An angular momentum approach to quadratic fourier transform, hadamard matrices, gauss sums, mutually unbiased bases, the unitary group and the pauli group

Journal of Physics A: Mathematical and Theoretical

Volumn 42, Issue 35, 2009, Pages

  Kibler, Maurice R ^a,b,c  

^a UNIVERSITÉ DE LYON   (France)

^b UNIVERSITÉ LYON 1   (France)

^c CENTRE DE CALCUL DE L INSTITUT NATIONAL DE PHYSIQUE NUCLEAIRE ET DE PHYSIQUE DES PARTICULES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70449495060     PISSN: 17518113     EISSN: 17518121     Source Type: Journal    
DOI: 10.1088/1751-8113/42/35/353001     Document Type: Article

References (162)

1. Biedenharn L C and Louck J D 1985 Angular momentum in quantum physics Encyclopedia of Mathematics and Its Applications vol 8 ed G-C Rota (Cambridge: Cambridge University Press)
2. Wigner E P 1941 Am. J. Math. 63 57
3. Wigner E P 1965 On the matrices which reduce the Kronecker products of representations of S R groups Quantum Theory of Angular Momentum ed L C Biedenharn and H van Dam (New York: Academic)
4. Racah G 1942 Phys. Rev. 62 438
5. Racah G 1949 Phys. Rev. 76 1352
6. Racah G 1961 Group theory and spectroscopy Report CERN 61-8 (Geneva: CERN)
7. Biedenharn L C and Louck J D 1985 The Racah-Wigner algebra in quantum theory Encyclopedia of Mathematics and Its Applications vol 9 ed G-C Rota (Cambridge: Cambridge University Press)
8. Sharp W T 1960 Report AECL-1098 (Chalk River, ON: Atomic Energy of Canada Ltd)
9. Derome J R and Sharp W T 1965 J. Math. Phys. 6 1584
10. Cottingham W N and Greenwood D A 2007 An Introduction to the Standard Model of Particle Physics (Cambridge: Cambridge University Press)
11. Sviridov D T and Smirnov Yu F 1977 Teoriya Opticheskikh Spektrov Perekhodnykh Metallov (Moscow: Nauka)
12. Tang A-C, Sun C-C, Kiang Y-S, Deng Z-H, Liu J-C, Chang C-E, Yan G-S, Goo Zien and Tai S-S 1979 Theoretical Method of the Ligand Field Theory (Peking: Science Press)
13. Kibler M R 1979 Finite symmetry adaptation in spectroscopy Recent Advances in Group Theory and Their Application to Spectroscopy ed J C Donini (New York: Plenum)
14. Ma E 2007 arXiv:0705.0327v4 [hep-ph]
15. Bonatsos D, Raychev P P, Roussev R P and Smirnov Yu F 1990 Chem. Phys. Lett. 175 300
16. Bonatsos D, Drenska S B, Raychev P P, Roussev R P and Smirnov Yu F 1991 J. Phys. G: Nucl. Part. Phys. 17 L67
17. Smirnov Yu F and Kibler M R 1993 Some aspects of q-boson calculus Symmetries in Science VI: From the Rotation Group to Quantum Algebras ed B Gruber (New York: Plenum)
18. 1,1) Symmetry Methods in Physics ed A N Sissakian, G S Pogosyan and S I Vinitsky (Dubna: Joint Institute for Nuclear Research)
19. Kibler M R, Asherova R M and Smirnov Yu F 1995 Some aspects of q- and qp-boson calculus Symmetries in Science VIII ed B Gruber (New York: Plenum)
20. 2 in a non-standard basis Symmetry and Structural Properties of Condensed Matter ed T Lulek, B Lulek and A Wal (Singapore: World Scientific)
21. Kibler M and Daoud M 2001 Recent Res. Dev. Quantum Chem. 2 91
22. Kibler M R 2005 Collect. Czech. Chem. Commun. 70 771
