Decoherence-insensitive quantum communication by optimal C*-encoding

IEEE Transactions on Information Theory

Volumn 53, Issue 12, 2007, Pages 4738-4749

  Bodmann, Bernhard G ^a   Kribs, David W ^a,b   Paulsen, Vern I ^c  

^a UNIVERSITY OF WATERLOO   (Canada)

^b UNIVERSITY OF GUELPH   (Canada)

^c University of Houston   (United States)

Author keywords

C* algebra embedding; Codes; MSC (2000): 81P68 (primary), 94A24, 81R15 (secondary); PACS numbers: 03.67.Hk, 03.67.Pp; Quantum communication

Indexed keywords

CODES; CONVEX COMBINATIONS; DECOHERENCE; DECOHERENCE-FREE SUBSPACES; ENCODINGS; GEOMETRIC CONDITIONS; MSC (2000): 81P68 (PRIMARY), 94A24, 81R15 (SECONDARY); NOISE MODELS; NOISELESS SUBSYSTEMS; OPTIMAL ENCODING; PACS NUMBERS: 03.67.HK, 03.67.PP; PHASE DAMPING; QUANTUM COMMUNICATION; QUANTUM INFORMATIONS; QUANTUM STATE; TRANSMISSION CHANNELS;

ALGEBRA; CELL CULTURE; DAMPING; DECODING; ENCODING (SYMBOLS); FLOWCHARTING;

QUANTUM OPTICS;

EID: 54149103374     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2007.909105     Document Type: Article

References (50)

