Entanglement in graph states and its applications

Proceedings of the International School of Physics "Enrico Fermi"

Volumn 162, Issue , 2006, Pages 115-218

  Hein, M ^a   Dur W ^a,b   Eisert, J ^c   Raussendorf, R ^d   Van Den Nest, M ^e   Briegel, H J ^a,b  

^a UNIVERSITY OF INNSBRUCK   (Austria)

^b ERICH SCHMID INSTITUTE OF MATERIALS SCIENCE   (Austria)

^c IMPERIAL COLLEGE LONDON   (United Kingdom)

^d CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^e UNIVERSITY OF LEUVEN   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

ENTANGLED STATE; ENTANGLEMENT PROPERTIES; ITS APPLICATIONS; MULTIPARTITE ENTANGLEMENTS; ONE-WAY QUANTUM COMPUTERS; QUANTUM ERROR CORRECTIONS; QUANTUM INFORMATION THEORY; STATE OF RESEARCH;

ALGORITHMS; ERROR CORRECTION; QUANTUM COMMUNICATION; QUANTUM COMPUTERS;

QUANTUM ENTANGLEMENT;

EID: 84861950556     PISSN: 0074784X     EISSN: 18798195     Source Type: Conference Proceeding    
DOI: 10.3254/1-58603-660-2-115     Document Type: Conference Paper

References (210)