23. Albouy O and Kibler M R 2007 SIGMA 3 076
24. Melvin M A 1956 Rev. Mod. Phys. 28 18
25. Altmann S L and Cracknell A P 1965 Rev. Mod. Phys. 37 19
26. Altmann S L and Bradley C J 1965 Rev. Mod. Phys. 37 33
27. Michelot F and Moret-Bailly J 1975 J. Phys. (Paris) 36 451
28. Champion J P, Pierre G, Michelot F and Moret-Bailly J 1977 Can. J. Phys. 55 512
29. Champion J P 1977 Can. J. Phys. 55 1802
30. Kibler M 1969 Int. J. Quantum Chem. 3 795
31. Kibler M and Grenet G 1986 Int. J. Quantum Chem. 29 485
32. Kibler M and Gcon J C 1989 Croat. Chem. Acta 62 783
33. Moret-Bailly J 1965 J. Mol. Spectrosc. 15 344
34. Kibler M 1968 J. Mol. Spectrosc. 26 111
35. Kibler M 1969 C. R. Acad. Sci., Paris B 268 1221
36. Patera J and Winternitz P 1973 J. Math. Phys. 14 1130
37. Patera J and Winternitz P 1976 J. Chem. Phys. 65 2725
38. Michel L 1977 Invariants polynomiaux des groupes de symétrie moléculaire et cristallographique Group Theoretical Methods in Physics ed R T Sharp and B Kolman (New York: Academic)
39. Grenet G, Zhilinskii B I and Kibler M 1980 Vestnik Moskovsk. Univ., Ser. Khim. 21 244
40. Vourdas A 2004 Rep. Prog. Phys. 67 267
41. Kibler M R 2008 J. Phys. A: Math. Theor. 41 375302
42. Di P 2005 J. Phys. A: Math. Gen. 37 5355
43. Bengtsson I, Bruzda W, Ericsson, Larsson J-, Tadej W and yckowski K 2007 J. Math. Phys. 48 052106
44. Brierley S and Weigert S 2009 Phys. Rev. A 79 052316
45. Tadej W and yczkowski K 2005 arXiv:quant-ph/0512154
46. Tadej W and yczkowski K 2006 Open Syst. Inf. Dyn. 13 133
47. Tadej W and yczkowski K 2009 On-line catalogue of known Hadamard matrices http://chaos.if.uj.edu.pl/∼karol/hadamard/
48. Kibler M R 2006 Int. J. Mod. Phys. B 20 1792
49. Kibler M R and Planat M 2006 Int. J. Mod. Phys. B 20 1802
50. Berndt B C and Evans R J 1981 Bull. Am. Math. Soc. 5 107
51. Berndt B C, Evans R J and Williams K S 1998 Gauss and Jacobi Sums (New York: Wiley)
52. Hannay J H and Berry M V 1980 Physica D 1 267
53. Matsutani S and nishi Y 2003 Found. Phys. Lett. 16 325
54. Rosu H C, Trevĩo J P, Cabrera H and Murguía J S 2006 Int. J. Mod. Phys. B 20 1860
55. Merkel W et al 2006 Int. J. Mod. Phys. B 20 1893
56. Delsarte P, Goethals J M and Seidel J J 1975 Philips Res. Rep. 30 91
57. Ivanović I D 1981 J. Phys. A: Math. Gen. 14 3241
58. Wootters W K 1986 Found. Phys. 16 391
59. Wootters W K 1987 Ann. Phys., NY 176 1
60. Wootters W K and Fields B D 1989 Ann. Phys., N Y 191 363
61. Wootters W K 2006 Found. Phys. 36 112
62. Calderbank A R, Cameron P J, Kantor W M and Seidel J J 1997 Proc. London Math. Soc. 75 436
63. Zauner G 1999 Quantendesigns: Grundzüge einer nichtcommutativen Designtheorie Diploma Thesis University of Wien
64. Grassl M 2004 arXiv:quant-ph/0406175
65. Grassl M 2005 Elec. Notes Discrete Math. 20 151
66. Brierley S and Weigert S 2008 Phys. Rev. A 78 042312
67. Heath R W, Strohmer T and Paulraj A J 2006 IEEE Trans. Inf. Theory 52 1217
68. Cerf N J, Bourennane M, Karlsson A and Gisin N 2002 Phys. Rev. Lett. 88 127902
69. Aharonov Y and Englert B-G 2001 Z. Naturforsch. 56a 16
70. Englert B-G and and Aharonov Y 2001 Phys. Lett. A 284 1
