Google Docs :Qutrits and Tensor Product of SU(3)

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Qutrits and Tensor Product of

By

SITI NURUL ‘AIN BINTI ZAITON

December 2009

Chair              : Associate Professor Hishamuddin Zainuddin, PhD

Institute          : Institute for Matematical Research

The simplest extension from qubit is qutrit. One qutrit can be represented as density matrices using generators of   i.e. the eight Gell Mann matrices and the unit   matrix. Two qutrits give rise to nine-level systems whose complete description involves 81 operators including the identity operator. The aim of this work is to obtain the Lie algebraic structure for two qutrits involving  .

By applying tensor product of Gell-Mann matrices, we construct explicit matrix representations of the generators for  . These results will be used to compute commutator Lie algebra of . The complete table of the commutator Lie algebra of   is constructed by simplifying their expressions in terms of the tensor product basis operators of   . We verify not all commutators can be simplified directly to single individual operators, i.e. they involve linear combination of tensor product basis operators of  .

In this work, the complete table of commutator algebra of   can serve the purpose of identifying Lie subalgebras. The trivial  algebra is easily identifiable but the nontrivial  and other subalgebras will require further work. We have also included the case for two qubit studied by Rao et al. in which the trivial and nontrivial subalgebras  are identified. We made small corrections to their results.