Generation of Universal Linear Optics by Any Beamsplitter

TL;DR

This paper proves that any nontrivial beamsplitter in linear optics can generate all unitary transformations on multiple modes, simplifying the realization of universal linear optical systems without tunable components.

Contribution

It establishes that any nontrivial beamsplitter is sufficient for universal linear optics, showing a dichotomy between trivial and universal sets of optical gates.

Findings

Any nontrivial beamsplitter densely generates the set of unitary transformations for three or more modes.

Universal linear optics can be achieved without tunable components, using any nontrivial beamsplitter.

No intermediate models of linear optical computation exist between trivial and universal sets.

Abstract

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. Theoretically, it means that one cannot produce "intermediate" models of linear optical computation (analogous to the Clifford group for qubits) by restricting the allowed beamsplitters and phaseshifters: there is a dichotomy; one either gets a trivial set or else a universal set. No similar classification theorem for gates acting on qubits is currently known. We leave open the problem of classifying optical gates that act on three or more modes.