Abstract
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, e.g., 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.
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