lett. 89 (2002) 167901.
37. W. Vogel, D.-G. Welsch, S. Wallentowitz, Quantum Optics, An Introduction, Wiley-VCH, Weinheim, 2001.
38. Arvind, B. Dutta, N. Mukunda, R. Simon, Phys. Rev. A 52 (1995) 1609.
39. K. Zyczkowski, P. Horodecki, A. Sanpera, M. Lewenstein, Phys. Rev. A 58 (1998) 883.
40. J. Eisert, M.B. Plenio, J. Mod. Opt. 46 (1999) 145.
41. It should be mentioned that the quantification of entanglement for infinite-dimensional quantum systems is a subtle issue [22-24]. The minimization necessary to evaluate most measures of entanglement is typically not feasible, and without further assumptions they are not even trace-norm continuous [22]. For those two-mode Gaussian states the sequence of states converges to in the weak sense, the entanglement of formation in the infinite setting [22] is available in principle: it has been shown in [231 that for the symmetric Gaussians that we encounter here, the entanglement of formation is identical to the Gaussian entanglement of formation. For the intermediate steps this clearly does not hold, however. In this paper, we take a pragmatic approach and choose the computable logarithmic negativity as the appropriate functional that quantifies the degree of entanglement.
42. J. Eisert, C. Simon, M. B. Plenio, J. Phys. A 3 5 (2002) 3911.
43. G. Giedke, M.M. Wolf, O. Krüger, R.F. Werner, J.I. Cirac, Phys. Rev. Lett. 91 (2003) 017901.
44. M. Keyl, D. Schlingemann, R.F. Werner, Quant. Inf. Comp. 3 (2003) 281.
45. R. Clifton, H. Halvorson, A. Kent, Phys. Rev, A 61 (2000) 042101.
46. J. Eisert, Ph.D. thesis, Potsdam, February 2001.
47. G. Vidal, R.F. Werner, Phys. Rev. A 65 (2002) 032314.
48. K. Audenaert, M.B. Plenio, J. Eisert, Phys. Rev. Lett. 90 (2003) 027901.
49. S. Scheel, K. Nemoto, W.J. Munro, P.L. Knight, Phys. Rev. A 68 (2003) 032310.
50. S. Scheel, D.-G. Welsch, Phys. Rev. A 64 (2001) 063811.
51. B. Demoen, P. Vanheuswijn, A. Verbeure, Lett. Math. Phys. 2 (1977) 161.
52. A.S. Holevo, R.F. Werner, Phys. Rev. A 63 (2001) 032312.
53. J. Harrington, J. Preskill, Phys. Rev. A 64 (2001) 062301.
54. J. Eisert, M.B. Plenio, Phys. Rev. Lett. 89 (2002) 097901.
55. K. Nemoto, S.L. Braunstein, Phys. Rev. A 66 (2002) 032306.
56. i,j,k,l = 0 if i + j + k + l is an odd number, as can again be shown by induction.