Orbit Classification of Qutrit via the Gram Matrix
B. A. Tay, Hishamuddin Zainuddin
Department of Physics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, MalaysiaLaboratory of Computational Science and Informatics, Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia
Orbit Classification of Qutrit via the Gram Matrix
B. A. Tay;Hishamuddin Zainuddin
Department of Physics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, MalaysiaLaboratory of Computational Science and Informatics, Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia
摘要We classify the orbits generated by unitary transformation on the density matrices of the three-state quantum systems (qutrits) via the Gram matrix. The Gram matrix is a real symmetric matrix formed from the Hilbert--Schmidt scalar products of the vectors lying in the tangent space to the orbits. The rank of the Gram matrix determines the dimensions of the orbits, which fall into three classes for qutrits.
Abstract：We classify the orbits generated by unitary transformation on the density matrices of the three-state quantum systems (qutrits) via the Gram matrix. The Gram matrix is a real symmetric matrix formed from the Hilbert--Schmidt scalar products of the vectors lying in the tangent space to the orbits. The rank of the Gram matrix determines the dimensions of the orbits, which fall into three classes for qutrits.
B. A. Tay;Hishamuddin Zainuddin. Orbit Classification of Qutrit via the Gram Matrix[J]. 中国物理快报, 2008, 25(6): 1923-1926.
B. A. Tay, Hishamuddin Zainuddin. Orbit Classification of Qutrit via the Gram Matrix. Chin. Phys. Lett., 2008, 25(6): 1923-1926.
[1] Nielsen M A and Chuang I L 2000 Quantum Computationand Quantum Information (New York: Cambridge University Press) [2] Mattle K, Weinfurter H, Kwiat P G and Zeilinger A 1996 Phys. Rev. Lett. 76 4656 [3] Bechmann-Pascuinucci H and Peres A 2000 Phys. Rev. Lett. 85 3313 [4] Bruss D, Machiavello C 2002 Phys. Rev. Lett. 88127901 [5] Mallesh K S and Mukunda N 1997 Pramana J. Phys. 49 371 Arvind A, Mallesh K S and Mukunda N 1997 J. Phys. A 302417 Khanna G, Mukhopadhyay S, Simon R, Mukunda N 1997 Ann. Phys 253 55 [6] Byrd M and Sudarshan E C G 1998 J. Phys. A 31 9255 Byrd M and Sudarshan E C G 1998 J. Math. Phys. 39 6125 [7] Caves M C and Milburn G J 2000 Opt. Comm. 179 439 [8] Boya L J, Byrd M, Mims M and Sudarshan E C G 1998 hep-utexas-98-21e-print quant-ph/9810084v1 [9] Kimura G 2003 Phys. Lett. A 314 339 [10] Byrd M and Khaneja N 2003 Phys. Rev. A 68 062322 [11] Mendas I P 2006 J. Phys. A 39 11313 [12] Burlakov A V, Chekhova M V, Karabutova O A, Klyshko D N andKulik S P 1999 Phys. Rev. A 60 R4209 Krivitskii L A, Kulik S P, Maslennikov G A and Chekhova M V 2005 J. Exp. Theor. Phys. 100 589 Bogdanov Y I et al 2004 Phy. Rev. A 70 042303 [13] Thew R, Acin A, Zbinden H and Gisin N 2004 Quant. Inf.Proc. 4 93 [14] Howell J C, Lamas-Linares A and Bouwmeester D 2002 Phys.Rev. Lett. 88 030401 [15] Gell-Mann M and Neeman Y 1964 The Eightfold Way (NewYork: Benjamin) de Swart J J 1963 Rev. Mod. Phys. 35 916 [16] Kus M and \.{Zyczkowski K 2001 Phys. Rev. A 63 032307 [17] Bloch F 1946 Phys. Rev. 70 460 [18] Carteret H A and Sudbery A 2000 J. Phys. A 33 4981 [19] Bengtsson I and \.Zyczkowski K 2006 Geometry ofQuantum States (New York: Cambridge University Press)
2008, Vol. 25 
