Simple factorization of unitary transformations

We demonstrate a method for general linear optical networks that allows one to factorize any $\mathrm{SU}(n$) matrix in terms of two $\mathrm{SU}(n-1)$ blocks coupled by an SU(2) entangling beam splitter. The process can be recursively continued in a straightforward way, ending in a tidy arrangement of SU(2) transformations. The method hinges only on a linear relationship between input and output states, and can thus be applied to a variety of scenarios, such as microwaves, acoustics, and quantum fields.

Figure 1

A schematic illustration of the factorization of an SU(3) transformation into a sequence of SU(2) transformations. Each mode is represented by a line. Transformations between modes are represented by boxes, into which the modes are fed. The number on each box indicates the number of parameters in the transformation; we use color for visual ease of distinguishing between transformations on the same number of modes, but differing numbers of parameters.

Figure 2

A schematic illustration of the factorization of an SU(4) transformation as a sequence of SU(3) transformations, each itself written as SU(2) blocks.

Figure 3

A schematic illustration of the factorization of an SU(5) transformation as a sequence of two SU(4) transformations, coupled by an SU(2). The bottom panel shows the final result in terms of SU(2) blocks. The shaded triangle is an equivalent SU(4) transformation.

Figure 4

Illustration of the factorization schemes of Clements et al. [22] (top) and Reck et al. [16] for the case of four modes.