Abstract
The intimate connection between the Banach space wavelet reconstruction method for each unitary representation of a given group and homogenous space, and the quantum entanglement description using group theory were both studied in our previous articles. Here, we present a universal description of quantum entanglement using group theory and non-commutative characteristic functions for homogenous space and projective representation of compact groups on Banach spaces for some well known examples, such as: Moyal representation for a spin; Dihedral and Permutation groups.
[1] A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777(1935) http://dx.doi.org/10.1103/PhysRev.47.77710.1103/PhysRev.47.777Search in Google Scholar
[2] M. N. Nielsen, I. L. Chuang, Quantum computation and quantum information (Cambridge University Press, Cambridge, 2000) Search in Google Scholar
[3] R.F. Werner, Phys. Rev. A 40, 4277 (1989) http://dx.doi.org/10.1103/PhysRevA.40.427710.1103/PhysRevA.40.4277Search in Google Scholar
[4] A. Peres, Phys. Rev. Lett. 77, 1413 (1996) http://dx.doi.org/10.1103/PhysRevLett.77.141310.1103/PhysRevLett.77.1413Search in Google Scholar
[5] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 223, 1 (1996) http://dx.doi.org/10.1016/S0375-9601(96)00706-210.1016/S0375-9601(96)00706-2Search in Google Scholar
[6] S. L. Woronowicz, Rep. Math. Phys. 10, 165 (1976) http://dx.doi.org/10.1016/0034-4877(76)90038-010.1016/0034-4877(76)90038-0Search in Google Scholar
[7] P. Horodecki, Phys. Lett. A 232, 333 (1997) http://dx.doi.org/10.1016/S0375-9601(97)00416-710.1016/S0375-9601(97)00416-7Search in Google Scholar
[8] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 80, 5239 (1998) http://dx.doi.org/10.1103/PhysRevLett.80.523910.1103/PhysRevLett.80.5239Search in Google Scholar
[9] M. A. Jafarizadeh, M. Rezaee, S. K. S. Yagoobi, Phys. Rev. A 72, 062106 (2005) http://dx.doi.org/10.1103/PhysRevA.72.06210610.1103/PhysRevA.72.062106Search in Google Scholar
[10] B. M. Terhal, Phys. Lett. A 271, 319 (2000) http://dx.doi.org/10.1016/S0375-9601(00)00401-110.1016/S0375-9601(00)00401-1Search in Google Scholar
[11] M. A. Jafarizadeh, M. Rezaee, S. Ahadpur, Phys. Rev. A 74, 042335 (2006) http://dx.doi.org/10.1103/PhysRevA.74.04233510.1103/PhysRevA.74.042335Search in Google Scholar
[12] M. A. Jafarizadeh, M. Mirzaee, M. Rezaee, Physica A 349, 459 (2005) http://dx.doi.org/10.1016/j.physa.2004.10.02810.1016/j.physa.2004.10.028Search in Google Scholar
[13] M. A. Jafarizadeh, M. Mirzaee, M. Rezaee, Int. J. QuantumInf. 3, 3 (2005) Search in Google Scholar
[14] M. A. Jafarizadeh, M. Mirzaee, M. Rezaee, Quantum Inf. Process. 4, 3 (2005) http://dx.doi.org/10.1007/s11128-005-5657-010.1007/s11128-005-5657-0Search in Google Scholar
[15] M. A. Jafarizadeh, M. Mirzaee, M. Rezaee, Int. J. QuantumInf. 2, 541 (2004) http://dx.doi.org/10.1142/S021974990400092410.1142/S0219749904000924Search in Google Scholar
[16] G. Folland, A course in Abstract Harmonic Analysis (CRC Press, Boca Raton, 1995) Search in Google Scholar
