A method to construct all representations of finite dimensional unitary matrices as the
product of Householder reflections is given. By arbitrarily severing the state space into
orthogonal subspaces, the method may, eg, identify the entangling and single-component
quantum operations that are required in the engineering of quantum states of composite
(multipartite) systems. Earlier constructions are shown to be extreme cases of the unifying
scheme that is presented here.