71. Aravind P K 2003 Z. Naturforsch. 58a 2212
72. Hayashi A, Horibe M and Hashimoto T 2005 Phys. Rev. A 71 052331
73. Paz J P, Roncaglia A J and Saraceno M 2005 Phys. Rev. A 72 012309
74. Wocjan P and Beth T 2005 Quantum Inf. Comput. 5 93
75. Durt T 2006 Int. J. Mod. Phys. B 20 1742
76. Svetlichny G 2007 arXiv:0708.3079 [quant-ph]
77. Tolar J and Chadzitaskos G 2009 J. Phys. A: Math. Theor. 42 245306
78. Weigert S and Wilkinson M 2008 Phys. Rev. A 78 020303
79. Bandyopadhyay S, Boykin P O, Roychowdhury V and Vatan F 2002 Algorithmica 34 512
80. Lawrence J, Brukner and Zeilinger A 2002 Phys. Rev. A 65 032320
81. Lawrence J 2004 Phys. Rev. A 70 012302
82. Chaturvedi S 2002 Phys. Rev. A 65 044301
83. Vlasov A Yu 2003 arXiv:0302064 [quant-ph]
84. Klappenecker A and Rötteler M 2004 Lect. Notes Comput. Sci. 2948 262
85. Gibbons K S, Hoffman M J and Wootters W K 2004 Phys. Rev. A 70 062101
86. Pittenger A O and Rubin M H 2004 Linear Algebra Appl. 390 255
87. Pittenger A O and Rubin M H 2005 J. Phys. A: Math. Gen. 38 6005
88. Durt T 2005 J. Phys. A: Math. Gen. 38 5267
89. Archer C 2005 J. Math. Phys. 46 022106
90. Aschbacher M, Childs A M and Wocjan P 2007 J. Algebr. Comb. 25 111
91. ulc P and Tolar J 2007 J. Phys. A: Math. Theor. 40 15099
92. Albouy O 2009 Discrete algebra and geometry applied to the Pauli group and mutually unbiased bases in quantum information theory Thesis Université de Lyon
93. Albouy O 2009 Determinantal measure for pure states entanglement J. Phys. A: Math. Theor. submitted
94. Havlicek H and Saniga M 2007 J. Phys. A: Math. Theor. 40 F943
95. Havlicek H and Saniga M 2008 J. Phys. A: Math. Theor. 41 015302
96. Planat M and Baboin A-C 2007 J. Phys. A: Math. Theor. 40 F1005
97. Weyl H 1931 The Theory of Groups and Quantum Mechanics (New York: Dover)
98. Schwinger J 1960 Proc. Natl Acad. Sci. USA 46 570
99. Sylvester J J 1883 C. R. Acad. Sci., Paris XCVII 1336
100. Morris A O 1967 Q. J. Math. 18 7
101. Morris A O 1968 Q. J. Math. 19 289
102. Balian R and Itzykson C 1986 C. R. Acad. Sci., Paris 303 773
103. Patera J and Zassenhaus H 1988 J. Math. Phys. 29 665
104. Galetti D and De Toledo Piza A F R 1988 Physica A 149 267
105. Knill E 1996 arXiv:quant-ph/9608048
106. Gottesman D 1999 Chaos Solitons Fractals 10 1749
107. Pittenger A O and Rubin M H 2000 Phys. Rev. A 62 032313
108. Gottesman D, Kitaev A and Preskill J 2001 Phys. Rev. A 64 012310
109. Bartlett S D, de Guise H and Sanders B C 2002 Phys. Rev. A 65 052316
110. Klimov A B, Snchez-Soto L L and de Guise H 2005 J. Phys. A: Math. Gen. 38 2747
111. Klimov A B, Snchez-Soto L L and de Guise H 2005 J. Opt. B: Quantum Semiclass. Opt. 7 283
112. Snchez-Soto L L, Klimov A B and de Guise H 2006 Int. J. Mod. Phys. B 20 1877
113. Romero J L, Björk G, Klimov A B and Snchez-Soto L L 2005 Phys. Rev. A 72 062310
114. Ne'eman Y 1961 Nucl. Phys. 26 222
115. Gell-Mann M 1962 Phys. Rev. 125 1067
116. Okubo S 1962 Prog. Theor. Phys. (Kyoto) 27 949
117. Glashow S L, Iliopoulos J and Maiani L 1970 Phys. Rev. D 2 1285
118. Georgi H and Glashow S L 1974 Phys. Rev. Lett. 32 438
119. Moffat J W 1975 Phys. Rev. D 12 288