1. EINSTEIN A., PODOLSKY B. and ROSEN N., Phys. Rev., 47 (1935) 777.
2. SCHRÖDINGER E., Naturwiss., 23 (1935) 807, 823, 844.
3. BÖHM D., Quantum Theory (Prentice-Hall, Englewood Cliffs, New York) 1951.
4. BELL J. S., Physics, Vol. 1 (1964) p. 195
5. Reprinted in: BELL J. S., Speakable and Unspeakable in Quantum Mechanics (Cambidge University Press) 1987.
6. ASPECT A., GRANGIER P. and ROGER G., Phys. Rev. Lett., 47 (1981) 460.
7. ZEILINGER A., Rev. Mod. Phys., 71 (1999) 288.
8. NIELSEN M. A. and CHUANG I. L., Quantum Computation and Information (Cambridge University Press, Cambridge) 2000.
9. BENNETT C. H. and BRASSARD G., Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore India, 175 (1984).
10. EKERT A. K., Phys. Rev. Lett., 67 (1991) 661.
11. SHOR P. W. and PRESKILL J., Phys. Rev. Lett, 85 (2000) 441.
12. CURTY M., LEWENSTEIN M. and LÜTKENHAUS N., Phys. Rev. Lett., 92 (2004) 217903.
13. BRAUNSTEIN S. L., CAVES C. M., JOZSA R., LINDEN N., POPESCU S. and SCHACK R., Phys. Rev. Lett, 83 (1999) 1054.
14. LLOYD S., Phys. Rev. A, 61 (2000) 010301.
15. JOZSA R. and LINDEN N., E-print quant-ph/0201143.
16. MIYAKE A. and WADATI M., Phys. Rev. A, 64 (2001) 042317.
17. VIDAL G., Phys. Rev. Lett, 91 (2003) 147902.
18. FEYNMAN R. P., Int. J. Theor. Phys., 21 (1982) 467.
19. DEUTSCH D., Proc. R. Soc. London, Ser. A, 400 (1985) 97.
20. SHOR P. W., SI A M J. Sci. Stat. Comput, 26 (1997) 1484.
21. GROVER L. K., Phys. Rev. Lett, 79 (1997) 325.
22. BENNETT C. H., DIVINCENZO D. P., SMOLIN J. A. and WOOTTERS W. K., Phys. Rev. A, 54 (1996) 3824.
23. GOTTESMAN D., Stabilizer Codes and Quantum Error Correction, PhD thesis, CalTech, Pasadena (1997).
24. WERNER R. F., Phys. Rev. A, 40 (1989) 4277.
25. OSTLUND S. and ROMMER S., Phys. Rev. Lett, 75 (1995) 3537
26. FANNES M., NACHTERGAELE B. and WERNER R. F., Commun. Math. Phys., 144 (1992) 443.
27. EISERT J. and PLENIO M. B., Int. J. Quantum Inf., 1 (2003) 479
28. BRAUNSTEIN S. L. and VAN LOOCK P., Rev. Mod. Phys., 77 (2005) 513.
29. WEST D. B., Introduction to Graph Theory (Prentice Hall, Upper Saddle River) 2001.
30. DIESTEL R., Graph Theory (Springer, Heidelberg) 2000.
31. O'CONNOR K. M. and WOOTTERS W. K., Phys. Rev. A, 63 (2001) 052302.
32. PLESCH M. and BUZEK V., Phys. Rev. A, 67 (2003) 012322
33. Phys. Rev. A, 68 (2003) 012313.
34. GIORDA P. and ZANARDI P., Phys. Rev. A, 68 (2003) 062108.
35. PARKER M. G. and RUMEN V., The Quantum Entanglement of Binary and Bipolar Sequences, in HELLESETH T., KUMAR P. V. and YANG K. (Editors), Sequences and Their Applications, SETA '01, Discrete Mathematics and Theoretical Computer Science Series (Springer) 2001.
36. RAUSSENDORF R. and BRIEGEL H.-J., Phys. Rev. Lett., 86 (2001) 5188.
37. RAUSSENDORF R. and BRIEGEL H.-J., Quantum Inf. Comput., 2 (2002) 443.