[17] J. K. Korbicz, M. Lewenstein, Phys. Rev. A 74, 022318 (2006) http://dx.doi.org/10.1103/PhysRevA.74.02231810.1103/PhysRevA.74.022318Search in Google Scholar
[18] M. Rezaee, M. A. Jafarizadeh, M. Mirzaee, Int. J. Quantum Inf. 5, 367 (2007) http://dx.doi.org/10.1142/S021974990700296710.1142/S0219749907002967Search in Google Scholar
[19] M. A. Man’ko, V. I. Man’ko, R. Vilela Mendes, J. Phys. A-Math. Gen. 34, 8321 (2001) http://dx.doi.org/10.1088/0305-4470/34/40/30910.1088/0305-4470/34/40/309Search in Google Scholar
[20] E. B. Davis, Quantum theory of open system (Academic Press, London, 1976) Search in Google Scholar
[21] O. Bratteli, D. Robinson, Operator Algebras and Quantum Statistical Mechanics, Volume I, II (Springer, New York, 1979, 1981) 10.1007/978-3-662-02313-6Search in Google Scholar
[22] M. Mirzaee, M. Rezaee, M. A. Jafarizadeh, Int. J. Theor. Phys. 46, 2326 (2007) http://dx.doi.org/10.1007/s10773-007-9351-010.1007/s10773-007-9351-0Search in Google Scholar
[23] M. Mirzaee, M. Rezaee, M. A. Jafarizadeh, Eur. Phys. J. B 60, 193 (2007) http://dx.doi.org/10.1140/epjb/e2007-00330-110.1140/epjb/e2007-00330-1Search in Google Scholar
[24] H. G. Feichtinger, K. H. Grochenig, J. Funct. Anal. 86, 308 (1989) http://dx.doi.org/10.1016/0022-1236(89)90055-410.1016/0022-1236(89)90055-4Search in Google Scholar
[25] Y. Gu, Phys. Rev. A 32, 1310 (1985) http://dx.doi.org/10.1103/PhysRevA.32.131010.1103/PhysRevA.32.1310Search in Google Scholar
[26] G. Giedke, B. Kraus, M. Lewenstein, J. I. Cirac, Phys. Rev. Lett. 87, 167904 (2001) http://dx.doi.org/10.1103/PhysRevLett.87.16790410.1103/PhysRevLett.87.167904Search in Google Scholar
[27] G. Cassinelli, G. M. D’Ariano, E. DeVito, A. Levrero, J. Math. Phys. 41, 7940 (2000) http://dx.doi.org/10.1063/1.132349710.1063/1.1323497Search in Google Scholar
[28] M. Paini, arXiv:quant-ph/0002078 Search in Google Scholar
[29] J. M. Bondia, J. C. Varilly, Ann. Phys. 190, 107 (1989) http://dx.doi.org/10.1016/0003-4916(89)90262-510.1016/0003-4916(89)90262-5Search in Google Scholar
[30] S. Heiss, S. Weigert, Phys. Rev. A 63, 012105 (2001) http://dx.doi.org/10.1103/PhysRevA.63.01210510.1103/PhysRevA.63.012105Search in Google Scholar
[31] I. Schur, J. Reine. Angew. Math. 139, 155 (1911) 10.1515/crll.1911.139.155Search in Google Scholar
[32] P. N. Hoffman, J. F. Humphereys, Projective Representation of the Symmetric Groups (Clarendon Press, Oxford, 1992) Search in Google Scholar
[33] R. Brauer, H. Weyl, Am. J. Math. 57, 425 (1935) http://dx.doi.org/10.2307/237121810.2307/2371218Search in Google Scholar
[34] F. Wilczek, arXiv:hep-th/9806228 Search in Google Scholar
[35] L. Dornhoff, Group Representation Theory, Volume A (Marcel Dekker, New York, 1971)MR50:458a Search in Google Scholar
[36] G. Karpilovesky, Projective Representation of Finite Groups (Marcel Dekker, Ink. New York and Basel, 1985) Search in Google Scholar
[37] G. James, M. Liebeck, Representations and Characters of Groups (Cambridge Mathematical Textbooks, Cambridge, 1995) Search in Google Scholar
© 2010 Versita Warsaw
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.