120. de Azcrraga J A and Macfarlane A J 2001 Int. J. Mod. Phys. A 16 1377
121. Planat M, Saniga M and Kibler M R 2006 SIGMA 2 066
122. Saniga M, Planat M, Pracna P and Havlicek H 2007 SIGMA 3 075
123. Planat M, Baboin A-C and Saniga M 2008 Int. J. Theor. Phys. 47 1127
124. Planat M and Saniga M 2008 Quantum Inf. Comput. 8 0127
125. Saniga M, Planat M and Pracna P 2008 Theor. Math. Phys. 155 463
126. Albouy O 2009 J. Phys. A: Math. Theor. 42 072001
127. Kostrikin A I, Kostrikin I A and Ufnarovskii V A 1981 Proc. Steklov Inst. Math. 158 105
128. Kostrikin A I, Kostrikin I A and Ufnarovskii V A 1981 Sov. Math. Dokl. 24 292
129. Kostrikin A I, Kostrikin I A and Ufnarovskii V A 1982 Sov. Math. Dokl. 25 2327
130. Kostrikin A I and Tiep P H 1994 Expositions in Mathematics vol 15 (Berlin: de Gruyter)
131. Boykin P O, Sitharam M, Tiep P H and Wocjan P 2007 Quantum Inf. Comput. 7 371
132. Bengtsson I 2006 quant-ph/0610216
133. Wolf K B and García A 1972 Rev. Mex. Fís. 21 191
134. Wolf K B 1975 The Heisenberg-Weyl ring in quantum mechanics Group Theory and its Applications 3 ed E M Loebl (New York: Academic)
135. García-Bullé M, Lassner W and Wolf K B 1986 J. Math. Phys. 27 29
136. Atakishiyev N M, Pogosyan G S and Wolf K B 2003 Int. J. Mod. Phys. A 18 317
137. Terras A 1996 Exp. Math. 5 15
138. Terras A 1999 Fourier Analysis on Finite Groups and Applications (Cambridge: Cambridge University Press)
139. Howe R 2005 Indag. Mathem., (N S) 16 553
140. iovíek P and Tolar J 1984 Rep. Math. Phys. 20 157
141. Perrine S 2009 De Frobenius Riedel: analyse des solutions de l'équation de Markoff (unpublished manuscript)
142. Vourdas A 2008 J. Phys. A: Math. Theor. 41 455303
143. Gamba A 1969 J. Math. Phys. 10 872
144. Nielsen M A and Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)
145. Jozsa R 2005 arXiv:0508124v2 [quant-ph]
146. Gottesman D 1997 arXiv:9705052 [quant-ph]
147. Calderbank A R, Rains E M, Schor P W and Sloane N J A 1998 IEEE Trans. Inf. Theory 44 1369
148. Clark S, Jozsa R and Linden N 2008 Quantum Inf. Comput. 8 106
149. Appleby D M 2005 J. Math. Phys. 46 052107
150. Flammia S T 2006 J. Phys. A: Math. Gen. 39 13483
151. Cormick C, Galvo E F, Gottesman D, Paz J P and Pittenger A O 2006 Phys. Rev. A 73 012301
152. Planat M and Jorrand Ph 2008 J. Phys. A: Math. Theor. 41 182001
153. Lambert D and Kibler M 1988 J. Phys. A: Math. Gen. 21 307
154. Marchiolli M A, Silva E C and Galetti D 2009 Phys. Rev. A 79 022114
155. Fujii K 2007 Int. G. Geom. Meth. Mod. Phys. 4 1205
156. Lévy-Leblond J-M 1973 Rev. Mex. Fís. 22 15
157. Vourdas A 1990 Phys. Rev. A 41 1653
158. Chaichian M and Ellinas D 1990 J. Phys. A: Math. Gen. 23 L291
159. Ellinas D 1991 J. Math. Phys. 32 135
160. Daoud M, Hassouni Y and Kibler M 1998 The k-fermions as objects interpolating between fermions and bosons Symmetries in Science X ed B Gruber and M Ramek (New York: Plenum)
161. Daoud M, Hassouni Y and Kibler M 1998 Phys. Atom. Nuclei 61 1821
162. Schwinger J 1965 On angular momentum Quantum Theory of Angular Momentum ed L C Biedenharn and H van Dam (New York: Academic)