38. RAUSSENDORF R., BROWNE D. E. and BRIEGEL H.-J., Phys. Rev. A, 68 (2003) 022312.
39. RAUSSENDORF R., Measurement-based quantum computation with cluster states, PhD thesis, LMU München, Germany (2003).
40. BRIEGEL H.-J. and RAUSSENDORF R., Phys. Rev. Lett, 86 (2001) 910.
41. HEIN M., EISERT J. and BRIEGEL H.-J., Phys. Rev. A, 69 (2004) 062311.
42. DUR W., ASCHAUER H. and BRIEGEL H. J., Phys. Rev. Lett, 91 (2003) 107903
43. ASCHAUER H., DUR W. and BRIEGEL H.-J., Phys. Rev. A, 71 (2005) 012319.
44. VAN DEN NEST M., DEHAENE J. and DE MOOR B., Phys. Rev. A, 69 (2004) 022316.
45. SCHLINGEMANN D. and WERNER R. F., Phys. Rev. A, 65 (2002) 012308.
46. SCHLINGEMANN D., Quantum information processing with graph states, Habilitationsschrift (Thesis), University of Braunschweig (2005).
47. DUR W., CIRAC J. I. and TARRACH R., Phys. Rev. Lett, 83 (1999) 3562
48. DÜR W. and CIRAC J. I., Phys. Rev. A, 61 (2000) 042314
49. Phys. Rev. A, 62 (2000) 022302.
50. LINDEN N. and POPESCU S., Fortsch. Phys., 46 (1998) 567.
51. GISIN N. and PERES A., Phys. Lett. A, 162 (1992) 15
52. POPESCU S. and ROHRLICH D., Phys. Lett. A, 166 (1992) 293.
53. THAPLIYAL A. V. and SMOLIN J. A., Phys. Rev. A, 68 (2003) 062324.
54. EISERT J. and GROSS D., E-print quant-ph/0505149.
55. CHEN K. and Lo H. K., E-print quant-ph/0404133.
56. DUR W., CALSAMIGLIA J. and BRIEGEL H.-J., Phys. Rev. A, 71 (2005) 042336.
57. DUR W. and BRIEGEL H.-J., Phys. Rev. Lett, 92 (2004) 180403
58. HEIN M., DÜR W. and BRIEGEL H.-J., Phys. Rev. A, 71 (2005) 032350.
59. GÜHNE O., TÓTH G., HYLLUS P. and BRIEGEL H. J., Phys. Rev. Lett, 95 (2005) 120405.
60. CALSAMIGLIA J., HARTMANN L., DÜR W. and BRIEGEL H.-J., Phys. Rev. Lett, 95 (2005) 180502; E-print quant-ph/0502017
61. HARTMANN L., CALSAMIGLIA J., DÜR W. and BRIEGEL H.-J., Phys. Rev. A, 72 (2005) 052107.
62. AUDENAERT K., EISERT J., PLENIO M. B. and WERNER R. F., Phys. Rev. A, 66 (2002) 042327.
63. PLENIO M. B., EISERT J., DREISSIG J. and CRAMER M., Phys. Rev. Lett, 94 (2005) 060503
64. CRAMER M., EISERT J., PLENIO M. B. and DREISSIG J., Phys. Rev. A, 73 (2006) 012309.
65. HARARY F. and PALMER E. M., Graphical Enumeration (Academic Press, New York) 1973, p. 240.
66. SCHLINGEMANN D., Quantum Inf. Comput, 2 (2002) 307.
67. GRASSL M., KLAPPENECKER A. and ROTTELER M., Proceedings of the 2002 IEEE International Symposium on Information Theory, Lausanne, Switzerland (2002), pp. 45.
68. AUDENAERT K. M. R. and PLENIO M. B., New J. Phys., 7 (2005) 170.
69. VAN DEN NEST M., DEHAENE J. and DE MOOR B., Phys. Rev. A, 70 (2004) 034302.
70. BOUCHET A., Combinatorica, 11 (1991) 315
71. Discrete Math., 114 (1993) 75.
72. GLYNN D. G., On self-dual quantum codes and graphs, Submitted to Electronic Journal of Combinatorics, Preprint (2002).
73. SCHLINGEMANN D., Quantum Inf. Comput, 4 (2004) 287
