The quotient-algebra partition, which consists of abelian subspaces obeying the quaternion condition,is a structure that can be universally constructed in every unitary Lie algebra. In this thesis, a retrospect of the foundation of quotient-algebra partition is presented first.
Then, applications demonstrated in this thesis are treated
by the scheme of quotient-algebra partition in the field of quantum information theory. As the first application,
the bipartite separability problem for a generalization of the well-known Bell-diagonal states, called Cartan eigen-mixtures, is thoroughly solved by the scheme.
The necessary and sufficient conditions of the separability for this class of states corresponds to a closed convex hull in a Hilbert space and may serve as an essential framework in analyzing the complexity of entanglement.
Besides, the scheme of quotient-algebra partition makes
possible an algorithm to systematically and exhaustively generate quantum error-correction codes, including both the additive and nonadditive types.
By virtue of the structure of quotient-algebra partition once more, the orthogonality condition, the most important rule to generate successful quantum codes, is fully equivalent to the distinguishability of conjugate-pair subspaces in this partition.
Meanwhile, a correspondence between the classical and the quantum codes is illustrated in the structure.
Finally, the scheme is also applied to a probe in the representation theory of Lie algebras, specifically the search of the unitary representations and the irreducible representations of Lie algebras.

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