Generation of Universal Linear Optics by Any Beamsplitter

Understand

In 1994, Reck et al.

• showed how to realize any unitary transformation on a single photon using a product of beamsplitters and phaseshifters.
• Here we show that any single beamsplitter that nontrivially mixes two modes, also densely generates the set of unitary transformations (or orthogonal transformations, in the real case) on the single-photon subspace with m>=3 modes.
• (We prove the same result for any two-mode real optical gate, and for any two-mode optical gate combined with a generic phaseshifter.) Experimentally, this means that one does not need tunable beamsplitters or phaseshifters for universality: any nontrivial beamsplitter is universal for linear optics.