74. J. Math. Phys., 45 (2004) 4322.
75. ZHOU D. L., ZENG B., XU Z. and SUN C. P., Phys. Rev. A, 68 (2003) 062303.
76. HOSTENS E., DEHAENE J. and DE MOOR B., Phys. Rev. A, 71 (2005) 042315.
77. GROSS D., Finite phase space methods in quantum information, Diploma thesis, University of Potsdam, Germany (2005).
78. WOLF M. M., VERSTRAETE F. and CIRAC J. I., Phys. Rev. Lett., 92 (2004) 087903.
79. ZHANG J., Phys. Rev. A, 73 (2006) 032318
80. ASOUDEH M. and KARIMIPUR V., Phys. Rev. A, 72 (2005) 032339.
81. CRAMER M. and EISERT J., New J. Phys., 8 (2006) 71.
82. KEYL M., SCHLINGEMANN D. and WERNER R. F., Quantum Inf. Comput., 3 (2002) 281.
83. CALDERBANK A. R., RAINS E. M., SHOR P. W. and SLOANE N. J. A., IEEE Trans. Inform. Theory., 44 (1998) 1369.
84. DANIELSEN L. E. and PARKER M. G., E-print math. CO/0504522
85. DANIELSEN L. E., On Self-Dual Quantum Codes, Graphs, and Boolean Functions, Master thesis, University of Bergen, Norway (2005), E-print quant-ph/0503236 (the database of self-dual quantum codes can also be found on the webpage http://www.ii.uib.no/~larsed/vncorbits/).
86. VERSTRAETE F. and CIRAC J. I., Phys. Rev. A, 70 (2004) 060302(R).
87. SCHOLLWÖCK U., Rev. Mod. Phys., 77 (2005) 259.
88. AFFLECK I., KENNEDY T., LIEB E. H. and TASAKI H., Commun. Math. Phys., 115 (1988) 477.
89. VERSTRAETE F. and CIRAC J. L, E-print cond-mat/0407066.
90. VERSTRAETE F., GARCIA-RIPOLL J. J. and CIRAC J. I., Phys. Rev. Lett, 93 (2004) 207204
91. VERSTRAETE F., PORRAS D. and CIRAC J. I., Phys. Rev. Lett, 93 (2004) 227205.
92. GOTTESMAN D., The Heisenberg Representation of Quantum Computers, in Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, Cambridge, MA, edited by CORNEY S. P. et al. (International Press) 1999.
93. AARONSON S. and GOTTESMAN D., Phys. Rev. A, 70 (2004) 052328.
94. ANDERS S. and BRIEGEL H.-J., Phys. Rev. A, 73 (2006) 022334.
95. GREENBERGER D. M., HORNE M. and ZEILINGER A., in Bell's Theorem, Quantum Theory, and Conceptions of the Universe, edited by KAFATOS E. (Kluwer, Dordrecht) 1989, pp. 69.
96. GISIN N. and BECHMANN-PASQUINUCCI H., Phys. Lett. A, 246 (1998) 1.
97. HUELGA S. F., MACCHIAVELLO C, PELLIZZARI T., EKERT A. K., PLENIO M. B. and CIRAC J. I., Phys. Rev. Lett, 79 (1997) 3865.
98. HILLERY M., BUZEK V. and BERTHIAUME A., Phys. Rev. A, 59 (1999) 1829
99. GOTTESMAN D., Phys. Rev. A, 61 (2000) 042311
100. CRÉPEAU C, GOTTESMAN D. and SMITH A., Proc. 34th ACMSTOC (New York) (2002) 643.
101. NIELSEN M. A., Rep. Math. Phys., 57 (2006) 147.
102. CHILDS A. M., LEUNG D. W. and NIELSEN M. A., Phys. Rev. A, 71 (2005) 032318
103. JORRAND P. and PERDRIX S., E-print quant-ph/0404125.
104. NIELSEN M. A. and DAWSON C. M., E-print quant-ph/0405134
105. RAUSSENDORF R., HARRINGTON J. and GOYAL K., E-print quant-ph/0510135
