A Scheme of Cartan Decomposition for su(N)

Details A Scheme of Cartan Decomposition for su(N) A Scheme of Cartan Decomposition for su(N)

2006-03-22

arxiv.org/abs/quant-ph/0603190v1

Physics

Quantum Physics

An extension of the talk in Annual Conf. PSROC 2006

Scientific paper

A scheme to perform the Cartan decomposition for the Lie algebra su(N) of arbitrary finite dimensions is introduced. The schme is based on two algebraic structures, the conjugate partition and the quotient algebra, that are easily generated by a Cartan subalgebra and generally exist in su(N). In particular, the Lie algebras su(2^p) and every su(2^{p-1} < N < 2^p) share the isomorphic structure of the quotient algebra. This structure enables an efficient algorithm for the recursive and exhaustive construction of Cartan decompositions. Further with the scheme, a unitary transformation in SU(N) can be recursively decomposed into a product of certain designated operators, e.g., local and nonlocal gates. Such a recursive decomposition of a transformation implies an evolution path on the manifold of the group.

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