Decompositions of unitary evolutions and dynamics of quantum systems

Decompositions of the Lie group of unitary matrices are very useful tools in the control and analysis of quantum dynamics. In this paper, we survey recent results on decompositions, concerning their significance in terms of symmetries, entanglement, and their applications to control and dynamics. Several decompositions can be obtained by recursively applying the Cartan classification of the symmetric spaces of the classical Lie groups. The emphasis is on a novel recursive procedure to decompose the unitary evolution of bipartite quantum systems of arbitrary dimensions, in simpler factors. This procedure makes transparent the contributions of the entangling and non entangling transformations