106. ALIFERIS P. and LEUNG D. W., Phys. Rev. A, 73 (2006) 032308.
107. KRUSZYNSKA C, MIYAKE A., BRIEGEL H. J. and DUR W., E-print: quant-ph/0606090, to be published in Phys. Rev. A.
108. MHALLA M. and PERDRIX S., E-print quant-ph/0412071.
109. BRIEGEL H.-J., RAUSSENDORF R. and SCHENZLE A., Optical lattices as a playground for studying multiparticle entanglement, in Laserphysics at the Limit, edited by FIGGER H., MESCHEDE D. and ZIMMERMANN C. (Springer) 2002, pp. 433-477.
110. JAKSCH D., BRIEGEL H.-J., CIRAC J. I., GARDINER C. W. and ZOLLER P., Phys. Rev. Lett, 82 (1999) 1975
111. BRIEGEL H.-J., CALARCO T., JAKSCH D., CIRAC J. I. and ZOLLER P., J. Mod. Opt., 47 (2000) 415.
112. JAKSCH D., BRUDER C, CIRAC J. I., GARDINER C. W. and ZOLLER P., Phys. Rev. Lett, 81 (1998) 3108.
113. DUAN L.-M., DEMLER E. and LUKIN M. D., Phys. Rev. Lett, 91 (2003) 090402.
114. GREINER M., MANDEL O., HÄNSCH T. W. and BLOCH I., Nature, 419 (2002) 51
115. GREINER M., MANDEL O., ESSLINGER T., HÄNSCH T. W. and BLOCH L, Nature, 415 (2002) 39.
116. MANDEL O., GREINER M., WIDERA A., ROM T., HÄNSCH T. W. and BLOCH I., Nature, 425 (2003) 937.
117. KAY A., PACHOS J. K. and ADAMS C. S., Phys. Rev. A, 73 (2006) 022310.
118. VOLLBRECHT K. G. H., SOLANO E. and CIRAC J. I., Phys. Rev. Lett, 93 (2004) 220502.
119. RABL P., DALEY A. J., FEDICHEV P. O., CIRAC J. I. and ZOLLER P., Phys. Rev. Lett., 91 (2003) 110403.
120. CLARK S. R., MOURA ALVES C. and JAKSCH D., New J. Phys., 7 (2005) 124.
121. BORHANI M. and Loss D., Phys. Rev. A, 71 (2005) 034308.
122. KNILL E., LAFLAMME R. and MILBURN G. J., Nature, 409 (2001) 46.
123. EISERT J., Phys. Rev. Lett, 95 (2005) 040502
124. SCHEEL S. and LÜTKENHAUS N., New J. Phys., 6 (2004) 51.
125. YORAN N. and REZNIK B., Phys. Rev. Lett, 91 (2003) 037903.
126. NIELSEN M. A., Phys. Rev. Lett, 93 (2004) 040503.
127. BROWNE D. E. and RUDOLPH T., Phys. Rev. Lett, 95 (2005) 010501.
128. RALPH T. C, HAYES A. J. F. and GILCHRIST A., Phys. Rev. Lett, 95 (2005) 100501.
129. PITTMAN T. B., JACOBS B. and FRANSON J., Phys. Rev. A, 65 (2001) 062311.
130. ZHANG A.-N., LU C.-Y., ZHOU X.-Q., CHEN Y.-A., ZHAO Z., YANG T. and PAN J.-W., Phys. Rev. A, 73 (2006) 022330.
131. VARNAVA M., BROWNE D. E. and RUDOLPH T., Phys. Rev. Lett, 97 (2006) 120501
132. DAWSON C. M., HASELGROVE H. L. and NIELSEN M. A., Phys. Rev. Lett., 96 (2006) 020501.
133. WALTHER P., RESCH K. J., RUDOLPH T., SCHENCK E., WEINFURTER H., VEDRAL V., ASPELMEYER M. and ZEILINGER A., Nature, 434 (2005) 169.
134. CABRILLO C, CIRAC J. I., GARCHIA-FERNDANDEZ P. and ZOLLER P., Phys. Rev. A, 59 (1999) 1025
135. BOSE S., KNIGHT P. L. and PLENIO M. B., Phys. Rev. Lett, 83 (1999) 5158
136. LEGERO T., WILK T., HENNRICH M., REMPE G. and KUHN A., Phys. Rev. Lett, 93 (2004) 070503.
137. CIRAC J. L, ZOLLER P., KIMBLE H. J. and MABUCHI H., Phys. Rev. Lett, 78 (1997) 3221
