Universality of beamsplitters

TL;DR

This paper proves that most nontrivial 2-mode and nearly all 3-mode beamsplitters are universal for constructing any real orthogonal operator on three or more modes, using control theory and rotation properties.

Contribution

It establishes the universality of certain beamsplitters on multiple modes, expanding understanding of quantum optics gate capabilities.

Findings

Most 2-mode beamsplitters are universal on 3 or more modes.

Nearly all 3-mode beamsplitters are universal on 3 or more modes.

Universality is proven using control theory and rotation properties.

Abstract

We consider the problem of building an arbitrary $N\times N$ real orthogonal operator using a finite set, $S$, of elementary quantum optics gates operating on $m\leq N$ 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 $m\geq3$ modes.