1. G. M. Palma, K.-A. Suominen, and A. Ekert, "Quantum computers and dissipation," Proc. Royal Soc. A, vol. 452, pp. 567-584, 1996.
2. L.-M. Duan and G.-C. Guo, "Preserving coherence in quantum computation by pairing quantum bits," Phys. Rev. Lett., vol. 79, pp. 1953-1956, 1997.
3. P. Zanardi and M. Rasetti, "Noiseless quantum codes," Phys. Rev. Lett., vol. 79, pp. 3306-3309, 1997.
4. D. A. Lidar, I. L. Chuang, and K. B. Whaley, "Decoherence free subspaces for quantum computation," Phys. Rev. Lett., vol. 81, pp. 2594-2597, 1998.
5. E. Knill, R. Laflamme, and L. Viola, "Theory of quantum error correction for general noise," Phys. Rev. Lett., vol. 84, pp. 2525-2528, 2000.
6. P. Zanardi, "Stabilizing quantum information," Phys. Rev. A, vol. 63, pp. 012301/1-4, 2001.
7. J. Kempe, D. Bacon, D. A. Lidar, and K. B. Whaley, "Theory of decoherence-free fault-tolerant universal quantum computation," Phys. Rev. A, vol. 63, pp. 042307/1-29, 2001.
8. P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. White, "A decoherence-free quantum memory using trapped ions," Science, vol. 290, pp. 498-501, 2000.
9. D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, "Experimental verification of decoherence-free subspaces," Science, vol. 291, pp. 1013-1015, 2001.
10. E. M. Fortunato, L. Viola, J. Hodges, G. Teklemariam, and D. G. Cory, "Implementation of universal control on a decoherence-free qubit," New J. Phys., vol. 4, pp. 5.1-5.20, 2002.
11. L. Viola, E. M. Fortunato, M. A. Pravia, E. Knill, R. Laflamme, and D. G. Cory, "Experimental realization of noiseless subsystems for quantum information processing," Science, vol. 293, pp. 2059-2063, 2003.
12. M. Carravetta, O. G. Johannessen, and M. H. Levitt, "Beyond the T 1 limit: Singlet nuclear spin states in low magnetic fields," Phys. Rev. Lett., vol. 92, pp. 153003/1-4, 2004.
13. N. Boulant, L. Viola, E. M. Fortunato, and D. G. Cory, "Experimental implementation of a concatenated quantum error-correcting code," Phys. Rev. Lett., vol. 94, pp. 130501/1-4, 2005.
14. S. D. Bartlett, T. Rudolph, and R. W. Spekkens, "Classical and quantum communication without a shared reference frame," Phys. Rev. Lett. vol. 91, pp. 027901/1-4, 2003.
15. J.-. Boileau, D. Gottesman, M. A. Pravia, R. Laflamme, D. Poulin, and R. W. Spekkens, "Robust polarization-based quantum key distribution over collective-noise channel," Phys. Rev. Lett., vol. 92, pp. 017901/1-4, 2004.
16. L. A. Wu, P. Zanardi, and D. A. Lidar, "Holonomic quantum computation in decoherence-free subspaces," Phys. Rev. Lett., vol. 95, pp. 130501/1-4, 2005.
17. O. Dreyer, F. Markopoulou, and L. Smolin, Symmetry and Entropy of Black Hole Horizons, E-print: Arxiv.org/hep-th/0409056.
18. D. W. Kribs and F. Markopoulou, Geometry From Quantum Particles, E-print: Arxiv.org/gr-qc/0510052.
19. S. O. Bilson-Thompson, F. Markopoulou, and L. Smolin, Quantum Gravity and the Standard Model, E-print: Arxiv.org/hep-th/0603022.
20. T. Konopka and F. Markopoulou, Constrained Mechanics and Noiseless Subsystems, E-print: Arxiv.org/gr-qc/0601028.
21. D. W. Kribs, R. Laflamme, and D. Poulin, "Unified and generalized approach to quantum error correction," Phys. Rev. Lett., vol. 94, pp. 180501/1-4, 2005.
22. D. W. Kribs, R. Laflamme, D. Poulin, and M. Lesosky, "Operator quantum error correction," Quantum Inf. and Comput., vol. 6, pp. 382-399, 2006.
23. D. A. Lidar and A. Shabani, "Theory of initialization-free decoherence-free subspaces and subsystems," Phys. Rev. A, vol. 72, pp. 042303/1-14, 2005.
24. M. A. Nielsen and D. Poulin, Algebraic and Information-Theoretic Conditions for Operator Quantum Error Correction, E-print: Arxiv.org/ quant-ph/0506069.
25. D. W. Kribs and M. D. Choi, "Method to find quantum noiseless subsystems," Phys. Rev. Lett., vol. 96, pp. 050501/1-4, 2006.
26. E. Knill, On Protected Realizations of Quantum Information, E-print: Arxiv.org/quant-ph/0603252.
27. D. W. Kribs and R. W. Spekkens, "Quantum error correcting subsystems are unitarily recoverable subsystems," Phys. Rev. A, vol. 74, pp. 042329/1-7, 2006.
28. T. Strohmer and R. W. Heath, Jr, "Grassmannian frames with applications to coding and communication," Appl. Comput. Harmon. Anal., vol. 14, pp. 257-275, 2003.
29. R. B. Holmes and V. I. Paulsen, "Optimal frames for erasures," Linear Algebra Appl., vol. 377, pp. 31-51, 2004.
30. B. G. Bodmann and V. I. Paulsen, "Frames, graphs and erasures," Linear Algebra Appl., vol. 404, pp. 118-146, 2005.
31. P. Xia, S. Zhou, and G. B. Giannakis, "Achieving the Welch bound with difference sets," IEEE Trans. Inf. Theory, vol. 51, no. 5, pp. 1900-1907, May 2005.
32. D. Kalra, Complex Equiangular Cyclic Frames and Erasures, E-print: arxiv.org/math.FA/0602342.
33. B. G. Bodmann, "Optimal linear transmission by loss-insensitive packet encoding," Appl. Comput. Harmon. Anal., to be published.
34. S. G. Hoggar, "64 lines from a quaternionic polytope," Geom. Dedicata, vol. 69, pp. 287-289, 1998.
35. G. Zauner, "Quantendesigns - Grundzüge Einer Nichtkommutativen Designtheorie," Ph.D. dissertation, Univ. Vienna, Vienna, Austria, 1999.
36. J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, "Symmetric informationally complete quantum measurements," J. Math. Phys., vol. 45, pp. 2171-2180, 2004.
37. D. M. Appleby, "SIC-POVMs and the extended clifford group," J. Math. Phys., vol. 46, pp. 052107/1-29, 2005.
38. S. T. Flammia, "On SIC-POVMs in prime dimensions," J. Phys. A: Math. Gen., vol. 39, pp. 13483-13493, 2006.
39. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information. Cambridge, U.K.: Cambridge Univ. Press, 2000.
40. V. Paulsen, Completely Bounded Maps and Operator Algebras. Cambridge, U.K.: Cambridge Univ. Press, 2002.
41. I. Bengtsson and K. Zyczkowski, Geometry of Quantum States. Cambridge, U.K/: Cambridge Univ. Press, 2006.
42. G. Kuperberg, "The capacity of hybrid quantum memory," IEEE Trans. Inf. Theory, vol. 49, no. 6, pp. 1465-1473, Jun. 2003.
43. C. Beny, A. Kempf, and D. W. Kribs, Generalization of Quantum Error Correction via the Heisenberg Picture, E-print: Arxiv.org/quant-ph/ 0608071.
44. K. R. Davidson, "C* -algebras by example," in Fields Institute Monographs. Providence, RI: AMS, 1996, vol. 6.
45. P. G. Casazza and G. Kutyniok, "Frames of subspaces," in Wavelets, Frames and Operator Theory, Contemporary Math. Providence, RI: AMS, 2004, vol. 345, pp. 87-113.
46. P. G. Casazza, G. Kutyniok, and S. Li, Fusion Frames and Distributed Processing, E-print: Arxiv.org/math.FA/0605374.
47. L. Welch, "Lower bounds on the maximum cross correlation of signals," IEEE Trans. Inf. Theory, vol. IT-20, no. 3, pp. 397-399, May 1974.
48. P. W. H. Lemmens and J. J. Seidel, "Equi-isoclinic subspaces of euclidean spaces," Indag. Math., vol. 35, pp. 98-107, 1973.
49. C. D. Godsil and A. D. Hensel, "Distance regular covers of the complete graph," J. Combin. Theory Ser. B, vol. 56, pp. 205-238, 1992.
50. C. Godsil and A. Roy, Equiangular Lines, Mutually Unbiased Bases and Spin Models, 2005, E-print: Arxiv.org/quant-ph/0511004.