138. VAN ENK S. J., KIMBLE H. J., CIRAC J. I. and ZOLLER P., Phys. Rev. A, 59 (1999) 2659.
139. BARRETT S. D. and KOK P., Phys. Rev. A, 71 (2005) 060310(R).
140. LIM Y. L., BEIGE A. and KWEK L. C, Phys. Rev. Lett, 95 (2005) 030505.
141. BENJAMIN S. C, EISERT J. and STACE T., New J. Phys, 7 (2005) 194.
142. BENJAMIN S. C, BROWNE D. E., FITZSIMONS J. and MORTON J. L., New J. Phys., 8 (2006) 141.
143. VAN DEN NEST M., DEHAENE J. and DE MOOR B., Internal Report 04-154, ESATSISTA, K. U. Leuven, Belgium (2004).
144. DUR W., VIDAL G. and CIRAC J. I., Phys. Rev. A, 62 (2000) 062314.
145. VERSTRAETE F., DEHAENE J., DE MOOR B. and VERSCHELDE H., Phys. Rev. A, 65 (2002) 052112.
146. VERSTRAETE F., DEHAENE J. and DE MOOR B., Phys. Rev. A, 68 (2003) 012103.
147. VAN DEN NEST M., DEHAENE J. and DE MOOR B., Phys. Rev. A, 71 (2005) 062323.
148. VAN DEN NEST M., DEHAENE J. and DE MOOR B., Phys. Rev. A, 70 (2004) 032323
149. Phys. Rev. A, 71 (2005) 022310
150. Phys. Rev. A, 72 (2005) 014307.
151. FON-DER-FLAASS D., Electr. J. Comb., 3 (1) (1996).
152. HÖHN G., Mathematische Annalen, 327 (2003) 227.
153. GLYNN D. G., GULLIVER T. A., MAKS J. G. and GUPTA M. K., The Geometry of Additive Quantum Codes (Springer) 2004.
154. On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/~njas/sequences/.
155. TÓTH G. and GÜHNE O., Phys. Rev. Lett, 94 (2005) 060501
156. AIP Conf. Proc, 734 (2004) 234.
157. POPESCU S., Phys. Rev. Lett, 74 (1995) 2619.
158. MERMIN N. D., Phys. Rev. Lett, 65 (1990) 1838.
159. WERNER R. F. and WOLF M. M., Phys. Rev. A, 64 (2001) 032112.
160. ZUKOWSKI M., BRUKNER C, LASKOWSKI W. and WIESNIAK M., Phys. Rev. Lett., 88 (2002) 210402.
161. PERES A., Found. Phys., 29 (1999) 589.
162. WERNER R. F. and WOLF M. M., Quantum Inf. Comput, 1 (2001) 1.
163. TERHAL B. M., Phys. Lett. A, 271 (2000) 319.
164. LEWENSTEIN M., KRAUS B., CIRAC J. I. and HORODECKI P., Phys. Rev. A, 62 (2000) 052310.
165. SCARANI V., ACIN A., SCHENCK E. and ASPELMEYER M., Phys. Rev. A, 71 (2005) 042325.
166. CLAUSER J. F., HÖRNE M. A., SHIMONY A. and HOLT R. A., Phys. Rev. Lett., 23 (1969) 880.
167. TÓTH G. and GÜHNE O., Phys. Rev. A, 72 (2005) 022340.
168. TÓTH G., GÜHNE O. and BRIEGEL H. J., Phys. Rev. A, 73 (2006) 022303.
169. MOURA ALVES C. and JAKSCH D., Phys. Rev. Lett, 93 (2004) 110501
170. PALMER R. N., MOURA ALVES C. and JAKSCH D., Phys. Rev. A, 72 (2005) 042335.
171. VERSTRAETE F., POPP M. and CIRAC J. I., Phys. Rev. Lett, 92 (2004) 027901.
172. VERSTRAETE F., MARTIN-DELGADO M. A. and CIRAC J. I., Phys. Rev. Lett, 92 (2004) 087201.
173. POPP M., VERSTRAETE F., MARTIN-DELGADO M. A. and CIRAC J. I., Phys. Rev. A, 71 (2005) 042306.
174. DIVINCENZO D. P., FUCHS C. A., MABUCHI H., SMOLIN J. A., THAPLIYAL A. and UHLMANN A., E-print quant-ph/9803033.
