Mixed-state entanglement and quantum error correction

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 54, Issue 5, 1996, Pages 3824-3851

  Bennett, Charles H ^a,b,c   DiVincenzo, David P ^a,b,c   Smolin, John A ^a,b,c   Wootters, William K ^a,b,c  

^a IBM RESEARCH   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c WILLIAMS COLLEGE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 4243216277     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.54.3824     Document Type: Article

References (55)

1. A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935) [reprinted in Quantum Theory and Measurement, edited by J. A. Wheeler and W. Z. Zurek (Princeton University Press, Princeton, 1983)].
2. J. S. Bell, Physics 1, 195 (1964) [reprinted in J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1987), p. 14].
3. J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Phys. Rev. Lett. 23, 880 (1980).
4. J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1981 (1978);
5. A. Aspect, P. Grangier and G. Roger, Phys. Rev. Lett. 47, 460 (1981);
6. P. G. Kwiat et al, Phys. Rev. Lett. 75, 4337 (1995).
7. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).
8. B. Schumacher, Phys. Rev. A 51, 2738 (1995).
9. C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992).
10. I. L. Chuang and R. Laflamme, Report No. quant-ph/9511003;
11. I. L. Chuang and Y. Yamamoto, Phys. Rev. Lett. 76, 4281 (1996).
12. P. W. Shor, Phys. Rev. A 52, 2493 (1995).
13. A. R. Calderbank and P. W. Shor, Phys. Rev. A 54, 1098 (1996).
14. A. M. Steane, Phys. Rev. Lett. 77, 793 (1996); Phys. Rev. A (to be published); Report No. quant-ph/9601029.
15. R. Laflamme, C. Miquel, J.-P. Paz and W. H. Zurek, Phys. Rev. Lett. 77, 198 (1996); Report No. quant-ph/9602019.
16. A. Ekert and C. Machiavello, Phys. Rev. Lett. 77, 2585 (1996); Report No. quant-ph/9602022.
17. S. L. Braunstein, Report No. quant-ph/9603024.
18. M. B. Plenio, V. Vedral, and P. L. Knight, Phys. Rev. A (to be published); Report No. quant-ph/9603022.
19. L. Vaidman, L. Goldenberg, and S. Wiesner, Report No. quant-ph/9603031.
20. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin and W. K. Wootters, Phys. Rev. Lett. 76, 722 (1996).
21. Omitted end note.
22. N. Gisin, Phys. Lett. A 154, 201 (1991).
23. C. H. Bennett, H. J. Bernstein, S. Popescu and B. Schumacher, Phys. Rev. A 53, 2046 (1996).
24. R. F. Werner, Phys. Rev. A 40, 4277 (1989).
25. S. Popescu, Phys. Rev. Lett. 72, 797 (1994).
26. S. Popescu, Phys. Rev. Lett. 74, 2619 (1995).
27. A. Wehrl, Rev. Mod. Phys. 50, 221 (1978), esp. p. 237.
28. This is an application of the Schmitt decomposition; see A. Peres, Quantum Theory: Concepts and Methods (Kluwer, Dordrecht, 1993), p. 123f.
29. A. Peres, Phys. Rev. Lett. 77, 1413 (1996).
30. M. Horodecki, P. Horodecki, and R. Horodecki, Report No. quant-ph/9605038.
31. R. Jozsa, J. Mod. Opt. 41, 2315 (1994).
32. Alice and Bob should choose a basis for the Bell states which maximizes the fully entangled fraction f, Eq. (19).
33. Reference 17 states incorrectly that a set of four discrete rotations is sufficient to simulate a random bilateral rotation; in fact, 12 are both necessary and sufficient. See Appendix A.
34. This follows from the fact that ensembles with the same density matrix are indistinguishable by measurements performed on members of the ensembles. (See, e.g., Ref. 25, p. 75.) (Such measurements could include executing a purification protocol followed by testing the quality of the resulting purified pairs.)
35. A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin and H. Weinfurter, Phys. Rev. A 52, 3457 (1995).
36. C. H. Bennett, Phys. Today 48, (10), 27 (1996).
37. C. Macchiavello (private communication)pointed out that the yield of the recurrence protocol could be increased by substituting a deterministic (Formula presented) operation for the random twirl (Formula presented).
38. See D. Deutsch et al, Phys. Rev. Lett. 77, 2818 (1996).
39. P. Shor and J. A. Smolin, Report No. quant-ph/9604006.
40. This purification strategy was worked out in a discussion with Richard Jozsa.
41. M. Horodecki, P. Horodecki, and R. Horodecki, Report No. quant-ph/9607009.
42. C. H. Bennett, G. Brassard, C. Crépeau and U. Mauer, IEEE Trans. Inf. Theory 41, 1915 (1995).
43. W. K. Wootters and W. Zurek, Nature (London) 299, 802 (1982).
44. E. Knill and R. Laflamme, Report No. quant-ph/9604034.
45. C. H. Bennett, G. Brassard and N. D. Mermin, Phys. Rev. Lett. 68, 557 (1992).
46. For a large-block code which only error-corrects to some high fidelity, the code may not need to properly treat the no-error case because this case is highly improbable. Such a case may also arise for certain peculiar block error models from which the no-error case is excluded.
47. Omitted end note.
48. D. Coppersmith and E. Grossman, SIAM J. Appl. Math. 29, 624 (1975).
49. H. Mabuchi and P. Zoller, Phys. Rev. Lett. 76, 3108 (1996).
50. Omitted end note.
51. D. P. DiVincenzo and P. W. Shor, Phys. Rev. Lett (to be published); Report No. quant-ph/9605031.
52. Omitted end note.
53. An important step has been taken in this characterization of (Formula presented) and E by Horodecki et al. 37, who show that for mixed state M of a pair of qubits, (Formula presented)(M)=0 if and only if E(M)=0.
54. See, e.g., M. Tinkham, Group Theory and Quantum Mechanics (Prentice-Hall, Englewood Cliffs, NJ, 1964).
55. Omitted end note.