On algebraic and quantum random walks

Quantum Probability And Infinite Dimensional Analysis: From Foundations To Appllications

Volumn , Issue , 2005, Pages 174-200

  Ellinas, Demosthenes ^a  

^a TECHNICAL UNIVERSITY OF CRETE   (Greece)

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[No Author keywords available]

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EID: 33646866763     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1142/9789812702104_0012     Document Type: Chapter

References (49)

1. P. A. Meyer, Quantum Probability for Probabilists (Lect. Notes Math. 1538), (Springer, Berlin 1993).
2. M. Schiirmann, White Noise on Bialgebras (Lect. Notes Math. 1544), (Springer, Berlin 1993).
3. S. Majid, Foundations of Quantum Groups Theory (Cambridge Univ. Press, 1955), ff. chapter 5.
4. U. Franz and R. Schott, Stochastic Processes and Operator Calculus on Quantum Groups, (Kluwer Academic Publishers, Dodrecht 1999).
5. A. Ambainis, E. Bach, A. Nayak, A. Vishwanath and J. Watrous, Proc. 33rd Annual Symp. Theory Computing (ACM Press, New York, 2001), p. 37.
6. D. Aharonov, A. Ambainis, J. Kempe and U. Vasirani, Proc. 33rd Annual Symp. Theory Computing (ACM Press, New York, 2001), p. 50.
7. A. Nayak and A. Vishwanath, arXive eprint quant-ph/0010117.
8. J. Kempe, Proc. 7th Int. Workshop, RANDOM’03, p. 354 (2003).
9. A. M. Childs, E. Farhi and S. Gutmann, Quantum Information Processing 1, 35 (2002).
10. B. C Travaglione and G. J. Milburn, Phys. Rev. A 65, 032310 (2002).
11. B. C. Sanders, S. D. Bartlett, B. Tregenna and P. L. Knight, Phys. Rev. A 67, 042305 (2003).
12. J. Kempe, Contemp. Phys. 44, 307 (2003).
13. A. J. Bracken, D. Ellinas and I. Tsohantjis, J. Phys. A: Math. Gen. 37, L91 (2004).
14. E. Abe, Hopf Algebras (CUP Cambridge 1997).
15. D. Ellinas, J. Comp. Appl. Math. 133, 341 (2001).
16. A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications (Academic Press, New York, 1979).
17. R. Bhatia, Matrix Analysis (Spinger-Verlag, New York, 1997).
18. P. M. Alberti and A. Uhlmann, Stochasticity and Partial Order: Double Stochastic Maps and Unitary Mixing (Dordecht, Boston, 1982).
19. M. A. Nielsen, An Introduction to Majorization and its Applications to Quantum Mechanics (unpublished notes).
20. D. Ellinas and E. Floratos, J. Phys. A: Math. Gen. 32, L63 (1999).
21. B. D. Hughes, Random Walks and Random Environments Vol. I, (Clarendon Press, Oxford 1995)
22. J. Gillis, Quarterly J. Math., (Oxford, 2nd series) 7, 144 (1956).
23. K. Lakatos-Lindenberg and K. E. Shuler, J. Math. Phys. 12, 633 (1971).
24. J. R. Klauder and B.-S. Skagerstam, Coherent States (World Scientific, Singapore (1986)
25. A. Perelomov, Generalized Coherent States and their Applications, (Springer-Verlag, Berlin 1986).
26. E. D. Davies, Quantum Theory of Open System, (Academic, New York, 1973).
27. G. Lindblad, Non-Equilibrium Entropy and Irreversibility, (Reidel, Dordrecht 1983).
28. M. O. Scully and M. S. Zubairy, Quantum Optics, (Cambridge Univ. Press, Cambridge 1997), p. 255, 448, 453.
29. S. Stenholm, Phys. Scripta T12 (1986).
30. J. Gea-Banacloche, Phys. Rev. Lett. 59, 543 (1987).
31. K. Kraus, States, effects and operations (Springer-Verlag, Berlin, 1983).
32. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge Univ. Press, Cambridge 2000).
33. S. Majid, Int. J. Mod. Phys. 8, 4521-4545 (1993).
34. S. Majid, M. J. Rodriguez-Plaza, J. Math. Phys. 33, 3753-3760 (1994).
35. P. Feinsilver and R. Schott, J. Theor. Prob. 5, 251 (1992).
36. P. Feinsilver, U. Franz and R. Schott, J. Theor. Prob. 10, 797 (1997).
37. U. Franz and R. Schott, J. Phys. A: Gen. Math. 31, 1395 (1998).
38. U. Franz and R. Schott, J. Math. Phys. 39, 2748 (1998).
39. D. Ellinas and I. Tsohantjis, J. Non-Lin. Math. Phys. 8Suppl. 93 (2001).
40. D. Ellinas and I. Tsohantjis, Inf. Dim. Anal.-Quant. Prob. 611 (2003).
41. G. H. Hardy, J. E. Littlewood and G. Polya, Messenger Math. 58, 145 (1929).
42. H. Sikic and M. V. Wickerhauser, Appl. Comp. Harm. Anal. 11, 147 (2001).
43. R. Gilmore, Lie Groups, Lie Algebras and Some of Their Applications, (Wiley, New York 1974).
44. D. Ellinas, et al, Work in progress.
45. P. J. Davis, Circulant Matrices, (Wiley, New York 1979).
46. M. R. Schroeder, Number Theory in Science and Communication, (Springer, Berlin 1997).
47. C. D. Cushen and R. L. Hudson, J. Appl. Prob. 8, 454 (1972).
48. This second line of the equation has the form a Lindblad type master equation taken in the large temperature limit, where for the average number of thermal photons we have n ≈ n + 1 = |γ|, c.f. Ref. 28.
49. H. Risken, The Fokker-Planck Equation, (Springer, Berlin 1996). 83 (1990).