175. LAUSTSEN T., VERSTRAETE F. and VAN ENK S. J., Quantum Inf. Comput., 3 (2003) 64.
176. EISERT J. and BRIEGEL H.-J., Phys. Rev. A, 64 (2001) 022306.
177. WEI T.-C. and GOLDBART P. M., Phys. Rev. A, 68 (2003) 042307.
178. COFFMAN V., KUNDU J. and WOOTTERS W. K., Phys. Rev. A, 61 (2000) 052306.
179. PLENIO M. B. and VEDRAL V., J. Phys. A, 34 (2001) 6997.
180. MEYER D. and WALLACH N., J. Math. Phys., 43 (2002) 4273.
181. BARNUM H. and LINDEN N., J. Phys. A, 34 (2001) 6787.
182. FATTAL D., CUBITT T. S., YAMAMOTO Y., BRAVYI S. and CHUANG I. L., E-print quant-ph/0406168 (2004).
183. HEIN M., Entanglement in Graph States, PhD thesis, University of Innsbruck, Austria (2005).
184. VIDAL G., Phys. Rev. Lett, 93 (2004) 040502.
185. DALEY A. J., KOLLATH C, SCHOLLWÖCK U. and VIDAL G., J. Stat. Mech., 04 (2004) P04005.
186. DUR W., HARTMANN L., HEIN M. and BRIEGEL H.-J., Phys. Rev. Lett., 94 (2005) 097203.
187. WOOTTERS W. K., Phys. Rev. Lett., 80 (1998) 2245.
188. BRENNEN G. K., Quantum Inf. Comput., 3 (2003) 619.
189. OSBORNE T. J. and NIELSEN M. A., Phys. Rev. A, 66 (2002) 032110.
190. OSTERLOH A., AMICO L., FALCI G. and FAZIO R., Nature, 416 (2002) 608.
191. LATORRE J. L, Rico E. and VIDAL G., Quantum Inf. Comput, 4 (2004) 048.
192. VIDAL G., LATORRE J. L, Rico E. and KITAEV A., Phys. Rev. Lett, 90 (2003) 227902.
193. PACHOS J. K. and PLENIO M. B., Phys. Rev. Lett., 93 (2004) 056402.
194. CIRAC J. I. and ZOLLER P., Nature, 404 (2000) 579.
195. JANÉ E., VIDAL G., DUR W., ZOLLER P. and CIRAC J. L, Quantum Inf. Comput., 3 (2003) 15.
196. ANDERS S., PLENIO M. B., DUR W. and BRIEGEL H.-J., Phys. Rev. Lett, 97 (2006) 107206.
197. GORINI V., KOSSAKOWSKI A. and SUDARSHAN E. C. G., L. Math. Phys., 17 (1976) 821
198. LINDBLAD G., Commun. Math. Phys., 48 (1976) 119.
199. DUR W., HEIN M., CIRAC J. I. and BRIEGEL H.-J., Phys. Rev. A, 72 (2005) 052326.
200. TOLKUNOV D., PRIVMAN V. and ARAVIND P. K., Phys. Rev. A, 71 (2005) 060308.
201. DUR W. and CIRAC J. L, J. Phys. A, 34 (2001) 6837.
202. CARVALHO A. R. R., MINTERT F. and BUCHLEITNER A., Phys. Rev. Lett, 93 (2004) 230501.
203. PERES A., Phys. Rev. Lett, 77 (1996) 1413.
204. HORODECKI M., HORODECKI P. and HORODECKI R., Phys. Rev. Lett., 78 (1997) 574.
205. HORODECKI M., HORODECKI P. and HORODECKI R., Phys. Lett A, 223 (1996) 8.
206. BANDYOPADHYAY S. and LIDAR D. A., Phys. Rev. A, 72 (2005) 042339.
207. SIMON C. and KEMPE J., Phys. Rev. A, 65 (2002) 052327.
208. RAUSSENDORF R., BRAVYI S. and HARRINGTON J., Phys. Rev. A, 71 (2005) 062313.
209. MURAO M., PLENIO M. B., POPESCU S., VEDRAL V., KNIGHT P. L., Phys. Rev. A, 57 (1998) R4075.
210. MANEVA E. N. and SMOLIN J. A., in SAMUEL J. and LOMONACO J. (Editors), Quantum Computation and Quantum Information, AMS Contemporary Mathematics, Vol. 305 (2002), E-print quant-ph/0003099.