Universality of beamsplitters

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We consider the problem of building an arbitrary N× N real orthogonal operator using a finite set, S, of elementary quantum optics gates operating on m≤ N modes - the problem of universality of S on N modes. In particular, we focus on the universality problem of an m-mode beamsplitter. Using methods of control theory and some properties of rotations in three dimensions, we prove that any nontrivial real 2-mode and "almost" any nontrivial real 3-mode beamsplitter is universal on m≥3 modes.